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export interface Chat{
nome: String;
menssagem: String;
dtHora: String;
}
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Elizabeth Boutell, geborene Davenport (* 1650er Jahre; † 1715) war eine englische Schauspielerin.
Leben
Sie wurde zu Beginn der 1650er Jahre als Tochter von Christopher Davenport und Frances Ridley geboren. Elizabeth heiratete nach 1669 Barnaby Boutell aus gutem Hause.
Als Teenager schloss sie sich zusammen mit ihrer Schwester Frances der Londoner Theatergruppe King's Company an. Sie waren zwei der ersten Frauen, denen es in England gestattet war, auf einer Bühne aufzutreten. Der erste dokumentierte Auftritt Elizabeths datiert vom Jahr 1663 oder 1664 im eben erst errichteten Theatre Royal, als sie als Estifania in Rule a Wife and Have a Wife erschien. Sie übernahm viele bedeutende Rollen, darunter die Benzayda in John Drydens The Conquest of Granada (Dezember 1670 und Januar 1671) und möglicherweise auch die der Rosalinda in Nathaniel Lees Sophonisbe (3. April 1675).
Neben anderen Charakteren "erschuf" sie die Melantha in Drydens Marriage à la mode (April 1672), die Cleopatra in Drydens All for Love und die Mrs. Termagant in Thomas Shadwells Squire of Alsatia. Ihre bekannteste Rolle hatte sie jedoch als liebevolle und treuherzige Königin Stateira in Nathaniel Lees Erfolgsstück The Rival Queens (Premiere am 17. März 1677).
Boutell bildete mit ihrer Kollegin Rebecca Marshall eine "bemerkenswerte schauspielerische Kombination". Erstmals konstituierte sich diese im August 1670, als eine der ersten Rollen Boutells an der Dury Lane, in dem Drama "The Roman Empress", dem einzigen Theaterstück William Joyners (1622–1706). Der Erfolg und das Verlangen nach mehr solcher "Women in Conflict"-Stücke (Frauen in Feindschaft) zog ähnliche Werke nach sich, in denen Boutell die tugendhafte Heldin gegen Marshalls dunklerem Antagonisten spielte. Diese Konstellation zeigte sich in The Conquest of Granada und The Tragedy of Nero von Lee im Jahr 1674; wie auch 1677 in John Crownes The Destruction of Jerusalem und Lees The Rival Queens.
Dieses erfolgreiche Schema ließ nicht lange auf Nachahmer warten. So trat noch im gleichen Jahrzehnt die konkurrierende Theatergesellschaft Duke's Company mit einem eigenen Paar an: Mary Saunderson (die Ehefrau Thomas Bettertons) und Mary Lee (später Mary Slingsby). In den 1680er und 1690er Jahren war es das Duo Elizabeth Barry und Anne Bracegirdle.
Boutell brillierte oft in Hosenrollen, wie der als Fidelia in William Wycherleys The Plain Dealer (11. Dezember 1676), der Margery Pinchwife in William Wycherleys Die Unschuld vom Lande (12. Januar 1675) oder als Constancia in der (nicht erfolgreichen) Komödie The She Gallants (von George Granville, 1695).
Edmund Curll (1675–1747) beschrieb Boutell in seinem Werk The History of the English Stage von 1741 als eine sehr bedeutende Schauspielerin, welche von geringer Statur war, sehr angenehme Gesichtszüge besaß sowie einen guten Teint, jedoch ein kindisches Aussehen. Ihre Stimme war schwach, aber sehr sanft; Sie verkörperte im Allgemeinen die junge, unschuldige Dame, in die alle Helden verliebt sind und die eine Favoritin der Stadt war.
Zwischen März 1678 und April 1688 sind von ihr keine Auftritte bekannt. Ihr Ehemann wurde ab 1681 zum Leutnant ernannt und das Biographical Dictionary of Actors mutmaßt, dass sie ihm in jenem Jahrzehnt auf den Kontinent gefolgt sein könnte. Jedoch ist auch eine rege Reisetätigkeit innerhalb Europas überliefert, oftmals zusammen mit ihrer Freundin, der Schauspielerin Elizabeth Price, welche sie auch in ihrem Testament bedachte. Ihre letzte Rolle hatte sie 1696 bei Thomas Bettertons Company im Lincoln's Inn Fields Theatre, wo sie die T(h)omyris in Cyrus the Great (von John Banks, siehe auch Kyros II.) spielte.
1697 ging sie mit ihrem Mann in die Niederlande und verbrachte dort vermutlich angenehme Jahre. Ihr Mann starb 1711; sie im Jahr 1715. Im Jahr zuvor machte sie ihr Testament, in welchem sie ein Vermögen von 800 Pfund hinterließ.
Rezeption
Während ihrer aktiven Karriere in den 1670er Jahren galt sie laut dem "Biographical Dictionary of Actors" allgemein als "sehr talentierte, beliebte, schöne und promiskuitive junge Frau". Das Promiskuitive wurde ihr in vielen Schmähschriften und Satiren – auch noch lange nach ihrem Tod – nachgesagt. Auch vor dem Wort "Hure" ("whore" wurde damals allerdings nicht im engeren Sinne der Prostitution verwandt, eher vergleichbar dem heutigen "Schlampe") wurde nicht zurück geschreckt, obgleich dabei mehrheitlich nur ein gängiges Vorurteil bedient wurde, unter dem Frauen des Theaters seit jeher litten. Auch dürften männliche sexuelle Projektionen auf die begehrten Stars der Bühne eine Rolle gespielt haben.
Davies bezeichnet in seinen "Dramatic Miscellanies" (Bd. II. S. 404) "Mrs. Boutel" als "gefeiert für die sanfteren Rollen in einer Tragödie, wie die der Aspatia in der Maid's Tragedy".
Ergänzungen
Eine Anekdote besagt, dass, nachdem Barry für die Rolle der Roxana vom Requisiteur einen Schleier erhielt, ein Streit zwischen ihr und Boutell entbrannte, da Boutell für dieses elegante Kostümteil das Recht der Älteren reklamierte. Während der folgenden Vorstellung der (nun wohl buchstäblichen) Rival Queens führte Barry dann, mit dem an dieser Stelle auch vorgesehenen Rollentext "Die, Sorceress, die! And all my wrongs die with thee!", einem kräftigen Hieb mit dem Theaterdolch seitlich gegen den Leib Boutells, so dass das – eigentlich stumpfe – Theaterrequisit ihr Korsett durchdrang und der Schauspielerin eine Wunde von über einem halben Zentimeter Länge zufügte ("quarter of an inch"). Zeitgenössische Gerüchte unterstellten der Boutell Eifersucht auf Barrys Beziehung zum Earl of Rochester.
In einem Gerichtsverfahren aufgrund Bigamie gegen Charles Knollys, 4. Earl of Banbury, sagte sie 1695 zugunsten ihrer Freundin Elizabeth Price aus. Aufgrund ihrer "flexiblen Moralvorstellungen" und der Unterstellung, dass sie doch mit verschiedenen Männern bereits Beziehungen hatte, verlor Price den Prozess und die Ehe wurde nicht anerkannt.
In Colley Cibber berühmten Memoiren "Apology for the Life of Colley Cibber" (1740), in welchen er so ziemlich alle Schauspielkollegen seiner Zeit erwähnte, taucht der Name Boutell allerdings nicht auf.
Literatur
Philip Jr, Burnim Highfill, A. Kalman, Edward Langhans: Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers and Other Stage Personnel in London, 1660–1800. 16 Bände. Southern Illinois University Press, Carbondale (1973–1993)
Elizabeth Howe (1992): The First English Actresses: Women and Drama, 1660–1700 Cambridge University Press, Cambridge 1992.
Judith Milhous: Elizabeth Bowtell and Elizabeth Davenport: some puzzles solved" in 'Theatre Notebook', 39, The Society for Theatre Research, Seiten 124–134, London 1985
Einzelnachweise
Theaterschauspieler
Engländer
Brite
Geboren im 17. Jahrhundert
Gestorben 1715
Frau
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DAVY KNOWLES - Charlotte Blues Society Christmas Bash
presented by Charlotte Blues Society & MaxxMusic
Sun, December 5, 2021, 8:00 PM EST
Doors at 7:00 PM
Neighborhood Theatre
Blues / Rock / R&B
Covid Policy: Venue entry requires valid ID and proof of a full course of COVID-19 vaccination or proof of a negative lab-conducted COVID-19 test taken within 72 hours of the show date for entry into any event. The final vaccination dose must occur at least 14 days prior to the event.
*City of Charlotte Mask Mandate - Everyone is required to wear a mask. Guests are allowed to remove masks in order to take a sip of beverage.*
Tickets: $18 adv/$20 dos (plus sales tax and service fee) *Tickets available online only*
18+ Valid ID required for entry into venue / Under 18 permitted with parent - Accepted forms of ID: State Issued ID or Driver's License, Military ID, Passport.
DAVY KNOWLES
On his aptly-titled Mascot Label Group' debut, What Happens Next, roots singer-songwriter and guitarist Davy Knowles boldly steps forward with timeless and cohesive songwriting; sleek modern production; and a lyrical, play-for-the-song guitar approach informed from soul, folk, rock, and blues. The 12-song album is just as influenced by The Black Keys, Fantastic Negrito, Gary Clark Jr., as it is Muddy Waters, Junior Kimbrough, and R.L. Burnside. It is a cohesive body of work rather than a collection of disparate songs.
"I've always loved songs. When it came time to record this album, it was all about supporting the songs as opposed to 'how quickly I can get to the guitar solo'," the Isle of Man-born, Chicago, Illinois-based artist says laughing. "Not that there's anything wrong with that—I've been there, done that!" He continues: "I'm not a purist, and the way for roots music to stay relevant is to adapt and progress. The 'field hollers' Alan Lomax recorded are worlds apart from Muddy Waters' music, but they come from the same place."
Knowles has earned plum endorsements from D'Addario strings, PRS guitars, Bludotone amps, and Celestion speakers. In 2010, he became the first musician in history to play live directly to the International Space Station from Mission Control in Houston. In addition, he's held his own sharing the stage with such heavyweight artists as The Who, Jeff Beck, Gov't Mule, Lynyrd Skynyrd, Kid Rock, Joe Bonamassa, Sonny Landreth, Peter Frampton, Joe Satriani, and the Sammy Hagar-fronted supergroup "Chickenfoot," among many others.
On What Happens Next, Knowles's poetic songwriting, and his soulfully emotive singing steal the show. The 12-song body of work offers forth a peaks-and-valleys album experience winding through brawny riffs, jazzy blues balladry, and vintage soul before concluding with one of Knowles's most personal songs released to date. Throughout it all, his guitar playing is brilliantly understated, his rhythm work is deft and dynamic—beefy on the rockers, and subtly supportive on the slower tunes—and his leads are economical but feature juicy blues bends and thick as molasses lead guitar tones.
North Carolina Events
Things to do in Charlotte, NC
Charlotte Performances
Charlotte Music Performances
#blues
#rnb
#blues_rock
#neighborhoodtheatre
Organizer MaxxMusic
Organizer of DAVY KNOWLES - Charlotte Blues Society Christmas Bash
DAVY KNOWLES - Charlotte Blues Society Christmas Bash at Neighborhood Theatre
511 East 36th Street, Charlotte, NC 28205, United States
MaxxMusic
Fri, Apr 22 2:45 PM
Digital Gardens Music & Arts Celebration 2022
Digital Gardens Music & Arts Celebration, Charlotte
RUSKO w/ G-Rex, Notixx, Devious & Joneses
Blackbox Theater, Charlotte
Fri, Feb 11 10:00 PM
Q PARKER of 112 and WILLIE TAYLOR of Day26 - Live in Charlotte! Feb 11th!!
The Tavern, Charlotte
$20 - $1,099.99
Diva Royale Drag Queen Show Charlotte, NC - Weekly Drag Queen Shows
Drag Queen Show Charlotte, Charlotte
STEPHEN MARLEY ACOUSTIC SOUL
Neighborhood Theatre, Charlotte
SHABLEEK 4PLAY 1ST THURSDAYS AT STUDIO 229
Studio 229 on Brevard, Charlotte
WAKA FLOCKA @PAPARAZZI CLT 1/29
Paparazzi CLT, Charlotte
Ivy Lab - Blackbox Theater Charlotte
Comedy Open Mic at the Common Market Southend
Common Market SouthEnd, Charlotte
Heathers: The Drag Musical
Petra's, Charlotte
Browse Charlotte Events
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A 1-cent increase in the sales tax for seven years would pay for roads, bridges, parks, vocational training and mental health facilities in Collier.
With just over a month to go before voters decide the issue of a one-cent increase in the sales tax for seven years to pay for infrastructure improvements in Collier County, proponents of the idea are going social.
The Greater Naples Chamber of Commerce, which initially investigated the sales tax idea as a way to pay for a backlog of road, bridge, park, mental health and other facilities, has formed a political action committee that is becoming more active as the vote looms.
The chamber has put $45,000 into the PAC, called One Collier, since it formed June 1, with individual supporters kicking in almost $5,000 more.
Most of the $27,000 spent so far has gone to Digital Savvy, a social and digital advertising firm based in Ocala.
Part of that effort is an active Facebook presence.
Local leaders, including former state Rep. Dudley Goodlette and Ed Morton, the former CEO of the NCH Healthcare System, have recorded short videos extolling the benefits of the proposed tax.
Goodlette explains how the sales tax would diversify the county's revenue stream, which now relies mostly on property taxes, with impact fees, other fees and borrowing making up the rest. He talks about the resident oversight committee that would monitor how the tax is being spent and the tax's sunset after seven years — sooner if the targeted amount of $490 million is raised in less time.
Mary Ann Guerra, director of quality improvement at the David Lawrence Center, also cut a video for One Collier. The center would likely partner with the county to add mental health facilities if the tax passes.
"I feel strongly that behavioral health is important to our overall health. The opportunity to support a sales tax is a no-brainer to me," she said.
The videos haven't always been warmly received.
But the comments appearing after Goodlette's video, for example, weren't what the PAC was hoping for.
"No thanks go back to the high tax state you came from," one commenter wrote to Goodlette, who has lived in Southwest Florida since the 1950s.
"Live within you means. Quit spending others money," wrote another.
While Goodlette and Garrett Richter, another One Collier supporter, are both Republicans who have served in the state Legislature, the local Republican Party is taking a dim view of the tax.
Ron Kezeske, chairman of Collier County Republican Executive Committee, has taken a lead role in opposing the ballot measure.
He argues the tax will hit small businesses that lease space especially hard, since the cap of $5,000 that limits the amount of additional tax on most large purchases wouldn't apply to leases.
Many of the projects on the county's list are "wants" rather than "needs," Kezeske and other opponents argue.
The best argument for the tax, one set forth in the videos and other material produced by One Collier, is that it offers an alternative means of paying for projects that are going to be built regardless.
Some, such as the eastward extension of Vanderbilt Beach Road, are needed to relieve traffic on other east-west routes and are firmly in the county's plans, albeit without full funding.
Others, such as a vocational training center and mental health facilities, would improve the quality of life.
Existing property owners will end up paying for most of the new infrastructure coming to Collier County through their property taxes.
The sales tax would shift a significant portion, about 30 percent, to visitors.
Paying for the same projects by borrowing also would shift some of the cost away from existing taxpayers — namely to future property owners who will help repay the bonds over time.
A key difference is that under the first scenario, the cost of the projects would be lower because it wouldn't be necessary to borrow money and pay interest over the long term.
Passage of any tax increase is difficult. For One Collier to win over a majority of Collier County voters, it will need all the Digital Savvy it can muster, and then some.
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The First Official Game of Thrones Season 8 Teaser Has Arrived
BY Natalie Zamora
HBO just loves to keep teasing us with what they know we so desperately want.
It's been confirmed that Game of Thrones will return for its eighth and final season in April 2019. And though there will only be six episodes, each one is set to be over one hour long, and will hopefully answer all of our burning questions.
Although a trailer has not been released today, and this actually shows no footage of the beloved show whatsoever, the first official Game of Thrones Season 8 teaser is out, and it's still enough to get us hyped.
The teaser, which debuted at Brazil Comic Con in São Paulo, shows the Painted Table at Dragonstone catching fire, symbolizing fire and ice. The 54-second clip ends with the Game of Thrones logo, followed by what we already know: "The Final Season April 2019."
This, we suppose, is one step closer to an actual trailer, as the previous Game of Thrones teaser included only footage from previous seasons, along with the announcement of when Season 8 would premiere. We'll take whatever progress we can get at this point.
entertainment Game of Thrones News Pop Culture Television tv
A Stranger Things Fan Is Selling Epic Demogorgon Dog Costumes on Etsy
BY Jessica Bowman
Joe Keery, Maya Hawke, Priah Ferguson, and Gaten Matarazzo in Stranger Things.
Stranger Things is great at placing the truly terrifying alongside the absolutely adorable. One minute we are gushing over Eleven and Mike's teen romance, and the next we're jumping off the couch at the sight of those possessed by the Mind Flayer.
No matter how seamless the Duffer Brothers' Netflix series is in weaving together these moments, it seems like it would be impossible to make the Demogorgon cute. But somehow, one crafty fan has done just that.
Etsy shop ThatCraftyFriendShop has created Demogorgon headpieces that fit perfectly on your dog's head.
Turn Your Dog Into an Adorable Demogorgon with Stranger Things Dog Costumes on Etsy https://t.co/sJgoD0OzeU
— People (@people) July 16, 2019
People reports that the headpieces range in size from extra small (for 5- to 10-pound dogs) all the way to extra large (for dogs over 75 pounds). Prices range from $25 to $75, depending on the size of your four-legged friend.
These wool and felt doggy costumes are perfect for Halloween, or even a Stranger Things watch party while you continue to binge and re-binge the third season—with a decked-out doggy by your side.
[h/t People]
Animals cute dogs entertainment Netflix News Pets Pop Culture shopping Smart Shopping Stranger Things Television tv
J.K. Rowling Reveals How San Francisco Inspired Major Harry Potter Location
Jamie McCarthy, Getty Images
The award-winning play Harry Potter and the Cursed Child is about to open at the Curran Theater in San Francisco. The two-part drama takes place 19 years after the events in Harry Potter and the Deathly Hallows and depicts Harry's life as his son, Albus, is about to begin school at Hogwarts.
J.K. Rowling has pointed out that San Francisco had a deep influence on the original Harry Potter novels, SFGate reports. In the video below, Rowling talks about how Alcatraz, the infamous former prison, inspired her creation of Azkaban.
"[San Francisco] is a very distinctive, special place—I love the feel of it, I love the architecture," Rowling said. "I've actually said this before, but Azkaban is a combination of Alcatraz and Abbadon, which is an old word for hell. I squeezed those words together. The idea of the rock in the middle of the ocean was directly inspired by a visit to Alcatraz."
With its mist and Gothic mood, it's no wonder this slice of San Francisco inspired a big part of the Harry Potter world.
[h/t SFGate]
books celebrities cities entertainment Harry Potter News Pop Culture theater bookcorner
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Learn more about our Southeast Asian language offerings, including Indonesian/Malay 101 and 102.
Read the Arches Magazine feature article on the experiential pedagogy behind the 2017 field school trip.
Explore video and photos from our 2017 LIASE Southeast Asia Symposium, including our keynote presentation by Abidin Kusno, student research panels, LIASE faculty presentations, and performances from Thai musical groups Chaopraya Ensemble, PongLaang RuamJai, and The Wayside.
Learn more about the 2019 LIASE field school in Thailand on Wednesday, October 10th at 12pm in McIntyre 309.
The 4th annual LIASE Southeast Asia Symposium focuses on the intersections of culture and sustainability in Indonesia and throughout Southeast Asia, and will feature internationally renowned scholar Abedin Kusno of York University as keynote speaker.
Explore video and photos from our 2016 LIASE Southeast Asia Symposium, including our keynote presentations, student research panels, Southeast by Northwest faculty panels, and a performance excerpt from Gamelan Pacifica.
This year's symposium features internationally renowned scholars, artists, musicians, and even craft workshops. Join us!
Learn to make your own batik, the Indonesian method of dying textiles and art, at one of our two workshops during the Southeast Asia Symposium.
The 2017 field school course, SOAN 312, will teach students about Indonesia in cultural context, both over the course of the spring semester, and with a three-week trip to Central Java in May and early June.
Philip Dearden of the University of Victoria is an esteemed scholar of marine conservation in Southeast Asia. His talk was titled "Highlands to Islands: Insights on Conservation and Culture from Thailand"
Learn more about Keynote Speaker Philip Dearden and the 2015 Southeast Asia Symposium, which was held October 23-24, 2015.
Elena Becker, a participant in our 2015 LIASE field-school course in Malaysia, describes her research on notions of authenticity in Sarawak's ethnic tourism.
Watch video of our 2014 keynote, browse student research presentations, and learn about our goals and vision for the Southeast Asia Symposium.
ALERT: To donate to victims of the recent earthquakes that have caused humanitarian crises in several areas of Indonesia, please visit the Indonesian Red Cross.
With generous support from the Henry Luce Foundation, the University of Puget Sound LIASE program focuses on initiatives designed to promote richer and deeper understandings of Southeast Asian cultural, linguistic, and environmental topics among students and faculty. These programs include field-school courses involving research and service in Southeast Asia, new Southeast Asian language courses taught on campus, and faculty field initiatives in the region. The Southeast Asia Symposium brings all of these programs together on the Puget Sound campus, allowing students and faculty to share their research and development with the broader community, while serving as a forum for international speakers, Northwest faculty specialists, and partner organization representatives from Southeast Asia.
Learn more about the LIASE program and our Southeast Asia focus.
The LIASE program is administered by University of Puget Sound faculty from a variety of departments and programs with expertise on Southeast Asian languages, cultural, and environmental topics. We would like to express our gratitude to the Henry Luce Foundation for their support of our programming.
For full contact information see About the Symposium page.
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Q: data.rolling().mean() works in console, but not when run I try to compute a moving average with the following code :
mav05 = dataT['close'].rolling(window=5).mean() on the 'close' values of a list of prices contained in data DataFrame. I intend then to use : dataTrad['mav05'] = pd.Series(mav05, index = dataTrad.index)
Both lines work perfectly in console. But when I run it, I get a SyntaxError: invalid syntax message.
I then tried dataT['mav05'] = dataT['close'].rolling(window=5).mean() to replace the two lines. and I get the exact same error message.
What do I do wrong ?
THX for your help ;-)
Full code :
import pandas as pd
import sqlite3 as sq
import numpy as np
from sqlalchemy import create_engine
# ==== Connecting to SQL DB and loading lists ============
connect = sq.connect('tradData.db')
cursor = connect.cursor()
tS = pd.read_sql_query("SELECT * FROM TRBel20", connect)
dataTrad = tS.rename(columns={"Price Open": "open", "Price Close": "close", "Price High": "high", "Price Low": "low", "Volume": "volume" })
dataTrad.drop(dataTrad[pd.isnull(dataTrad["close"])].index, inplace=True) # clean DB by dropping all rows where closing price is null
dataTrad.drop(dataTrad[pd.isnull(dataTrad["open"])].index, inplace=True) # clean DB by dropping all rows where closing price is null
# ==== Adding technical parameters ============
# moving average
mav05 = dataTrad['close'].rolling(window=5).mean()
mav20 = dataTrad['close'].rolling(window=20).mean()
mav60 = dataTrad['close'].rolling(window=60).mean()
# bollinger band
rstd20 = dataTrad['close'].rolling(window=20).std()
bollUpperband = mav20 + 2 * rstd20
bollLowerband = mav20 - 2 * rstd20
dataTrad['mav05'] = pd.Series(mav05, index = dataTrad.index)
dataTrad['mav20'] = pd.Series(mav20, index = dataTrad.index)
dataTrad['BollUp'] = pd.Series(bollUpperband, index = dataTrad.index)
dataTrad['BollDown'] = pd.Series(bollLowerband, index = dataTrad.index)
A: A question sent to me made things clear... I had misplaced parenthesis above and the code stopped not because of wrong rolling.mean() syntax but because of previous error.
Thanks so much for the wise private question and sorry to have bothered you ;-)
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{
"redpajama_set_name": "RedPajamaStackExchange"
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"""Test the plugit proxy from an external point of view (Using the standalone proxy and a simple plugit service)"""
from test_proxy_with_service import *
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{
"redpajama_set_name": "RedPajamaGithub"
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On the back of our successes providing CDM client advisory services on other Cat A office fit-outs, we were appointed by CES to the Atlantic House scheme.
The works comprised of a comprehensive refurbishment to a Grade A specification with a construction value of £1,260,890; including reception, common areas and floor plates of Atlantic House at Birchwood Point. M&E installations were upgraded throughout with external works to improve the car park.
The refurbishment commenced in July 2017 and were completed in November 2017 providing 20,676 sq ft of new office space.
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\section{Introduction}
Parabolic vector bundles over a compact Riemann surface $\Sigma$ with $n$
marked points are holomorphic
vector bundles over $\Sigma$ with a weighted flag structure over each
of the marked
points. They were introduced by Seshadri, \cite{S}, and are of interest for
many
reasons. For instance, there is a natural bijective correspondence between
the isomorphism classes of polystable parabolic bundles of parabolic degree
zero and the equivalence classes of unitary representations of the fundamental
group of the $n$-punctured surface.
Parabolic Higgs bundles are pairs of the form $(E,\Phi)$, where $E$ is a
parabolic vector bundle on $\Sigma$
and $\Phi$ is a meromorphic $End(E)$-valued $1$-form
holomorphic outside the $n$ marked points such that $\Phi$
has at most a simple pole with nilpotent residue (with respect to the flag)
at each of the marked points.
There is a natural relationship between the polystable parabolic Higgs bundles
of parabolic degree zero and the representations
of the fundamental group of
the $n$-punctured surface in the general linear groups \cite{Si}. Parabolic
Higgs bundles have been studied in other works such as \cite{BY,N1,K1,GM}.
We will be particularly interested in the case of parabolic Higgs
bundles of rank two over a $n$-pointed Riemann surface of genus zero.
Consider the split real form ${\rm PGL}(2,\ensuremath{\mathbb R})$ of ${\rm PGL}(2,\ensuremath{\mathbb C})$ defined by the
involution $A\, \longmapsto\, \overline{A}$. It produces
the anti-holomorphic involution on the moduli space of representations
corresponding to the holomorphic involution
\begin{equation}\label{eq:0.1}
\sigma:(E\, ,\Phi) \,\longmapsto\, (E\, ,-\Phi)
\end{equation}
of the moduli space of parabolic Higgs bundles \cite{Hi1}.
Note that $\sigma$ is the restriction to $-1$ of the $\text{U}(1)= S^1$-action
on the moduli space of parabolic Higgs bundles defined by
$$
\lambda\cdot (E, \Phi)\, =\, (E,\lambda \Phi), \quad \lambda \,\in\,
S^1\, .
$$
The isomorphism classes of stable parabolic Higgs bundles fixed by this
involution correspond to ${\rm SU}(2)$ or ${\rm SL}(2,\ensuremath{\mathbb R})$ representations, the
former corresponding to parabolic Higgs bundles with zero Higgs field;
see \cite{Hi1}.
We study the fixed points in the special case where the
underlying vector bundle is holomorphically trivial.
Let $\mathcal{H}(\beta)$ be the moduli space of
parabolic Higgs bundles $(E,\Phi)$, where $E$ is a
holomorphically trivial vector bundle over $\ensuremath{\mathbb C}
\ensuremath{\mathbb P}^1$ of rank two with a weighted complete flag structure over each of
the $n$
marked points $x_1,\cdots,x_n$
\begin{align*}
E_{x_i,1}\,\supsetneq\, E_{x_i,2} \,\supsetneq\, 0\, , \\
0 \,\leq\, \beta_1(x_i) \,< \,\beta_2(x_i) \,<\,1\, .
\end{align*}
As shown in \cite{GM},
there is an isomorphism between $\mathcal{H}(\beta)$
and the hyperpolygon space $X(\alpha)$, with
$\alpha_i\,=\,\beta_2(x_i)-\beta_1(x_i)$, defined as a hyper-K\"{a}hler
quotient
of $T^* \ensuremath{\mathbb C}^{2n}$ by
$$K \,:= \,\Big({\rm U}(2) \times {\rm U}
(1)^n\Big)/{\rm U}(1)\,=\, \Big({\rm SU}(2) \times
{\rm U}(1)^n\Big)/ (\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z})\, ,$$
where $\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}$ acts by multiplication of each factor by $-1$.
(See also Sections~\ref{hyperpolygons} and \ref{PHBs} for details.)
Using this correspondence between the two moduli spaces, we study in
Section~\ref{sec:inv} the fixed point set of the corresponding involution
of $X(\alpha)$ defined by
\begin{equation}\label{eq:0.2}
\sigma\,:\,[p,q] \,\longmapsto\, [-p,q]\, ,
\end{equation}
with $(p,q)\in T^* \ensuremath{\mathbb C}^{2n}$. We show that this fixed-point set is formed
by $M(\alpha)$, the space of polygons in
$\ensuremath{\mathbb R}^3$ obtained when $p=0$, and several other connected components
$Z_S$, where $S$ runs over all subsets of $\{1\, , \cdots \, ,n\}$ with
$\lvert S\rvert \geq 2$ and
\begin{equation}\label{eq:0short}
\sum_{i\in S} \alpha_i \,<\, \sum_{i\in S^c} \alpha_i
\end{equation}
(the complement of $S$ is denoted by $S^c$).
These components $Z_S$ are all non-compact except when $\lvert S \rvert \,=\,n-1$, in
which case $Z_S\,=\,\ensuremath{\mathbb C} \ensuremath{\mathbb P}^{n-2}$ and $M(\alpha)$ is empty. Let $\mathcal{S}'(\alpha)$ be the collection of all subsets of $\{1\, , \cdots \, ,n\}$ with
$\lvert S\rvert \geq 2$ satisfying \eqref{eq:0short}.
We describe these sets $Z_S$ and the corresponding
components ${\mathcal Z}_S$ of the fixed point set of the involution of
${\mathcal H}(\beta)$ defined in \eqref{eq:0.1}; the following
theorem is proved (see Section~\ref{sec:inv}).
\begin{theorem}\label{thm:0.1}
The fixed-point set of the involution in \eqref{eq:0.1} of the space of
parabolic Higgs bundles $\mathcal{H}(\beta)$ is
$$ \mathcal{H}(\beta)^{\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}}\,=\, \mathcal{M}_{\beta,2,0} \cup \bigcup_{S \in
\mathcal S'(\alpha)} \mathcal{Z}_S\, ,$$
with $\alpha_i=\beta_2(x_i)-\beta_1(x_i)$, where
$\mathcal{M}_{\beta,2,0}$ is the space of rank two degree zero
parabolic vector bundles over $\ensuremath{\mathbb C}{\mathbb P}^1$, and where
$\mathcal{Z}_S\,\subset\, \mathcal{H}(\beta)$ is formed by parabolic Higgs
bundles $\mathbf{E}\,=\,(E,\Phi)\in \mathcal{H}(\beta)$ such that
\begin{enumerate}
\item[(i)] the parabolic vector bundle $E$ admits
a direct sum decomposition $E\,=\,L_0\oplus L_1$,
where $L_0$ and $L_1$ are parabolic line bundles where the
parabolic weight of $L_0$ (respectively, $L_1$) at
$x_i\,\in\, S^c$ is $\beta_2(x_i)$ (respectively, $\beta_1(x_i)$), and
the parabolic weight of $L_0$ (respectively, $L_1$) at
$x_i\,\in\, S$ is $\beta_1(x_i)$ (respectively, $\beta_2(x_i)$);
\item[(ii)] the residues of the Higgs field $\Phi$ at the parabolic points $x_i$
are either upper or lower triangular with respect to the above decomposition, according to whether $i$ is in $S$ or in $S^c$.
\end{enumerate}
Moreover, $\mathcal{Z}_S$ is a non-compact manifold of dimension $2(n-3)$ except
when $\lvert S\rvert \,= \,n-1$, in which case $\mathcal{Z}_S\,=\,
\mathcal{M}_S$ is compact and diffeomorphic to $\ensuremath{\mathbb C} \ensuremath{\mathbb P}^{n-3}$. In all cases, $\mathcal{H}(\beta)^{\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}}$ has $2^{(n-1)}-(n+1)$ non-compact components and one compact component.
\end{theorem}
\begin{rem}
\mbox{}
\begin{itemize}
\item {\rm Since the vector bundle underlying $E$ is holomorphically trivial, it follows that the
holomorphic line bundles underlying $L_0$ and $L_1$ are both holomorphically
trivial. }
\item {\rm Statement (i) in Theorem \ref{thm:0.1} means that if
\begin{align*}
E_{x_i,1} & \supset E_{x_i,2} \supset 0\\
0 \leq \beta_{1}(x_i) & < \beta_{2}(x_i) < 1
\end{align*}
is the parabolic structure, then
$E_{x_i,2}=E_{x_j,2}$ whenever $i,j \,\in\, S$ or $i,j \,\in\,
S^c$. Note that this condition is independent of the choice of the
trivialization of $E$.}
\end{itemize}
\end{rem}
In Section~\ref{sec:Mink}, we show that for any
$S\,\in \mathcal S'(\alpha)$, the corresponding component of the fixed
point sets of the involution of $X(\alpha)$ (or of $\mathcal{H}(\beta)$) is
diffeomorphic to a moduli space
of polygons in Minkowski $3$-space, meaning $\ensuremath{\mathbb R}^3$ equipped with the Minkowski inner
product
$$
v \circ w \,=\, -x_1x_2-y_1y_2+t_1t_2\, ,
$$
for $v=(x_1,y_1,t_1)$ and $w\,=\,(x_2,y_2,t_2)$.
The surface $S_R$ in $\ensuremath{\mathbb R}^3$ defined by the equation $-x^2-y^2+t^2=R^2$ (a
\emph{pseudosphere} of radius $R$) has two connected components: $S_R^+$,
corresponding to $t>0$, which is called a \emph{future pseudosphere}, and
$S_R^-$,
corresponding to $t<0$, which is called a \emph{past pseudosphere}. The group
${\rm SU}(1,1)$ acts transitively on $S_R^+$ (respectively,
$S_R^-$) since one can think of $\ensuremath{\mathbb R}^3$
as $\mathfrak{su}(1,1)^*$ with
$S_R^+$ (respectively, $S_R^-$)
being an elliptic coadjoint orbit. Consequently, both
$S_R^+$ and $S_R^-$ have the ${\rm SU}(1,1)$--invariant Kostant--Kirillov symplectic structure
of a coadjoint orbit. Fixing
two positive integers $k_1,k_2$ with $k_1+k_2\,=\,n$, we consider closed
polygons in Minkowski $3$-space with the first $k_1$ sides lying in future
pseudospheres of radii $\alpha_1, \cdots,\alpha_{k_1}$ and the last $k_2$
sides lying in past pseudospheres of radii $\alpha_{k_1+1}, \cdots ,
\alpha_n$. The space of all such closed polygons can be
identified with the zero level set of the moment map
\begin{equation*}
\mu\,:\,\mathcal{O}_1\times \cdots \times \mathcal{O}_n\,\longrightarrow\,
\mathfrak{su}(1,1)^*
\end{equation*}
for the diagonal ${\rm SU}(1,1)$-action, where the coadjoint ${\rm SU}
(1,1)$--orbit $\mathcal{O}_i\,\cong\, S_{\alpha_i}^+$, $1 \leq i \leq k_1$,
is a future pseudosphere of radius $\alpha_i$, and
$\mathcal{O}_i\,\cong\, S_{\alpha_i}^-$, $k_1+1 \leq i \leq n$, is a
past pseudosphere of radius $\alpha_i$, equipped with its Kostant--Kirillov
symplectic structure \cite{F}. Then the corresponding moduli space of polygons is defined as the symplectic quotient
$$
M^{k_1,k_2}(\alpha)\,:=\,\mu^{-1}(0)/{\rm SU}(1,1)\, .
$$
We have the following result.
\begin{theorem}
For any $S\,\in\, \mathcal{S}^\prime(\alpha)$, the components $\mathcal{Z}_S$
and $Z_S$, of the fixed-point sets of the involutions in \eqref{eq:0.1} and
\eqref{eq:0.2} respectively, are diffeomorphic to the moduli space
$$
M^{\lvert S\rvert,\lvert S^c \rvert}(\alpha)
$$
of closed polygons in Minkowski $3$-space.
\end{theorem}
This interpretation allows us to see the fixed-point set of the above
involutions as a moduli space of another related problem, thus helping us to
understand many of its geometrical properties as seen in the example of Section~\ref{sec:ex}.
\medskip
\noindent
\textbf{Acknowledgements.}\, We thank O. Garc\'{i}a-Prada for suggesting the study
of the hyperpolygon description of the fixed-point set of
the natural involution of the moduli space of parabolic Higgs bundles.
\section{Hyperpolygon spaces} \label{hyperpolygons}
Let $\mathcal Q$ be the star-shaped quiver with vertices parametrized by $I\cup
\{0\} \,=\, \{1\, ,\cdots\, ,n\}\cup \{0\}$ and the arrows parametrized by $I$
such that, for any $i\, \in\, I$, the tail and the head of the corresponding
arrow are $i$ and $0$ respectively. Consider all representations of $\mathcal
Q$ with $V_i \,=\, \ensuremath{\mathbb C} $, for $i \,\in\, I$, and $V_0\,=\,
\ensuremath{\mathbb C}^2$. They are parametrized by
$$
E(\mathcal Q, V)\,:=\, \bigoplus_{i\in I}
\textrm{Hom} (V_i, V_0) \,=\, \ensuremath{\mathbb C}^{2n}\, .
$$
Using the actions of ${\rm U}(1)$ and ${\rm U}(2)$ on $\ensuremath{\mathbb C}$ and $\ensuremath{\mathbb C}^2$
respectively, we construct an action of ${\rm U}(2)\times{\rm U}(1)^n$ on
$E(\mathcal Q, V)$. This action produces an action of ${\rm U}(2)\times{\rm
U}(1)^n$ on the cotangent bundle $T^*E(\mathcal Q, V)\,=\, T^*\ensuremath{\mathbb C}^{2n}$. One
gets a hyper-K\"ahler quiver variety by performing the hyper-K\"ahler reduction
on $T^*E(\mathcal Q, V)$ for this action of ${\rm U}(2)\times{\rm U}(1)^n$.
Since the diagonal circle
$$
\{(c\cdot \text{Id}_{\ensuremath{\mathbb C}^{2}}, c,\cdots ,c)\, \mid\, ~
|c|\,=\, 1\}\,\subset\, {\rm U}(2) \times {\rm U}(1)^n
$$
acts trivially on $T^*E(\mathcal Q, V)$, the action factors through the
quotient group
$$K \,:=\, \Big( {\rm U}(2) \times {\rm U}(1)^n\Big)/{\rm U}
(1)\,=\, \Big({\rm SU}(2) \times
{\rm U}(1)^n\Big)/(\ensuremath{\mathbb Z} /2\ensuremath{\mathbb Z})\, ,$$
where $\ensuremath{\mathbb Z} /2\ensuremath{\mathbb Z}$ acts as multiplication by $-1$ on each factor.
As $T^* \ensuremath{\mathbb C}^2 \,=\, (\ensuremath{\mathbb C}^{2})^* \times \ensuremath{\mathbb C}^2$ can be identified with
the space of quaternions, the cotangent bundle $T^*E(\mathcal Q, V) \,=\, T^* \ensuremath{\mathbb C}^{2n}$ has
a natural hyper-K\"ahler structure (see for example \cite{K2, H}).
The hyper-K\"ahler quotient of $T^* \ensuremath{\mathbb C}^{2n}$
by $K $ can be explicitly described as follows. Let
$(p,q)$ be coordinates on $T^* \ensuremath{\mathbb C}^{2n}$, where $p=(p_1, \cdots, p_n)$
is the $n$-tuple of row vectors $p_i =\left( \begin{array}{ll} a_i &
b_i\end{array}\right) \in (\ensuremath{\mathbb C}^2)^*$ and
$q= (q_1, \cdots, q_n)$ is the $n$-tuple of column vectors
$q_i =\Big( \begin{array}{c}
c_i \\ d_i
\end{array} \Big) \in \ensuremath{\mathbb C}^2$. In terms of these coordinates,
the action of $K$ on $ T^*\ensuremath{\mathbb C}^{2n}$ is given by
$$ (p,q) \cdot [A; e_1, \cdots, e_n]=
\Big( (e_1^{-1}p_1 A, \cdots, e_n^{-1}p_n A), ( A^{-1} q_1 e_1, \cdots, A^{-1}
q_n e_n ) \Big).$$
This action is hyper-Hamiltonian with hyper-K\"ahler moment map
$$ \mu_{HK}:= \mu_{\ensuremath{\mathbb R}} \oplus \mu_{\ensuremath{\mathbb C}} : T^* \ensuremath{\mathbb C}^{2n} \longrightarrow
\big(\mathfrak{su}(2)^* \oplus (\ensuremath{\mathbb R}^n)^*\big) \oplus \big(\mathfrak{sl}(2,
\ensuremath{\mathbb C})^* \oplus (\ensuremath{\mathbb C}^n)^*\big)\, ,$$
\cite{K2}, where the real moment map $\mu_{\ensuremath{\mathbb R}}$ is given by
\begin{equation} \label{real}
\mu_{\ensuremath{\mathbb R}} (p,q)\,=\,\frac{\sqrt{-1}}{2} \sum_{i=1}^n (q_i q_i^* -p_i^* p_i )_0
\oplus \Big(\frac{1}{2} (|q_1|^2 -|p_1|^2), \cdots,
\frac{1}{2} (|q_n|^2 -|p_n|^2) \Big)\, ,
\end{equation}
and the complex moment map $\mu_{\ensuremath{\mathbb C}}$ is given by
\begin{equation} \label{complex}
\mu_{\ensuremath{\mathbb C}} (p,q)\,=\,- \sum_{i=1}^n (q_i p_i)_0 \oplus (\sqrt{-1}p_1 q_1,
\cdots,\sqrt{-1} p_n q_n)\, .
\end{equation}
The hyperpolygon space $X(\alpha)$ is then defined to be the hyper-K\"ahler
quotient
\begin{equation}\label{icd}
X(\alpha)\,:=\,T^*\ensuremath{\mathbb C}^{2n} / \!\! / \!\!/ \!\!/_{\alpha} K\,:=\,
\Big( \mu_{\ensuremath{\mathbb R}}^{-1} (0, \alpha) \cap \mu_{\ensuremath{\mathbb C}}^{-1} (0,0) \Big) / K
\end{equation}
for $\alpha\,=\,(\alpha_1, \cdots, \alpha_n )\,\in\,\ensuremath{\mathbb R}^n_+$.
An element $(p,q)\,\in\,T^*\ensuremath{\mathbb C}^{2n}$ is in $\mu_{\ensuremath{\mathbb C}}^{-1} (0,0)$ if
and only if
$$
p_i q_i\,=\,0 \quad \text{and} \quad \sum_{i=1}^n (q_i p_i)_0\, =\, 0\, .
$$
In other words, an element $(p,q)$ of $T^*\ensuremath{\mathbb C}^{2n}$ is in $\mu_{\ensuremath{\mathbb C}}^{-1}
(0,0)$ if and only if
\begin{equation}\label{complex1}
a_i c_i + b_i d_i = 0
\end{equation}
and
\begin{equation}\label{complex2}
\sum_{i=1}^n a_i c_i - b_i d_i =0, \quad
\sum_{i=1}^n a_i d_i =0, \quad
\sum_{i=1}^n b_i c_i =0\, .
\end{equation}
Similarly, $(p,q)$ is in $\mu_{\ensuremath{\mathbb R}}^{-1} (0, \alpha) $ if and only if
$$
\frac{1}{2} \big(|q_i|^2 -|p_i|^2\big) \,=\, \alpha_i \quad \text{and} \quad
\sum_{i=1}^n \big(q_i q_i^* -p_i^* p_i \big)_0 \,=\,0\, ,
$$
i.e., if and only if
\begin{equation}\label{real1}
|c_i|^2 +|d_i|^2 - |a_i|^2 - |b_i|^2 \,=\, 2 \alpha_i
\end{equation}
and
\begin{equation}\label{real2}
\sum_{i=1}^n |c_i|^2 - |a_i|^2 + |b_i|^2 - |d_i|^2 \,=\,0, \quad
\sum_{i=1}^n a_i \bar{b_i} - \bar{c_i} d_i \,=\,0\, .
\end{equation}
An element $\alpha\,=\,(\alpha_1, \cdots, \alpha_n )\,\in\,\ensuremath{\mathbb R}^n_+$ is said to
be
\emph{generic} if and only if
\begin{equation}
\label{eq:epsilon}
\varepsilon_S(\alpha)\,:= \,\sum_{i \in S} \alpha_{i} - \sum_{i \in S^c}
\alpha_{i} \,\neq\, 0
\end{equation}
for every subset $S\,\subset\,\{1, \cdots, n\}$. For a generic $\alpha$, the
hyperpolygon space $X(\alpha)$ is a
non-empty complex manifold of complex dimension $2(n-3)$ (see \cite{K2} for details).
Hyperpolygon spaces can be described from an algebro-geometric point of view as geometric
invariant theoretic quotients. To elaborate this, we need the
stability criterion, developed by
Nakajima \cite{N2,N3} for quiver varieties and adapted by Konno \cite{K2} to hyperpolygon spaces. We will recall this below.
Let $\alpha$ be generic. A subset $S\,\subset\,\{1, \cdots, n\}$ is called
\emph{short} if
\begin{equation}\label{eq:short}
\varepsilon_S(\alpha) \,<\,0
\end{equation}
and \emph{long} otherwise (see \eqref{eq:epsilon} for the definition
of $\varepsilon_S(\alpha)$).
Given $(p,q)\,\in\, T^* \ensuremath{\mathbb C}^{2n}$ and a subset $S\,\subset\,\{1, \cdots, n\}$, we
say that $S$ is \emph{straight} at $(p,q)$ if $q_i$ is proportional to $q_j$
for all $i, j \,\in\, S$.
\begin{theorem}[\cite{K2}]\label{alpha-stability}
Let $\alpha \in \ensuremath{\mathbb R}^n_{+}$ be generic. A point $(p,q)\,\in \,T^* \ensuremath{\mathbb C}^{2n}$ is
\emph{$\alpha$-stable}
if and only if the following two conditions hold:
\begin{itemize}
\item[(i)] $q_i\,\neq\, 0$ for all $i$, and
\item[(ii)] if $S\,\subset\, \{1, \cdots, n \}$ is straight at $(p,q)$ and $p_j
\,=\,0$ for all $j \in S^c$, then $S$ is short.
\end{itemize}
\end{theorem}
\begin{rem}\label{maximalstraight}
{\rm Note that it is enough to verify (ii) in Theorem \ref{alpha-stability} for all
maximal straight sets,
that is, for those that are not contained in any other straight set at $(p,q)$.}
\end{rem}
Let $\mu_{\ensuremath{\mathbb C}}^{-1} (0,0)^{\alpha\text{-st}}$ denote the set of points in
$\mu_{\ensuremath{\mathbb C}}^{-1} (0,0)$ that are
$\alpha$-stable, and let $$K^{\ensuremath{\mathbb C}}\,:=\, ({\rm SL}(2,\ensuremath{\mathbb C}) \times (\ensuremath{\mathbb C}^*)^n) /(\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z})$$
be the complexification of $K$.
\begin{proposition}[\cite{K2}]\label{git}
Let $\alpha\,\in\,\ensuremath{\mathbb R}^n_{+}$ be generic. Then
$$\mu_{HK}^{-1} \big( (0, \alpha),(0,0) \big) \,\subset\, \mu_{\ensuremath{\mathbb C}}^{-1}
(0,0)^{\alpha\text{-{\rm st}}}\, ,$$
and there exists a natural bijection
$$ \iota\,:\,\mu_{HK}^{-1} \big( (0, \alpha),(0,0) \big)/K\,\longrightarrow\,
\mu_{\ensuremath{\mathbb C}}^{-1} (0,0)^{\alpha\text{-{\rm st}}}/ K^{\ensuremath{\mathbb C}}.$$
\end{proposition}
{}From Proposition \ref{git} and the definition in \eqref{icd} it follows
that
$$X(\alpha) \,=\, \mu_{\ensuremath{\mathbb C}}^{-1} (0,0)^{\alpha\text{-st}} / K^{\ensuremath{\mathbb C}}\, . $$
Following \cite{HP}, we denote the elements in $\mu_{\ensuremath{\mathbb C}}^{-1}
(0,0)^{\alpha\text{-st}}/ K^{\ensuremath{\mathbb C}}$ by
$[p,q]_{\alpha\text{-st}}$, and denote by $[p,q]_{\ensuremath{\mathbb R}}$ the elements in
$\mu_{HK}^{-1}
\big( (0, \alpha),(0,0) \big)/K$, when we need to make an explicit use of one of the two constructions. In all other cases, we will simply write $[p,q]$ for a hyperpolygon in $X(\alpha)$.
\subsection{A circle action}
Consider the $S^1$-action on $X(\alpha)$ defined by
\begin{equation}\label{action}
\lambda \cdot [p,q] \,=\, [\lambda \, p, q]\, .
\end{equation}
This action is Hamiltonian with respect to
symplectic structure on $X(\alpha)$; the associated moment map
$\phi\,:\,X(\alpha) \,\longrightarrow\, \ensuremath{\mathbb R}$ is given by
\begin{equation}\label{eq:phi}
\phi([p,q]) \,= \,\frac{1}{2}\sum_{i=1}^n |p_i|^2\, .
\end{equation}
This $\phi$ is a Morse-Bott function that is proper and bounded from
bellow. Following Konno \cite{K2}, let us consider $\mathcal S(\alpha)$,
namely the collection of short sets for $\alpha$, and its subset
$$ \mathcal S' (\alpha)\,:=\, \big\{ S \subset \{1, \cdots,n \}\,\mid\, S
\text{ is } \alpha\text{-short}, |S| \geq 2 \big\}\, . $$
For any $S\,\in\,\mathcal S'(\alpha)$, define
$$ X_S \,:=\, \big\{[p,q] \in X(\alpha)\,\mid\, S \text{ and } S^c\,
\text{ are straight at } (p,q) \text{ and }\, p_j=0\, \,
\forall\, \, j \in S^c \big\}\,.$$
Then the fixed-point set of the circle action on $X(\alpha)$ is
the following.
\begin{theorem}[\cite{K2}]\label{thm:fixedpoints}
The fixed point set for the $S^1$-action in \eqref{action} is
$$ X(\alpha)^{S^1}\,=\, M(\alpha) \cup \bigcup_{S \in \mathcal S'(\alpha)}
X_S\, .$$
The fixed-point set component $X_S$ is diffeomorphic to $\ensuremath{\mathbb C}\ensuremath{\mathbb P}^{|S|-2}$, and it
has index $2(n-1-|S|)$.
\end{theorem}
Let us now determine the isotropy weights of the circle action in \eqref{action}
at different fixed points.
For $S\,\in\, \mathcal S'(\alpha)$, let us fix
$[p^\prime,q^\prime]_{\alpha\text{-st}}\,\in\, X_S$. We may assume that for
each $i\,\in\, S$,
$$
q_i^\prime=\left( \begin{array}{c} c_i \\ 0 \end{array}\right)\, \text{ and }\quad p_i^\prime=\left(\begin{array}{cc} 0 & b_i \end{array}\right), \, \text{for $i \in S$}
$$
and for each $i\,\in\, S^c$,
$$
\quad q_i^\prime=\left( \begin{array}{c} 0 \\ d_i \end{array}\right)
\, \text{ and } \quad p_i^\prime=\left(\begin{array}{cc} 0 & 0 \end{array}\right), \, \text{for $i \in S^c$}.
$$
Moreover, we can assume that $S\,=\,\{1, \cdots, l \}$ and that
$b_1,b_2\,\neq\,
0$. Since $c_i,d_i \,\neq\, 0$ for all $i$, there exists a unique element $h
\,\in\, K^\ensuremath{\mathbb C}$ such that $(p^\prime,q^\prime)h\,=\,(p^0,q^0)\,\in\,
\mu^{-1}_\ensuremath{\mathbb C}(0,0)^{\alpha\text{-st}}$, where for each $i\,\in\, S$,
$$
q_i^0=\left( \begin{array}{c} 1\\ 0 \end{array}\right)\,
\text{ and } \quad p_i^0= \left(\begin{array}{cc} 0 & b_i^0\end{array}
\right)\, ,
$$
and for $i\,\in\, S^c$,
$$
q_i^0=\left( \begin{array}{c} 0 \\ 1\end{array}\right)\,
\text{ and }
\quad p_i^0=\left(\begin{array}{cc} 0 & 0\end{array}\right)\, ,
$$
with $b_1^0\,=\,1$ and $b_2^0 \,\neq\, 0$. There exists an open
neighborhood
$U$
of $(p^0,q^0)$ in $T^*\ensuremath{\mathbb C}^{2n}$ such that for all $(p,q)\,\in\, U \cap
\mu^{-1}_\ensuremath{\mathbb C}(0,0)$, there is a unique element $[A;e_1,\cdots,e_n]\,\in\,
K^\ensuremath{\mathbb C}$ satisfying the conditions that
$$
A^{-1} q_i e_i = \left\{ \begin{array}{l l} \left(\begin{array}{cc} 1 & 0 \end{array}\right)^t, & \text{if}\,\, i=1 \\
\left(\begin{array}{cc} 1 & r_1 \end{array}\right)^t, & \text{if}\,\, i=2 \\
\left(\begin{array}{cc} 1 & w_i \end{array}\right)^t, & \text{if}\,\, i=3,
\cdots, l \\
\left(\begin{array}{cc} w_i & 1 \end{array}\right)^t, & \text{if}\,\,
i=l+1,\cdots,n-1\\
\left(\begin{array}{cc} 0& 1 \end{array}\right)^t, & \text{if}\,\, i=n \end{array}\right.
$$
and
$$
e_i^{-1}p_i A = \left\{ \begin{array}{l l} \left(\begin{array}{cc} 0& 1 \end{array}\right), & \text{if}\,\, i=1 \\
\left(\begin{array}{cc} -r_1r_2 & r_2 \end{array}\right), &\text{if}\,\, i=2 \\
\left(\begin{array}{cc} -z_i w_i & z_i \end{array}\right), & \text{if}\,\,
i=3, \cdots, l \\
\left(\begin{array}{cc} z_i & -z_i w_i \end{array}\right), & \text{if}\,\,
i=l+1,\cdots,n-1\\
\left(\begin{array}{cc} r_3 & 0 \end{array}\right), & \text{if}\,\, i=n, \end{array}\right.
$$
where $r_1,r_2,r_3$ are uniquely determined by
\begin{equation}
\label{eq:coord1}
\{z_i,w_i\,\mid\, i=3,\cdots,n-1\}\, ;
\end{equation}
so the functions in \eqref{eq:coord1} define
a local coordinate system in $X(\alpha)$ around
$[p^\prime,q^\prime]_{\alpha\text{-st}}$. Indeed,
$\alpha$-stability is an open condition so that any $(p,q)\,\in\,
\mu_{\ensuremath{\mathbb C}}^{-1}(0,0)^{\alpha\text{-st}}$ sufficiently close to
$(p^\prime,q^\prime)$ will be $\alpha$-stable. Moreover, there exist unique,
up to multiplication by $\pm I$, $$e_1,e_n\,\in\, \ensuremath{\mathbb C}\setminus\{0\}\,
\, \text{ and }\, \, A\,\in\, {\rm SL}(2,\ensuremath{\mathbb C})\, $$ such that
$$
A^{-1}q_1 e_1 = \left(\begin{array}{c} 1 \\ 0 \end{array}\right), \quad \quad A^{-1}q_n e_n = \left(\begin{array}{c} 0 \\ 1 \end{array}\right)
$$
and
$$
e_1^{-1}p_1 A \,= \left(\begin{array}{cc} 0 & 1 \end{array}\right).
$$
Then one can uniquely determine $e_2,\cdots,e_{n-1}$ such that
$$
A^{-1} q_i e_i = \left(\begin{array}{c} 1 \\ * \end{array}\right), \text{ for
$i=2,\cdots, l$}, \quad \quad A^{-1}q_i e_i = \left(\begin{array}{c} * \\ 1
\end{array}\right), \text{ for $i=l+1,\cdots, n-1$}.
$$
Now it can be easily shown that the $S^1$-action (constructed in
\eqref{action}) in these local coordinates is given by
\begin{equation}
\label{eq:loccirc1}
\lambda \cdot (z_i,w_i)= \left\{ \begin{array}{ll} (z_i, \lambda w_i), &
\text{if $i=3,\cdots,l$} \\ \\ (\lambda^2z_i, \lambda^{-1}w_i), & \text{if
$i=l+1,\cdots, n-1$}. \end{array}\right.
\end{equation}
Let us now consider a fixed point $[0,q^\prime]_{\alpha\text{-st}}\,\in\,
M(\alpha)$. Then we may assume that
\begin{align*}
q^\prime_1 & = \left( \begin{array}{c} c_1 \\ 0 \end{array}\right), \,\, \text{with}\,\, c_1\neq 0, \quad q^\prime_2= \left( \begin{array}{c} c_2 \\ d_2 \end{array}\right), \, \text{with}\,\, c_2,d_2\neq 0, \\
q_3^\prime &=\left( \begin{array}{c} c_3 \\ d_3 \end{array}\right), \,\text{with} \,\, d_3\neq 0, \\
q_i^\prime & =\left( \begin{array}{c} c_i \\ d_i \end{array}\right), \,
\text{with} \,\, c_i\neq 0, \,\, \text{for} \,\, i=4,\cdots,n-1, \\
q_n^\prime & =\left( \begin{array}{c} 0 \\ d_n \end{array}\right), \, \text{with}\,\, d_n\neq 0,
\end{align*}
since $[0,q^\prime]_{\alpha\text{-st}}$ is not in any of the sets $X_S$.
As $c_i\,\neq\, 0$ for all $i\neq 3,n$, and $d_3\,\neq\, 0\, \neq\, d_n$, there
exists a
unique element $h\,\in\, K^\ensuremath{\mathbb C}$ such that $(0,q^\prime)h\,=\,(0,q^0)\,\in\,
\mu^{-1}_\ensuremath{\mathbb C}(0,0)^{\alpha\text{-st}}$, where
\begin{align*}
q^0_1 & = \left( \begin{array}{c} 1 \\ 0 \end{array}\right), \quad q^0_2= \left( \begin{array}{c}1 \\ 1 \end{array}\right), \quad
q_3^0 =\left( \begin{array}{c} w_3 \\ 1 \end{array}\right), \\
q_i^0 & =\left( \begin{array}{c} 1 \\ w_i \end{array}\right), \,\, \text{for}
\,\, i=4,\cdots,n-1, \\
q_n^0 & =\left( \begin{array}{c} 0 \\ 1 \end{array}\right).
\end{align*}
Then there exists an open neighborhood $U$ of $(0,q^0)$ in $T^*\ensuremath{\mathbb C}^{2n}$ such
that for all $(p,q)\in U \cap \mu^{-1}_\ensuremath{\mathbb C}(0,0)$, there is a unique element
$[A;e_1,\cdots, e_n]\in K^\ensuremath{\mathbb C}$ such that
$$
A^{-1} q_i e_i = \left\{ \begin{array}{l l}
\left(\begin{array}{cc} 1 & 0 \end{array}\right)^t, & \text{if}\,\, i=1 \\
\left(\begin{array}{cc} 1 & 1 \end{array}\right)^t, & \text{if}\,\, i=2 \\
\left(\begin{array}{cc} w_3 & 1 \end{array}\right)^t, & \text{if}\,\, i=3 \\
\left(\begin{array}{cc} 1 & w_i \end{array}\right)^t, & \text{if}\,\, i=4,
\cdots, n-1 \\
\left(\begin{array}{cc} 0& 1 \end{array}\right)^t, & \text{if}\,\, i=n \end{array}\right.
$$
and
$$
e_i^{-1}p_i A = \left\{ \begin{array}{l l}
\left(\begin{array}{cc} 0& r_1 \end{array}\right), & \text{if}\,\, i=1 \\
\left(\begin{array}{cc} -r_2 & r_2 \end{array}\right), & \text{if}\,\, i=2 \\
\left(\begin{array}{cc} z_3 & - z_3 w_3 \end{array}\right), &
\text{if}\,\, i=3, \cdots, l \\
\left(\begin{array}{cc} -z_i w_i & z_i \end{array}\right), & \text{if}\,\,
i=4,\cdots,n-1\\
\left(\begin{array}{cc} r_3 & 0 \end{array}\right), & \text{if}\,\, i=n, \end{array}\right.
$$
where $r_1,r_2,r_3$ are uniquely determined by
\begin{equation}
\label{eq:coord}
\{z_i,w_i\mid i=3,\cdots,n-1\}\, ;
\end{equation}
so \eqref{eq:coord} is a local coordinate system around
$[0,q^\prime]_{\alpha\text{-st}}$ in $X(\alpha)$. Indeed, $\alpha$-stability is
an open condition and, moreover, there exist unique, up to multiplication by
$\pm I$,
$$A\,\in\, {\rm SL}(2,\ensuremath{\mathbb C}) \,\, \text{ and }\,\, e_1,e_2,e_n\,\in\,
\ensuremath{\mathbb C}\setminus\{0\}$$ such
that
$$
A^{-1}q_1 e_1 = \left(\begin{array}{c} 1 \\ 0 \end{array}\right), \quad A^{-1}q_2
e_2 = \left(\begin{array}{c} 1 \\ 1\end{array}\right), \quad \text{and} \quad A^{-1}q_n e_n = \left(\begin{array}{c} 0 \\ 1 \end{array}\right).
$$
Then one can uniquely determine $e_3,\cdots,e_{n-1}$ such that
$$
A^{-1}q_3 e_3 = \left(\begin{array}{c} * \\ 1 \end{array}\right) \quad
\text{and} \quad A^{-1} q_i e_i = \left(\begin{array}{c} 1 \\ *
\end{array}\right), \text{ for $i\,=\,4,\cdots, n-1$}.
$$
It is straight-forward to check that the circle action (see \eqref{action}) in these
local coordinates is given by
\begin{equation}
\label{eq:loccirc2}
\lambda \cdot (z_i,w_i)\,= \,(\lambda z_i, w_i)\,\,\text{for}\,\,
i\,=\,3,\cdots,n-1\, .
\end{equation}
Using \eqref{eq:loccirc1} and \eqref{eq:loccirc2} we obtain the following result.
\begin{theorem}\label{thm:weights}
Let $[p,q]_{\alpha\text{-{\rm st}}}$ be a point in $X_S$. Then the non-zero
isotropy weights of the $S^1$-representation on
$T_{[p,q]_{\alpha\text{-{\rm st}}}} X(\alpha)$ are
\begin{itemize}
\item $+1$ with multiplicity $\lvert S \rvert - 2$;
\item $-1$ with multiplicity $(n-1)-\lvert S \rvert$;
\item $+2$ with multiplicity $(n-1)- \lvert S \rvert$.
\end{itemize}
Let $[0,q]_{\alpha\text{-{\rm st}}}$ be a point of
the space $M(\alpha)$. Then the
non-zero isotropy weights of the $S^1$--representation on
$T_{[0,q]_{\alpha\text{-{\rm st}}}} X(\alpha)$ are
\begin{itemize}
\item $+1$ with multiplicity $(n-1)-\lvert S \rvert$.
\end{itemize}
\end{theorem}
\section{Spaces of Parabolic Higgs bundles}\label{PHBs}
Let $\Sigma$ be $\ensuremath{\mathbb C} \ensuremath{\mathbb P}^1$ with $n$ ordered marked points $D\, =\,
\{x_1,\cdots,x_n\}$ and let $E$
be a parabolic vector bundle of rank two over $\Sigma$ with parabolic structure
\begin{align*}
& E_x \, :=\, E_{x,1}\,\supset\, E_{x,2}\,\supset\, 0\, , \\
0 \leq & \beta_1(x) < \beta_2(x) <1
\end{align*}
over each point of $D$. Its \emph{parabolic degree} is
then
$$
\text{par-deg}(E)\,:=\,{\rm degree}(E) + \sum_{x \in D} \left( \beta_1(x) +
\beta_2(x)\right)\, .
$$
We recall that $E$ is said to be \emph{stable} if
$\text{par-}\mu(E)\,>\,\text{par-}\mu(L)$
for every line subbundle $L$ of $E$ equipped with the
induced parabolic structure, where, for any parabolic vector bundle $F$, the
\emph{slope} $\text{par-}\mu(F)$ is defined as $\text{par-deg}(F)/\text{rank}(F)$.
Now if $L$ is a parabolic line subbundle of $E$, its induced parabolic structure is given
by the trivial flag over each point $x$ of $D$,
$$
L_{x} \supset 0,
$$
with weights
$$
\beta^L(x) = \left\{ \begin{array}{l} \beta_1(x), \quad \text{if} \quad L_x \cap E_{x,2} = \{ 0 \}, \\ \\ \beta_2(x), \quad \text{if} \quad L_x \cap E_{x,2} = \mathbb{C}, \end{array} \right.
$$
and so it has parabolic degree
$$
\text{par-deg}(L)\,=\,{\rm degree}(L) +
\sum_{i\in S_L} \beta_2(x_i) + \sum_{i\in S_L^c} \beta_1(x_i),
$$
where
\begin{equation}\label{eq:setpb}
S_L:=\{ i \in \{1, \cdots, n\} \mid\, \beta^L(x_i)=\beta_2(x_i)\}.
\end{equation}
Hence, $E$ is stable if and only if every parabolic line subbundle $L$
satisfies the inequality
\begin{equation}
\label{eq:stable}
{\rm degree}(E) - 2\cdot {\rm degree}(L) \,>\, \sum_{i \in S_L}
\big(\beta_2(x_i) -\beta_1(x_i)\big) - \sum_{i \in S_L^c} \big(\beta_2(x_i) -\beta_1(x_i)\big).
\end{equation}
The holomorphic cotangent bundle of the Riemann surface $\Sigma$ will be
denoted by $K_\Sigma$. The line bundle on $\Sigma$ defined by the divisor $D$
will be denoted by ${\mathcal O}_{\Sigma}(D)$. A \emph{parabolic Higgs bundle}
of rank two is a pair ${\bf E} \,:=\, (E, \Phi)$, where $E$ is a parabolic
vector bundle over $\Sigma$ of rank two, and
$$\Phi \in H^0(\Sigma, S Par End(E) \otimes K_\Sigma(D))$$
is a \emph{Higgs field} on $E$. Here $S Par End(E)$ denotes the
subsheaf of $End(E)$ formed by strongly parabolic endomorphisms $\varphi\,:\,E
\,\longrightarrow\, E$, which, in this situation, simply means that
$$
\varphi(E_{x,1}) \subset E_{x,2} \quad \text{and} \quad \varphi(E_{x,2})=0, \quad \text{for all $x\in D$}.
$$
Note that $\Phi$ is then a meromorphic endomorphism-valued one-form with
simple poles along $D$ whose residue at each $x\,\in\, D$ is nilpotent with respect to the
flag, i.e.,
$$
(\text{Res}_x \Phi) (E_{x,i}) \,\subset\, E_{x,i+1}
$$
for all $i=1,2$ and $x\in D$, with $E_{x,3}=0$. The definition of stability
extends to Higgs bundles: a parabolic Higgs bundle ${\bf E}= (E,\Phi)$ is
\emph{stable} if
$\text{par-}\mu(E) \,>\, \text{par-}\mu(L)$ for all parabolic line
subbundles $L\subset E$ which are preserved by $\Phi$.
Let $\mathcal{H}(\beta)$ be the moduli space of parabolic Higgs bundles of rank
two such that the underlying holomorphic vector bundle is holomorphically
trivial. In \cite{GM} it is shown that $\mathcal{H}(\beta)$ is diffeomorphic to
the space of hyperpolygons $X(\alpha)$ with $\alpha_i=\beta_2(x_i)-\beta_1(x_i)$. The correspondence between these two spaces is given by the map
\begin{equation}
\label{eq:isom}
\begin{array}{rl}
\mathcal{I}:X(\alpha) \longrightarrow & \mathcal H(\beta)\\ \\
{[p,q]}_{\alpha \textnormal{-st}} \longmapsto & (E_{(p,q)}\, , {\Phi}_{(p,q)}) =:
\mathbf E_{(p,q)} \\
\end{array}
\end{equation}
where $E_{(p,q)}$ is the trivial vector bundle $\ensuremath{\mathbb C}\ensuremath{\mathbb P}^1\times \ensuremath{\mathbb C}^2
\,\longrightarrow\, \ensuremath{\mathbb C}\ensuremath{\mathbb P}^1$ with the parabolic structure consisting of weighted flags
\begin{align*}\label{flag}
\ensuremath{\mathbb C}^2 & \supset \langle q_i\rangle \supset 0\\
0 \leq \beta_{1}(x_i) & < \beta_{2}(x_i) < 1
\end{align*}
over the $n$ marked points $\{x_1,\cdots,x_n\} \,=\, D \,\subset\, \ensuremath{\mathbb C} \ensuremath{\mathbb P}^1$ with $\beta_i(x_j)$
satisfying
\begin{equation}\label{dif}
\beta_2(x_i)-\beta_1(x_i)\,=\,\alpha_i\, ,
\end{equation}
and $\Phi_{[p,q]} \,\in\,
H^0\big(S Par End(E_{(p,q)}) \otimes K_{\ensuremath{\mathbb C}\ensuremath{\mathbb P}^1}(D)\big)$
is the Higgs field uniquely determined by the following condition
on the residue:
\begin{equation}
\label{eq:res}
\text{Res}_{x_i} \Phi \,:=\, (q_i p_i)_0
\end{equation}
at each $x_i\,\in\, D$.
In particular, the polygon space $M(\alpha)$ (obtained when $p=0$) is mapped to
the moduli space $\mathcal{M}_{\beta,2,0}$ of parabolic vector
bundles of rank two over $\Sigma$ such that the underlying holomorphic
vector bundle is trivial (this map is obtained by setting $\Phi=0$).
This isomorphism is equivariant with respect to the circle
action on $X(\alpha)$ (see \eqref{action}) and the circle action on $\mathcal{H}(\beta)$ defined by
\begin{equation}
\label{eq:actionHiggs}
\lambda \cdot (E,\Phi) = (E, \lambda \Phi), \quad \text{for $\lambda \in S^1$}.
\end{equation}
Each connected component $X_S$ of the fixed point set of
the circle action on $X(\alpha)$ is mapped to a manifold $\mathcal{M}_S$
formed by the trivial holomorphic bundle $E$ over $\Sigma$ equipped with
weighted flag structures
\begin{align*}
\ensuremath{\mathbb C}^2 & \supset E_{x_i,2} \supset 0\\
0 \leq \beta_{1}(x_i) & < \beta_{2}(x_i) < 1
\end{align*}
such that $E_{x_i,2}\,=\,E_{x_j,2}$ whenever $i,j \in S$ or $i,j \in S^c$, and an
Higgs
field with zero residue at all points $x_i$ with $i \in S^c$. Note that this
description of the critical sets agrees with the one given by Simpson in
\cite{Si}. Indeed, the bundles in $\mathcal{M}_S$ have a direct sum
decomposition $E\,=\,L_0\oplus L_1$ as parabolic bundles, where the
parabolic weight of $L_0$ (respectively, $L_1$) at
$x_i\,\in\, S^c$ is $\beta_2(x_i)$ (respectively, $\beta_1(x_i)$), and
the parabolic weight of $L_0$ (respectively, $L_1$) at
$x_i\,\in\, S$ is $\beta_1(x_i)$ (respectively, $\beta_2(x_i)$). (Note that
the holomorphic line bundles underlying $L_0$ and $L_1$ are trivial.)
Moreover, $\Phi$, being lower triangular with
respect to this decomposition, preserves $L_1$ and induces a nonzero strongly
parabolic homomorphism $\Phi_{|_{L_0}}\,:\,L_0\,\longrightarrow\, L_1\otimes
K_\Sigma(D)$; that this strongly parabolic homomorphism is nonzero
follows from the fact that $X_S \cap M(\alpha) \,=\,\emptyset$.
\section{An involution}\label{sec:inv}
As before, let $\mathcal{H}(\beta)$ be the moduli space of $\beta$-stable
parabolic Higgs bundles of rank two such that the underlying holomorphic vector
bundle is trivial. In this section we will
restrict ourselves to the action of $\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}\, \subset\, S^1$ on
$\mathcal{H}(\beta)$ giving the involution
\begin{equation}\label{eq:invPHB}
(E, \Phi) \,\longmapsto\, (E, -\Phi)\, ,
\end{equation}
and we will study the fixed-point set of this involution.
The parabolic Higgs bundles with zero Higgs field are clearly fixed by the
involution in \eqref{eq:invPHB}, and so the moduli space
$\mathcal{M}_{\beta,2,0}$ of $\beta$-stable rank two holomorphically trivial
parabolic vector bundles over $\ensuremath{\mathbb C} \ensuremath{\mathbb P}^1$ is contained in the fixed point set of the
involution.
For the remaining fixed points, we will use the isomorphism in
\eqref{eq:isom} and study the fixed point set of the corresponding involution
on the hyperpolygon space $X(\alpha)$ with
$\alpha\,=\,(\alpha_1,\cdots,\alpha_n)$ satisfying \eqref{dif}. The fixed-point
set of the involution
\begin{equation}
\label{eq:inv}
(p,q) \,\longmapsto\, (-p,q)\, ,
\end{equation}
on the hyperpolygon space $X(\alpha)$ is the set of points $X(\alpha)^{\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}}$
that are fixed by the action of ${\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}}\, \subset\, S^1$ in \eqref{action}.
As before, $M(\alpha)$ denotes the moduli space of polygons in $\ensuremath{\mathbb R}^3$.
Theorem \ref{thm:invfixedpoints} describes the fixed-point set of the
$\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}$-action in \eqref{eq:inv}.
For each element $S$ of $\mathcal S'(\alpha)$,
$$ Z_S \,:= \,\big\{[p,q] \in X(\alpha) \,\mid\,~ S \, \text{ and } S^c\,
\text{ are straight at }\, (p,q) \big\}\,.$$
\begin{theorem}
\label{thm:invfixedpoints}
The fixed-point set of the involution in \eqref{eq:inv} is
$$ X(\alpha)^{\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}}\,=\, M(\alpha) \cup \bigcup_{S \in \mathcal S'(\alpha)}
Z_S\, ,$$
where $Z_S$ is defined above.
Moreover, $Z_S$ is a non-compact manifold of dimension $2(n-3)$ except when
$\lvert S\rvert = n-1$, in which case $Z_S=X_S$ is compact and diffeomorphic
to $\ensuremath{\mathbb C} \ensuremath{\mathbb P}^{n-3}$. In all cases, $X(\alpha)^{\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}}$ has $2^{n-1}-(n+1)$ non-compact components and one compact component.
\end{theorem}
\begin{proof}
{}From the isotropy weights of the $S^1$--action given in Theorem
\ref{thm:weights} it follows immediately that, if $M(\alpha)$ is nonempty, then
it is a connected component of the fixed point set of the $\ensuremath{\mathbb Z}_2/2\ensuremath{\mathbb Z}$--action.
Furthermore, the connected components of the complement $X(\alpha)^{\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}}
\setminus M(\alpha)$ are parametrized by the elements $S$ of $S'(\alpha)$ and have dimension
$$
\dim Z_S \,=\, 2((n-1)-\lvert S \rvert )+ \dim X_S \,=\, 2(n-3)\, .
$$
Therefore, it remains to show that each connected component $Z_S$ of
$X(\alpha)^{\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}}\setminus M(\alpha)$ can be described as
$$ Z_S \,=\, \big\{[p,q] \in X(\alpha) \mid S \text{ and } S^c\, \text{are
straight at }\, (p,q) \big\}\, .$$
Suppose that $[p,q]\,\in\, X(\alpha)^{\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}}\setminus M(\alpha)$. Then there
exists
an element $$[A;e_1,\cdots,e_n]\,\in\, K\setminus \{I\}$$ such that
$$
e_i^{-1} p_i A \,=\, - p_i \quad \text{and}\quad A^{-1}q_i e_i \,=\,q_i,
\,\,\text{for} \,\, i\,=\,1,\cdots, n\, ,
$$
and so
$$
A p_i^* \,=\, -e_i p_i^* \quad \text{and} \quad A q_i \,=\,e_i
q_i\,\,\text{for}
\,\,
i\,=\,1,\cdots, n\, .
$$
Since $\lvert q_i\rvert^2 - \lvert p_i \rvert^2=2\alpha_i$, we have $q_i\neq 0$
for all $i=1,\cdots,n$ and so $q_i$ is an eigenvector of $A$ with eigenvalue
$e_i$. Moreover, since $[p,q]\in X(\alpha)^{\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}}\setminus M(\alpha)$, there
exists an integer $i_0\,\in\, \{1,\cdots, n \}$ such that $p_{i_0}\neq 0$, and so
$p_{i_0}^*$ is
an eigenvector of $A$ with eigenvalue $-e_{i_0}$. Hence, assuming that $A$ is diagonal, we have
$$
A\,=\,\left( \begin{array}{cc} e_{i_0} & 0 \\ 0 & -e_{i_0} \end{array}
\right)
$$
with $e_{i_0}=\pm \sqrt{-1}$. We conclude that there exists an index set
$S\subset \{1,\cdots, n \}$ with $i_0 \in S$ such that
\begin{align} \label{eq:Z0}
p_i & = \left( \begin{array}{ll} 0 & b_i \end{array}\right), \quad q_i= \left(\begin{array}{c} c_i \\ 0 \end{array} \right), \forall i\in S \\ \nonumber
p_i & = \left( \begin{array}{ll} a_i & 0 \end{array}\right), \quad q_i= \left(\begin{array}{c} 0 \\ d_i \end{array} \right),\forall i \in S^c.
\end{align}
Since $\lvert q_i\rvert^2 - \lvert p_i\rvert^2=2\alpha_i$, we conclude that
\begin{equation}
\label{eq:Z}
\lvert c_i\rvert^2 - \lvert b_i\rvert^2=2\alpha_i\,\,\text{for all $i\in S$} \quad \text{and} \quad \lvert d_i\rvert^2 - \lvert a_i\rvert^2=2\alpha_i \,\, \text{for all $i\in S^c$}.
\end{equation}
Moreover, since $\sum_{i=1}^n ( q_i q_i^* - p_i^*p_i)_0 =0$, we obtain that
\begin{equation}
\label{eq:Z0nn}
\sum_{i\in S} \left(\lvert c_i\rvert^2 + \lvert b_i\rvert^2\right) - \sum_{i
\in S^c} \left( \lvert d_i\rvert^2 + \lvert a_i\rvert^2\right)\,=\, 0
\end{equation}
and so, using \eqref{eq:Z}, we get that
\begin{equation}\label{eq:Z2}
\sum_{i \in S} \alpha_i - \sum_{i\in S^c} \alpha_i \,=\,\sum_{i\in S^c}\lvert
a_i\rvert^2 -\sum_{i\in S} \lvert b_i \rvert^2\, .
\end{equation}
On the other hand, since $\sum_{i=1}^n (q_ip_i)_0\,=\,0$, we have that
\begin{equation}\label{eq:zero}
\sum_{i \in S} b_i c_i \,=\,\sum_{i \in S^c} a_i d_i \,=\, 0\, .
\end{equation}
If $S$ is short, then we work with $S$ and, in particular, since $c_i\neq 0$ and
there exists an $i_0\in S$ such that $b_{i_0}\neq 0$, from \eqref{eq:zero}
it follows that there exists another $i_1 \in S$ such that $b_{i_1}\neq 0$ and
we obtain that $S$ has cardinality at least two. If, on the other hand, $S$ is
long, we consider $S^c$ instead (which is now short). Since $S$ is long, from
\eqref{eq:Z2} it follows that there is at least one $i_1\,\in\, S^c$ such that
$p_{i_1}\neq 0$ and then, since $d_{i_1}\neq 0$, from \eqref{eq:zero} it follows
that there is
another element $i_2$ in $S^c$ with $p_{i_2}\neq 0$, implying that the short set
$S^c$ has cardinality at least two.
Finally, since for every subset $S\subset \{1,\cdots,n\}$, we have that either $S$ or $S^c$ is short, the number of short sets for $\alpha$ is
$$
\frac{1}{2} \sum_{k=1}^{n-1} \left( \begin{array}{l} n \\ k \end{array} \right) = 2^{(n-1)}-1.
$$
If there is no short set of cardinality $n-1$, then there are exactly $n$ short sets of cardinality $1$ and so
$$
\lvert \mathcal{S}^\prime(\alpha) \rvert = 2^{(n-1)}-(n+1).
$$
Moreover, in this case, all the components $Z_S$ are non-compact. If, on the other hand, there is a short set $\widetilde{S}$ of cardinality $n-1$, then there are only $n-1$ short sets of cardinality $1$ and then the number of elements in $\mathcal{S}^\prime(\alpha)$ is
$$
\lvert \mathcal{S}^\prime(\alpha) \rvert = 2^{(n-1)}-n.
$$
However, in this case, $M(\alpha)$ is empty and $Z_{\widetilde{S}}$ is compact. We conclude that, in both cases, the number of non-compact components of $X(\alpha)^{S^1}$ is $2^{(n-1)}-(n+1)$ and that there is exactly one compact component (which is either $M(\alpha)$ or $Z_{\widetilde{S}}$).
\end{proof}
Each manifold $Z_S$, being a component of $X(\alpha)^{\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}}$, is symplectic and
invariant under the circle action in \eqref{action}. Hence, whenever $\lvert
S\rvert \neq n-1$, we obtain an effective Hamiltonian circle action on $Z_S$
(the action factors through the quotient of $S^1$ by $\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}$). The corresponding moment map then
coincides with the restriction of $\frac{1}{2}\phi$ to $Z_S$. The only
critical submanifold of this map is $X_S$ where it attains its minimum value. Consequently, we have the following results.
\begin{theorem} \label{thm:defret}
Each manifold $X_S\,\cong\, \ensuremath{\mathbb C} \ensuremath{\mathbb P}^{\lvert S \rvert -2}$ is a deformation
retraction of $Z_S$. In particular, $Z_S$ is simply connected.
\end{theorem}
\begin{theorem}\label{thm:Poinc}
Let $P_t(M)$ be the Poincar\'{e} polynomial of $M$. Then
$$
P_t(Z_S)\,=\,P_t(X_S)\,=\,P_t(\ensuremath{\mathbb C} \ensuremath{\mathbb P}^{\lvert S \rvert-2})\,=\, 1+t+\cdots +
t^{2(\lvert S \rvert-2)}\, .
$$
\end{theorem}
Going back to the space of parabolic Higgs bundles $\mathcal{H}(\beta)$
and using the isomorphism in \eqref{eq:isom}, we obtain
from \eqref{eq:res} and \eqref{eq:Z0} that the fixed-point set
$\mathcal{H}(\beta)^{\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}}$ of the involution in \eqref{eq:invPHB} is described
as in Theorem~\ref{thm:0.1}.
\section{Polygons in Minkowski $3$-space}
Let us consider the \emph{Minkowski inner product} on $\ensuremath{\mathbb R}^3$
$$
v \circ w \,=\, -v_1w_1-v_2w_2+v_3w_3\, ,
$$
for $v\,=\,(v_1,v_2,v_3)$ and $w\,=\,(w_1,w_2,w_3)$.
The inner product space consisting of $\ensuremath{\mathbb R}^3$ together with this signature
$(-,-,+)$-inner product will be denoted by $\ensuremath{\mathbb R}^{2,1}$; it is called
the \emph{Minkowski $3$-space}. The \emph{Minkowski norm} of a vector $v\in
\ensuremath{\mathbb R}^{2,1}$ is then defined to be
$$
\lvert\lvert v\rvert\rvert_{2,1}\,=\, \sqrt{\lvert v\circ v \rvert}\, .
$$
All elements $v$ of $R^{2,1}$ are classified according to the sign of $v \circ v
$. The set of all $v$ such that $v\circ v =0$ is called the \emph{light cone}
of $\ensuremath{\mathbb R}^3$; any vector $v$ with $v\circ v =0$ is said to be
\emph{light-like}. If $ v\circ v \,>\, 0$, then
$v$ is called \emph{time-like}, and if $v\circ v \,<\,0$, then it is
called \emph{space-like}. A time-like vector is said to be lying in \emph{future}
(respectively, \emph{past}) if $v_3\,>\,0$ (respectively, $v_3<0$). Note that the
exterior of the light cone consists of all space-like vectors, while its interior
consists of all time-like vectors. From now on we will write any $v\,\in\,\ensuremath{\mathbb R}^{2,1}$
as $v\,=\,(x,y,t)$.
Moduli spaces of polygons in $\ensuremath{\mathbb R}^{2,1}$ were described by Foth, \cite{F}, as
follows. Consider the surface $S_R$ in $\ensuremath{\mathbb R}^3$ defined by the equation
$-x^2-y^2+t^2\,=\,R^2$, which is called a \emph{pseudosphere}. The
Minkowski metric on $\ensuremath{\mathbb R}^{2,1}$ restricts to a constant curvature Riemannian metric on
$S_R$. It is an hyperboloid of two sheets. The connected
component $S_R^+\, \subset\, S_R$, corresponding to $t\,>\,0$, is called a
\emph{future pseudosphere}, and the connected component $S_R^-\, \subset\, S_R$, corresponding to
$t\,<\, 0$, is called a \emph{past
pseudosphere}. The group ${\rm SU}(1,1)$ acts transitively on each connected
component since one can think of $\ensuremath{\mathbb R}^{2,1}\,\cong\,\ensuremath{\mathbb R}^3$ as
$\mathfrak{su}(1,1)^*$
with $S_R^+$ and $S_R^-$ being elliptic coadjoint orbits. Consequently, both $S_R^+$ and
$S_R^-$ have a natural invariant symplectic structure (the Kostant--Kirillov
form on a coadjoint orbit). The Minkowski metric is
also invariant (since ${\rm SU}(1,1)$ acts by isometries) and both connected
components are K\"{a}hler manifolds; they are in fact isomorphic to the hyperbolic
plane ${\rm SU}(1,1)/{\rm U}(1)$. We will study the geometry of the symplectic
quotients of the products of several future and past pseudospheres
with respect to
the diagonal ${\rm SU}(1,1)$--action.
Let $\alpha\,=\,(\alpha_1,\cdots,\alpha_n)$ be an $n$-tuple of
positive real numbers. Let us fix two positive integers $k_1,k_2$ with
$k_1+k_2\,=\,n$. We will consider polygons in Minkowski $3$-space that have the
first $k_1$ edges in the future time-like cone and the last $k_2$ edges in the past
time-like cone, such that the Minkowski length of the $i$-th edge is $\alpha_i$. A
closed
polygon will then be one whose sum of the first $k_1$ sides in the future time-like cone coincides with the negative of the sum of the last $k_2$ sides in the past time-like cone. The space of all such closed polygons can be identified with the zero level set of the moment map
\begin{equation}
\label{eq:mmhyp}
\begin{array}{ccc}
\mu: \mathcal{O}_1\times \cdots \times \mathcal{O}_n & \longrightarrow &
\mathfrak{su}(1,1)^* \\
(u_1, \cdots, u_n) & \longmapsto & u_1 + \cdots + u_n
\end{array}
\end{equation}
for the diagonal ${\rm SU}(1,1)$--action, where $\mathcal{O}_i\cong
S_{\alpha_i}^+$
is a future pseudosphere of radius $\alpha_i$ if $1 \leq i \leq k_1$, and
$\mathcal{O}_i\cong S_{\alpha_i}^-$ is a past pseudosphere if $k_1+1 \leq i \leq
n$, equipped with the Kostant-Kirillov symplectic form on coadjoint orbits.
Hence,
$$
M^{k_1,k_2}(\alpha)\,:=\,\mu^{-1}(0)/{\rm SU}(1,1)\, ,
$$
which is a quotient of a non-compact space by a non-compact Lie group. For a
generic choice of $\alpha$, meaning $M^{k_1,k_2}(\alpha)$ is
non-empty with $\sum_{i=1}^{k_1}\alpha_i\,\neq\,\sum_{i=k_1+1}^{n}\alpha_i$, every
point in $M^{k_1,k_2}(\alpha)$ represents a polygon with a trivial stabilizer.
In that situation, the group ${\rm SU}(1,1)$ acts freely and properly on $\mu^{-1}(0)$.
Moreover, $0$ is a regular value of the moment map $\mu$ and so the quotient space $M^{k_1,k_2}(\alpha)$ is, for a generic $\alpha$, a smooth symplectic manifold of dimension $2(n-3)$. Note that the spaces
$$
M^{k_1,k_2}(\alpha_1,\cdots,\alpha_{k_1},\alpha_{k_1+1},\cdots,\alpha_{n})\quad
\text{and} \quad
M^{k_2,k_1}(\alpha_{k_1+1},\cdots,\alpha_{n},\alpha_{1},\cdots,\alpha_{k_1})
$$
are symplectomorphic by the isomorphism given by the involution of
${\mathbb R}^{2,1}$ defined by $(x,y,t)\,\longmapsto\, (-x,-y,-t)$.
\begin{theorem}[\cite{F}]\label{thF}
The space $M^{k_1,k_2}(\alpha)$ is non-compact, unless $k_1=1$ or $k_2=1$ in which case it
is compact.
\end{theorem}
We give a brief outline of an argument for Theorem \ref{thF}.
Let us first assume that $k_2=1$. The last side of the polygon can be
represented (after being acted on by an element of ${\rm SU}(1,1)$) by a vector in
$\ensuremath{\mathbb R}^{2,1}$ with coordinates $(0,0,-\alpha_n)$. Hence, the sum of the first $n-1$ future
time-like sides of the polygon is $(0,0,\alpha_n)$. The only symmetry
left is the circle rotation around the $t$-axis. Therefore, this space
of polygons is clearly bounded
and closed and therefore compact.
The space $M^{k_1,k_2}(\alpha)$ is non-compact if $k_2\,>\,1$. For
example, let us consider the simple case where $k_1=k_2=2$ and
$\alpha_1=\alpha_2=\alpha_4=1$. Again we can assume that the last side is $(0,0,-1)$ and the only symmetry left is the circle rotation around the $t$-axis. Let $x_n$ be the closed polygon with sides
\begin{align*}
u_1& =(-1,0,\sqrt{2}), \quad u_2=(1-P(n),Q(n),n-\sqrt{2}), \\ u_3 &=(P(n),-Q(n),1-n)\quad \text{and} \quad u_4=(0,0,-1),
\end{align*}
where
$$
P(n)=\frac{1}{2}(3+2(\sqrt{2}-1)n) \quad \text{and} \quad Q(n)=\sqrt{8(\sqrt{2}-1)n^2-4(3\sqrt{2}-1)n-9} \, .
$$
The sequence $\{x_n\}$ has no limit point in $M^{2,2}(1,1,2,1)$ and so this
space is not compact.
Let us now describe the symplectic structure on $M^{k_1,k_2}(\alpha)$. For that,
define the \emph{Minkowski cross product} $\dot{\times}$ as
$$
v\dot{\times} w\, :=\, \det \left( \begin{array}{rrr} -e_1 & -e_2 & e_3 \\ v_1 &
v_2 & v_3 \\ w_1 & w_2 & w_3 \end{array}\right),
$$
where $v=(v_1,v_2,v_3)$ and $w=(w_1,w_2,w_3)$ with $e_1,e_2,e_3$
being the standard
unit vectors in $\ensuremath{\mathbb R}^3$. This cross product satisfies the usual properties:
\begin{align*}
& v \dot{\times} w \,=\, - w \dot{\times} v\\
& (u\dot{\times} v) \dot{\times} w + (v \dot{\times} w)\dot{\times} u + (w
\dot{\times} u)\dot{\times} v\,=\, 0
\end{align*}
and so $(\ensuremath{\mathbb R}^3,\dot{\times})$ is a Lie algebra. Moreover, it is isomorphic to
$\mathfrak{su}(1,1)$ via the map
$$
\left(\begin{array}{l} x \\ y \\ t \end{array} \right)\,\longmapsto\,\frac{1}{2}
\left( \begin{array}{cc} -\sqrt{-1} t & x + \sqrt{-1} y \\ x - \sqrt{-1} y &
\sqrt{-1} t \end{array} \right)\, .
$$
Under this identification, the Minkowski inner product $\circ$ corresponds to $(A,B)\,
\longmapsto\, -2 \cdot \mathrm{trace}(AB)$.
The symplectic form on the pseudosphere $S_R$ is then given by
$$
\omega_u(v,w)\,=\,\frac{1}{R^2}\,u \circ (v\dot{\times} w)\, ,
$$
where $u\,\in\, S_R$ and $v,w\,\in \,T_u S_R$ (we think of $T_u S_R$ as the
linear
subspace of $\ensuremath{\mathbb R}^{2,1}$ orthogonal to $u$ with respect to the Minkowski inner product), and the map
in \eqref{eq:mmhyp} is the moment map for the diagonal ${\rm SU}(1,1)$-action and
the product symplectic structure.
\section{Back to the involution}
\label{sec:Mink}
Let us go back to Section~\ref{sec:inv} and consider a point $[p,q]$ in some $Z_S$. Let $u_i\in \ensuremath{\mathbb R}^3$ be the vector
$$
u_i \,:=\, \frac{\sqrt{-1}}{2} (p^*_ip_i -q_i q^*_i)_0 +\frac{1}{2} (p_i^*q_i^* + q_i p_i)_0\, ,
$$
where we use the identifications, $\mathfrak{su}(2)^*\cong( \ensuremath{\mathbb R}^3)^*
\cong \mathfrak{su}(1,1)$.
If $i \in S$ then $p_i=\left( \begin{array}{ll}0 & b_i\end{array}\right)$ and $q_i=\left( \begin{array}{l} c_i \\ 0 \end{array}\right)$, implying that
\begin{equation}
\label{eq:uS}
u_i\,=\,\left( \operatorname{Re}{(b_i c_i)}\, , \operatorname{Im}{(b_i
c_i)}\, ,\frac{\lvert b_i\rvert^2+\lvert c_i \rvert^2}{2}\right)
\end{equation}
with
$$
u_i\circ u_i \,=\, (\lvert c_i\rvert^2-\lvert b_i \rvert^2)^2/4= \alpha_i^2
$$
($u_i$ has Minkowski norm $\alpha_i$).
Similarly, if $i \in S^c$, we have $p_i=\left( \begin{array}{ll}a_i & 0 \end{array}\right)$ and $q_i=\left( \begin{array}{l} 0 \\ d_i \end{array}\right)$, yielding
\begin{equation}
\label{eq:uSc}
u_i=\left( \operatorname{Re}{(a_i d_i)},- \operatorname{Im}{(a_i d_i)},-\frac{\lvert a_i\rvert^2+\lvert d_i \rvert^2}{2}\right)
\end{equation}
and
$$
u_i\circ u_i \,=\, (\lvert a_i\rvert^2-\lvert d_i \rvert^2)^2/4\,=\,
\alpha_i^2\, .
$$
Moreover, by \eqref{eq:Z0nn} and \eqref{eq:zero} we have that
$$
\sum_{i=1}^n u_i\,=\,0\, .
$$
So the vectors $u_i$ form a closed polygon in Minkowski $3$-space with the first
$\lvert S\rvert$ sides in the positive time-like cone and the last $n-\lvert
S\rvert$ sides in the past, with the $i$-th side being of Minkowski length $\alpha_i$.
\begin{theorem}\label{thm:polMink}
For any $S\,\in\, \mathcal{S}^\prime(\alpha)$, the components $\mathcal{Z}_S$
and $Z_S$, of the fixed-point sets of the involutions in \eqref{eq:invPHB} and
\eqref{eq:inv} respectively, are diffeomorphic to the moduli space
$$
M^{\lvert S\rvert,\lvert S^c \rvert}(\alpha)
$$
of closed polygons in Minkowski $3$-space.
\end{theorem}
\begin{proof} Let $S$ be a short set of cardinality at least two.
Consider the map $\varphi:Z_S \longrightarrow M^{\lvert S\rvert,\lvert S^c
\rvert}(\alpha)$ defined above, that is, $\varphi([p,q]_\ensuremath{\mathbb R})$ is the element of
$M^{\lvert S\rvert,\lvert S^c \rvert}(\alpha)$ represented by the polygon whose
sides are the vectors $u_i$ given by \eqref{eq:uS} and \eqref{eq:uSc}
for $i$ in $S$ and $S^c$ respectively.
Note that the pseudo-unitary group ${\rm SU}(1,1)$ is generated by the following
orientation preserving isometries of the pseudosphere: $A_\theta$
and $T_\phi$, where
$$
A_\theta =\left(\begin{array}{ccc} \cos{\theta} & -\sin{\theta} & 0 \\ \sin{\theta} & \cos{\theta} & 0 \\ 0 &0 & 1\end{array} \right),
$$
is an Euclidean \emph{rotation} by an angle $\theta$ in the $(x,y)$-plane, and
$$
T_\phi =\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cosh{\phi} & \sinh{\phi} \\ 0 & \sinh{\phi} & \cosh{\phi} \end{array} \right)
$$
is a \emph{boost} of rapidity $\phi$ along the $y$-direction\footnote{In special relativity, the rapidity parameter $\phi$ is defined by $\tanh{\phi}=v/c$,
where $v$ is the velocity.} (cf. \cite{BV} for the details).
Let us first see that $\varphi$ is well defined. For that, consider two
representatives $(p,q)$ and $(p^\prime,q^\prime)$ of the same element $[p,q]_\ensuremath{\mathbb R}$ in
$Z_S$. Then there exists $[A;e_1,\cdots,e_n]\,\in\, K$ such that
$$
e_i^{-1}p_i A= p_i ^\prime \quad \text{and} \quad A^{-1} q_i e_i =q_i^{\prime},
\quad i=1,\cdots,n.
$$
Since $p_i=\left( \begin{array}{ll} a_i & b_i\end{array}\right)$, $p_i^\prime=\left(\begin{array}{ll}a_i^\prime & b_i^\prime\end{array}\right)$ with $a_i=a_i^\prime=0$ for $i \in S$ and $b_i=b_i^\prime=0$ for $i \in S^c$, while
$q_i=\left( \begin{array}{ll} c_i & d_i \end{array}\right)^t$, $q_i^\prime=\left( \begin{array}{ll} c_i^\prime & d_i^\prime \end{array}\right)^t$
with $d_i\,=\,d_i^\prime\,=\,0$ for $i \in S$ and $c_i\,=\,c_i^\prime=0$ for $i
\,\in\, S^c$, we conclude that
$$
A= \left(\begin{array}{cc}\alpha & 0 \\ 0 & \overline{\alpha} \end{array}\right)
$$
with $\alpha=e^{\sqrt{-1}\,\theta_0} \in S^1$. Then we have
$$
\left(\begin{array}{l} \operatorname{Re}{(b^\prime_i c^\prime_i)} \\ \\
\operatorname{Im}{(b^\prime_i c^\prime_i)} \\ \\ \frac{\lvert b^\prime_i\rvert^2+\lvert c^\prime_i \rvert^2}{2} \end{array}\right) = A_{-2\theta_0} \left(\begin{array}{l}\operatorname{Re}{(b_i c_i)} \\ \\ \operatorname{Im}{(b_i c_i)} \\ \\ \frac{\lvert b_i\rvert^2+\lvert c_i \rvert^2}{2} \end{array}\right)
\,\,\text{for $i \in S$,}
$$
and
$$
\left(\begin{array}{r} \operatorname{Re}{(a^\prime_i d^\prime_i)} \\ \\ - \operatorname{Im}{(a^\prime_i d^\prime_i)} \\ \\ - \frac{\lvert a^\prime_i \rvert^2+ \lvert d^\prime_i \rvert^2}{2} \end{array}\right) = A_{-2\theta_o} \left(
\begin{array}{r} \operatorname{Re}{(a_i d_i)} \\ \\ - \operatorname{Im}{(a_i d_i)} \\ \\ - \frac{\lvert a_i\rvert^2+\lvert d_i \rvert^2}{2} \end{array} \right)\,\, \text{for $i \in S^c$},
$$
where $A_{-2\theta_0}$ is a rotation in ${\rm SU}(1,1)$. Therefore, it follows
that $\varphi$ is well-defined.
To show that $\varphi$ is injective, let us consider two points $[p,q]_\ensuremath{\mathbb R},
[p^\prime,q^\prime]_\ensuremath{\mathbb R} \,\in\, Z_S$ with
$\varphi([p,q]_\ensuremath{\mathbb R})\,=\,\varphi([p^\prime,q^\prime]_\ensuremath{\mathbb R}) $. Then, writing
$$ p_i=\left( \begin{array}{ll}0 & b_i\end{array}\right), \,\, p_i^\prime=\left( \begin{array}{ll}0 & b_i^\prime \end{array}\right)\quad \text{and} \quad q_i=\left( \begin{array}{l} c_i \\ 0 \end{array}\right),\,\, q_i^\prime=\left( \begin{array}{l} c_i^\prime \\ 0 \end{array}\right), \,\, \text{for $i\in S$}
$$
with $\sum_{i\in S} b_i c_i=\sum_{i\in S} b^\prime_i c^\prime_i=0$ (cf. \eqref{eq:zero}),
and
$$ p_i=\left( \begin{array}{ll} a_i & 0 \end{array}\right), \,\, p_i^\prime=\left( \begin{array}{ll} a_i^\prime & 0 \end{array}\right)\quad \text{and} \quad q_i=\left( \begin{array}{l} 0 \\ d_i \end{array}\right),\,\, q_i^\prime=\left( \begin{array}{l} 0 \\ d_i^\prime \end{array}\right), \,\, \text{for $i\in S^c$},
$$
with $\sum_{i\in S^c} a_i d_i=\sum_{i\in S^c} a^\prime_i d^\prime_i=0$, there exists an Euclidean rotation $A_{\theta_0}$ by an angle $\theta_0$ on the $(x,y)$-plane
such that
\begin{equation}\label{eq:prime1}
\left(\begin{array}{l} \operatorname{Re}{(b^\prime_i c^\prime_i)} \\ \\ \operatorname{Im}{(b^\prime_i c^\prime_i)} \\ \\ \frac{\lvert b^\prime_i\rvert^2+\lvert c^\prime_i \rvert^2}{2} \end{array}\right) = A_{\theta_0} \left(\begin{array}{l}\operatorname{Re}{(b_i c_i)} \\ \\ \operatorname{Im}{(b_i c_i)} \\ \\ \frac{\lvert b_i\rvert^2+\lvert c_i \rvert^2}{2} \end{array}\right)
\,\,\text{for $i \in S$,}
\end{equation}
and
\begin{equation}\label{eq:prime2}
\left(\begin{array}{r} \operatorname{Re}{(a^\prime_i d^\prime_i)} \\ \\ - \operatorname{Im}{(a^\prime_i d^\prime_i)} \\ \\ - \frac{\lvert a^\prime_i \rvert^2+ \lvert d^\prime_i \rvert^2}{2} \end{array}\right) = A_{\theta_0} \left(
\begin{array}{r} \operatorname{Re}{(a_i d_i)} \\ \\ - \operatorname{Im}{(a_i d_i)} \\ \\ - \frac{\lvert a_i\rvert^2+\lvert d_i \rvert^2}{2} \end{array} \right)\,\, \text{for $i \in S^c$}.
\end{equation}
Indeed, if the two vectors on the left-hand side of \eqref{eq:prime1} and
\eqref{eq:prime2} were not obtained from the corresponding vectors on the
right-hand side by an Euclidean rotation, but by an element of ${\rm SU}(1,1)$
involving a boost, they would fail to satisfy the condition
$$
\sum_{i\in S} b_i^\prime c_i^\prime \,=\, \sum_{i\in S^c} a_i^\prime d_i^\prime
\,=\,0\, .
$$
We conclude that
$$
b_i^\prime c_i^\prime \,= \,e^{\sqrt{-1}\, \theta_0} b_i c_i, \quad \text{and} \quad
\lvert b_i^\prime \rvert^2 + \lvert c_i^\prime \rvert^2 = \lvert b_i \rvert^2 + \lvert c_i\rvert^2, \quad \text{for $i\in S$},
$$
while
$$
a_i^\prime d_i^\prime \,=\, e^{-\sqrt{-1}\, \theta_0} a_i d_i, \quad \text{and} \quad
\lvert a_i^\prime \rvert^2 + \lvert d_i^\prime \rvert^2 = \lvert a_i \rvert^2 + \lvert d_i\rvert^2, \quad \text{for $i\in S^c$},
$$
and so
$$
p_i^\prime \,= \,p_i A \quad \text{and} \quad q_i^\prime = A^{-1}q_i, \quad
i\,=\,1,\cdots,n
$$
with $A\,= \left( \begin{array}{cc} e^{-\sqrt{-1}\, \theta_0/2 } & 0 \\ 0 & e^{\sqrt{-1}\,
\theta_0/2 } \end{array}\right)$, implying that $[p,q]_\ensuremath{\mathbb R}=[p^\prime,q^\prime]_\ensuremath{\mathbb R}$.
Let us now see that $\varphi$ is surjective. For that, take any element
$[v]\in M^{\lvert S\rvert,\lvert S^c \rvert}(\alpha)$. Using the ${\rm
SU}(1,1)$--action, the $(k_1+1)$-th vertex can be placed
on the $t$-axis (so that $\sum_{i=1}^{k_1} v_i$ is a vector along the $t$-axis). Therefore,
we may assume that $[v]$ is represented by a polygon with the first $\lvert S\rvert$ sides being
$(x_i,y_i,t_i)$ with $t_i>0$, on the positive time-like cone and the
last $n-\lvert S\rvert$ sides being $(x_i,y_i,-t_i)$ with $t_i>0$, in the past, satisfying the additional conditions
$$
\sum_{i=1}^{k_1} x_i=\sum_{i=1}^{k_1} y_i =\sum_{i=k_1+1}^n x_i = \sum_{i=k_1+1}^n y_i=0.
$$
Then $[v]$ is the image of the hyperpolygon $[p,q]_\ensuremath{\mathbb R}$, where
$$
p_i= \left( \begin{array}{ll}0 & \frac{1}{l_i}(x_i+\sqrt{-1}\, y_i)\end{array}\right), \quad q_i= \left(\begin{array}{l} l_i \\ 0 \end{array}\right) \,\, \text{for $i\in S$},
$$
and
$$
p_i= \left( \begin{array}{ll} \frac{1}{l_i} (x_i-\sqrt{-1} y_i) & 0 \end{array}\right), \quad q_i=\left( \begin{array}{c} 0 \\ l_i \end{array}\right)\,\, \text{for $i\in S^c$},
$$
with
$$
l_i \,=\, \sqrt{\alpha_i +\sqrt{\alpha^2+\lvert x_i +\sqrt{-1}\, y_i \rvert^2}}
\,=\, \sqrt{\alpha_i+t},\quad i\,=\,1,\cdots, n\, .
$$
Here $[p,q]_\ensuremath{\mathbb R}\in Z_S$ since
$$
\sum_{i\in S} b_i c_i \,=\,\sum_{i=1}^{k_1} (x_i + \sqrt{-1}\, y_i )\,= \,0,
\quad \sum_{i \in S^c} a_i d_i= \sum_{i=k_1+1}^n ( x_i - \sqrt{-1}\, y_i )\, ,
$$
and
$$
\lvert c_i \rvert^2 - \lvert b_i\rvert^2 =2 \alpha_i \,\,\text{for all $i \in S$} \quad \text{while} \quad \lvert d_i\rvert^2 - \lvert a_i\rvert^2 =2 \alpha_i \,\,\text{for all $i \in S^c$} ,
$$
where as usual we write $p_i= \left( \begin{array}{ll} a_i& b_i
\end{array}\right)$ and $q_i= \left( \begin{array}{ll} c_i& d_i \end{array}\right)^t$, for $i=1,\cdots,n$.
Note that clearly $\varphi$ and its inverse are differentiable and the theorem
follows.
\end{proof}
\begin{rem}
{\rm Note that when $\lvert S\rvert=n-1$ we obtain that the space $M^{n-1,1}(\alpha)$,
which we already knew is compact, is, in fact, diffeomorphic to $\ensuremath{\mathbb C}\ensuremath{\mathbb P}^{n-3}$.}
\end{rem}
Theorem~\ref{thm:polMink} allows us draw several conclusions on the polygon
spaces in Minkowski $3$-space which are immediate consequences of Theorem
\ref{thm:defret} and Theorem \ref{thm:Poinc}.
\begin{theorem}
Let $M^{k_1,k_2}(\alpha)$ be the moduli space of closed polygons in Minkowski
$3$-space that have the first $k_1$ sides in the future time-like cone and the
last $k_2$ in the past, such that the Minkowski length of the $i$-th side is
$\alpha_i$. Assume without loss of generality that $\sum_{i=1}^{k_1} \alpha_i\, <
\,\sum_{i=k_1+1}^n \alpha_i$. Then,
\begin{enumerate}
\item[(i)] $M^{k_1,k_2}(\alpha)$ admits a deformation retraction to $\ensuremath{\mathbb C} \ensuremath{\mathbb P}^{k_1-2}$, and
\item[(ii)] the Poincar\'{e} polynomial of $M^{k_1,k_2}(\alpha)$ is
$$
P_t(M^{k_1,k_2}(\alpha))\,=\,P_t(\ensuremath{\mathbb C} \ensuremath{\mathbb P}^{k_1-2})=1+t+\cdots+t^{2(\lvert S\rvert
-2)}\, .
$$
\end{enumerate}
\end{theorem}
\section{An Example}\label{sec:ex}
As an example, we consider the case where $n=4$. Let $\mathcal{H}(\beta)$ be
the moduli space of parabolic Higgs bundles $(E,\Phi)$ of rank two over $\ensuremath{\mathbb C} \ensuremath{\mathbb P}^1$ with
four parabolic points, where the underlying holomorphic vector bundle is trivial
and $\beta$ is generic. Let
$$
\alpha_i\,:=\,\beta_2(x_i)-\beta_1(x_i), \quad i=1,\cdots,4\, .
$$
Since a subset of $\{1,2,3,4\}$ is either short or long, we know that there are
exactly three short sets of cardinality two for any value of
$\alpha=(\alpha_1,\cdots,\alpha_4)$. Let us denote these sets by $S_1$, $S_2$ and
$S_3$. Then the fixed point set of the involution in \eqref{eq:invPHB} has
exactly $4$ connected components
$$
\mathcal{M}_{\beta,2,0}, \mathcal{Z}_{S_1}, \mathcal{Z}_{S_2}, \mathcal{Z}_{S_3} \quad \text{or} \quad \mathcal{Z}_{S_1}, \mathcal{Z}_{S_2}, \mathcal{Z}_{S_3}, \mathcal{Z}_{\widetilde{S}},
$$
according to whether $\mathcal{M}_{\beta,2,0}$ is empty or nonempty, where
$\widetilde{S}$ is a short set of cardinality $3$ which we know exists exactly
when $\mathcal{M}_{\beta,2,0}\,=\,\emptyset$ \cite{GM,BY}.
If $\mathcal{M}_{\beta,2,0}\,\neq\, \emptyset$, then this space
$\mathcal{M}_{\beta,2,0}$ is a compact
toric
manifold of dimension two, therefore diffeomorphic to $\ensuremath{\mathbb C} \ensuremath{\mathbb P}^1$. Indeed, let us
assume without loss of generality that $\alpha_1\neq \alpha_2$ (note that
$\alpha$ is generic) and consider the diagonal $d_2:=u_1+u_2$ connecting the
origin to the third vertex of the polygon. For each intermediate value of the
length of $d_2$, we have a circle of possible classes of polygons obtained by
rotating the first two sides of the polygon around the diagonal, while fixing
the other two. The minimum and maximum values of this length are
$$
\max \left\{\lvert \alpha_1-\alpha_2\rvert\, , \lvert \alpha_3- \alpha_4\rvert
\right\} \quad \text{and} \quad \min\left\{ \alpha_1+\alpha_2\, ,
\alpha_3+\alpha_4 \right\}
$$
respectively, in which cases we only have one possible polygon. Note that this
length is the
moment map for the bending flow obtained by rotating the first two sides of the polygon around the diagonal.
If $\mathcal{M}_{\beta,2,0}\,=\,\emptyset$ then, since $\lvert \widetilde{S}
\rvert =3$, we have that
$\mathcal{Z}_{\widetilde{S}}=\mathcal{M}_{\widetilde{S}}$ is a connected
component of $\mathcal{H}(\beta)^{\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}}$ diffeomorphic to $\ensuremath{\mathbb C} \ensuremath{\mathbb P}^1$.
Let us now consider $\mathcal{Z}_{S_i}$. By Theorem~\ref{thm:polMink} we know
that this space is diffeomorphic to $M^{2,2}(\alpha)$ formed by classes of
closed polygons in Minkowski $3$-space with the first two sides $u_1,u_2$ in
the future time-like cone and the last two, namely $u_3$ and $u_4$, in the past, where each
side $u_i$ has Minkowski length $\alpha_i$. Let us again consider the diagonal
$d_2=u_1+u_2$ connecting the origin to the third vertex of the polygon. This
vector is also a future time-like vector and we can consider its Minkowski
length $\ell$. Note that if we place the first vertex at the origin and use the
${\rm SU}(1,1)$--action to place the third vertex on the $t$-axis, then the
bending flow can be described as a rotation of the vectors $u_1,u_2$ around the
$t$-axis with a constant angular speed while fixing the other two vectors. Hence,
$\mathcal{Z}_{S_i}$ is a non-compact toric manifold with moment map $\ell$. By
the reversed triangle inequality we have that $\ell$ has the minimum value
$$\max \left\{ \alpha_1+\alpha_2\, ,\alpha_3+\alpha_4\right\}$$
which is attained at just one point (the polygon with two sides aligned along the $t$-axis) and has no other critical value. We conclude that $\mathcal{Z}_{S_i}$ is diffeomorphic to $\ensuremath{\mathbb C}$.
In all cases we conclude that $\mathcal{H}(\beta)^{\ensuremath{\mathbb Z}/2\ensuremath{\mathbb Z}}$ has one compact
connected component diffeomorphic to $\ensuremath{\mathbb C} \ensuremath{\mathbb P}^1$ and three non-compact components diffeomorphic to $\ensuremath{\mathbb C}$.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,886
|
{"url":"https:\/\/www.jiskha.com\/questions\/413724\/Find-the-area-of-the-curve-y-1-x-3-from-x-1-to-x-t-and-evaluate-it-for-t","text":"# Calculus ll - Improper Integrals\n\nFind the area of the curve y = 1\/(x^3) from x = 1 to x = t and evaluate it for t = 10, 100, and 1000. Then find the the total area under this curve for x \u2265 1.\n\nI'm not sure how to do the last part of question (\"find the the total area under this curve for x \u2265 1.\")\n\nFor the area of the curve, I found that the integral from 1 to t is (1\/2)-[1\/(2t^2)].\n\nI used the equation I found and substituted t for 10, 100, and 1000, and got 0.495, 0.49995, and 0.4999995 respectively.\n\n1. area = LIM [integral]1\/x^3 dx from x = 1 to x -- infin.\n= lim (-(1\/2)(1\/x^2) from x=1 to x-- inf.\n= lim{ -(1\/2)\/x^2 as x--inf} - lim {(-1\/2)\/x^2 as x=1\n= 0 - (-1\/2) = 1\/2\n\nposted by Reiny\n2. or\n\n1\/2 - 1\/(2t^2)\n\nas t ---> infinity, doesn't 1\/(2t^2) approach 0 ?\n\nso you are left with 1\/2 -0\n= 1\/2 , as your approximations suggested.\n\nposted by Reiny\n3. Oh! I understand now. The \"total area\" part kind of threw me off.\n\nThank you so much!\n\nposted by Alyssa\n\nFirst Name\n\n## Similar Questions\n\n1. ### Calculus\n\nIntegrals: When we solve for area under a curve, we must consider when the curve is under the axis. We would have to split the integral using the zeros that intersect with the axis. Would this be for all integrals? What if we just\n2. ### Statistics\n\nA certain curve (that is NOT the normal curve) is shown below. The curve is symmetric around 0, and the total area under the curve is 100%. The area between -1 and 1 is 58%. What is the area to the right of 1? I do not know how to\n3. ### calculus showed work\n\nfind the area of the rgion bounded by the graphs of y=x^3-2x and g(x)=-x i drew the graph and half of the graph is above the xaxis and the other half is below the axis. so the integrals i came up with are two because i broke them\n4. ### Calculus\n\nFind the area cut off by x+y=3 from xy=2. I have proceeded as under: y=x\/2. Substituting this value we get x+x\/2=3 Or x+x\/2-3=0 Or x^2-3x+2=0 Or (x-1)(x-2)=0, hence x=1 and x=2 are the points of intersection of the curve xy=2 and\n5. ### math, calculus 2\n\nConsider the area between the graphs x+y=16 and x+4= (y^2). This area can be computed in two different ways using integrals. First of all it can be computed as a sum of two integrals integrate from a to b of f(x)dx + integrate\n6. ### Calculus Area between curves\n\nConsider the area between the graphs x+6y=8 and x+8=y2. This area can be computed in two different ways using integrals First of all it can be computed as a sum of two integrals where a= , b=, c= and f(x)= g(x)= I found a, but not\n7. ### calculus II\n\nWe're doing areas by integrals now, with 2 eqns. I have a few questions. 1. Sketch the region in the xy-plane defined by the inequalities x-2(y^2)> 0 and 1-x-abs(y)>0. and find its area. Would it be best to solve for x, then\n8. ### Calculus\n\nCalculate the Riemann sum of the area under the curve of f(x)=9-x^2 between x=-2 and x=3 The answer I come up with is 10\/3, but when I solve using integrals, the answer I get is 100\/3. Am I doing something wrong?\n9. ### brief calc\n\nCalculate the total area of the region described. Do not count area beneath the x-axis as negative. Bounded by the curve y = square root of x the x-axis, and the lines x = 0 and x = 16 This is under Integrals, i don't know what\n10. ### calculus(Lab)\n\nWell, first graph the graph of f(x)=-1\/10x^2 + 3 2. We are going to approximate the area between f and the x-axis from x = 0 to x = 4 using rectangles (the method of Riemann sums). This is not the entire area in the first\n\nMore Similar Questions","date":"2018-08-16 12:45:36","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\": 0.8586432337760925, \"perplexity\": 736.4230918246681}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-34\/segments\/1534221210735.11\/warc\/CC-MAIN-20180816113217-20180816133217-00614.warc.gz\"}"}
| null | null |
\section{Introduction}
The so-called participation factor \cite{PF:ori} and contribution one \cite{CF}
quantify mutual impacts between state
variables and modes in multi-degree-of-freedom linear dynamical systems.
Many groups of researches have studied definitions and interpretations of these factors from
viewpoints of systems theory and physics (see, e.g., \cite{PF:garofalo, PF:new}).
Their generalizations to nonlinear systems are also reported in \cite{PF:nonlinear, PF:marcos}.
Such generalizations are of fundamental concern in
nonlinear systems theory and of technological significance in applications such as analysis and control of electric power systems \cite{PF:book}.
In this paper, as a novel proposal, we introduce the mode-in-state contribution factor to a class of nonlinear dynamical systems.
The mode-in-state contribution factor depends on (initial) states and is thus expected to extract global properties of nonlinear systems far from attractors.
Our theory is based on spectral theory of nonlinear dynamical systems, precisely speaking, spectral properties of the Koopman operators \cite{KMD:2005} and the idea of sensitivity analysis \cite{Kundur, PF:algebra}.
As introduced in Section~\ref{sec:KMD}, the Koopman operator is a linear infinite-dimensional operator defined for a nonlinear dynamical system and keeps full information on the nonlinear system: see, e.g., \cite{budisic12, ssk16, KMD:book}.
The spectrum of the linear operator is the mathematical foundation of Koopman Mode Decomposition (KMD) \cite{KMD:2005, KMD:rowley}.
Based on the KMD and the idea of sensitivity analysis, we introduce a data-driven mode-in-state contribution factor for a class of nonlinear systems.
It should be noted that a KMD-based ``participation" factor has been proposed in \cite{PF:marcos}, which does not depend on states whereas our contribution factor does.
The introduced contribution factor is applied to the numerical analysis of large-signal simulations in an interconnected AC/Multi-Terminal DC (MTDC) power system \cite{Ohashi,KWMT2019,ISGT,KWMT2020}
in order to show its technological potential.
This paper is a substantially enhanced version of \cite{TKMC-NLP20,TKMC-NSW20}, former of which introduces a different definition of the mode-in-state contribution factor based on KMD.
\section{Proposal}
This section describes a new proposal of the mode-in-state contribution factor for nonlinear systems.
For this, we introduce the Koopman operator and its spectrum leading to signal representation, called KMD (Koopman Mode Decomposition).
\subsection{Koopman operator and Koopman mode decomposition
(KMD)
}
\label{sec:KMD}
Motivated by the application to power system analysis in this paper, we now consider continuous-time dynamical systems described by the following Differential-Algebraic Equation (DAE):
\begin{equation}
\frac{d{\bm x}}{dt}={\dot{\bm x}} = {\bm F} ({\bm x} , {\bm y}), \qquad
{\bm 0} = {\bm G} ({\bm x}, {\bm y}),
\label{DAE}
\end{equation}%
where ${\bm x} \in {\mathbb R}^n$ is the vector of differential variables,
${\bm y} \in {\mathbb R}^m$ is the vector of algebraic variables, and
${\bm F}: {\mathbb R}^{n+m} \rightarrow {\mathbb R}^n$ and ${\bm G}: {\mathbb R}^{n+m} \rightarrow {\mathbb R}^m$ are
given nonlinear vector-valued functions.
We make the so-called regularity assumption \cite{SSK21} such that there exists a unique solution of DAE \eqref{DAE}, for which we address the dynamics in the set $\mathsf{R}$ defined as
\begin{equation}
\mathsf{R}:=\{({\bm x},{\bm y})\in\mathbb{R}^n\times\mathbb{R}^m~|~
{\bm G}({\bm x},{\bm y})={\bm 0},~
\mathrm{det}({\rm D}_y{\bm G}({\bm x},{\bm y}))\neq 0\},
\end{equation}
where ${\rm D}_y{\bm G}$ is the Jacobian of $\bm G$ with respect to $\bm y$.
It is shown in \cite{SSK21} that the Koopman operator is defined for DAE \eqref{DAE} with an asymptotically stable Equilibrium Point (EP).
To see this, we denote by ${\bm S}_t({\bm x},{\bm y})$ the solution in $\mathsf{R}$ starting at $({\bm x},{\bm y})$ and converging to the stable EP (with a non-empty basin of attraction $\mathsf{A}\subset\mathsf{R}$) as $t\to\infty$.
Then, the Koopman operator ${\bf U}_t$ is defined as a composition operator acting on a scalar-valued continuous function $g: {\mathsf A
\rightarrow {\mathbb C}$,
given by
\begin{gather}
{\bf U}_t {g} := {g} \circ {\bm S_t}, \quad t\geq 0.
\end{gather}
The function $g$ is called \emph{observable}.
Although DAE \eqref{DAE} and $\bm S_t$ are finite-dimensional and nonlinear, ${\bf U}_t$ is an infinite-dimensional but \emph{linear} operator acting on the space of functions.
The linearity is utilized in the so-called Koopman operator framework for analysis and synthesis of nonlinear dynamical systems: see, e.g., \cite{budisic12,ssk16,KMD:book}.
One successful method in the framework is the KMD, in which signals generated by nonlinear systems are represented in terms of spectral properties of the Koopman operator.
For this, the pair of \emph{Koopman eigenvalue} $\lambda\in\mathbb{C}$ and \emph{Koopman eigenfunction} $\phi_\lambda$ (a non-zero function defined on $\mathsf{A}$) is defined for $\mathbf{U}_t$ as follows:
\begin{equation}
{\bf U}_t \phi_\lambda =
{\rm e}^{\lambda t} \phi_\lambda, \quad t\geq 0.
\end{equation}
As proven in \cite{SSK21} under certain conditions, we expand a $K$-dimensional vector-valued observable ${\bm g}: \mathsf{A}\to\mathbb{R}^K$ in terms of Koopman eigenfunctions $\phi_j$, labeled by integer numbers $j$, and decompose the associated multivariate signal as follows:
\begin{equation}
{\bm g}(\bm S_t({\bm x},{\bm y}))
= \sum_{j=1}^{\infty} {\rm e}^{\lambda_j t} \phi_j({\bm x},{\bm y}) {\bm V_j},
\qquad t\geq 0,
\quad ({\bm x},{\bm y})\in\mathsf{A},
\end{equation}
where $\lambda_j$ are the $j$-th Koopman eigenvalues, and ${\bm V_j}$ are called \emph{Koopman modes} for the expansion.
If we can take $\bm g(\bm x) = \bm x$, then the time evolution of the differential variables, denoted as $\bm x(t)=\bm S_t({\bm x},{\bm y})|_{\mathbb{R}^n}$, is decomposed as follows:
\begin{equation}
\bm x(t) = \sum_{j=1}^{\infty}
{\rm e}^{\lambda_j t}\phi_j ({\bm x},{\bm y}) {\bm V_j},
\quad t\geq 0.
\label{KMD_x}
\end{equation}
Here, the $\bm y$ dependence in $\phi_j$ is resolved with the algebraic equation $\bm G(\bm x,\bm y)=\bm 0$.
Precisely speaking, by introducing a transformation $\bm \varphi$ satisfying $\bm G(\bm x,\bm y=\bm \varphi(\bm x))=\bm 0$, in which its existence is guaranteed in $\mathsf{R}$ according to the implicit function theorem, it is possible to remove the $\bm y$ dependence as $\phi_j ({\bm x},{\bm \varphi}(\bm x))$ and to simply rewrite it as $\phi_j ({\bm x})$.
For applications, it is numerically important to estimate the Koopman eigenvalues and eigenfunctions, and their computation is generally termed as the Dynamic Mode Decomposition (DMD) \cite{Kutz}.
In particular, the so-called Extended DMD (EDMD) \cite{EDMD
} is widely used for approximately deriving the Koopman eigenfunction as $\phi_j({\bm x}) \approx {\bm u}_j^\top {\bm \gamma}({\bm x})$,
where $\bm u_j\in\mathbb{C}^m$ are related to left eigenvectors of an approximate
matrix representation
of the underlying Koopman operator, computed directly from time-series data of ${\bm x}(t)$.
The $m$-dimensional vector-valued observable $\bm \gamma({\bm x})$ need to be designed
for the computation.
\subsection{KMD-based mode-in-state contribution factor}
As a novel point of this paper, we introduce a mode-in-state contribution factor based on the KMD.
This is based on the idea of sensitivity like \cite{Kundur, PF:algebra}.
Here, the term \emph{mode-in-state} indicates \emph{the contribution of modes in the time evolution of a single state, denoted by $x_k$, which are excited by a small change of initial value on the same state $x_k$}. For this purpose, we consider an infinitesimal change of the state evolution $x_k(t)$, expressed as
\begin{equation}
{\rm d}{x_k}(t) = \sum_{i=1}^n \frac{\partial x_k(t)}{\partial x_i}{\rm d}x_i,
\label{CF1}
\end{equation}
where ${\rm d}x_i$ is a small change of the initial state $x_i$.
Now, to quantify the effect of the excitation of state $x_k$, we set ${\rm d}x_i = 0~(i \neq k)$.
By substituting \eqref{KMD_x} into \eqref{CF1} with $\bm x=\bm x(0)$, \eqref{CF1} is formally re-written
as follows:
\begin{equation}
{\rm d}{x_k}(t) = \sum_{j=1}^\infty {\rm e}^{\lambda_j t} \frac{\partial \phi_j}{\partial x_k} ({\bm x}(0)) V_{j,k} {\rm d}x_k,
\label{CF2}
\end{equation}
where $V_{j,k}$ is the $k$-th element of ${\bm V}_j$.
Here, from \eqref{CF2}, we define $\omega_{k,j}$ as the mode-in-state contribution factor between $j$-th mode and $k$-th state as follows:
\begin{equation}
\omega_{k,j}:=\displaystyle \frac{\partial \phi_j}{\partial x_k} ({\bm x}(0))V_{j,k}.
\label{CF3}
\end{equation}
This factor $\omega_{k,j}$ is dependent on the initial state ${\bm x}(0)$ and hence affected by the nonlinearity of the DAE system \eqref{DAE} through the Koopman eigenfunction $\phi_j({\bm x})$.
Numerically, using the estimated Koopman modes $\bm v_j$
and Koopman eigenfunctions ${\bm u_j}^\top {\bm \gamma}({\bm x})$, $\omega_{k,j}$ is approximately computed as follows:
\begin{equation}
\left. \omega_{k,j} \approx v_{j,k} \sum_{\ell=1}^m u_{j,\ell} \frac{\partial \gamma_\ell}{\partial x_k}\right|_{{\bm x}={\bm x}(0)}.
\label{CF4}
\end{equation}
For the linear case, $\omega_{k,j}$ includes
the mode-in-state participation factor in \cite{PF:ori}.
Consider the linear DAE with $\bm F(\bm x, \bm y)={\bm A}_1{\bm x}+{\bm A}_2{\bm y}$ and $\bm G(\bm x, \bm y)={\bm A}_3{\bm x}+{\bm A}_4{\bm y}$ ( ${\bm A}_1\in\mathbb{R}^{n\times n}, {\bm A}_2\in\mathbb{R}^{n\times m}, {\bm A}_3\in\mathbb{R}^{m\times n}, {\bm A}_4\in\mathbb{R}^{m\times m}$).
It can be obtained by calculating the Jacobian matrices of the terms on the right-hand sides of the nonlinear DAE \eqref{DAE}.
If $\mathrm{det}(\mathrm{D}_y\bm G)=\mathrm{det}(\bm A_4)\neq 0$, then the dynamics of the differential variables are represented by the linear ordinary differential equation as
\begin{equation}
\dot{\bm x}=(\bm A_1-\bm A_2\bm A^{-1}_4\bm A_3)\bm x
=:\bm A\bm x.
\label{eqn:linearODE}
\end{equation}
For this linear system, by assuming distinct $n$ eigenvalues $\lambda_j$ of $\bm A$ and associated left- (or right-) eigenvectors $\bm u_j$ (or $\bm v_j$), it is shown in \cite{Mezic13} that $\lambda_j$ and $\phi_j(\bm x)=\bm u^\top_j\bm x$ are principal eigenvalues and eigenfunctions of the Koopman operator.
Since the Koopman mode $\bm V_j$ corresponds to $\bm v_j$ for the expansion of $\bm x$ in the linear system \cite{KMD:book},
\eqref{CF3} is written as
\begin{equation}
\omega_{k,j}=u_{j,k}v_{j,k},
\label{PF_linear}
\end{equation}
where it coincides with the participation factor \cite{PF:ori} derived for the linear system \eqref{eqn:linearODE}.
\section{Application to power system analysis}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=1.0\textwidth]{ACMTDC.eps}
\caption{Single-line diagram of interconnected AC/multi-terminal DC power system in \cite{ISGT}
}
\label{ACMTDC}
\end{center}
\end{figure*}
In this section, we apply the KMD-based mode-in-state contribution factor to the analysis of an interconnected AC/MTDC (Multi-Terminal DC) power system in \cite{Ohashi,KWMT2019,ISGT,KWMT2020}.
The single-line diagram of the AC/MTDC system is depicted in Fig.~\ref{ACMTDC}.
As shown in \cite{KWMT2019,ISGT}, the mathematical model of the system is represented with DAE
\eqref{DAE} with 12 differential variables $\bm x$ and 7 algebraic ones $\bm y$.
The $\bm x$ includes the angular position $\delta$ of the synchronous generator, the deviation $\omega$ of rotor speed relative to a nominal angular frequency, the voltage $e'_q$ behind transient reactance in the generator, the DC current $I_{{\rm dc}ij}$ between DC bus $i$ and bus $j$, the DC voltage $v_{{\rm dc}i}$ at bus $i$, and the control inputs $u_i$ at Voltage Source Converter (VSC) $i$.
Also, $\bm y$ includes the amplitudes and arguments (angles $\theta_i$) of voltage phasor at AC bus $i$.
The variables focused in this paper are shown in Fig.~\ref{ACMTDC}, and the details of the model are described in \cite{Ohashi,KWMT2019,ISGT}.
The model is intended for large-signal analysis of the AC/MTDC system, in which we need to evaluate the system dynamics beyond a neighborhood of a stable EP, that is to say, affected by the nonlinearity of the mathematical model.
The KMD-based mode-in-state contribution factor is now evaluated for numerical simulations of the model against disturbances in the generator voltage $e'_q$.
The simulations are necessary for estimating the Koopman eigenvalues and eigenfunctions with
EDMD.
We generate $401$ initial conditions of $e'_q$ as $e'_q = e_q^{\prime \ast} + 0.0005\ell$ (for $\ell = 0,1,\ldots,400$), where $e_q^{\prime \ast}$ is the value at a stable EP, and we fix the initial conditions of the other differential variables in $\bm x$ at the stable EP.
Then, for each initial condition, we simulate the time series of $\bm x$ with 300
numbers of snapshots and sampling period 0.005\,s.
Therefore, 120300 snapshots are used for the EDMD.
In addition, based on \cite{Marcos2020},
the 22
observables are selected as
$\bm\gamma (\bm x)=[{\bm x}^\top, \cos(\delta-\theta_1), \sin(\delta-\theta_1), \cos2(\delta-\theta_1), \sin2(\delta-\theta_1), \omega \cos(\delta-\theta_1), \omega \sin(\delta-\theta_1), \omega \cos2(\delta-\theta_1), \omega \sin2(\delta-\theta_1), e_q^\prime \cos(\delta-\theta_1), e_q^\prime \sin(\delta-\theta_1)]^\top$.
The parameters are basically the same values as used in \cite{ISGT} although they do
not show their complete list.
In this paper there
is no space here to show the complete set due to the limitation of space, but which can be provided upon request and will be shown in an archive.
Some of Koopman eigenvalues estimated by EDMD and eigevalues of the matrix $\bm A$ of \eqref{eqn:linearODE} derived via linearization of
the mathematical model around a stable EP are presented in Table \ref{table:Eigenvalues}, which are related to the dynamics of the synchronous generator in the AC system.
In Table \ref{table:Eigenvalues},
the estimated Koopman eigenvalues $\lambda_1$, $\lambda_{2,3}$ are close to the
eigenvalues $\lambda_1^\prime$, $\lambda_{2,3}^\prime$ for the linearized model.
This is valid in terms of the spectral characterization of the DAE with stable EP in \cite{SSK21}.
Furthermore, the Koopman eigenvalues $\lambda_{4,5}$ are likely the linear combination as $\lambda_1 + \lambda_{2,3} \times 2 = -3.06 \pm 27.2{\rm i}$.
The linear combination is known as the algebraic property of Koopman eigenvalues (see, e.g., \cite{SSK21}), hence $\lambda_{4,5}$ are generated by the nonlinearity of the model.
Finally, the KMD-based mode-in-state contribution factors are evaluated for different initial conditions.
For clear comparison, motivated by \cite{Endegnanew}, we introduce the normalized magnitude of contribution factor based on the absolute values of $\omega_{k,j}$ as follows:
\begin{equation}
\omega_{k,j}^\ast := \frac{|\omega_{k,j}|}{\sum_{j=1}^{m}|\omega_{k,j}|}.
\label{CF_normal}
\end{equation}
Fig.~\ref{CF_ALL} shows the normalized magnitudes of contribution and participation factors for the four variables $I_{\rm dc23}, \delta, e'_q$, and $\omega$.
The reason why the variables are chosen is that we emphasize the meaning and the relevance of the proposed factors as explained below.
The mode-in-state participation factors with the conventional linear modal analysis calculated by \eqref{PF_linear} are presented in Fig.~\ref{CF_ALL}(a).
The KMD-based mode-in-state contribution factors are presented in Figs.~\ref{CF_ALL}(b) and (c).
The difference between Figs.~\ref{CF_ALL}(b) and (c) is that of the initial condition: $e'_q(0)=e^{\prime\ast}_q+0.01$ for (b) and $e'_q(0)=e^{\prime\ast}_q+0.2$ for (c).
In the figures, multiple colors are used to show the contributions of multiple modes, where some of the eigenvalues in Table~\ref{table:Eigenvalues} are explicitly shown.
A clear difference in the contribution factors of $\delta$ among Figs.~\ref{CF_ALL}(a)-(c) is observed.
In particular, the dependence on initial conditions in the figures (b) and (c) is caused by the nonlinearity of the model.
In Fig.~\ref{CF_ALL}(b) with the small excitation, the linear mode $\lambda_{2,3}$ has a large contribution, whereas
$\lambda_1$ has a small contribution.
In Fig.~\ref{CF_ALL}(c) with the large excitation, $\lambda_{2,3}$ has lower a contribution and $\lambda_1$ has a larger contribution to $\delta$.
Also, the contribution factors for $I_{\rm dc23}$ in the MTDC system do not change when the change of initial conditions in the AC system happens.
This implies that disturbances that have strong impacts on the AC system do not affect the DC system remarkably.
This computation is relevant in terms of control systems of VSC in the interconnected AC/MTDC system.
This is because the power system is controlled by the VSC in which dynamical interactions between the AC and MTDC systems are minimized.
\begin{table}[t]
\caption
Eigenvalues for linearized model and estimated Koopman eigenvalues, which are related to the dynamics of the synchronous generator in the AC system}
\vspace*{2mm}
\label{table:Eigenvalues}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
\multicolumn{2}{|c|}
Eigenvalues for Linearized Model} & \multicolumn{2}{|c|}{Koopman Eigenvalues}\\
\hline
${\lambda}_1^\prime$ & $-1.64$ & $\lambda_1$ & $-1.80$\\
${\lambda}_{2,3}^\prime$ & $-0.64$ $\pm$ 13.5i & $\lambda_{2,3}$ & $-0.63$ $\pm$ $13.6$i \\
& & $\lambda_{4,5}$ & $-2.98$ $\pm$ $26.2$i\\
\hline
\end{tabular}
\end{table}
\begin{figure}[t]
\begin{minipage}[b]{0.32\linewidth}
\centering
\includegraphics[width=1.1\textwidth]{PF_linear_1210}
\subcaption{linearization}
\label{PF}
\end{minipage}
\begin{minipage}[b]{0.32\linewidth}
\centering
\includegraphics[width=1.1\textwidth]{CF_001_1210}
\subcaption{KMD-based}
\label{CF001}
\end{minipage}
%
\begin{minipage}[b]{0.32\linewidth}
\centering
\includegraphics[width=1.1\textwidth]{CF_020_1210}
\subcaption{KMD-based}
\label{CF020}
\end{minipage}
%
\caption
Computation results on mode-in-state contribution and participation factors $\omega^\ast_{k,j}$ for the four variables.
\tkmc{The difference between Figs.~\ref{CF_ALL}(b) and (c) is that of the initial condition: $e'_q(0)=e^{\prime\ast}_q+0.01$ for (b) and $e'_q(0)=e^{\prime\ast}_q+0.2$ for (c). }
Multiple colors are used to show the contributions of multiple modes (eigenvalues), where some of the eigenvalues in Table~\ref{table:Eigenvalues} are explicitly shown.}
\label{CF_ALL}
\end{figure}
\section{Conclusion}
A KMD-based mode-in-state contribution factor guided by the sensitivity analysis was proposed and applied to the analysis of an interconnected AC/MTDC power system.
Numerical results show that the proposed factor can capture the nonlinearity of the model and the physical property of the target power system.
Several follow-up studies on this paper are possible.
A KMD-based state-in-mode contribution factor should be pursued to quantify the mutual impact between one mode and each state.
Also, an application of the proposed factor to a multi-machine power system exhibiting an inter-area oscillation is interesting and important in practical viewpoints.
\section*{Acknowledgments}
The work was partially supported by JST-PRESTO Grant Number JPMJPR1926 and JST-SICORP Grant Number JPMJSC17C2.
\section*{Appendix \red{to the Archival Manuscript}}
We show the
parameter values \red{for numerical simulations} of the AC/multi-terminal DC system used in Section 3.
The same parameters and mathematical model are used as in the simulation\red{s} in \cite{ISGT} although they do not show their complete parameter list.
The parameters on the AC and DC grids are mainly based on \cite{Ohashi}, and the parameters on the \red{synchronous} generator are mainly based on \cite{KWMT2019}.
Table \ref{table:Parameters} shows the descriptions and values of the parameters on the AC grid, DC grid, and generator.
\red{For per-unit system,}
we \red{use the following base quantities}:
Base power of the generator is 2800\,MVA;
Base power \red{and voltage} of the AC \red{grid}
is 1000\,MVA and
is 500\,kV;
Base power \red{and voltage} of the DC \red{grid}
is 1000\,MVA and
500\,kV; and
Base angular frequency of \red{both} the AC and DC \red{grids}
is $50 \times 2\pi$ radian \red{per second}.
\newpage
\begin{table}[t]
\caption{
Descriptions and values of the parameters on \red{the} AC grid, DC grid, and
generator.
The parameters are mainly based on \cite{Ohashi, KWMT2019, ISGT}.
}
\vspace*{2mm}
\label{table:Parameters}
\centering
\begin{tabular}{ccc}
\hline
\hline
Parameter & Description & Value\\
\hline
$r_{\rm AC}$ & Resistance of AC \red{line}
per length [\si{\ohm}/km] & 0.0261\\ \hline
$l_{\rm AC}$ & Reactance of AC \red{line}
per length [mH/km] & 0.83\\ \hline
$r_{\rm DC}$ & Resistance of DC \red{line}
per length [\si{\ohm}/km] & 0.0192\\ \hline
$l_{\rm DC}$ & Reactance of DC \red{line}
per length [mH/km] & 0.24\\ \hline
$c_{\rm DC}$ & Capacitance of DC \red{line}
per length [$\mu$F/km] & 0.152\\ \hline
$c_{\rm DC conv}$ & Smoothing capacitance \red{for VSC} [$\mu$F/km] & 75\\ \hline
$K_V$ & \red{Constant for voltage conversion}
& 2\\ \hline
$\sqrt{3}K_I$ & \red{Constant for current conversion}
& 1\\ \hline
$G$ & \red{Gain constant of controller for VSC}
& --1\\ \hline
$T$ & Time constant \red{of controller for VSC}
& 0.001\\ \hline
$P_{\rm DC(ref)1}$ & Reference power at VSC1 & 0.2\\ \hline
$V_{\rm DC(ref)2}$ & Reference voltage at VSC2 & 1.0\\ \hline
$P_{\rm DC(ref)3}$ & Reference power at VSC3 & --0.3\\ \hline
$X_{d}$ &
$d$-axis synchronous reactance \red{in generator}
& 1.79\\ \hline
$X_{q}$ &
$q$-axis synchronous reactance \red{in generator}
& 1.77\\ \hline
$X^{\prime}_{d}$ &
$d$-axis transient reactance \red{in generator}
& 0.3\\ \hline
$E_{fd}$ & Constant voltage behind $d$-axis synchronous reactance & 1.70\\ \hline
$P_{m}$ &
Mechanical power injection \red{to generator} & 0.5\\ \hline
$D$ &
Damping coefficient \red{in generator} & 1.0\\ \hline
$H$ & Inertia constant of rotor \red{in generator} & $0.89 \times 100\pi$\\ \hline
$T^{\prime}_{d0}$ & $d$-axis transient open-circuit time constant & $1.2 \times 100\pi$\\ \hline
\hline
\end{tabular}
\end{table}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 8,518
|
{"url":"http:\/\/math.tutorcircle.com\/analytical-geometry\/how-to-find-the-equation-for-locus-of-a-point-whose-distance-from-a-point-equal-its-distance-from-y-axis.html","text":"Sales Toll Free No: 1-855-666-7446\n\n# How to Find the Equation for Locus of a Point whose Distance from a Point equal its Distance from y-Axis?\n\nTopLocus is defined as a curve that is achieved when some Point moves in the space according to certain conditions. So, the curve we obtain can be called as a collection of points each of which satisfying the given conditions or we can also say that every point satisfying the given conditions are supposed to lie on this curve. For any expression or relation H (x, y) = 0 such that every point lying on the locus satisfies this equation, then it is called as the equation of the locus. So, let us learn how to find the locus of any point in the space through an example:\n\nExample: How to find the equation for locus of a point whose distance from a point equals its distance from y axis?\n\nSolution: Given condition to find the locus of a point whose distance from y \u2013 axis equals to the distance from other point. It is clear from the given condition that the point whose locus has to be found lies between the y \u2013 axis and other point. To maintain an equal distance from the y \u2013 axis and the other point, required point must lie in a line such that its distance from the y \u2013 axis and the point to the right or to the left remains constant\n\nAlso this line has to be parallel to the y \u2013 axis and perpendicular to the x \u2013 axis. So, this will make an intercept on the x \u2013 axis and its equation can be given as: x = c, where 'c' is the distance that the point maintains from y \u2013 axis and other point. Suppose the distance between the line parallel to y- axis is 4 units then equation of line can be given as x = 4.","date":"2016-09-30 03:11:01","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\": 0.811419665813446, \"perplexity\": 127.75234359823567}, \"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-2016-40\/segments\/1474738662018.69\/warc\/CC-MAIN-20160924173742-00265-ip-10-143-35-109.ec2.internal.warc.gz\"}"}
| null | null |
How Long Is 'Bandersnatch'? You Can Literally Get Lost In This 'Black Mirror' Movie
By Leah Marilla Thomas
Black Mirror has returned with an interactive and very special "choose your own adventure" episode that fans have already spent hours combing through. Whether you're just about to jump down the rabbit hole or have already chosen some "wrong paths," knowing exactly how long Black Mirror: Bandersnatch actually is is valuable information for your mental health. But asking how long this thing really takes is a complicated question, with some unsatisfactory answers. Much like Bandersnatch itself.
According to Netflix, the run time for default choices (which Bustle believes means choosing the left hand option every single time) is 90 minutes. The first time I went through, I had reached a few possible endings and their variations in just under that amount of time, so it checks out.
But you can easily get stuck in a loop — or start the episode over to try and reach a different ending, and that could ultimately take hours. According to Netflix, there are about five main endings, with variations on each one, and there are theoretically trillions of possibilities (but there are not a trillion paths or endings). Spoiler alert: one thing that means is that some of the choices don't actually affect the plot, but they are choices and do affect your viewing experience.
There's no status bar at the bottom of the film like there usually are on Netflix shows and movies (you know, the red bar that tells you how much longer you have left), and there no time stamps that tell you how much you've watched... because Netflix doesn't know! That's up to you. Unless you're at a decision point, you can pause the episode or do a 15 second rewind and fast forward. Other than that there's no way to track where you are in the story at any given time. Certain "endings" allow you to escape to the credits, or return to a fork in the story to take another path.
If you do pause, or stop the episode yourself, Netflix will remember your place and your choices. There will also be a small thumbnail status bar, which you could use to indicate how much time you have left, but given the number of paths you can take and endings you may want to see, that's not really relevant.
A leak before Bandersnatch was announced suggested that the total runtime was 312 minutes. Does that mean that this are over five hours of content? Keep in mind that it took 35 days to shoot, which is a lot for television but well under the average Hollywood film. By my rough estimate, the only way that runtime is correct is if you include all the rewatching and recapping one must complete in order to catch every unique minute. But then again, I could have it all wrong.
Here's the bottom line: carve out at least 90 minutes for yourself to watch and play the new Black Mirror. Plan for two hours or more if you really want to get lost in it, or keep coming to unsatisfactory conclusions. As one of the characters insists, time is just a construct anyway.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,647
|
Q: How to add content-type to WordPress favicon Good afternoon, there is a standard favicon output -
<link rel="icon" href="/wp-content/uploads/2016/04/MyLogo.png" sizes="32x32" />
Is there an example function and hook that would add type=" desired type" to the output in the link
<link rel="icon" type="image/png">
A: You can programatically change this through functions.php
function myfavicon() {
$path = get_bloginfo('wpurl') . "/wp-content/uploads/2016/04/MyLogo.png";
echo '<link rel="icon" type="image/png" href="' . $path . '" sizes="32x32" />';
}
add_action('wp_head', 'myfavicon');
I also have added the absolute path because for IE you need a fully qualified URL instead of a relative url.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,408
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\section{Introduction}
\subsection{Previous work}
Pipelines for automating the detection and processing of moving objects have been proposed and implemented \citep{mops-Allekotte} in the past. Some, such as \citet{mops-Bekte}, have focused on detection only, with little subsequent processing. Due to the computationally intensive tasks associated with detection, more recently such systems have been implemented in the cloud \citep{mops-a24n}. Of these pipelines, very few (and none from the major surveys of minor planets, except Pan-STARRS \cite{Magnier_2020} \cite{panstarss_ipp}) are known to be open-source.\\
Among asteroid detection methods, synthetic tracking \citep{syntrack-2005} has the best detection properties, however, it requires significant processing power and fast camera readout, thus only recently \citep{syntrack-Shao_2014} has become accessible to surveys \citep{syntrack-zhai2018technical}. Some faster alternatives to synthetic tracking have been suggested by \citet{trails-brown}, albeit with worse noise rejection (and still requiring fast imaging equipment). Therefore, efforts were mainly concentrated on the blink method.\\
Among detection algorithms for the blink method, notable are maximum likelihood techniques, as described in \citet{dawson2016blind}, which, given a set of assumptions, yield an optimal algorithm. Such detection methods are non-trivial to implement, especially since image defects can be hard to model accurately, and the optimization nature of the task can make it computationally expensive compared to simpler algorithms, such as flood fill.\\
One project, \citet{Waszczak_2017}, targeting asteroid trails using the Palomar Transient Factory camera has focused on applying machine learning to a set of morphological features of the detections to separate the latter from noise. Notably, this project has documented well (through a rather extensive graphical enumeration) the failure modes of a simple detection pipeline. From the authors' experience, these occur on other detection pipelines too, but most can be solved through detection post-processing (the approach taken by Umbrella), significantly improving rejection rate of false positives and loss rate of true positives. However, this project uses the flood fill algorithm, which is less efficient at detecting trails, and the software is not known to be open-source.
\subsection{History of the project}
The version presented in this paper is an early preview of the third version of the Umbrella software. The Umbrella project started in June 2015, as a volunteer summer research project that has extended and expanded intermittently over the past 5 years, targeting automated asteroid detection using the blink method (requiring a minimum of 3 images per field). Umbrella is now part of the larger EURONEAR project\footnote{\url{www.euronear.org}}. The following is an approximate timeline of the development of the software.\\
At the start of the project, Source Extractor was considered for detecting light sources in the image; the catalogs generated were to be compared and only the transient objects were to be kept. The method proved to be difficult due to the large number of bad pixels and cosmic rays that plagued the images. It was decided that the moving object detection and processing software would use its own detection and filtering algorithms. Within the next year, the resulting desktop application, named SourceUmbrella, peaked at 80 \% detection rate (compared to human blink methods) on the sample images provided (from the EURONEAR archive \citep{euronear-discoveries1}\citep{euronear-discoveries2}) for development (training / development set) and reached 20-50 \% detection rate on new images (test set). However, it had much worse performance in detecting trailed objects (such as Near Earth Asteroids). It was also very rigid in accepting FITS files, which made the use of other image sources difficult. The architecture of the application also proved unsustainable for the wide array of variations in the pipeline.\\
Beginning in December 2017, the development resumed on a new design, an open-source modular library, called Umbrella2, that would be much more flexible and overcome the shortcomings of SourceUmbrella. In particular, due to the low detection rate of Umbrella2 on trailed objects, and taking into account that another similar software developed for the EURONEAR project, NEARBY \citep{nearby-conference}\citep{nearby-unpub}, would not detect trailed objects either, the development focused more on trail object detection. In that sense, in February 2018, a new line segment detection algorithm was designed, based on the Hough Transform \citep{hough-dudahart}, to detect long and faint NEA trails (SV-AFAV-HT, i.e. State Variable Auxiliary Feature Augmented Voting Hough Transform, paper by Stănescu in prep - see Annex A for a working description). By November 2018 a desktop application building on the Umbrella library was ready for processing images and had an early live test during the November 2018 EURONEAR survey; using the images with astrometry (more precisely, the FITS World Coordinate System headers) resolved by the NEARBY pipeline, as described in \citet{pipeline-denisa} (using AstrOmatic software: Source Extractor \citep{sExtractor}, SCAMP \citep{scamp} and SWarp \citep{swarp-terapix}). Since then, many components of the library have been improved and many bugs have been resolved.\\
In the second half of 2019, the focus shifted to the development of a web-based deployment of Umbrella. This resulted in the Webrella\footnote{\url{http://s141.central.ucv.ro:49239/}} server, a web interface to the configurable pipeline also used by the desktop version. The web server was ready by the end of October (2019), and had been tested with images from previous runs. It also had a very limited live test during the October-November survey, using images captured with the Wide Field Camera (WFC) on the Isaac Newton Telescope (INT). Unfortunately, some crashes in the NEARBY pipeline (installed on the same server) that provided WCS-reduced images to Webrella hindered live testing of the web version during the next INT survey. Nevertheless, the desktop version (which acted as a backup) has been used successfully for near real-time in the survey, using the images from the remaining NEARBY server.
\subsection{Project source code and technology stack}
The entire Umbrella suite is developed on top of Mono / .NET Framework, with the library and the ViaNearby pipeline written entirely in C\#. The server backend for Webrella is built on top of the Nancy Framework, with plain HTML on the browser side. The desktop GUI is implemented using WinForms. The software should be portable across operating systems (Windows, Linux, MacOS, BSD) and architectures (x86, ARM, etc.), with minor caveats mostly depending on the underlying Mono/.NET implementation (for example WinForms is not implemented on 64-bit MacOS, so the desktop application is unavailable on that platform). The authors have tested it only on Windows, Linux and MacOS on x86. It should be noted that the custom quickselect algorithm (presented below in this paper) is by default compiled to require 64-bit support.\\
The source code of the core library is available on Github, at \url{https://github.com/mostanes/umbrella2}. The code is documented in-line using XML, covering almost all classes, properties, fields and methods, both public and private. This documentation is available both to IDEs (typically available when writing the code, through a similar mechanism to autocomplete) and Umbrella itself (for example, when inspecting internal data structures with the property inspector, see Figure \ref{fig:vn-objprop}). Stable versions are available on NuGet\footnote{\url{https://www.nuget.org/packages/umbrella2/}}. The source code of ViaNearby is available at \url{https://github.com/mostanes/umbrella2-euronear}. The ViaNearby pipeline code has scarce in-line documentation. Besides the documentation available in the source code, releases of the ViaNearby pipeline come with usage and reference manuals.
\begin{figure*}
\begin{subfigure}{\textwidth}
\centering
\includegraphics[width=.99\linewidth]{webrella-detection.png}
\caption{A valid detection as a reducer would see it}
\label{fig:wbr-detect}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=.9\linewidth]{webrella-loadfield.png}
\caption{Uploading a field. It is possible to either directly upload the files or pull them from an existing NEARBY deployment residing on the same server.}
\label{fig:wbr-load}
\end{subfigure}
\hspace{0.1\textwidth}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=.9\linewidth]{webrella-userperm.png}
\caption{Adjusting the permissions of a Webrella user for a survey}
\label{fig:wbr-perm}
\end{subfigure}
\caption{Webrella interface}
\label{fig:webrella}
\end{figure*}
\section{Umbrella2 core platform provisions}
Umbrella2 has a set of algorithms and interfaces on which the rest of the platform is based. They are enumerated below:
\subsection{Shared components}
A common library provides the types for Cartesian coordinates on 2D images, Projection plane coordinates (as in the FITS 3.0 standard \citep{fitsStandard}), and Equatorial coordinates, as well as the corresponding velocity vectors. Also present in this common library are the interface definitions for transforming between the different coordinate types. A converter to and from Minor Planet Center optical report format \cite{mpcobsformat} of the coordinates is also provided.\\
The common library also provides a plugin system (lightweight dependency-inversion), which can scan loaded modules and automatically use them in extensible components (such as the WCS projection types). Another extensibility feature present in the common library is the property model, a type-safe mechanism for dynamically attaching new data (which is usually either optional or user-defined) to objects implementing the property model (such as detections and tracklets).
\subsection{Image I/O}
Umbrella2 provides support for generic image types -- besides the I/O implementation for FITS files included in the I/O library, users may write support for other image formats of their own convenience. The FITS implementation in Umbrella2 handles the images according to the FITS Standard Version 3.0 \citep{fitsStandard}, including multi-image variants (MEF) and all 6 types of data representation (BITPIX), with extensible support for parsing headers and WCS projection types. There are however a few known missing features:\\
\begin{itemize}
\item Currently the only the gnomonic (TAN) projection type has an implementation. (however, other projection types can be implemented in user plugins).
\item No support for BSCALE and BZERO flux scaling.
\item No support for quoting via consecutive single quotes in the headers.
\end{itemize}
The last two in particular make Umbrella2 non-compliant to the FITS standard. These shortcomings may be addressed in future releases.\\
A noteworthy detail is that Umbrella2 accepts FITS files either via memory-mapped files or from non-seekable streams, where the contents are being copied into memory, so that files may be streamed over networks if deemed necessary. For performance considerations, all images are by design read in chunks, with thread-safe accesses synchronized by a readers-writers lock allowing independent access to non-overlapping portions of the image. The lock can be disabled when access is proxied and the proxy provides the locking facility.
\subsection{Algorithms Framework}
To maintain a clean, high-level implementation of all image processing tasks while retaining high performance levels, Umbrella provides a framework that abstracts away image I/O and algorithm scheduling. This framework is, of course, exposed to library users, so that users may run their own algorithms on par with built-in ones. Included in the framework are also common definitions and interfaces that provide a high degree of modularity and seamless integration of different algorithms in pipelines.
\subsubsection{Detections and Tracklets}
The algorithms framework offers the types representing detections on individual images and tracklets (sets of detections that correspond to the same object on consecutive images). Both of these types are extensible through the property model. Very common properties are provided in the library: object identity, photometry measurements (flux and magnitude), the coordinates on the image (including all coordinates of the detection blob in the X-Y plane), shape and size information, pairing information, and velocity regression data.
\subsubsection{Object identity}
Umbrella2 can match MPC names of asteroids to tracklets, or create its own provisional names (given a field name, tracklet number, and optionally a CCD number). It supports both numbered and temporary designations for objects, as well as the packed form of the two (provisional names created by Umbrella are given in the form of packed temporary designations, where the first 4 characters are the field name). Name matching is implemented with a scoring algorithm, which gives scores normalized as integers from 0 to 100 (minimum score 1). The score takes into account the distance between the estimated and measured positions of the object and decreases exponentially with the number of detections that are very far from the estimated positions:
\[ \textbf{Score} = \frac{2^{\sqrt{n}}}{\bar{d} + 1} \]
normalized to $100$, where $n$ is the number of detections and $\bar{d}$ is the average distance between the detection and the SkyBoT prediction in arcseconds.
\begin{figure*}
\centering
\includegraphics[width=.9\linewidth]{trackviewer2.png}
\caption[Tracklet Viewer]{An object that is not in the SkyBoT database.}
\label{fig:vn-trackview-unk}
\end{figure*}
\subsubsection{Scheduler for high-level image algorithms}
Due to the amounts of data to be processed, a goal of Umbrella is for algorithms to make use of all available computing resources. This is achieved by a scheduler for the algorithms that runs the computation kernel in parallel over multiple threads, defaulting to one per logical core, on the CPU. Future versions of Umbrella may allow support for scheduling the algorithms on GPGPUs (General Purpose Graphics Processing Units) too. A notable feature of the scheduler is that it also separates the I/O layer from the algorithms, which allows algorithm developers to focus on the algorithms themselves.\\
It should be noted that the current CPU scheduler implementation comes with some limitations regarding input images where WCS crops are significant (such as when input images have lower overlap) as well as image chunk size and the level of parallelism.
\subsubsection{Miscellaneous}
Several convenience math routines are also provided with the algorithms framework for: linear regression, intersection of lines, and intersection of semilines with rectangles. Besides the math routines, there are also three data structures available for algorithms: a quad tree, a multi-threaded resource pool, and a graph structure for finding connected components.
\subsection{Visualization components}
\subsubsection{FITS Viewer component}
To present sections of FITS images within Umbrella, the software package includes a FITS viewer widget that can display images centered around a given coordinate. The viewer also supports highlighting a given set of pixels. Data is displayed via a scaling algorithm; currently, only a linear scale is implemented.
\subsubsection{Tracklet viewer}
The results of a pipeline using Umbrella can be displayed using a "Tracklet viewer" window (Figure \ref{fig:vn-trackview-hq}). This window offers visual and parametric inspection of the tracklets (and their individual detections). Visual inspection may be performed on any of the input or intermediate images (which have been tagged by the pipeline as corresponding to the same original image). The displayed parameters are selected from those computed in the pipeline. Wherever possible, names of the tracklets are obtained from their ObjectIdentity property(see Figure \ref{fig:vn-trackview-unk}). To better cope with the cases where the pipeline filtering might break down (due to missing or incorrect badpixel files, very bright stars, or broken optics correction), portions of the WCS or image coordinates can be manually filtered out from the results (see Figure \ref{fig:vn-trackview-filter}). Also for the convenience of the reducers, common operations (navigation, blinking) can be directly performed using the keyboard.
\begin{figure*}
\centering
\includegraphics[width=.9\linewidth]{trackviewer.png}
\caption[Tracklet Viewer]{Example of a high quality input image. Detections on each image are shown individually. Important information about the detection is available at a glance.}
\label{fig:vn-trackview-hq}
\end{figure*}
\section{Algorithms provided in the core repository}
Umbrella provides algorithms for all steps in the detection and processing of the moving objects:
\subsection{Noise removal}
Noise removal is performed using multiple algorithms for different purposes. Input images can be masked with a badpixel file. Initial per-image denoising can be performed with a trimmed mean filter (of which there exists a variant combined with the badpixel filter, which yields significantly better results). Cross-image denoising for creating a static objects mask can be done with a median filter. A further deep smoothing can be performed to use a much faster version of the AFAV long trail detection algorithm. There is also a simple implementation of image normalization.
\subsection{Object detection}
The library comes with two built-in object detection algorithms. The first is a simple threshold method, using two thresholds in a hysteresis flood fill. Historically, this is the algorithm used in the first version of Umbrella; it is referred to as the blob detection algorithm. The other detection algorithm uses a State Variable Auxiliary Feature Augmented Voting Hough Transform (SV-AFAV-HT, a technique based on the Hough Transform, paper by Stănescu in prep; a working description also presented in Annex A) on connectivity, which combines local and global properties using a Markov variable along the Radon line basis, to identify potential long trails. These potential long trails are further detected through another flood fill algorithm, the blobs being combined to yield more accurate trail detections. It is also possible to import Source Extractor \citep{sExtractor} catalog files.\\
There is also a variant of the blob algorithm for detection recovery on original images, which estimates positions from the tracklet velocity regression and can work with lower thresholds.
\subsection{Object pairing \& filtering}
Per-image detections are combined into tracklets through a pairing algorithm. There are two implementations currently available in Umbrella, one newer and which is expected to give better results (but has not had extensive testing on long trails) and an older one, better tested, but with a weaker design (which is why it is considered obsolete).\\
The library has a few built-in detection and tracklet filters. Currently, they are boolean, but this is expected to change in the future, so that the parameters can be automatically tuned through machine learning.
\begin{figure*}
\centering
\includegraphics[width=.8\linewidth]{trackviewer-filter.png}
\caption[Tracklet Viewer]{On inputs where image artifacts greatly increase false detection rate, manual filtering is possible.}
\label{fig:vn-trackview-filter}
\end{figure*}
\subsection{External APIs and tools}
Umbrella can take advantage of existing platforms in reducing images. Currently, it hosts methods for accessing SkyBoT \citep{skyBot} and VizieR \citep{vizier} APIs and algorithms to integrate the query results into the pipeline: there are functions to match objects to SkyBoT name lists, stars to VizieR star lists and for calibrating the zero-point magnitude. Furthermore, there are functions for running the Digest2 \citep{digest2} software locally on the resulting MPC reports, so that objects of interest can be spotted quickly.
\section{Detailed description of the algorithms}
\begin{figure*}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{mainwindow.png}
\caption{Example from the SURA1 field. Input folder is always validated before calling the pipeline. Example of the software running on Windows.}
\label{fig:vn-suramain}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{vn-field.png}
\caption{Fields named with the conventional EURONEAR name (Enff) are recognized as such by ViaNearby and can make use of additional defaults (such as default names compatible with the MPC Optical Report format for new detections). Example of the software running on Linux.}
\label{fig:vn-namedfield}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{vn-conf2.png}
\caption{ViaNearby options. If properly set up, all form fields are filled automatically with the first field not yet run, and the user only has to tab-cycle them for validation before running the pipeline.}
\label{fig:vn-conf2}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{vn-conf.png}
\caption{Configuration of the pipeline parameters.}
\label{fig:vn-conf}
\end{subfigure}
\caption{ViaNearby desktop pipeline}
\label{fig:vn-shots}
\end{figure*}
\subsection{Noise removal}
All the trimmed mean and median algorithms, except for the single-image median, use the built-in sorting functions of the environment \citep{arraySort} (insertion sort for small array and quicksort for larger ones). The single-image median uses a custom quickselect, where initial pivots are estimated from the mean of the input pixels and the last output pixels.\\
Initial pivot choice algorithm:
\begin{itemize}
\item A linear median estimate is taken from the last 2 median values
\item The previous estimate is averaged with the mean of the kernel window
\item Upper pivot is estimate + $0.3$ standard deviations
\item Lower pivot is estimate - $0.35$ standard deviations
\item If median is not in range, take upper or lower pivot at estimate $\pm 1$ standard deviation
\end{itemize}
The parameters were obtained via a quick empirical benchmark on typical Umbrella input images.
\subsection{Normalization}
The current normalization algorithm is a quick workaround for removing the halos of very bright stars; it is expected that this algorithm will be deprecated and replaced by a better one in the future, as it has obvious flaws.\\
The algorithm segments the image into square regions and computes the local median. This is later subtracted from the image using the following formulas, where $\Omega = \left\{ (-,-), (-,+), (+,-), (+,+) \right\}$ is the set of the 4 surrounding regions:
\[ d_\textbf{Sum} = \sum_{\iota \in \Omega} d_\iota \]
\[ M_\textbf{Interpolated} = \sum_{\iota \in \Omega} (d_\textbf{Sum} - d_\iota) \times M_\iota \]
\[ O = I - M_\textbf{Interpolated} \]
$M_\iota$ is the local median of the region, and $d_\iota$ is the distance in pixels from the current pixel to the region.
\subsection{Detection algorithms}
\subsubsection{Blob}
The blob detection algorithm performs a standard flood fill, with a 4-way connected component recognition, and using a 2-threshold hysteresis for robustness.
\subsubsection{Long Trails}
The long trail detection algorithm is based on the Auxiliary Feature Augmented Voting Hough Transform (a detailed description is to be published in a separate paper). The auxiliary feature used for this algorithm is the local connected-ness of the trails, which is computed efficiently using a state variable along the Radon voting basis. The vote table is then thresholded and high scores are searched with another blob-like algorithm, which can also pair neighboring disjoint detections.\\
The default settings for scheduling the algorithm are to split the input image into $300 \text{px} \times 300 \text{px}$ tiles that overlap with $50 \text{px}$ margins (on each side). These settings may be used as-is for scheduling or may be adapted by the pipeline. It should be noted that the reference pipeline uses the default values and does not expose these settings to the end user.
\subsection{Detection pairing}
The newer pairing algorithm provided in Umbrella works by linearly estimating the positions of detections and pairing accordingly. This is achieved by putting all detections in a quad tree, and for each pair of detections, performing the following operations:
\begin{itemize}
\item Performs a quick check for pair compatibility\\
Here it is checked that the detections are from different images and that the distance between them is not too large (given their size on input images, the exposure time and the time difference between images).
\item Estimates positions of the supposed object on all images, along with an estimate of the position error
\item Queries the quad-tree for all detections around the estimated position that are within the position error at the correct time
\item For each detection that matches the query, performs a linear fit of the coordinates of the initial pair and the match against time (this is done via ordinary least squares, as time is assumed to be accurately measured and thus exogenous to the model)
\item If detections are matched on at least one other image (thus 3 images in total), creates a tracklet.
\end{itemize}
\subsection{Filtering}
Filtering is done through individual pass/drop filters, applied both to image detections and tracklets. Several filters are included in the Umbrella2 library. Another collection of filters is implemented in the reference pipeline.
\section{The reference pipeline}
The current deployments of Umbrella-based software include a standalone desktop application (screenshots in Figure \ref{fig:vn-shots}) and a web-based server solution (screenshots in Figure \ref{fig:webrella}). Both implementations currently use a common pipeline, but this is expected to change in the future. The following describes the 7 steps performed in this reference pipeline (also shown in Figure \ref{fig:pipeline-flowchart}):
\begin{figure*}
\centering
\includegraphics[width=.8\linewidth]{brella-ref-pipe.png}
\caption{Flowchart of the Umbrella reference pipeline}
\label{fig:pipeline-flowchart}
\end{figure*}
\subsection{Initialization}
On calling the pipeline, the first steps are:
\begin{itemize}
\item Enabling or disabling scaling the image brightness according to SWarp headers.
\item Checking whether the temporary directory exists (and ensuring it does)
\item Generating the kernels used in the image processing functions ahead
\item Checking whether a badpixel file is provided and reading the badpixel data
\end{itemize}
\subsection{Image processing}
There are three steps of image processing in the reference pipeline. These are:
\subsubsection{Initial smoothing step}
In the initial smoothing step, the images are first noise filtered with a trimmed mean filter (either the usual one or the badpixel-aware one, depending on whether the badpixel file is provided). Optionally, a normalization step is performed directly after.
\subsubsection{Median combine}
A median image is generated by stacking the previously processed images. The median image is used to compute the required parameters for the detection algorithms (such as thresholds). A list of fixed stars is also generated at this step.
\subsubsection{Removal of fixed stars}
The images processed in the pre-median step are masked with the list of fixed stars in the median. Following the masking, an optional deep smoothing step is performed, which is designed to provide smooth data for connected-ness recognition in the long trail algorithm.
\subsection{Detection and filtering}
The reference pipeline currently uses three detection algorithms, two internal (Blob and Long trail) and one external (Source Extractor, with catalogs imported by Umbrella). The resulting detections are filtered, paired, and filtered again. Finally, the tracklets are recovered on the input images (using the estimated positions from a linear fit) and displayed to the reducers.
\section{Results}
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{NEASUR-comparison.png}
\caption{Umbrella compared to other automated detection software}
\label{fig:defSettings}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=.9\linewidth]{objprop-new.png}
\caption[Tracklet properties]{Almost all information built by the pipeline regarding a given object is available in the property viewer. Explanations about the various properties and their fields are pulled directly from source code documentation (including user-supplied properties). Here, one may see how the SkyBoT name (2002 CT256, score 41) takes priority over the default provisional tracklet name (E114301, i.e. night 1, field 14, ccd 3, tracklet 1).}
\label{fig:vn-objprop}
\end{figure*}
\subsection{Test setup}
A high quality dataset created for the similar NEARBY project (also within EURONEAR) was used to assess the detection rate of Umbrella. The exposures were taken with the Wide Field Camera on the Isaac Newton Telescope, close to opposition. The average seeing of the exposures is approx. 1.5". The results are from the latest release of ViaNearby at the time of comparison.
The results from Figure \ref{fig:defSettings} use the default settings, which are presented in Table \ref{tab:defSettings}. These settings have been selected over time to offer an even balance between false negative and false positive results on most images captured by the EURONEAR surveys.
\subsection{Summary of results}
Umbrella has detected 378 detections in the 15 WFC fields (4 sq. deg.), compared to 357 detected by Astrometrica (106\%), 384 detected by NEARBY (98\%) and 430 by manual blink (88 \%). Despite the lower detection rate, it has detected 13 asteroids not detected by NEARBY, one of which missed also by manual blink.\\
There have been 1826 false positives, for a total true positive rate of 17\%. However, this rate has shown tremendous variation between different fields and CCDs; for example, the SURA1 CCD2 has had 6 true positives and no false positives, while SURD0 CCD3 has had 2 true positives and 106 false positives.
\subsection{Detection hijacking}
It has been noticed that on the test data (from the logs of the object identification component) that on some fields up to an additional 10 \% (compared to the detection rate shown in this paper) known objects (from the SkyBoT database) should have been detected if not for detection hijacking, described below. It is unknown how many detections that do not show up in the SkyBoT database have been missed in this way.\\
One of the design objectives of the current pairing algorithm was that tracklets shall not be duplicated. Therefore, the earliest steps in the pairing algorithm check if the chosen pair have been already included into a tracklet. When a large number of false detections are present, it becomes more likely that true positive are paired with false positives. Let us consider a field on which an object was detected on $n$ exposures. If more than $n-2$ of these detections are paired with false detections before the true positives are paired, then the true detections will no longer be paired, and the object is lost by Umbrella.\\
For the 4-exposure fields of the NEASUR dataset described here, if 2 detections are paired with a false detection, or more commonly, when the object is detected only on 3 images (which can happen due to poor badpixel removal or involvment with the PSF of a star), if 1 (of any 3) detection is paired incorrectly, the object is lost. This effect is particularly relevant when images have extended bad areas (which is the case for one of the CCDs in the NEASUR dataset) and less than ideal badpixel removal methods, where bad areas not only can obscure a detection, but also introduce artifacts that can hijack the other detections.\\
Finaly, this phenomenon is also relevant when operating at high sensitivity (low SNR threshold for detection), such as when one desires a more "brute force" approach to asteroid detection. In such cases, lowering the thresholds for detection quickly increases false positives, which then results in greatly increased chances of hijacking. This in fact creates a barrier to the number of achievable object discoveries for a given image.
\subsection{Comment on results}
The settings used to reduce the images are the software default, also used during EURONEAR real-time surveys. While not optimized for these particular images and not selected for the maximum detection rate, these settings do yield most of the detections currently achievable with Umbrella (likely above 80 \% maximum detection rate). In particular, due to detection hijacking, greatly increasing false positives can have a detrimental effect on detecting true positives.\\
It should be noted that these results represent a snapshot of the performance of the current implementation of the pipelines, with adjustments to algorithms, pipeline and settings being common. To automate this process, the authors have envisioned a software tool that can track detections (known or unknown to Umbrella) across the pipeline on a large scale and provide statistics and visualizations such that weaknesses in algorithms can be identified and corrected, while detection settings can be tweaked more appropriately. Partial support for such a tool has been added to the reference pipeline implementation (OutputRemovedDetections in Core/Enabled operations - see Table \ref{tab:defSettings}) and it is hoped that the tool is ready in the near future.\\
\section{Acknowledgments}
The authors would like to thank Daniel Berteșteanu and Costin Boldea for testing Webrella and suggesting improvements to the interface, Alexandru Georoceanu for testing the desktop version (especially for reporting that the desktop interface is broken on 64-bit MacOS), and Elisabeta Petrescu for cross-validating difficult detections in early live tests.
\begin{table*}
\centering
\texttt{
\begin{tabular}{|l|c|}
\hline
\textbf{Setting} & \textbf{Value}\\
\hline
Star masking / Star masking threshold & 3.5; 2\\
Star masking / Extra mask radius & 1\\
Star masking / Mask radius multiplier & 1.1\\
Blob detection / Blob detector threshold & 5; 2.5\\
Blob detection / Blob min pixels & 15\\
Blob detection / Non-representative threshold & 50\\
General pipeline properties / Standard BITPIX & -32\\
General pipeline properties / Maximum algorithm detections & 1000\\
Core / Shot noise radius & 3\\
Core / Use CoreFilter & false\\
Core / Skip CCD 2 & false\\
Deep smoothing / Second median radius & 5\\
Input / Correct SWARP & true\\
Normalization / Normalization Mesh size & 40\\
Core / PSF Diameter & 5\\
Long trails / RLHT Threshold & 10\\
Long trails / Segment selection threshold & 5; 2.5\\
Long trails / Maximum interblob distance & 40\\
Long trails / Minimum trail pixels & 100\\
Filtering / Max line thickness & 15\\
Filtering / Pairwise matching max distance & 40\\
Filtering / Pairwise matching mix pixels & 10\\
Filtering / Star-crossing radius multiplier & 1.55\\
Filtering / Star-crossing minimum star flux & 10000\\
Pairing / Same-object separation & 0.1\\
Pairing / Max residual sum & 2\\
Pairing / Extra search radius & 3.5\\
Core / Enabled operations & Norm, Mask, BD, ORD, SE\footnotemark\\
Original image recovery / Detection threshold & 1.75\\
Original image recovery / Recovery radius & 10\\
Reporting / Observatory code & 950\\
Reporting / Magnitude Band & R\\
Reporting / SkyBoT Radius & 5\\
\hline
\end{tabular}
}
\caption{Default Umbrella settings}
\label{tab:defSettings}
\end{table*}
\footnotetext{Normalization, Masking, BlobDetector, OutputRemovedDetections, SourceExtractor}
\pagebreak
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,976
|
Manny's Blue Room Lounge in Robbins on Monday, Nov. 12, 2018. A police officer fatally shot an armed security guard while responding to an early-morning shooting at a suburban Chicago bar, investigators said. Officers responded shortly after 4 a.m. Sunday to a call of shots fired at Manny's Blue Room in Robbins, just south of Chicago. An officer from nearby Midlothian shot the security guard, who was later identified as 26-year-old Jemel Roberson, according to Cook County Sheriff's Office spokeswoman Sophia Ansari.
It began in a way gun advocates have suggested would curtail violence. A gun comes out. Shots are fired. A "good guy with a gun" steps in to help before police can respond.
The theory turned to grim reality at Manny's Blue Room Bar in Robbins, Illinois, outside Chicago early Sunday.
That debate has gained urgency during the past year, as President Donald Trump and others have repeatedly said security guards — specifically armed ones — could have prevented the nation's mass shootings; earlier this year, Trump tweeted his support for the controversial idea of arming teachers.
Roberson had a valid gun owner's license but did not have a concealed carry permit, WGN reported. In Minnesota in 2016, Philando Castile was killed by an officer during a traffic stop seconds after he told an officer there was a weapon in the car.
The incident began with a confrontation involving several men, and a man left to retrieve a gun. He returned to the bar and opened fire, striking several people, the Chicago Tribune reported, citing remarks from Robbins Police Chief Roy Wells.
The Cook County Sheriff's Office and Robbins Police Department, neither of which responded to requests for comment, are investigating the shooting that first drew the police to the scene, Delaney said.
At least 173 of those shot and killed by police this year – 21 percent – were black. The U.S. population is about 13 percent black.
More than half of those killed – 459 people, including Roberson – were said to have a gun when police killed them.
Roberson was a musician who played drums and keyboards at churches in the area, Patricia Hill, a pastor at Chicago's Purposed Church, told WGN.
But Hill said Roberson had a greater ambition: To one day become a police officer.
Justin Jouvenal contributed to this report.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,508
|
To use our smartphone giving app, text a numerical amount (50 for fifty dollars, 200 for 200 dollars, etc.) to 952-479-5800. This number is dedicated to Colonial Church. The first time you text you will be prompted for your name and debit/credit card. Subsequent giving will only require a numerical amount. You will receive an email receipt. To donate online, please scroll down to the donation form.
Being a member of church is different from being a donor to a charity. You are each part of the body of Christ. That's what member means, part of a family of God. Together as a community we depend upon each other for our life together. Our giving is the primary source of support for the operation, programming and mission of Colonial Church. Our faithful giving underwrites the cost of worship, education and discipleship, children's and youth ministries, pastoral care, local and international mission, staff salaries, printing, mailings, building maintenance and supplies, utilities and so much more. Colonial isn't Colonial without each of us doing our part.
Since humans first loved the Lord, we've expressed our devotion through sacrifice, with gifts and offerings of tribute. Inasmuch as our treasure reveals our heart (Matthew 6:21) giving has always been an expression of worship. Giving as worship displays our belief in the worthiness of God to receive our praise and own costly sacrifice due to his sacrificial love for us. Offering is an integral part of our worship services, signaling our intention to place all that we have under the Lordship of Christ. We pass offering bags and then bring them to the front, placing them symbolically before the Lord with prayers of thanksgiving.
As Jesus parables insist over and over, giving is good stewardship and not an option for followers of Christ. We give out of obedience and devotion, and with joy because giving helps other and frees us from the spiritual dangers of greed and consumption. All that we have is gift from God. Our own giving shows our gratitude as well as our response to God's invitation to participate in his work in the world. If you're trying to figure out how much to give, the Bible constantly suggests 10% as the appropriate tithe—but what matters more is how you give. God loves a cheerful giver.
Lastly, giving to church is not the same as donating to charity. Churches are not charities and church members are to donors. We are each part of the body of Christ, that's what member means, being part of a family of God. We give as community. Our life of processing faith together depends on the generosity of each. We give and receive, we participate and contribute with money and with time and with prayer and with love.
Online giving is easy, secure, convenient, and benefits both you and Colonial. Take a quick moment to set up an online payment using the website and you won't need to remember if you gave this month or sent in your offering if you are out of town. You won't be distracted by writing your contribution check during the service. And online contributions require less processing time from our administrative staff.
Is it Safe To Make Online Contributions?
Absolutely! In many ways giving online is safer than writing a check, because an electronic gift cannot be lost or stolen. Colonial uses payment methods that are among the safest on the internet. The security of the system is continually managed by our merchant providers.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 856
|
package sbt.impl
import org.scalatest._
/**The test runner for ScalaTest suites. It is compiled in a second step after the rest of sbt.*/
class ScalaTestRunner1_0(val log: Logger, val listeners: Seq[TestReportListener],
val testLoader: ClassLoader) extends BasicTestRunner
{
import java.lang.reflect.Modifier
def runTest(testClassName: String): Result.Value = {
val testClass = Class.forName(testClassName, true, testLoader).asSubclass(classOf[Suite])
if( isAccessibleSuite(testClass)){
val test = testClass.newInstance
val reporter = new ScalaTestReporter
test.run(None, reporter, new Stopper {}, Filter(), Map.empty, None, new Tracker)
if (reporter.succeeded) Result.Passed else Result.Failed
}
else
Result.Passed
}
private val emptyClassArray = new Array[java.lang.Class[T] forSome { type T }](0)
private def isAccessibleSuite(clazz: java.lang.Class[_]): Boolean = {
try {
classOf[Suite].isAssignableFrom(clazz) &&
Modifier.isPublic(clazz.getModifiers) &&
!Modifier.isAbstract(clazz.getModifiers) &&
Modifier.isPublic(clazz.getConstructor(emptyClassArray: _*).getModifiers)
} catch {
case nsme: NoSuchMethodException => false
case se: SecurityException => false
}
}
/**An implementation of Reporter for ScalaTest. */
private class ScalaTestReporter extends Reporter with NotNull
{
import org.scalatest.events._
var succeeded = true
def apply(event: Event) {
event match {
// why log.info sometimes and fire(MessageEvent...) other times?
case rc: RunCompleted => log.info("Run completed.")
case rs: RunStarting => fire(MessageEvent("Run starting"))
case rs: RunStopped => {succeeded = false; fire(ErrorEvent("Run stopped"))}
case ra: RunAborted => {succeeded = false; fire(ErrorEvent("Run aborted"))}
// this one seems to be working really well
case ts: TestStarting => fire(TypedEvent(ts.testName, "Test Starting", None)(None))
// not sure what to do here at all
case tp: TestPending =>
case tf: TestFailed => {succeeded = false; fire(TypedErrorEvent(tf.testName, "Test Failed", None, tf.throwable)(Some(Result.Failed)))}
// this one also seems to be working really well
case ts: TestSucceeded => fire(TypedEvent(ts.testName, "Test Succeeded", None)(Some(Result.Passed)))
// need to check if there is a reason why this test was ignored
case ti: TestIgnored => fire(IgnoredEvent(ti.testName, Some("test ignored")))
case sc: SuiteCompleted => fire(TypedEvent(sc.suiteName, "Suite Completed", None)(None))
// why not sure Some(Result.Failed) here?
// also, why not say succeeded = false?
// seems like i should do both if the suite is aborted.
case sa: SuiteAborted => fire(TypedErrorEvent(sa.suiteName, "Suite Aborted", Some(sa.message), sa.throwable)(None))
case ss: SuiteStarting => fire(TypedEvent(ss.suiteName, "Suite Starting", None)(None))
// not actually sure if this is what i should do here...info provided is really just...some random extra info provided by a test, like a log statement
case ip: InfoProvided => fire(MessageEvent(ip.message))
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,493
|
Q: Can I Archive NSData objects inside another NSData without corrupting data? I have two NSData objects I want to store within a third NSData object. The idea is I want to make it easy when I later decode the larger object, to get at the two smaller objects independently of one another, without worrying about their relative sizes or datatypes.
It appears the best way to do this is to use NSKeyedArchiver to create a sort of root-level key-value structure within the larger NSData object, in which I can store the two smaller objects within separate keys. That's what I've attempted to do here:
NSData *data1Before = [@"abcdefgh" dataUsingEncoding:NSUTF8StringEncoding];
NSData *data2Before = [@"ijklmnop" dataUsingEncoding:NSUTF8StringEncoding];
NSMutableData *allData = [[NSMutableData alloc] init];
NSKeyedArchiver *archiver = [[NSKeyedArchiver alloc] initForWritingWithMutableData:allData];
[archiver encodeObject:data1Before forKey:@"key1"];
[archiver encodeObject:data2Before forKey:@"key2"];
[archiver finishEncoding];
NSKeyedUnarchiver *unarchiver = [[NSKeyedUnarchiver alloc] initForReadingWithData:allData];
NSData *data1After = [unarchiver decodeObjectForKey:@"key1"];
NSData *data2After = [unarchiver decodeObjectForKey:@"key2"];
[unarchiver finishDecoding];
NSString *string1After = [NSString stringWithUTF8String:[data1After bytes]];
NSString *string2After = [NSString stringWithUTF8String:[data2After bytes]];
NSLog(@"after1: %@",string1After);
NSLog(@"after2: %@",string2After);
The problem is, when you run this code over and over, you get all sorts of different results coming from the NSLog statements- someetimes special characters get appended to the end of the strings, sometimes they're just NULL.
It appears this corruption has something to do with this "double-encoding" process I'm using. When I modify the code so that the NSKeyedArchiver just calls encodeObject directly on NSStrings, rather than on NSData objects, I can later use decodeObjectForKey and get at those strings without any problems- no corruption at all.
Is there a better way of doing this than using NSKeyedArchiver? Or am I using it incorrectly?
A: Thanks to rmaddy for his answer above- I just needed to replace this:
[NSString stringWithUTF8String:[data1After bytes]];
with this:
[[NSString alloc] initWithData:data1After encoding: NSUTF8StringEncoding];
and that fixed it.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,097
|
Mad Men's Christina Hendricks: Hottest Woman Ever?
Curvy star of the hit AMC series graces the cover of May's Esquire magazine
By Peter Gicas Apr 19, 2010 5:45 PMTags
HotMagazinesSexMad Men
The men over at Esquire are obviously mad about Christina Hendricks.
Of course, given the seductive pose on the mag's May cover, it's not hard to see why.
But the voluptuous star is quick to point out there's more here than meets the eye.
"We remember everything you say about our bodies, be it good or bad," the Mad Men actress writes in an open letter to fellas in the issue. "Those things are stored away in a steel box, and we remember these things verbatim."
Fortunately, Esquire readers just voted her the hottest woman around. Surely that's something Christina won't forget.
See who else isn't afraid to reveal a little bit of themselves in our Celebrity Skin gallery.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 590
|
{"url":"https:\/\/stats.stackexchange.com\/questions\/223997\/tail-probability-for-heavy-tailed-distributions","text":"# Tail probability for heavy tailed distributions\n\nFor some data (where I have the mean and standard deviation) I currently estimate the probability of getting samples greater than some x by using the Q function; i.e., I'm calculating the tail probabilities. But this assumes a normal (Gaussian) distribution of my data, and I may be better off assuming a heavy tailed distribution, like log-normal or Cauchy. How can I calculate the tail probabilities for heavy tailed distributions?\n\n\u2022 Why can't you use the q-function for the relevant distribution (eg, Cauchy)? \u2013\u00a0gung - Reinstate Monica Jul 15 '16 at 18:53\n\u2022 @gung because the Q-function assumes a Gaussian distribution. I'm looking for the similar calculation that doesn't make this assumption, or assumes a different distribution. \u2013\u00a0BoltzmannBrain Jul 15 '16 at 19:11\n\u2022 Note that the Cauchy doesn't have a mean or variance \u2013\u00a0Glen_b -Reinstate Monica Jul 16 '16 at 7:22\n\u2022 Q-function appears to be a nonstandard term for survival function \u2013\u00a0Glen_b -Reinstate Monica Jul 16 '16 at 7:23\n\nThe Wikipedia article you linked shows that the Q-function is just 1 minus the cumulative distribution function (CDF) of the normal distribution. Every distribution has its own CDF. So just use the CDF for the distribution in question. In R, for example, 1 - pcauchy(5) tells you the probability of getting 5 or more from a standard Cauchy distribution.\n\n\u2022 That's precisely what I'm looking for, but do you know if there is the same easy calculation in Python? \u2013\u00a0BoltzmannBrain Jul 15 '16 at 21:17\n\u2022 Try SciPy: import scipy.stats; print 1 - scipy.stats.cauchy.cdf(5)' \u2013\u00a0Kodiologist Jul 15 '16 at 21:53\n\nOne approach is to estimate a tail index and then use that value to plug in one of the Tweedie extreme-value distributions. There are many approaches to estimating this index such as Hill's method or Pickand's estimator. These tend to be fairly expensive computationally. An easily built and widely used heuristic involves OLS estimation as described by Xavier Gabaix in his paper Rank-1\/2: A Simple Way to Improve the OLS Estimation of Tail Exponents. Here's the abstract:\n\nDespite the availability of more sophisticated methods, a popular way to estimate a Pareto exponent is still to run an OLS regression: log (Rank) = a\u2212b log (Size), and take b as an estimate of the Pareto exponent. The reason for this popularity is arguably the simplicity and robustness of this method. Unfortunately, this procedure is strongly biased in small samples. We provide a simple practical remedy for this bias, and propose that, if one wants to use an OLS regression, one should use the Rank \u22121\/2, and run log (Rank \u2212 1\/2) = a \u2212 b log (Size). The shift of 1\/2 is optimal, and reduces the bias to a leading order. The standard error on the Pareto exponent is not the OLS standard error, but is asymptotically (2\/n)1\/2\u0010. Numerical results demonstrate the advantage of the proposed approach over the standard OLS estimation procedures and indicate that it performs well under dependent heavy-tailed processes exhibiting deviations from power laws. The estimation procedures considered are illustrated using an empirical application to Zipf\u2019s law for the U.S. city size distribution.\n\nFor an opposing view, see Cosma Shalizi's presentation So, You Think You Have a Power Law, Do You? Well Isn't That Special? which states that relying on OLS estimators such as Gabaix proposes is \"bad practice.\" For more mathematical rigor see Clauset, Shalizi, Newman Power-Law Distributions in Empirical Data.\n\nWiki has a good review of Tweedie distributions which are based on the domain of the tail index. https:\/\/en.wikipedia.org\/wiki\/Tweedie_distribution\n\nIn finance what you're trying to do is called \"value-at-risk\" (VaR). There are many different ways to do VaR, for instance check out Jorion's book, which is a standard reference.\n\nThe approach you have taken is called \"parametric VaR\", i.e. you fit a distribution into your observations, then based on the parameters calculate $F^{-1}(1-\\alpha)$, where $F^{-1}$ is the inverse cumulative distribution (CDF) of profits and losses (P&L) and $\\alpha$ - significance. For instance, if you were to fit normal distribution and calculate 95% VaR then it would be approximately equal to $\\mu-2\\sigma$.\n\nSo, if you want to use another distribution, then you fit it to data and simply get the inverse CDF of the fitted distribution at a given significance. The popular choice is Student t (location scale) distribution. You have to be careful with Cauchy. As you know it doesn't have moments, which could be an issue in certain applications, such as portfolio optimization.\n\nNow, there are other approaches, e.g. \"historical VaR\", which is basically a nonparametric (empirical distribution) method. All you need is to get the corresponding percentile of your data set. There's no need to assume or fit a distribution.","date":"2020-06-04 15:00:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.7250630259513855, \"perplexity\": 684.6060191663331}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347441088.63\/warc\/CC-MAIN-20200604125947-20200604155947-00399.warc.gz\"}"}
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Q: Dependency Injection: setting and sharing properties of scoped service in ASP.NET Core I want to set property in some injected service in controller for later abusing it when this service will be injected in other place during the same request, so I expect this property not to change as far as service is injected as Scoped.
Is that safe approach or how would you suggest to achive such a behaviour?
MyController(IServiceWithProperty serviceWithProperty) {
_serviceWithProperty = serviceWithProperty;
}
public IActionResult Get(int settingToSet) {
_serviceWithProperty.SetProperty(settingToSet);
return Ok(_anotherService.GetSomething());
}
And as I said AnotherService injects ServiceWithProperty as well.
public class AnotherService : IAnotherService {
public AnotherService(IServiceWithProperty serviceWithProperty) {
_serviceWithProperty = serviceWithProperty;
}
public string GetSomething() {
int prop = _serviceWithProperty.GetProperty(); //here I expect to get property which has been set in controller, would that work and is it fine to do it like that?
}
}
A: The interaction between those two services is going to be hard to follow once the size of the code base grows. Even given this simple example (good job of reducing the problem to its essence, BTW), I had to look at the code for a few minutes to understand what's going on.
Besides, a design like this seems to be close to violating the Liskov Substitution Principle (LSP), because it'll only work when specific, concrete implementations are in use. According to the LSP, you should be able to exchange one subtype with another without changing the correctness of the system. Is this possible here? Can you replace the IServiceWithProperty implementation you have in mind with another implementation that does nothing?
Instead, follow the Dependency Inversion Principle, from which it follows that clients should define the abstraction. So, if MyController requires an abstraction that can translate an int to something else, then that's the abstraction it requires:
MyController(ITranslator translator) {
_translator = translator;
}
public IActionResult Get(int setting) {
return Ok(_translator.Translate(setting));
}
The next step, then, is to figure out how to implement the ITranslator interface.
A: It'll work because both services are singletons within the lifetime of a request, so the property will hold its value also.
However i would recommend to share those values not as properties rather as arguments of your service functions like:
public IActionResult Get(int settingToSet)
{
return Ok(_anotherService.GetSomething(settingToSet));
}
public class AnotherService : IAnotherService
{
public AnotherService(ISomeAnotherService service)
{
_service = service;
}
public string GetSomething(int inputValue)
{
int serviceResult = _service.GetSomething(inputValue);
}
}
This makes the code much clearer, and lower your chances that you will end up 5 months later investigating where and whom was that property set to that value you don't expect at that time.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,704
|
\section{Introduction}
The study of the spectral properties of a graph is a popular subject in the graph theory.
The relation among the rank of the adjacent matrix and other topological structure parameters of a graph
has been studied extensively by many researchers.
Recently there has been a growing study of the rank of the adjacent matrix associated to
signed graphs and mixed graphs. In this paper we characterize the properties of the rank of a complex unit
gain graph. We refer to \cite{BONDY} for undefined terminologies and notation.
In this paper, we only consider the simple and finite graphs.
Let $G$ be an undirected graph with vertex set $V(G)=\{ v_{1}, v_{2}, \cdots, v_{n} \}$. The $degree$ of a vertex $u \in V(G)$, denote by $d_{G}(u)$, is the number of vertices
which are adjacent to $u$. A vertex of $G$ is called a {\it pendant vertex} if it is a vertex of degree one in $G$, whereas a vertex of $G$ is called a {\it quasi-pendant vertex} if it is adjacent to
a pendant vertex in $G$ unless it is a pendant vertex.
Denote by $P_n$ and $C_n$ a path and cycle on $n$ vertices, respectively.
The {\it adjacency matrix} $A(G)$ of $G$ is the $n \times n$ matrix whose $(i, j)$-entry equals to
1 if vertices $v_{i}$ and $v_j$ are adjacent and 0 otherwise.
A $complex$ $unit$ $gain$ $graph$ (or $\mathbb{T}$-gain graph) is a graph with the additional structure that
each orientation of an edge is given a complex unit, called a $gain$, which is the inverse of the
complex unit assigned to the opposite orientation. For a simple graph $G$ with order $n$,
let $\overrightarrow{E}$ be the set of oriented edges, it is obvious that this set contains two copies of each edge with opposite directions. We write $e_{ij}$ for the oriented edge from $v_{i}$ to $v_{j}$. The circle group, which is denoted by $\mathbb{T}= \{ z \in C:|z|=1 \} $, is a subgroup of the multiplicative group of all
nonzero complex numbers $\mathbb{C}^{\times}$. A complex unit gain graph is
a triple $\Phi=(G, \mathbb{T}, \varphi)$ consisting of a graph $G$, $\mathbb{T}= \{ z \in C:|z|=1 \} $ is a subgroup of the multiplicative group of all nonzero complex numbers $\mathbb{C}^{\times}$ and a gain function $\varphi: \overrightarrow{E} \rightarrow \mathbb{T}$, where $G$ is the underlying graph of $\Phi$ and $\varphi(e_{ij})=\varphi(e_{ji})^{-1}=\overline{\varphi(e_{ji})}$.
For convenience, we write $(G, \varphi)$ for a complex unit gain graph $\Phi=(G, \mathbb{T}, \varphi)$ in this paper. The adjacency matrix associated to the complex unit gain graph $(G, \varphi)$ is the $n \times n$ complex matrix $A(G, \varphi)=a_{ij}$, where $a_{ij}=\varphi(e_{ij})$ if $v_{i}$ is adjacent to $v_{j}$, otherwise $a_{ij}=0$. It is obvious to see that $A(G, \varphi)$ is Hermitian and its eigenvalues are real. If the gain of every edge is 1 in $(G, \varphi)$, then the adjacency matrix $A(G, \varphi)$ is exactly the adjacency matrix $A(G)$ of the underlying graph $G$. It is obvious that a simple graph is assumed as a
complex unit gain graph with all positive gain 1's.
The $positive$ $inertia$ $index$, denoted by $p^{+}(G, \varphi)$, and the $negative$ $inertia$ $index$, denoted by $n^{-}(G, \varphi)$, of a complex unit gain graph $(G, \varphi)$ are defined to be the number of positive eigenvalues and negative eigenvalues of $A(G, \varphi)$ including multiplicities, respectively.
The $rank$ of a complex unit gain graph $(G, \varphi)$,
written as $r(G, \varphi)$, is defined to be the rank of $A(G, \varphi)$.
Obviously, $r(G, \varphi)=p^{+}(G, \varphi)+n^{-}(G, \varphi)$.
For an induced subgraph $H$ of a graph $G$,
denote by $G-H$, the subgraph obtained from $G$ by deleting all vertices of $H$ and all incident edges.
For a subset $X$ of $V(G)$, $G-X$ is the induced subgraph obtained from $G$ by deleting all vertices in $X$ and
all incident edges. In particular, $G-\{ x \}$ is usually written as $G-x$ for simplicity.
For an induced subgraph $H$ and a vertex $u$
outside $H$, the induced subgraph of $G$ with vertex set $V(H) \cup \{ u \}$ is simply written as $H+u$.
For a graph $G$, let $c(G)$ be the {\it cyclomatic number} of $G$,
that is $c(G)=|E(G)|-|V(G)|+\omega(G)$, where $\omega(G)$ is the number of connected components of $G$.
Two vertices of a graph $G$ are said to be $independent$ if they are not adjacent. A subset $I$ of
$V(G)$ is called an $independent$ $set$ if any two vertices of $I$ are independent in $G$. An independent set $I$ is $maximum$ if $G$ has no independent set $I'$ with $|I'|> |I'|$. The number
of vertices in a maximum independent set of $G$ is called the $independence$ $number$
of $G$ and is denoted by $\alpha(G)$.
For a complex unit gain graph $(G, \varphi)$, the independence number and cyclomatic number
of $(G, \varphi)$ are defined to be the independence number and cyclomatic number of its underlying graph, respectively.
Let $G$ be a graph with pairwise vertex-disjoints cycles (if any) and $\mathscr{C}_{G}$ be the set of all cycles of $G$.
$T_{G}$ is an acyclic graph obtained from $G$ by contracting each cycle of $G$ into a vertex (called a {\it cyclic vertex}).
Denoted by $\mathscr{O}_{G}$ the set of all cyclic vertex of $G$.
Moreover, denoted by $[T_{G}]$ the subgraph of $T_{G}$
induced by all non-cyclic vertices. It is obviously that $[T_{G}]=T_{G}-\mathscr{O}_{G}$.
The rank of graphs have been discussed intensively by many researchers.
There are some papers focused on the study on the rank of graphs in terms of other topological structure parameters.
Wang and Wong characterized the bounds for the matching number, the edge chromatic number and the
independence number of a graph in terms of rank in \cite{WANGLONG}.
Gutman and Sciriha \cite{GUT} studied the nullity of line graphs of trees.
Guo et al. \cite{MOHAR} and Liu et al. \cite{LXL} introduced the Hermitian adjacency matrix of a mixed graph and presented some basic properties of the rank of the mixed graphs independently.
In \cite{FYZ1}, the rank of the signed unicyclic graph was discussed by Fan et al.
He et al. characterized the relation among the rank,
the matching number and the cyclomatic number of a signed graph in \cite{HSJ}.
Chen et al. \cite{LSC} investigated the relation between the $H$-rank of a mixed graph and the
matching number of its underlying graph.
For other research of the rank of a graph one may be referred to those in \cite{BEVI,HOU,MAH2,MOHAR2,WANGXIN}.
Recently, the study of the properties of complex unit gain graphs has attracted increased attention.
Reff extended some fundamental concepts from spectral graph theory to complex unit gain graphs and
defined the adjacency, incidence and Laplacian matrices of them in \cite{REFF}.
Yu et al. \cite{YGH} investigated some properties of inertia of complex unit gain graphs and discussed the inertia index of a complex unit gain cycle.
In \cite{FYZUNIT}, Wang et al. provided a combinatorial description of the determinant of the Laplacian matrix of a complex unit gain graph which generalized that for the determinant of the Laplacian matrix of a signed graph.
Lu et al. \cite{LUY} studied the complex unit gain unicyclic graphs with small positive or
negative index and characterized the complex unit gain bicyclic graphs with rank 2, 3 or 4.
In \cite{WLG}, the relation among the rank of a complex unit gain graph and the rank of its underlying graph and the cyclomatic number was investigated by Lu et al.
In this paper, the upper and lower bounds of the rank of a complex unit gain graph $(G, \varphi)$ with order $n$
in terms of the cyclomatic number and the independence number of its underlying graph are investigated.
Moreover, the properties of the extremal graphs which attended the lower bound are identified.
The following Theorems \ref{T30} and \ref{T50} are our main results.
\begin{theorem}\label{T30}
Let $(G, \varphi)$ be a complex unit gain graph with order $n$. Then
$$2n-2c(G)-2\alpha(G) \leq r(G, \varphi) \leq 2n-2\alpha(G).$$
\end{theorem}
\begin{theorem}\label{T50}
Let $(G, \varphi)$ be a complex unit gain graph with order $n$. Then $ r(G, \varphi)= 2n-2c(G)-2\alpha(G)$
if and only if all the following conditions hold for $(G, \varphi)$:
{\em(i)} the cycles (if any) of $(G, \varphi)$ are pairwise vertex-disjoint;
{\em(ii)} for each cycle (if any) $(C_{l}, \varphi)$ of $(G, \varphi)$, either $\varphi(C_{l}, \varphi)=(-1)^{\frac{l}{2}}$ and $l$ is even or $Re((-1)^{\frac{l-1}{2}}\varphi(C_{l}, \varphi))=0$ and $l$ is odd;
{\em(iii)} $\alpha(T_{G})=\alpha([T_{G}])+c(G)$.
\end{theorem}
The rest of this paper is organized as follows. Prior to showing our main results, in Section 2,
we list some known elementary lemmas and results which will be useful.
In Section 3, we give the proof of the Theorem \ref{T30}. In Section 4, the properties of the extremal signed graphs which attained the lower bound of Theorem \ref{T30} are identified, and the proof of the Theorem \ref{T50} is presented.
\section{Preliminaries}
In this section, some known results and useful lemmas which will be
used in the proofs of our main results are listed.
\begin{lemma} \label{L16}{\rm\cite{YGH}}
Let $(G, \varphi)$ be a complex unit gain graph.
{\em(i)} If $(H, \varphi)$ is an induced subgraph of $(G, \varphi)$, then $r(H, \varphi) \leq r(G, \varphi)$.
{\em(ii)} If $(G_{1}, \varphi), (G_{2}, \varphi), \cdots, (G_{t}, \varphi)$ are all the connected components of $(G, \varphi)$, then $r(G, \varphi)=\sum_{i=1}^{t}r(G_{i}, \varphi)$.
{\em(iii)} $r(G, \varphi)\geq 0$ with equality if and only if $(G, \varphi)$ is an empty graph.
\end{lemma}
\begin{definition} \label{L053}{\rm\cite{LUY}}
Let $(C_{n}, \varphi)$ ($n \geq 3$) be a complex unit gain cycle and
$$\varphi(C_{n}, \varphi)=\varphi(v_{1}v_{2} \cdots v_{n}v_{1} )=\varphi(v_{1}v_{2})\varphi(v_{2}v_{3}) \cdots \varphi(v_{n-1}v_{n})\varphi(v_{n}v_{1}).$$
Then $(C_{n}, \varphi)$ is said to be one of the following five Types:
$$\left\{
\begin{array}{ll}
\rm{Type~A}, & \hbox{if $\varphi(C_{n}, \varphi)=(-1)^{\frac{n}{2}}$ and $n$ is even;} \\
\rm{Type~B}, & \hbox{if $\varphi(C_{n}, \varphi) \neq (-1)^{\frac{n}{2}}$ and $n$ is even;} \\
\rm{Type~C}, & \hbox{if $Re((-1)^{\frac{n-1}{2}}\varphi(C_{n}, \varphi))>0$ and $n$ is odd;} \\
\rm{Type~D}, & \hbox{if $Re((-1)^{\frac{n-1}{2}}\varphi(C_{n}, \varphi))<0$ and $n$ is odd;} \\
\rm{Type~E}, & \hbox{if $Re((-1)^{\frac{n-1}{2}}\varphi(C_{n}, \varphi))=0$ and $n$ is odd.}
\end{array}
\right.
$$
Where $Re(\cdot)$ is the real part of a complex number.
\end{definition}
\begin{lemma} \label{L12}{\rm\cite{YGH}}
Let $(C_{n}, \varphi)$ be a complex unit gain cycle of order $n$. Then
$$(p^{+}(C_{n}, \varphi), n^{-}(C_{n}, \varphi))=\left\{
\begin{array}{ll}
(\frac{n-2}{2}, \frac{n-2}{2}), & \hbox{if $(C_{n}, \varphi)$ is of \rm{Type~A};} \\
(\frac{n}{2}, \frac{n}{2}), & \hbox{if $(C_{n}, \varphi)$ is of \rm{Type~B};} \\
(\frac{n+1}{2}, \frac{n-1}{2}), & \hbox{if $(C_{n}, \varphi)$ is of \rm{Type~C};} \\
(\frac{n-1}{2}, \frac{n+1}{2}), & \hbox{if $(C_{n}, \varphi)$ is of \rm{Type~D};} \\
(\frac{n-1}{2}, \frac{n-1}{2}), & \hbox{if $(C_{n}, \varphi)$ is of \rm{Type~E}.}
\end{array}
\right.
$$
\end{lemma}
\begin{lemma} \label{L15}{\rm\cite{YGH}}
Let $(T, \varphi)$ be an acyclic complex unit gain graph.
Then $r(T, \varphi)= r(T)$.
\end{lemma}
From Lemma \ref{L15}, we have the following Lemma \ref{LPN} directly.
\begin{lemma} \label{LPN}
Let $(P_{n}, \varphi)$ be a complex unit gain path with order $n$.
Then
$$ r(P_{n}, \varphi)=\left\{
\begin{array}{ll}
n-1, & \hbox{if $n$ is odd;} \\
n, & \hbox{if $n$ is even.}
\end{array}
\right.
$$
\end{lemma}
\begin{lemma} \label{L051}{\rm\cite{BONDY}}
Let $T$ be a acyclic graph with order $n$.
Then $r(T)=2m(T)$ and $\alpha(T)+m(T)=n$.
\end{lemma}
Obviously, by Lemmas \ref{L15} and \ref{L051}, the following Lemma \ref{L052} can be obtained.
\begin{lemma} \label{L052}
Let $(T, \varphi)$ be an acyclic complex unit gain graph with order $n$.
Then $r(T, \varphi)+2\alpha(T)=2n$.
\end{lemma}
\begin{lemma} \label{L13}{\rm\cite{YGH}}
Let $y$ be a pendant vertex of a complex unit gain graph $(G, \varphi)$ and $x$ is the neighbour of $y$.
Then $r(G, \varphi)=r((G, \varphi)- \{ x, y \} )+2$.
\end{lemma}
\begin{lemma} \label{L14}{\rm\cite{YGH}}
Let $x$ be a vertex of a complex unit gain graph $(G, \varphi)$.
Then $r(G, \varphi)-2 \leq r((G, \varphi)-x) \leq r(G, \varphi)$.
\end{lemma}
\begin{lemma} \label{L053}{\rm\cite{HLSC}}
Let $y$ be a pendant vertex of a graph $G$ and $x$ is the neighbour of $y$.
Then $\alpha(G)=\alpha(G-x)=\alpha(G-\{ x, y \})+1$.
\end{lemma}
\begin{lemma} \label{L23}{\rm\cite{WDY}}
Let $G$ be a graph with $x \in V(G)$.
{\em(i)} $c(G)=c(G-x)$ if $x$ lies outside any cycle of $G$;
{\em(ii)} $c(G-x) \leq c(G)-1$ if $x$ lies on a cycle of $G$;
{\em(iii)} $c(G-x) \leq c(G)-2$ if $x$ is a common vertex of distinct cycles of $G$.
\end{lemma}
\begin{lemma} \label{L054}{\rm\cite{HLSC}}
Let $G$ be a graph. Then
{\em(i)} $\alpha(G)-1 \leq \alpha(G-x) \leq \alpha(G) $ for any vertex $x \in V(G)$;
{\em(ii)} $\alpha(G-e) \geq \alpha(G)$ for any edge $e \in E(G)$.
\end{lemma}
\begin{lemma} \label{L055}{\rm\cite{HLSC}}
Let $T$ be a tree with at least one edge and $T_{0}$ be the subtree obtained
from $T$ by deleting all pendant vertices of $T$.
{\em(i)} $\alpha(T) \leq \alpha(T_{0}) +p(T) $, where $p(T)$ is the number of pendent vertices of $T$;
{\em(ii)} If $\alpha(T) = \alpha(T-D)+|D|$ for a subset $D$ of $V(T)$, then there is a pendant vertex $x$ such
that $x \notin D$.
\end{lemma}
\section{Proof of Theorem \ref{T30}}
In this section, the proof for Theorem \ref{T30} is presented.
\noindent
{\bf The proof of Theorem \ref{T30}.}
Firstly, we show that $ r(G, \varphi)\leq 2n -2\alpha(G)$.
Let $I$ be a maximum independent set of $G$, i.e., $|I|=\alpha(G)$. Then
$$A(G, \varphi)=\left(
\begin{array}{cc}
\mathbf{0} & \boldsymbol{B} \\
\boldsymbol{B}^{\top} & \boldsymbol{A} \\
\end{array}
\right)
$$
where $\boldsymbol{B}$ is a submatrix of $A(G, \varphi)$ with row indexed by $I$ and column indexed by $V(G)-I$,
$\boldsymbol{B}^{\top}$ refers to the transpose of $\boldsymbol{B}$ and $\boldsymbol{A}$ is the adjacency matrix of the induced subgraph $G-I$. Then it can be checked that
$$r(G, \varphi)\leq r(\mathbf{0}, \boldsymbol{B})+r(\boldsymbol{B}^{\top}, \boldsymbol{A})\leq n-\alpha(G)+n-\alpha(G)=2n-2\alpha(G).$$
Thus,
$$r(G, \varphi)\leq 2n-2\alpha(G).$$
Next, we argue by induction on $c(G)$ to show that $2n-2c(G) \leq r(G, \varphi)+2\alpha(G) $.
If $c(G)=0$, then $(G, \varphi)$ is a complex unit gain tree, and so result follows from Lemma \ref{L052}.
Hence one can assume that $c(G) \geq 1$. Let
$u$ be a vertex on some cycle of $(G, \varphi)$ and $(G', \varphi)=(G, \varphi)-u$.
Let $(G_{1}, \varphi), (G_{2}, \varphi), \cdots, (G_{l}, \varphi)$ be all
connected components of $(G', \varphi)$.
By Lemma \ref{L23}, we have
\begin{equation} \label{E1}
\sum_{i=1}^{l}c(G_{i})=c(G') \leq c(G)-1.
\end{equation}
By the induction hypothesis, one has
\begin{equation} \label{E2}
2(n-1)-2c(G') \leq r(G', \varphi)+2\alpha(G').
\end{equation}
By Lemmas \ref{L054} and \ref{L14}, we have
\begin{equation} \label{E3}
\sum_{i=1}^{l}\alpha(G_{i})=\alpha(G') \leq \alpha(G)
\end{equation}
and
\begin{equation} \label{E4}
\sum\limits_{i=1}^{l}r(G_{i}, \varphi)=r(G', \varphi) \leq r(G, \varphi).
\end{equation}
Thus the desired inequality now follows by combining (\ref{E1}), (\ref{E2}), (\ref{E3}) and (\ref{E4}),
\begin{eqnarray} \label{E0}
r(G, \varphi)+2\alpha(G) & \ge & r(G', \varphi)+2\alpha(G')
\\ \nonumber
& \ge & 2(n-1)-2c(G')
\\ \nonumber
& \geq & 2(n - 1) - 2(c(G) - 1) = 2n-2c(G),
\end{eqnarray}
as desired.
This completes the proof of Theorem \ref{T30}.
$\square$
\section{Proof of Theorem \ref{T50}.}
A complex unit gain graph $(G, \varphi)$ with order $n$ is called {\it lower-optimal} if $r(G, \varphi)=2n-2c(G)-2\alpha(G)$,
or equivalently, the complex unit gain graph which attain the lower bound in Theorem \ref{T30}.
In this section, we characterize the properties of the complex unit gain graphs which are lower-optimal,
and then we give the proof for Theorem \ref{T50}.
\begin{lemma} \label{L001}
Let $u$ be a cut vertex of a complex unit gain graph $(G, \varphi)$ and $(H, \varphi)$ be a component
of $(G, \varphi)-u$. If $r(H, \varphi)=r((H, \varphi)+u)$, then $r(G, \varphi)=r(H, \varphi)+r((G, \varphi)-(H, \varphi))$.
\end{lemma}
\begin{proof}
Let $|V(H, \varphi)|=k$ and
$$A(G, \varphi) = \left(
\begin{array}{ccc}
\boldsymbol{A} & \mathbf{\beta} & \mathbf{0} \\
\mathbf{\overline{\beta}^{\top}} & 0 & \mathbf{\gamma} \\
\boldsymbol{0} & \mathbf{\overline{\gamma}^{\top}} & \boldsymbol{B} \\
\end{array}
\right),
$$
where $\boldsymbol{A}$ and $\boldsymbol{B}$ are the Hermitian adjacency matrices of $(H, \varphi)$ and $(G, \varphi)-(H, \varphi)-u$, respectively. $\mathbf{\overline{\beta}^{\top}}$ refers to the conjugate transpose of $\mathbf{\beta}$. Since $r(H, \varphi)=r((H, \varphi)+u)$, the linear equation
$\boldsymbol{A}X=\mathbf{\beta}$ has solutions. Let $\xi$ be a solution of $\boldsymbol{A}X=\mathbf{\beta}$, and put
$$Q = \left(
\begin{array}{ccc}
\boldsymbol{E_{k}} & -\xi & \mathbf{0} \\
\mathbf{0} & 1 & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \boldsymbol{I_{n-k-1}} \\
\end{array}
\right),
$$
where $\boldsymbol{I_{k}}$ denotes a $k \times k$ identity matrix. By directly calculation, we have
$$\overline{Q}^{\top}A(G, \varphi)Q = \left(
\begin{array}{ccc}
\boldsymbol{A} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & -\overline{\mathbf{\beta}}^{\top}\xi & \mathbf{\gamma} \\
\mathbf{0} & \overline{\mathbf{\gamma}}^{\top} & \boldsymbol{B} \\
\end{array}
\right).
$$
Since $r(H, \varphi)=r((H, \varphi)+u)$, we have $-\overline{\mathbf{\beta}}^{\top}\xi=0$. Thus we have
$r(G, \varphi)=r(H, \varphi)+r((G, \varphi)-(H, \varphi))$.
\end{proof}
\begin{lemma} \label{L002}
Let $(C_{l}, \varphi)$ be a pendant complex unit gain cycle of a complex unit gain graph $(G, \varphi)$
with $u$ be the only vertex of $(C_{l}, \varphi)$ of degree 3. Let $(H, \varphi)=(G, \varphi)-(C_{l}, \varphi)$ and $(G', \varphi)=(H, \varphi)+u$. If $Re((-1)^{\frac{l-1}{2}}\varphi(C_{l}, \varphi))=0$ and $l$ is odd, then
$$r(G, \varphi)=r(G', \varphi)+l-1.$$
\end{lemma}
\begin{proof} Note that $u$ is a cut vertex of $(G, \varphi)$ and $(P_{l-1}, \varphi)$ is a complex unit gain path as a component of $(G, \varphi)-u$.
By the fact that $Re((-1)^{\frac{l-1}{2}}\varphi(C_{l}, \varphi))=0$ and $l$ is odd, then by Lemmas \ref{LPN} and \ref{L12} one has that
$$r(P_{l-1}, \varphi)=r(C_{l}, \varphi)=l-1.$$
Then, by Lemma \ref{L001}, we have
$$r(G, \varphi)=r(G', \varphi)+r(P_{l-1}, \varphi)=r(G', \varphi)+l-1.$$
\end{proof}
By Lemma \ref{L12}, the following Lemma \ref{L50} can be obtained directly.
\begin{lemma} \label{L50}
The complex unit gain cycle $(C_{q}, \varphi)$ is lower-optimal if and only if either $\varphi(C_{q}, \varphi)=(-1)^{\frac{q}{2}}$ and $q$ is even or $Re((-1)^{\frac{q-1}{2}}\varphi(C_{q}, \varphi))=0$ and $q$ is odd.
\end{lemma}
\begin{lemma} \label{L000} Let $(G, \varphi)$ be a complex unit gain graph and $u$ be a vertex of $(G, \varphi)$ lying on a complex unit gain cycle. If $r(G, \varphi)= 2n-2c(G)-2\alpha(G)$, then each of the following holds.
\\
{\em(i)} $r(G, \varphi)=r((G, \varphi)-u)$;
\\
{\em(ii)} $(G, \varphi)-u$ is lower-optimal;
\\
{\em(iii)} $c(G)=c(G-u)+1$;
\\
{\em(iv)} $\alpha(G)=\alpha(G-u)$;
\\
{\em(v)} $u$ lies on just one complex unit gain cycle of $(G, \varphi)$ and $u$ is not a quasi-pendant vertex of $(G, \varphi)$.
\end{lemma}
\begin{proof}
In the proof arguments of Theorem \ref{T30} that justifies $r(G, \varphi)+2\alpha(G) \geq 2n-2c(G)$.
If both ends of (\ref{E0}) in the proof of Theorem \ref{T30} are the same, then all inequalities in (\ref{E0}) must be equalities, and so Lemma \ref{L000} (i)-(iv) are observed.
To prove (v). By Lemma \ref{L000} (iii) and Lemma \ref{L23}, we conclude that $u$ lies on just one
complex unit gain cycle of $(G, \varphi)$.
Suppose to the contrary that $u$ is a quasi-pendant vertex which adjacent to a pendant vertex $v$.
Then by Lemma \ref{L13}, we have
$$r((G, \varphi)-u)=r((G, \varphi)- \{ u, v \})=r(G, \varphi)-2,$$
which is a contradiction to (i). This completes the proof of the lemma.
\end{proof}
\begin{lemma} \label{L51}
Let $(G, \varphi)$ be a complex unit gain graph and $(G_{1}, \varphi), (G_{2}, \varphi), \cdots, (G_{k}, \varphi)$ be all connected components of $(G, \varphi)$. Then $(G, \varphi)$ is lower-optimal if and only if $(G_{j}, \varphi)$ is lower-optimal for
each $j \in \{1, 2, \cdots, k \}$.
\end{lemma}
\begin{proof} (Sufficiency.) For each $i \in \{ 1, 2, \cdots, k \}$, one has that
$$r(G_{i}, \varphi)+2\alpha(G_{i})= 2|V(G_{i})|-2c(G_{i}).$$
Then, one has that
\begin{eqnarray*}
r(G, \varphi)&=&\sum\limits_{j=1}^{k}r(G_{j}, \varphi)\\
&=&\sum\limits_{j=1}^{k}[2|V(G_{i})|-2c(G_{i})-2\alpha(G_{i})]\\
&=&2|V(G)|-2c(G)-2\alpha(G).
\end{eqnarray*}
(Necessity.) Suppose to the contrary that there is a connected component of $(G, \varphi)$, say $(G_{1}, \varphi)$, which is not lower-optimal. By Theorem \ref{T30}, one has that
$$r(G_{1}, \varphi)+2\alpha(G_{1}) > 2|V(G_{1})|-2c(G_{1})$$
and for each $ j \in \{ 2, 3, \cdots, k \}$, we have
$$r(G_{j}, \varphi) +2\alpha(G_{j}) \geq 2|V(G_{j})|-2c(G_{j}).$$
Thus, one has that
$$r(G, \varphi)+2\alpha(G) > 2|V(G)|-2c(G),$$
a contradiction.
\end{proof}
\begin{lemma} \label{L56}
Let $u$ be a pendant vertex of a complex unit gain graph $(G, \varphi)$ and $v$ be the vertex which adjacent to $u$. Let $(G_{0}, \varphi)=(G, \varphi)-\{ u, v \}$. Then $(G, \varphi)$ is lower-optimal if and only if $v$ is not on any complex unit gain cycle of $(G, \varphi)$ and $(G_{0}, \varphi)$ is lower-optimal.
\end{lemma}
\begin{proof}
(Sufficiency.) Since $v$ is not on any complex unit gain cycle, by Lemma \ref{L23}, we have
$c(G)=c(G_{0})$.
By Lemmas \ref{L13} and \ref{L053}, one has that
$$r(G, \varphi)=r(G_{0}, \varphi)+2, \alpha(G)=\alpha(G_{0})+1.$$
Thus, one can get $(G, \varphi)$ is lower-optimal by the condition that $(G_{0}, \varphi)$ is lower-optimal.
(Necessity.) By Lemmas \ref{L13} and \ref{L000} and the condition that $(G, \varphi)$ is lower-optimal,
it can be checked that
$$r(G_{0}, \varphi)+2\alpha(G_{0})=2|V(G_{0})|-2c(G).$$
It follows from Theorem \ref{T30} that one has
$$r(G_{0}, \varphi)+2\alpha(G_{0}) \geq 2|V(G_{0})|-2c(G_{0}).$$
By the fact that $c(G_{0}) \leq c(G)$, then we have
$$c(G)=c(G_{0}), r(G_{0}, \varphi)+2\alpha(G_{0})=2|V(G_{0})|-2c(G_{0}).$$
Thus $(G_{0}, \varphi)$ is also lower-optimal and $v$ is not on any complex unit gain cycle of $(G, \varphi)$.
\end{proof}
\begin{lemma} \label{L55}
Let $(G, \varphi)$ be a complex unit gain graph obtained by joining a vertex $x$ of a complex unit gain cycle $(C_{l}, \varphi)$ by
an edge to a vertex $y$ of a complex unit gain connected graph $(K, \varphi)$. If $(G, \varphi)$ is lower-optimal, then the following properties hold for $(G, \varphi)$.
{\em(i)} For each complex unit gain cycle $(C_{q}, \varphi)$ of $(G, \varphi)$, either $\varphi(C_{q}, \varphi)=(-1)^{\frac{q}{2}}$ and $q$ is even or $Re((-1)^{\frac{q-1}{2}}\varphi(C_{q}, \varphi))=0$ and $q$ is odd;
{\em(ii)} If $\varphi(C_{l}, \varphi)=(-1)^{\frac{l}{2}}$ and $l$ is even, then $r(G, \varphi)=l-2+r(K, \varphi)$ and $\alpha(G)=\frac{l}{2}+\alpha(K)$; if $Re((-1)^{\frac{l-1}{2}}\varphi(C_{l}, \varphi))=0$ and $l$ is odd, then
$r(G, \varphi)=l-1+r(K, \varphi)$ and $\alpha(G)=\frac{l-1}{2}+\alpha(K)$.
{\em(iii)} $(K, \varphi)$ is lower-optimal;
{\em(iv)} Let $(G', \varphi)$ be the induced complex unit gain subgraph of $(G, \varphi)$ with vertex set $V(K)\cup \{ x\}$. Then $(G', \varphi)$ is also lower-optimal;
{\em(v)} $\alpha(G')=\alpha(K)+1$ and $r(G', \varphi)=r(K, \varphi)$.
\end{lemma}
\begin{proof}
{\bf (i):} We show (i) by induction on the order $n$ of $(G, \varphi)$. By Lemma \ref{L000}, $x$ can not
be a quasi-pendant vertex of $(G, \varphi)$, then $y$ is not an isolated vertex of $(G, \varphi)$. Then, $(K, \varphi)$ contains at least two vertices, i.e., $n\geq l+2$. If $n=l+2$, then $(K, \varphi)$ contains exactly
two vertices, without loss of generality, assume them be $y$ and $z$. Thus, one has that $(C_{l}, \varphi)=(G, \varphi)-\{ y, z \}$. By Lemma \ref{L56}, we have $(C_{l}, \varphi)$ is lower-optimal.
Then (i) follows from Lemma \ref{L50} directly.
Next, we consider the case of $n \geq l+3$. Suppose that (i) holds for every lower-optimal complex unit gain graph with order smaller than $n$. If $(K, \varphi)$ is a forest. Then $(G, \varphi)$ contains at least one pendant vertex. Let $u$ be
a pendant vertex of $(G, \varphi)$ and $v$ be the vertex which adjacent to $u$. By Lemma \ref{L000}, $v$ is not on $(C_{l}, \varphi)$. By Lemma \ref{L56}, one has that $(G, \varphi)-\{ u, v \}$ is lower-optimal.
By induction hypothesis to $(G, \varphi)-\{ u, v \}$, we have either $\varphi(C_{l}, \varphi)=(-1)^{\frac{l}{2}}$ and $l$ is even or $Re((-1)^{\frac{l-1}{2}}\varphi(C_{l}, \varphi))=0$ and $l$ is odd. Then (i) follows in this case.
If $(K, \varphi)$ contains cycles. Let $g$ be a vertex lying on a cycle of $(K, \varphi)$. By Lemma \ref{L000}, $(G, \varphi)-g$ is lower-optimal. Then, the induction hypothesis to $(G, \varphi)-g$ implies that either $\varphi(C_{l}, \varphi)=(-1)^{\frac{l}{2}}$ and $l$ is even or $Re((-1)^{\frac{l-1}{2}}\varphi(C_{l}, \varphi))=0$ and $l$ is odd. Let $s$ be a vertex lying on $(C_{l}, \varphi)$. By Lemma \ref{L000}, $(G, \varphi)-s$ is lower-optimal. Then, the induction hypothesis to $(G, \varphi)-s$ implies that for each cycle $(C_{q}, \varphi)$ of $(K, \varphi)$ either $\varphi(C_{q}, \varphi)=(-1)^{\frac{q}{2}}$ and $q$ is even or $Re((-1)^{\frac{q-1}{2}}\varphi(C_{q}, \varphi))=0$ and $q$ is odd. This completes the proof of (i).
Next we show (ii)-(v) according to the following two possible cases.
{\bf Case 1.} $\varphi(C_{l}, \varphi)=(-1)^{\frac{l}{2}}$ and $l$ is even.
{\bf (ii):}
Since $x$ lies on a cycle of $(G, \varphi)$, by Lemmas \ref{L000}, \ref{LPN} and \ref{L053}, one has that
\begin{equation} \label{E6}
r(G, \varphi)=r((G, \varphi)-x)= r(P_{l-1}, \varphi)+r(K, \varphi)=l-2+r(K, \varphi)
\end{equation}
and
\begin{equation} \label{E7}
\alpha(G)=\alpha(G-x)= \alpha(P_{l-1})+\alpha(K)=\frac{l}{2}+\alpha(K).
\end{equation}
{\bf (iii):} As $(C_{l}, \varphi)$ is a pendant cycle of $(G, \varphi)$, one has that
\begin{equation} \label{E8}
c(K)=c(G)-1.
\end{equation}
By (\ref{E6})-(\ref{E8}), we have
\begin{equation} \label{E9}
r(K, \varphi)+2\alpha(K)=2(n-l)-2c(K).
\end{equation}
{\bf (iv):} Let $s$ be a vertex of $(C_{l}, \varphi)$ which adjacent to $x$. Then, by Lemmas \ref{L000}, \ref{L13} and \ref{L053}, we have
\begin{equation} \label{E10}
r(G, \varphi)=r((G, \varphi)-s)= l-2+r(G', \varphi)
\end{equation}
and
\begin{equation} \label{E11}
\alpha(G)=\alpha(G-s)= \frac{l-2}{2}+\alpha(G').
\end{equation}
It is obvious that $c(G)=c(G')+1$. Then from (\ref{E10})-(\ref{E11}), we have
\begin{eqnarray*}
r(G', \varphi)+2\alpha(G')&=&r(G, \varphi)+2\alpha(G)-2(l-2)\\
&=&2n-2c(G)-2(l-2)\\
&=&2(n-l+1)-2c(G').
\end{eqnarray*}
{\bf (v):} Combining (\ref{E6}) and (\ref{E10}), one has that
$$r(K, \varphi)=r(G', \varphi).$$
From (\ref{E7}) and (\ref{E11}), we have
$$\alpha(K)+1=\alpha(G').$$
{\bf Case 2.} $Re((-1)^{\frac{l-1}{2}}\varphi(C_{l}, \varphi))=0$ and $l$ is odd.
{\bf (ii):}
Since $x$ lies on a cycle of $(G, \varphi)$, by Lemmas \ref{L000}, \ref{L13} and \ref{L053}, one has that
\begin{equation} \label{E06}
r(G, \varphi)=r((G, \varphi)-x)= r(P_{l-1}, \varphi)+r(K, \varphi)=l-1+r(K, \varphi)
\end{equation}
and
\begin{equation} \label{E07}
\alpha(G)=\alpha(G-x)= \alpha(P_{l-1})+\alpha(K)=\frac{l-1}{2}+\alpha(K).
\end{equation}
{\bf (iii):} As $C_{l}$ is a pendant cycle of $(G, \varphi)$, one has that
\begin{equation} \label{E08}
c(K)=c(G)-1.
\end{equation}
By (\ref{E06})-(\ref{E08}), we have
\begin{equation} \label{E09}
r(K, \varphi)+2\alpha(K)=2(n-l)-2c(K).
\end{equation}
{\bf (iv)} and {\bf (v):} By Lemma \ref{L002}, we have
\begin{equation} \label{E010}
r(G, \varphi)= l-1+r(G', \varphi).
\end{equation}
Then, by (\ref{E06}) and (\ref{E010}) we have
\begin{equation} \label{E011}
r(G', \varphi)=r(K, \varphi).
\end{equation}
By (\ref{E09}) and Theorem \ref{T30}, one has that
\begin{eqnarray*}
2\alpha(K)&=&2(n-l)-r(K, \varphi)-2c(K)\\
&=&2(n-l+1)-r(G', \varphi)-2c(K)-2\\
&=&2(n-l+1)-r(G', \varphi)-2c(G')-2\\
&\leq&2\alpha(G')-2.
\end{eqnarray*}
Thus, we have $\alpha(K)\leq \alpha(G')-1$.
On the other hand, by Lemma \ref{L054}, we have $\alpha(K) \geq \alpha(G')-1$.
Hence,
\begin{equation} \label{E012}
\alpha(K) = \alpha(G')-1.
\end{equation}
It is obvious that $c(G')=c(K)$. Combing (\ref{E09}), (\ref{E011}) and (\ref{E012}), one has that
$$r(G', \varphi)+2\alpha(G')=2(n-l+1)-2c(G').$$
This implies (iv). Moreover, equalities (\ref{E011}) and (\ref{E012}) implies (v).
This completes the proof.
\end{proof}
\begin{lemma} \label{L58}
Let $(G, \varphi)$ be a lower-optimal complex unit gain graph. Then
$\alpha(G)=\alpha(T_{G})+\sum_{C \in \mathscr{C}_{G}}\lfloor\frac{|V(C)|}{2}\rfloor-c(G)$.
\end{lemma}
\begin{proof}
We argue by induction on the order $n$ of $G$ to show the lemma. If $n=1$, then the lemma
holds trivially. Next, we consider the case of $n \geq 2$. Suppose that the result holds for every lower-optimal complex unit gain graph with order smaller than $n$.
If $E(T_{G})=0$, i.e., $T_{G}$ is an empty graph, then each component of $(G, \varphi)$ is a cycle or an isolated vertex. For each cycle $C_{l}$, it is routine to check that
$\alpha(C_{l})= \lfloor \frac{l}{2} \rfloor$. Then the lemma follows.
If $E(T_{G}) \geq 1$. Then $T_{G}$ contains at least one pendant vertex, say $x$. If $x$ is also a pendant vertex in $(G, \varphi)$, then $(G, \varphi)$ contains a pendant vertex. If $x$ is a vertex obtained by contracting a cycle of $(G, \varphi)$, then $(G, \varphi)$ contains a pendant cycle.
Then we will deal with the following two cases.
{\bf Case 1.} $x$ is also a pendant vertex in $(G, \varphi)$.
Let $y$ be the unique neighbour of $x$ and
$(G_{0}, \varphi)= (G, \varphi)-\{ x, y \}$. By Lemma \ref{L56}, one has that $y$ is not on any cycle of $(G, \varphi)$
and $(G_{0}, \varphi)$ is lower-optimal. Furthermore, it is obvious that $c(G)=c(G_{0})$. By induction hypothesis, we have
(a) $\alpha(G_{0})=\alpha(T_{G_{0}})+\sum_{C \in \mathscr{C}_{G_{0}}}\lfloor\frac{|V(C)|}{2}\rfloor-c(G_{0})$.
Sine $x$ is a pendant vertex of $(G, \varphi)$ and $y$ is a quasi-pendant vertex which is not in any cycle of $(G, \varphi)$, $x$ is a pendant vertex of $T_{G}$ and $y$ is a quasi-pendant vertex of $T_{G}$. Moreover,
$T_{G_{0}}=T_{G}-\{ x, y \}$. Thus, by Lemma \ref{L053} and assertion (a), we have
\begin{eqnarray*}
\alpha(G)&=&\alpha(G_{0})+1\\
&=&\alpha(T_{G_{0}})+\sum_{C \in \mathscr{C}_{G_{0}}}\lfloor\frac{|V(C)|}{2}\rfloor-c(G_{0})+1\\
&=&\alpha(T_{G})-1+\sum_{C \in \mathscr{C}_{G_{0}}}\lfloor\frac{|V(C)|}{2}\rfloor-c(G_{0})+1\\
&=&\alpha(T_{G})+\sum_{C \in \mathscr{C}_{G}}\lfloor\frac{|V(C)|}{2}\rfloor-c(G).
\end{eqnarray*}
Thus, the result holds in this case.
{\bf Case 2.} $x$ lies on a pendant cycle.
Let $x$ lies on a pendant cycle $C_{q}$.
In this case, one can suppose that $x$ is the unique vertex of $C_{q}$ of degree 3. Let $K=G-C_{q}$ and $(G_{1}, \varphi)$ be the induced complex unit gain subgraph of $(G, \varphi)$ with vertex set $V(K)\cup \{ x\}$. By Lemma \ref{L55} (iv), one has that $(G_{1}, \varphi)$ is lower-optimal. By induction hypothesis, we have
(c) $\alpha(G_{1})=\alpha(T_{G_{1}})+\sum_{C \in \mathscr{C}_{G_{1}}}\lfloor\frac{|V(C)|}{2}\rfloor-c(G_{1})$.
It can be checked that
$$\mathscr{C}_{G}=\mathscr{C}_{G_{1}} \cup C_{q}=\mathscr{C}_{K} \cup C_{q}.$$
Moreover, one has that
\begin{equation} \label{E13}
\sum_{C \in \mathscr{C}_{G}}\lfloor\frac{|V(C)|}{2}\rfloor=\sum_{C \in \mathscr{C}_{G_{1}}}\lfloor\frac{|V(C)|}{2}\rfloor+\lfloor\frac{q}{2}\rfloor=\sum_{C \in \mathscr{C}_{K}}\lfloor\frac{|V(C)|}{2}\rfloor+\lfloor\frac{q}{2}\rfloor.
\end{equation}
Since $C_{q}$ is a pendant cycle of $(G, \varphi)$, it is obvious that
\begin{equation} \label{E14}
c(G_{1})=c(K)=c(G)-1.
\end{equation}
By Lemma \ref{L55} (v), one has that
\begin{equation} \label{E15}
\alpha(G_{1})=\alpha(K)+1.
\end{equation}
Note that
\begin{equation} \label{E150}
T_{G_{1}}=T_{G}.
\end{equation}
By Lemma \ref{L55} (ii) and (\ref{E13})-(\ref{E150}), one has that
\begin{eqnarray*}
\alpha(G)&=&\alpha(K)+\lfloor\frac{p}{2}\rfloor\\
&=&\alpha(G_{1})+\lfloor\frac{p}{2}\rfloor-1\\
&=&\alpha(T_{G_{1}})+\sum_{C \in \mathscr{C}_{G_{1}}}\lfloor\frac{|V(C)|}{2}\rfloor-c(G_{1})+\lfloor\frac{p}{2}\rfloor-1\\
&=&\alpha(T_{G})+\sum_{C \in \mathscr{C}_{G}}\lfloor\frac{|V(C)|}{2}\rfloor-c(G_{1})-1\\
&=&\alpha(T_{G})+\sum_{C \in \mathscr{C}_{G}}\lfloor\frac{|V(C)|}{2}\rfloor-c(G).
\end{eqnarray*}
This completes the proof.
\end{proof}
\noindent
{\bf The proof of Theorem \ref{T50}.}
(Sufficiency.) We proceed by induction on the order $n$ of $(G, \varphi)$. If $n=1$, then the result holds trivially. Therefore we assume that $(G, \varphi)$ is a complex unit gain graph with order $n \geq 2$ and satisfies (i)-(iii). Suppose that any complex unit gain graph of order smaller than $n$ which satisfes (i)-(iii) is
lower-optimal. Since the cycles (if any) of $(G, \varphi)$ are pairwise vertex-disjoint, $(G, \varphi)$ has
exactly $c(G)$ cycles, i.e., $|\mathscr{O}_{G}|=c(G)$.
If $E(T_{G})=0$, i.e., $T_{G}$ is an empty graph, then each component of $(G, \varphi)$ is a cycle or an isolated vertex. By (ii) and Lemma \ref{L50}, we have $(G, \varphi)$ is lower-optimal.
If $E(T_{G}) \geq 1$. Then $T_{G}$ contains at least one pendant vertex. By (iii), one has that
$$\alpha(T_{G})=\alpha([T_{G}])+c(G)=\alpha(T_{G}-\mathscr{O}_{G})+c(G)=\alpha(T_{G}-\mathscr{O}_{G})+|\mathscr{O}_{G}|.$$
Thus, by Lemma \ref{L055} (ii), there exists a pendent vertex of $T_{G}$ which is not in $\mathscr{O}_{G}$. Then, $(G, \varphi)$ contains at least one pendant vertex, say $u$. Let $v$ be the unique neighbour of $u$ and let
$(G_{0}, \varphi)=(G, \varphi)-\{ u, v \}$. It is obvious that $u$ is a pendant vertex of $T_{G}$ adjacent to $v$ and
$T_{G_{0}}=T_{G}-\{ u, v \}$. By Lemma \ref{L053}, one has that
$$\alpha(T_{G})=\alpha(T_{G}-v)=\alpha(T_{G}-\{ u, v \})+1.$$
{\bf Claim.} $v$ does not lie on any cycle of $(G, \varphi)$.
By contradiction, assume that $v$ lies on a cycle of $(G, \varphi)$. Then $v$ is in $\mathscr{O}_{G}$. Note that the size of $\mathscr{O}_{G}$ is $c(G)$. Then, $H:=(T_{G}-v) \cup K_{1}$ is a spanning subgraph of $T_{G}$.
Delete all the edges $e$ in $H$ such that $e$ contains at least one end-vertex in $\mathscr{O}_{G} \backslash \{ v \}$.
Thus, the resulting graph is $[T_{G}] \cup c(G)K_{1}$. By Lemma \ref{L054}, one has that
$$\alpha([T_{G}] \cup c(G)K_{1}) \geq \alpha((T_{G}-v) \cup K_{1}),$$
that is,
$$\alpha([T_{G}])+c(G) \geq \alpha(T_{G}-v)+1.$$
Then, we have
$$\alpha([T_{G}]) \geq \alpha(T_{G}-v)+1-c(G)=\alpha(T_{G})+1-c(G),$$
a contradiction to (iii). This completes the proof of the claim.
Thus, $v$ does not lie on any cycle of $(G, \varphi)$. Moreover, $u$ is also a pendant vertex of $[T_{G}]$ which adjacent to $v$ and $[T_{G_{0}}]=[T_{G}]-\{ u, v \}$. By Lemma \ref{L053}, one has that
$$\alpha([T_{G}])=\alpha([T_{G_{0}}])+1.$$
It is routine to checked that $c(G)=c(G_{0}).$
Thus,
\begin{eqnarray*}
\alpha(T_{G_{0}})&=&\alpha(T_{G})-1\\
&=&\alpha([T_{G}])+c(G)-1\\
&=&\alpha([T_{G_{0}}])+1+c(G)-1\\
&=&\alpha([T_{G_{0}}])+c(G_{0}).
\end{eqnarray*}
Combining the fact that all cycles of $(G, \varphi)$ belong to $(G_{0}, \varphi)$, one has that
$(G_{0}, \varphi)$ satisfies all the conditions (i)-(iii). By induction hypothesis, we have $(G_{0}, \varphi)$
is lower-optimal. By Lemma \ref{L56}, we have $(G, \varphi)$ is lower-optimal.
(Necessity.) Let $(G, \varphi)$ be a lower-optimal complex unit gain graph. If $(G, \varphi)$ is a complex unit gain acyclic graph, then
(i)-(iii) holds directly. So one can suppose that $(G, \varphi)$ contains cycles. By Lemma \ref{L000} (v) and \ref{L55} (i), one has that the cycles (if any) of $(G, \varphi)$ are pairwise vertex-disjoint and for each cycle $(C_{l}, \varphi)$ of $(G, \varphi)$, either $\varphi(C_{l}, \varphi)=(-1)^{\frac{l}{2}}$ and $l$ is even or $Re((-1)^{\frac{l-1}{2}}\varphi(C_{l}, \varphi))=0$ and $l$ is odd. This completes the proof of (i) and (ii).
Next, we argue by induction on the order $n$ of $(G, \varphi)$ to show (iii). Since $(G, \varphi)$ contains cycles, $n \geq 3$. If $n=3$, then $(G, \varphi)$ is a 3-cycle and (iii) holds trivially. Therefore we assume that $(G, \varphi)$ is a lower-optimal complex unit gain graph with order $n \geq 4$.
Suppose that (iii) holds for all lower-optimal complex unit gain graphs of order smaller than $n$.
If $E(T_{G})=0$, i.e., $T_{G}$ is an empty graph, then each component of $(G, \varphi)$ is a cycle or an isolated vertex. Then, (iii) follows.
If $E(T_{G}) \geq 1$. Then $T_{G}$ contains at least one pendant vertex, say $x$. If $x$ is also a pendant vertex in $(G, \varphi)$, then $(G, \varphi)$ contains a pendant vertex. If $x$ is a vertex obtained by contracting a cycle of $(G, \varphi)$, then $(G, \varphi)$ contains a pendant cycle.
Then we will deal with (iii) with the following two cases.
{\bf Case 1.} $x$ is a pendant vertex of $(G, \varphi)$.
Let $y$ be the unique neighbour of $x$ and
$(G_{1}, \varphi)= (G, \varphi)-\{ x, y \}$. By Lemma \ref{L56}, one has that $y$ is not on any cycle of $(G, \varphi)$
and $(G_{1}, \varphi)$ is lower-optimal. By induction hypothesis, we have
$$\alpha(T_{G_{1}})=\alpha([T_{G_{1}}])+c(G_{1}).$$
Note that $x$ is also a pendant vertex of $T_{G}$ which adjacent to $y$, then $T_{G_{1}}=T_{G}-\{ x, y \}$,
$[T_{G_{1}}]=[T_{G}]-\{ x, y \}$ and $c(G)=c(G_{1})$.
By Lemma \ref{L053}, it can be checked that
$$\alpha(T_{G})=\alpha([T_{G}])+c(G).$$
The result follows.
{\bf Case 2.} $(G, \varphi)$ contains a pendant cycle.
Let $(C_{q}, \varphi)$ be a pendant complex unit gain cycle of $(G, \varphi)$ and $(K, \varphi)=(G, \varphi)-(C_{q}, \varphi)$. By Lemma \ref{L55} (ii), one has that
$(K, \varphi)$ is lower-optimal. By induction hypothesis, we have
\begin{equation} \label{E20}
\alpha(T_{K})=\alpha([T_{K}])+c(K).
\end{equation}
In view of Lemma \ref{L55} (ii), one has
\begin{equation} \label{E21}
\alpha(G)=\alpha(K)+\lfloor\frac{q}{2}\rfloor.
\end{equation}
Since $\mathscr{C}_{G}=\mathscr{C}_{K} \cup C_{q}$.
Then, we have
\begin{equation} \label{E22}
\sum_{C \in \mathscr{C}_{G}}\lfloor\frac{|V(C)|}{2}\rfloor=\sum_{C \in \mathscr{C}_{K}}\lfloor\frac{|V(C)|}{2}\rfloor+\lfloor\frac{q}{2}\rfloor.
\end{equation}
Since $(G, \varphi)$ and $(K, \varphi)$ are lower-optimal, by Lemma \ref{L58}, we have
\begin{equation} \label{E23}
\alpha(T_{G})=\alpha(G)-\sum_{C \in \mathscr{C}_{G}}\lfloor\frac{|V(C)|}{2}\rfloor+c(G)
\end{equation}
and
\begin{equation} \label{E24}
\alpha(T_{K})=\alpha(K)-\sum_{C \in \mathscr{C}_{K}}\lfloor\frac{|V(C)|}{2}\rfloor+c(K).
\end{equation}
It is routine to check that $c(G)=c(K)+1$. Then combining (\ref{E20})-(\ref{E24}), we have
\begin{eqnarray*}
\alpha(T_{G})&=&\alpha(G)-\sum_{C \in \mathscr{C}_{G}}\lfloor\frac{|V(C)|}{2}\rfloor+c(G)\\
&=&\alpha(K)+\lfloor\frac{q}{2}\rfloor-\sum_{C \in \mathscr{C}_{G}}\lfloor\frac{|V(C)|}{2}\rfloor+c(G)\\
&=&\alpha(K)-\sum_{C \in \mathscr{C}_{K}}\lfloor\frac{|V(C)|}{2}\rfloor+c(G)\\
&=&\alpha(K)-\sum_{C \in \mathscr{C}_{K}}\lfloor\frac{|V(C)|}{2}\rfloor+c(K)+1\\
&=&\alpha(T_{K})+1.
\end{eqnarray*}
Note that
\begin{equation} \label{E26}
[T_{G}]\cong [T_{K}].
\end{equation}
Then, in view of (\ref{E20}) and (\ref{E26}), one has that
\begin{eqnarray*}
\alpha(T_{G})&=&\alpha(T_{K})+1\\
&=&\alpha([T_{K}])+c(K)+1\\
&=&\alpha([T_{G}])+c(G).\\
\end{eqnarray*}
This completes the proof.
$\square$
\section*{Acknowledgments}
This work was supported by the National Natural Science Foundation of China
(No. 11731002), the Fundamental Research Funds for the Central Universities (No. 2016JBZ012) and the 111 Project of China (B16002).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,426
|
\section{Introduction}
The X-Ray Telescope (XRT) on board the SWIFT satellite has for the first time
allowed a follow-up
of the X-ray afterglows of GRBs starting within one minute of the BAT
trigger (Burrows et al. 2005a).
These early afterglow observations
have revealed several surprising features which cannot be
easily understood
in terms of the usual interpretation where the afterglow comes from
dissipation in a forward
shock propagating through the source environment. At very early times
immediately after the
burst prompt emission, the afterglow first exhibits a steep
decrease of temporal slope
$\alpha_1 \sim 3$ - 5 ($F_{\rm X} \propto t^{-\alpha}$)
(Tagliaferri et al. 2005). It is often followed
by a much shallower part with $0.2 < \alpha_2 < 0.8$ which can last for
several hours until a more standard slope $1 < \alpha_3 < 1.5$ is finally
observed (Nousek et al. 2005). Moreover flares with sharp rise and decay times are often present,
superimposed on the
power-law evolution (Burrows et al. 2005b). In most cases, the spectrum remains essentially
constant through the breaks
which may indicate that a single physical process is responsible
for the whole X-ray emission.
The forward shock could be such
a process but it seems unable,
at least in its simplest version, to account for the early slopes
$\alpha_1$ and $\alpha_2$. It has been
suggested that the shallow part of the light curve could still be produced
by the forward shock if it is continuously feeded in energy by the
central source (Panaitescu et al. 2005; Zhang et al. 2005).
Another possibility would be to delay the rise of the forward shock
contribution as a result of viewing angle effects (Eichler and Granot, 2005).
These two proposals
would however strengthen the constraint on the
efficiency of the prompt gamma-ray
emission which is already a potential problem for the internal shock scenario
(see however Fan \& Piran, 2006 and Zhang et al. 2006).
For the initial steep decay,
curvature effects of the emitting shell have been invoked (Nousek et al.
2005) while
flares are usually explained by a late activity of the central
source (Zhang et al. 2005; Fan \& Wei, 2005).
In this paper we do not consider the origin of flares but rather
focus on the evolution of the early X-ray afterglow.
We propose that it could be accounted for by a contribution from the
reverse shock. We develop a simple model which allows us to
follow the internal, reverse and forward shocks in a consistent way.
We compute the energy dissipated in the reverse shock and show that,
for some specific initial distribution of the Lorentz factor in the
flow, it is possible to reproduce the
succession of the three slopes $\alpha_1$, $\alpha_2$ and $\alpha_3$.
We then discuss under which conditions part of this dissipated energy can be
radiated in the X-ray range, providing an alternative explanation for
the early X-ray afterglows of GRBs. We also obtain the optical
emission of the reverse shock and show that chromatic breaks can be
observed in some cases.
The paper is organized as follows: in Sect.2 we present the simplified model
we use to follow the dynamics of internal shocks. We explain in Sect.3
how it is extended to include the interaction with the environment and
we compute the power dissipated in the reverse shock. We consider in Sect.4
the possibility for this power to be partially radiated in the X-ray range.
We discuss in Sect.5 the relative importance of the reverse and forward shock
contributions and present X-ray and optical afterglow light curves
produced by the reverse shock alone.
Sect.6 is our conclusion.
\section{The origin of GRB pulses}
In the context of the internal shock model for the prompt emission of GRBs
the pulses observed in the light curve are produced when
fast moving material catches up with slower one previously
ejected by the central source (Rees \& Meszaros, 1994).
This process has often
been represented by the collision of two ``shells'' of negligible
thickness. However the central source probably does not release
individual shells but a continuous relativistic outflow with a
varying Lorentz factor. For this reason the shape of the pulses
is largely
dominated by hydrodynamical effects (Daigne \& Mochkovitch, 2003) while
high latitude emission
(curvature effect) only becomes important at late times.
Soderberg and Fenimore (2001)
have for example found
that the decay of pulses differs
from what would be expected if it was controlled by
the curvature effect alone.
A hydrodynamical study
of the relativistic flow emerging from the central engine therefore
appears necessary for a detailed description of the physics of pulses
but it is naturally quite expensive in computing time (Daigne \&
Mochkovitch, 2000).
Fortunately it can often be replaced by a simplified model where the
flow is represented by a large number of regularly ejected
shells which interact by
direct collision only (Daigne \&
Mochkovitch, 1998). This neglects pressure waves but
this is a good approximation since kinetic energy strongly dominates over
internal energy of the flow. This approach implies to use
many shells (from $10^3$ to $10^4$) to represent accurately the distribution
of mass and Lorentz factor. It is different
from the even more simplified description where the numbers of shells
essentially corresponds to the number of pulses to be produced and where
the temporal profiles are then entirely fixed by the curvature effect
(Kobayashi, Piran \& Sari, 1997).
\\
The position $R_i$ of each shell of mass $M_i$ and Lorentz factor $\Gamma_i$
is followed as a function of time $t$ (in the source frame). When shell
$i$ catches up with shell $i+1$ a shock occurs at time $t_{\rm s}$ and
radius $R_{s}$. The two shells merge
and the resulting Lorentz factor after the collision is given by
\begin{equation}
\Gamma_{\rm r}=\sqrt{\Gamma_i\Gamma_{i+1}{m_i\Gamma_i+m_{i+1}\Gamma_{i+1}
\over m_i\Gamma_{i+1}+m_{i+1}\Gamma_i}}\ .
\end{equation}
If the the released energy
\begin{equation}
E=m_i\Gamma_i c^2+m_{i+1}\Gamma_{i+1} c^2-(m_i+m_{i+1})\Gamma_{\rm r} c^2
\end{equation}
can be efficiently radiated it will be received by the observer at a time
\begin{equation}
t_{\rm obs}=t_{\rm s}-{R_{\rm s}\over c}
\end{equation}
and for a typical duration
\begin{equation}
\Delta t_{\rm obs}= {R_{\rm s}\over 2c \Gamma_{\rm r}^2}
\end{equation}
under the condition that the radiative time is
much smaller than the dynamical time (fast cooling regime).
The burst bolometric luminosity can then be obtained from the sum of
all the elementary shock contributions, the dynamical evolution
being terminated
when all the shells have their Lorentz factor decreasing downstream
so that no new internal shock can form.
For an accurate description of the pulse profile at late times,
the contribution $\ell(t)$ of each elementary shock must include
the curvature effect of the emitting shell which yields
\begin{equation}
\ell(t)={2E\over \Delta t_{\rm obs}
(1+{t-t_{\rm obs}\over \Delta t_{\rm obs}})^3}
\end{equation}
for $t_{\rm obs}<t<t_{\rm obs}+\left(1-{\rm cos}
\Delta \theta\right)R_{\rm s}/c$, where $\Delta \theta$ is the opening
angle of the jet here supposed to be seen on axis
(Granot, Piran \& Sari, 1999; Woods \& Loeb, 1999).
The luminosity in a given energy band depends on some additional
(and uncertain) assumptions on the post-shock magnetic field and Lorentz factor
of the electrons which are discussed in Sect. 4 and 5 while in Sect. 3
we restrict
ourselves to the bolometric
emission only.
To produce a single pulse burst (or a pulse as a building block of
a more complex burst) we have often used in previous works
(Daigne \& Mochkovitch, 1998, 2000)
an initial distribution of the Lorentz factor of the form
\begin{equation}
\Gamma(t)={\Gamma_{\rm max}+\Gamma_{\rm min}\over 2}
-{\Gamma_{\rm max}-\Gamma_{\rm min}\over 2}\,{\rm cos}\,\Big(\pi {t\over
0.2\,t_{\rm W}}\Big)
\end{equation}
if $t< 0.2\,t_{\rm W}$ and $\Gamma(t)=\Gamma_{\rm max}$ if
$t> 0.2\,t_{\rm W}$; $\Gamma_{\rm max}$ and $\Gamma_{\rm min}$
are the maximum and minimum values of the Lorentz factor and $t_{\rm W}$ the
duration of the relativistic wind emission (the first shell is then ejected at
$t=0$ and the last one at $t=t_{\rm W}$).
This Lorentz factor distribution
where a rapid part of the flow is decelerated by a slower part placed
ahead of it,
has been represented in Fig.1a for $\Gamma_{\rm max}=200$,
$\Gamma_{\rm min}=50$ and $t_{\rm W}=10$ s.
The resulting bolometric
profile from dissipation by internal shocks is shown
in Fig.1b for a total (isotropic) radiated energy
$E_{\rm rad}=10^{53}$ erg.
\begin{figure*}{}
\begin{center}
\begin{tabular}{cc}
\resizebox{0.49\hsize}{!}{\includegraphics{fig1a.eps}}&
\resizebox{0.49\hsize}{!}{\includegraphics{fig1b.eps}}\\
\end{tabular}
\end{center}
\caption{A single pulse burst. (a): Initial distribution of the Lorentz factor in the relativistic
flow as a function of ejection time and distance $D$ to the source (in
light-seconds) according to Eq.(6) with $\Gamma_{\rm max}=200$,
$\Gamma_{\rm min}=50$ and $t_{\rm W}=10$ s (thick full line) and
to Eq.(7)
with $\delta=1$ and $\Gamma_{\rm f}=2 $ (thin full line);
(b): Bolometric profile for the distribution given by Eq.(6) (full line)
together
with the corresponding temporal slope $\alpha$ (dashed line).
After maximum, the profile is first controlled by the dynamics of
internal shocks
before the curvature effect eventually dominates after
$t_{\rm obs}\sim 20$ s.}
\end{figure*}
The decline of the pulse after maximum is
first controlled by the dynamics of internal shocks.
This
would lead to an asymptotic behavior
$L(t)\propto t^{-3/2}$ if it was not interrupted at a time
$\tau\sim t_{\rm W}$
when all the fast material of the ejecta has been shocked.
Daigne and Mochkovitch (2003) have shown that the gamma-ray profiles
which can be obtained from this first part of the bolometric
light curve are gene\-ral\-ly
in good agreement with the
early decline following maximum count rate in observed GRBs
(Ryde \& Svensson, 2000).
After all the ejecta has been shocked the pulse evolution
becomes fixed by geometry, the contribution of each
shocked shell being given by Eq.(5).
At large times $t\gg \tau$,
all the $\ell(t)$ and therefore the global profile $L(t)$
asymptotically behave as $t^{-3}$.
However at
early times $t\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} \tau$,
a steeper decline can be obtained (Nousek et al. 2005) as
illustrated in Fig.1b where the temporal slope $\alpha$
has been plotted together with the profile.
It has a maximum of 3.65 just at the end of the
internal shock phase before relaxing to 3 after a few $\tau$.
\section{The reverse shock}
\subsection{Physical description}
The profile calculated above
corresponds to a ``naked GRB'' (Kumar \& Panaitescu, 2000) and would be the only component observed
in the absence of external medium. The burst environment will however
interact with the ejecta, leading to a forward shock propagating
through the circumstellar medium
and a reverse shock sweeping back into the ejecta.
We compute below (Sect.3.2) the power dissipated in the reverse
shock and discuss its possible contribution to the early X-ray emission
of GRBs. With the initial distribution of the Lorentz factor given by Eq.(6)
the reverse shock crosses the ejecta in a short time and
cannot explain an emission
lasting for several days.
The situation is however very different if a slight change is made in the
initial distribution of the Lorentz factor.
We expect that the central source will not stop ejecting
relativistic material abruptly at $t=t_{\rm W}$.
We instead propose that $\Gamma$ will progressively decrease
until it reaches
a small value (possibly close to unity) at $t_{\rm W}$. Since $\Gamma(t)$
is given by the ratio $\dot E/\dot M$ of the energy to mass injection rates,
a small $\Gamma$ can be a consequence of ({\it i}) a decrease of $\dot E$,
less and less energy
becoming available from the source
to accelerate a given baryon load or/and ({\it ii}) a catastrophic increase of
$\dot M$. Case ({\it i}) appears more natural during the
late stages of source activity and has been adopted
in presenting our results in Sect.3.2.
We have then introduced
a new distribution of the Lorentz factor where, for $t>0.5\,t_{\rm W}$,
$\Gamma(t)$ decreases to a final value
$\Gamma_{\rm f}$
\begin{equation}
\Gamma(t)=\Gamma_{\rm f}
+\left({\Gamma_{\rm max}-\Gamma_{\rm f}}\right)\,
\left[{1-t/t_{\rm W}\over 0.5}\right]^{\delta}
\end{equation}
while for $t<0.5\,t_{\rm W}$, $\Gamma(t)$ is still given by Eq.(6).
This mo\-di\-fied Lorentz factor is plotted in Fig.1a for
$\Gamma_{\rm max}=200$, $\Gamma_{\rm f}=2$, $t_{\rm W}=10$ s
and $\delta=1$.
With this new distribution,
the duration of source activity remains unchanged but
the reverse shock will be present
for a much longer time, until all the ejecta has been decelerated to
$\Gamma\sim \Gamma_{\rm f}$.
The forward shock also remains feeded in energy
as slow material from the ejecta is continuously catching up but
the resulting effect is too small in this case to account
for the shallow part of the
light curve (we assumed that equal amounts of kinetic energy
are injected before and after $t=0.5\,t_{\rm W}$).
To compute the energy dissipated in the reverse shock
we had to implement
in our shell model the interaction with the burst environment.
This was done by considering the contact
discontinuity which separates the ejecta and the shocked
external medium. In our simple description it is represented
by two shells moving at the same Lorentz factor $\Gamma$.
The first one corresponds to the mass $M_{\rm ej}$ of the
ejecta already crossed by the reverse shock, which
carries a total e\-ner\-gy $\Gamma M_{\rm ej} c^2$,
and the second
to the shocked external medium of mass $M_{\rm ex}$. If the forward
shock moves quasi-adiabatically (slow cooling regime),
this shell
keeps its internal energy (since $pdV$ work is
neglected in our simple model) so that its
total energy is $\Gamma \Gamma_i M_{\rm ex} c^2$
where $(\Gamma_i-1)c^2$
is the dissipated energy (per unit mass) in
the fluid rest frame.
Two processes will affect this two shell structure
at the contact discontinuity : it will collide either with shells of the
external medium at rest, or with rapid shells of the relativistic ejecta
catching up.
This represents both the forward and reverse shock in our simplified
picture.
\vskip 0.3cm\noindent
\textit{Forward shock:}
the interaction with
the external medium is discretized by assuming that
a collision occurs each time
the contact discontinuity has travelled from a radius $R$
to a radius $R^{\,\prime}$ so that the swept-up mass is
\begin{equation}
m_{\rm ex}=\int_R^{R^{\,\prime}} 4\pi r^2 \rho(r) dr=q\,{M\over
\Gamma}
\end{equation}
where $M=M_{\rm ej}+M_{\rm ex}$, $\rho(r)$ is the density of the external
medium (for which we
adopted either a constant or a stellar wind distribution) and
$q\ll 1$ (we take in practice
$q=10^{-2}$). Writing the conservation of energy-momentum for this collision,
we obtain the new Lorentz factor $\Gamma_r$
at the contact discontinuity
\begin{equation}
\Gamma_r=\left[{(M_{\rm ej}+M_{\rm ex}\Gamma_i)\Gamma^2+m_{\rm ex}\Gamma\over
(M_{\rm ej}+M_{\rm ex}\Gamma_i)+2 m_{\rm ex}\Gamma}\right]^{1/2}
\end{equation}
and also the new Lorentz factor $\Gamma_i^{\prime}$
for internal motions after the collision
\begin{equation}
\Gamma_i^{\prime}={(M_{\rm ej}+M_{\rm ex} \Gamma_i)\Gamma+m_{\rm ex}-M_{\rm ej}\Gamma_r\over
(M_{\rm ex}+m_{\rm ex})\Gamma_r}\ .
\end{equation}
It should be noted that the above equations assume
that material
in the burst environment is at rest. This neglects the pair-loading
process resulting from the initial flash of gamma-rays
which pre-accelerates the circumstellar medium (Madau \& Thompson, 2000;
Thompson \& Madau, 2000; Beloborodov, 2002) out to a radius
\begin{equation}
R_{\rm acc}\sim 7\,10^{15}\,E_{\gamma,\,53}^{1/2}\ \ {\rm cm}
\end{equation}
where $E_{\gamma,\,53}$ is the isotropic gamma-ray energy of the flash in units of
$10^{53}$ erg. For this reason, the deceleration
by the external medium will be delayed by
\begin{equation}
\Delta t_{\rm acc}\sim {R_{\rm acc}\over 2c \Gamma^2}=
12\,E_{\gamma,\,53}^{1/2}\,\Gamma_2^{-2}\ \ {\rm s}
\end{equation}
where $\Gamma_2$ is the average Lorentz factor of the ejecta in units of
$10^2$.
Therefore the initial dynamical evolution will be that
of a naked GRB but the effect will last more than one minute only for the
most extreme bursts with $E_{\gamma,\,53}>10$. \\
\noindent
\textit{Reverse shock:}
as the Lorentz factor at the contact discontinuity decreases,
new shells from the ejecta become able to catch up.
Writing again the conservation of energy-momentum for these collisions,
we obtain the
change in Lorentz factor
\begin{equation}
\Gamma_r=\sqrt{\Gamma\gamma_{\rm ej}}\left[{(M_{\rm ej}+M_{\rm ex}\Gamma_i)\Gamma+m_{\rm ej}\gamma_{\rm ej}
\over (M_{\rm ej}+M_{\rm ex}\Gamma_i)\gamma_{\rm ej}+m_{\rm ej}\Gamma}\right]^{1/2}
\end{equation}
and the related dissipated energy
\begin{equation}
E_{\rm diss}=(M_{\rm ej}+M_{\rm ex}\Gamma_i)\Gamma c^2+
m_{\rm ej}\gamma_{\rm ej} c^2
-(M_{\rm ej}+m_{\rm ej}+M_{\rm ex} \Gamma_i)\Gamma_r c^2
\end{equation}
where $m_{\rm ej}$ and $\gamma_{\rm ej}$ are respectively the mass and Lorentz factor of the
colliding material from the ejecta.
\subsection{The dissipated power}
Using this simplified model for the interaction of the ejecta with
its environment we can describe the deceleration of the front shell and the
propagation of the reverse shock. We have obtained the dissipated power in the
reverse shock for different burst environments (uniform medium or wind).
In the wind case, we considered three values of the
parameter $A_*$: 0.5, 0.1 and 0.05 (such as
$\rho(r)=5\,10^{11}\,A_*/r^2$ g.cm$^{-3}$ with $A_*=1$
for a wind mass loss
rate $\dot M_{\rm w}=10^{-5}$ M$_\odot$.yr$^{-1}$ and a terminal velocity
$v_{\infty}=1000$ km.s$^{-1}$). In the constant density case, we
also tried three values of $n$: 1000, 10 and 0.1 cm$^{-3}$.
The resulting profiles are shown in Fig.2 for $\Gamma_{\rm f}=2$
and $\delta=1$ in Eq.(7) but we checked that they remain
essentially unchanged when $\Gamma_{\rm f}$ is varied between 1 and 10
and $\delta$ between 0.5 and 2.
\begin{figure*}{}
\begin{center}
\begin{tabular}{cc}
\resizebox{0.49\hsize}{!}{\includegraphics{f2a.eps}}&
\resizebox{0.49\hsize}{!}{\includegraphics{f2b.eps}}\\
\end{tabular}
\end{center}
\caption{Dissipated power as a function of observer time during
reverse shock propagation when the Lorentz factor is given
by Eq.(7) with $\Gamma_{\rm f}=2$ and $\delta=1$.
(a): wind case with $A_*=0.5$ (dashed line),
$A_*=0.1$ (full line) and $A_*=0.05$ (dotted line);
(b): uniform density case with
$n=1000$ (dashed line), 10 (full line) and 0.1 cm$^{-3}$ (dotted line).
In each panel the thin full line represents the naked burst.
}
\end{figure*}
The curves in Fig.2 show a striking
resemblance with the early X-ray afterglows observed by SWIFT.
After about 100 s the
reverse shock component
dominates over the tail of the
of the burst prompt emission
computed in the last section.
At late times the decline follows a constant
slope $\alpha\sim 1.5$. The shape of the intermediate region
is most sensitive to the density of the
burst environment. At high density it is nearly suppressed, the constant
slope $\alpha \sim 1.5$ following directly the initial steep
decrease. Conversely at low density, it
can become completely flat and even fall to a temporary minimum.
For comparison, we have also represented in Fig.2 the profile
corresponding to the
naked GRB. Without an external medium and
for the distribution of Lorentz factor given by Eq.(7)
(with $\Gamma_{\rm f}=2$ and $\delta=1$) we find that a large fraction
of the ejecta (all slow material with $\Gamma<140$)
remains unaffected by internal shocks.
With an external medium the reverse shock
propagates throught this material which produces the additional power
at late times.
\\
To elucidate the behavior of the reverse shock contribution
we have considered the following
simplified case which can be handled analytically:
the ejecta is supposed to be made of a rapid single
shell of mass $M_0$ and initial
Lorentz factor $\Gamma_0$ (representing the fast material
where the prompt emission takes place) followed by a slower tail
of the form
\begin{equation}
\Gamma(M)=\Gamma_{\rm f}+(\Gamma_0-\Gamma_{\rm f})\left({M\over M_{\rm s}}
\right)^
{\delta}
\end{equation}
where $\Gamma_{\rm f}$ is the final Lorentz factor at the end of the tail
and $M_{\rm s}$ is the
total mass of the slow material
($M=0$ corresponds to the last
emitted shell and $\delta$ allows to vary the tail shape). Notice
that this expression of $\Gamma(M)$ directly results from Eq.(7)
if $\dot M$ is constant.
Such a distribution of $\Gamma$ skips the prompt phase
and the resulting dissipated power $P_{\rm diss}$
comes from the
reverse shock only.
The reverse shock contribution is maximum at a time
close to the deceleration time of the front shell
\begin{equation}
t_{\rm dec}={R_{\rm dec}\over 2 c\,\Gamma_0^2}\ \ \
{\rm with}\ \ \ R_{\rm dec}=\left[{M_0\,(3-s)\over 4\pi A \Gamma_0}
\right]^{1\over 3-s}
\end{equation}
where we have used the notation $\rho=A r^{-s}$ with $A=\rho$ and
$s=0$ for a uniform medium and $s=2$ for a stellar wind.
A full analytical solution can be obtained for the reverse shock
contribution but we only derive below its
asymptotic behavior
assuming that the front shell
essentially follows the Blandford-McKee solution, i.e. it is
only weakly affected by the additional energy coming from the slow material
progressively catching up. We have checked this approximation with
the numerical simulations and it is satisfied to an accuracy of about
25\%.
We can write the power dissipated in the
reverse shock as
\begin{equation}
P_{\rm diss}=- {dM\over d\Gamma}{d\Gamma\over dt}\,
\Gamma\, e c^2
\end{equation}
where $t$ is the observer time when a shell of Lorentz factor
$\Gamma$ catches up with the front shell.
The fraction $e$ of the in\-co\-ming material
kinetic energy
dissipated in the collision can be obtained from Eq.(13) and (14)
with $m_{\rm ej}\ll M_{\rm ej}+M_{\rm ex}\Gamma_i$, leading to
\begin{equation}
e={1\over 2}\left[1-\left({\Gamma_{\rm fs}\over \Gamma}\right)\right]^2
\end{equation}
where $\Gamma_{\rm fs}$ is the Lorentz factor of the front shell given
by the Blandford-McKee solution
\begin{equation}
\Gamma_{\rm fs}\simeq \Gamma_0\left({R\over R_{\rm dec}}\right)^{-\lambda}
\end{equation}
with $\lambda={3-s\over 2}=3/2$ (resp.
$1/2$) for a uniform medium (resp. a stellar wind). Using
$dR/dt=2c\,\Gamma_{\rm fs}^2$ we then get the relation between shock radius and
observer time
\begin{equation}
{t\over t_{\rm dec}}={1\over 2\lambda +1}\left({R\over R_{\rm dec}}\right)
^{2\lambda+1}\ .
\end{equation}
With our assumed distribution (Eq.(15)) of the Lorentz factor in the
slow material which
is steadily increasing outwards, each shell moves independently
at the constant Lorentz factor $\Gamma$ until it catches up with
the forward shock.
We moreover neglect the fact that the slow material is emitted over
a certain duration and we write the position of each shell as a function
of observer time as
\begin{equation}
{R\over R_{\rm dec}}={t\over t_{\rm dec}}\left({\Gamma\over \Gamma_0}\right)^2
\ .
\end{equation}
Notice that if the Lorentz factor is not initially monotonic in the
slow material, internal shocks will take place which, when they are completed,
will leave a new distribution of $\Gamma$, now steadily increasing outwards.
For observing times long compared to the time of internal shocks
Eq.(21) will therefore still hold.
Eliminating the radius between Eq.(20) and (21) gives the time when
a shell of Lorentz factor $\Gamma$ catches up with the forward shock
\begin{equation}
{t\over t_{\rm dec}}=(2\lambda+1)^{1/2\lambda}
\left({\Gamma\over \Gamma_0}\right)^{-{2\lambda+1\over \lambda}}\ .
\end{equation}
Now from Eq.(19), (20) and (22) we can obtain the Lorentz factor $\Gamma_{\rm fs}$
of the forward shock when the slow shell of Lorentz factor $\Gamma$
catches up
\begin{equation}
\Gamma_{\rm fs}=\Gamma\,(2\lambda+1)^{-1/2}\ .
\end{equation}
To end with a simple power law expression for the dissipated power
we write
\begin{equation}
{dM\over d\Gamma}={M_{\rm s}\over \delta\, \Gamma_0}
\left({\Gamma\over \Gamma_0}\right)^{{1-\delta\over \delta}}
\end{equation}
i.e. we use Eq.(15) with $\Gamma_{\rm f}=0$ which is obviously
uncorrect but does not change the behavior of the solution
in the re\-la\-tivistic regime.
Using Eq.(22), (23) and (24) we finally get
\begin{equation}
P_{\rm diss}(t)=\Phi(\lambda,\delta)
\,{\Gamma_0 M_{\rm s}c^2\over \delta\,t_{\rm dec}}
\left({t\over t_{\rm dec}}\right)^{-\left[{3\lambda+1+\lambda/\delta\over
(2\lambda+1)}\right]}
\end{equation}
where
\begin{equation}
\Phi(\lambda,\delta)={\lambda\over 2}\,
\left[1-\left(2\lambda+1\right)^{-1/2}\right]^2
\times \left(2\lambda+1\right)^{{1-\delta(1+4\lambda)\over
2\delta(2\lambda+1)}}\ .
\end{equation}
For $\delta=1$ and the two values of interest for $\lambda$, Eq.(25) becomes
\begin{eqnarray}
{P_{\rm diss}\over {\Gamma_0 M_{\rm s}c^2/t_{\rm dec}}}& = &
6.6\,10^{-2} \left({t\over t_{\rm dec}}\right)^{-7/4}\ \ (\lambda=3/2)\nonumber\\
& = & 1.5\,10^{-2}\left({t\over t_{\rm dec}}\right)^{-3/2}\ \ (\lambda=1/2)
\end{eqnarray}
which is in good agreement with the asymptotic behavior of the light curves
shown in Fig.2.
\begin{figure*}
\begin{center}
\begin{tabular}{cc}
\resizebox{0.49\hsize}{!}{\includegraphics{fv.eps}}&
\resizebox{0.49\hsize}{!}{\includegraphics{fu.eps}}\\
\end{tabular}
\end{center}
\caption{Synthetic X-ray light curves in the 0.3 - 10 keV range
for the models of Fig.2. The adopted post-shock energy redistribution
parameters are $\epsilon_e=\epsilon_B=1/3$ and $\zeta=10^{-2}$. The
assumed burst redshift is $z=2$; The different cases considered
in Fig.2 are represented by the same full, dotted and dashed lines.}
\end{figure*}
\section{Can the reverse shock contribute in X-rays?}
Despite their similarity with the SWIFT observations,
it must remain clear that the profiles shown in Fig.2 only trace
the power dissipated in the reverse shock. With the assumptions
ordinary made to compute the reverse shock contribution in GRBs
it should manifest itself mainly in the visible/IR range (Sari \& Piran, 1999).
Moreover most of
the emission would generally take place in the slow cooling regime so that the
observed light curve will not necessarily trace the instantaneous energy release.
We therefore investigated whether, under some specific conditions,
a substantial fraction of the dissipated power can be
({\it i}) radiated in the X-ray range and ({\it ii}) in the fast cooling regime.
If the reverse shock contribution originates from synchrotron radiation of
shock accelerated electrons, the characteristic
synchrotron energy and cooling time behave as
\begin{equation}
E_{\rm s}\propto B \Gamma_e^2\ \ \ \ {\rm and}\ \ \ \
t_{\rm s}\propto B^{-2}\Gamma_e^{-1}
\end{equation}
in the rest frame of the shocked material.
Both the post-shock
magnetic field $B$ and typical electron Lorentz factor $\Gamma_e$ have
therefore to be large to produce an emission at high energy and on a
short time scale $t_{\rm s}<t_{\rm dyn}$. An estimate of $\Gamma_e$ is usually obtained assuming that
a fraction $\epsilon_e$ of the dissipated energy is injected into a
fraction $\zeta$ of the electrons so that
\begin{equation}
\Gamma_e\simeq {\epsilon_e\over \zeta}{m_p\over m_e}\,e
\end{equation}
where $m_p$ and $m_e$ are the proton and electron masses and $e c^2$ is the
energy dissipated per unit mass in the comoving frame. Similarly the
post-shock magnetic field can be expressed as
\begin{equation}
B=\left(8\pi\epsilon_B\,\rho\,e c^2 \right)^{1/2}
\end{equation}
where $\rho$ is the comoving density and $\epsilon_B$ the fraction of the
dissipated energy tranferred to the magnetic field.
To have large
$B$ and $\Gamma_e$ values behind the shock we first
supposed that a complete equipartion is
established between the electronic, magnetic and baryonic components
so that $\epsilon_e=\epsilon_B=\epsilon_{\rm baryon}=1/3$.
We also assumed that only a small fraction $\zeta\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 10^{-2}$
of the electron population is accelerated in the shock. Adopting
$\zeta= 10^{-2}$ increases $\Gamma_e$
by a factor of 100 and hence $E_{\rm s}$ by a factor
$10^4$ and decreases $t_{\rm s}$ by $10^2$
compared to the standard $\zeta=1$ case.
The possibility to have only a small fraction of electrons being
accelerated has already been considered by Bykov \& Meszaros (1996)
and also by Eichler \&
Waxman (2005) in the context
of GRB afterglows. They showed that $\zeta$ is not well constrained by
the observations and, even if $\zeta\sim 1$ appears slightly favored,
they included the whole interval $m_e/m_p<\zeta<1$ in their analysis.
In internal shocks, which are very similar to the reverse shock (both
take place in the burst ejecta and are mildly relativistic)
a large $\epsilon_e$ is required to maintain a reasonable global efficiency
since the
fraction of the total energy dissipated by internal shocks hardly exceeds 10\%.
A small $\zeta$ is also favored to insure that the emission takes place
in the gamma-ray range as shown by Daigne and Mochkovitch (1998) and
more recently by Lee et al. (2005) in the context of the short hard burst
GRB 050509b.
Examples of synthetic light curves in the XRT band 0.3 - 10 keV are
shown in Fig.3 for the cases already consi\-de\-red in Fig.2. They have
been obtained with $\epsilon_e=\epsilon_B=1/3$, $\zeta=10^{-2}$,
a slope $p=2.5$ for the electron energy distribution and an assumed
redshift $z=2$, typical of the SWIFT burst population. Especially in the
wind case, they seem able to reproduce many of the observed XRT light
curves. Conversely in the uniform density case we often obtain a depressed minimum
followed by a bump rather than a continuous shallow evolution
\section{Discussion}
Our proposal to explain the early X-ray afterglow of GRBs
by a contribution of the reverse shock relies on three well
defined assumptions: ({\it i}) the Lorentz factor of the material
ejected at late times by the source has to decrease
to small values, $\Gamma_{\rm f} < 10$;
({\it ii}) the shock dissipated energy must be transferred to only a small
fraction of the electron population; and ({\it iii})
the forward shock contribution should lie
below that of the reverse shock, at least during the first hours following burst
trigger.
This last condition requires
an ineffective transfer of energy to electrons
($\epsilon_e\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 10^{-2}$) or/and ma\-gne\-tic field
($\epsilon_B\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 10^{-5}$) in the material crossed by the
forward shock. Difficulties to produce a sufficiently large
ma\-gne\-tic field extending over the emitting region of GRB afterglows has
for example been recently emphasized by Milosav\-ljevi\'c \& Nakar (2006).
Then, if the reverse shock dominates in X-rays, what is the
situation in the visible? We have checked that in most cases,
taking small values of $\epsilon_e$ or/and $\epsilon_B$ in the forward
shock, equally implies that the reverse shock
dominates in the visible.
The consistency of our proposal must therefore be checked not only
with X-ray observations
but also at lower wavelengths.
To better understand the multiwavelength behavior of the reverse
shock contribution, we have computed the peak flux $F_{\rm max}$
and the characteristic synchrotron and cooling frequencies $\nu_m$
and $\nu_c$ (Sari, Piran \& Narayan, 1998). These three quantities
depend on $t$ (observer time), $N_e$ (total number of shock
accelerated electrons), $B$ (magnetic field in shocked material),
$\Gamma_e$ (typical electron Lorentz factor) and $\Gamma$ (Lorentz factor
of the emitting material) in the following way
\begin{equation}
\begin{array}{l}
F_{\rm max}\propto \Gamma B N_e \\
\nu_m\propto \Gamma\, \Gamma_e^2 B \\
\nu_c\propto \Gamma^{-1} B^{-3} t^{-2}
\end{array}
\end{equation}
We consider their temporal evolution in the asymptotic regime already
described in Sect.3.2, assuming a wind environment.
The evolution of $\Gamma$ is given by Eq.(22) which can be
reexpressed as
\begin{equation}
\Gamma(t)=\Gamma_0 \left({t\over 2\,t_{\rm dec}}\right)^{-1/4}
=119\ \Gamma_2 \left({t\over t_{\rm dec}}\right)^{-1/4}
\end{equation}
where $\Gamma_2=\Gamma_0/100$.
Eq.(18), (23) and (29) show that $\Gamma_e$ reaches a constant value
\begin{equation}
\Gamma_e=4.3\,10^{-2}\,{m_p\over m_e}\,{\epsilon_e\over \zeta}\,
{p-2\over p-1}=79\,{\epsilon_e\over \zeta}\,{p-2\over p-1}
\end{equation}
where we have added the normalizing factor ${p-2\over p-1}$ ($p$
being the slope of the relativistic electron distribution) which was not
present in Eq.(29). For the magnetic field, instead of Eq.(30) it is
easier to use the continuity of the energy density at the contact
discontinuity which yields
\begin{equation}
B=(32\pi \epsilon_B c^2 A)^{1/2}{\Gamma\over R}
\end{equation}
where $A$ is the wind constant such as $\rho(R)=A/R^2$ ($A=5\,10^{11}A_*$).
With Eq.(20) for $R$ and Eq.(32) for $\Gamma$ we get
\begin{equation}
B(t)=3\,10^4\;{\left(\epsilon_B\,A_*\right)^{1/2}
\over t_{\rm dec}\,\Gamma_2}
\left({t\over t_{\rm dec}}\right)^{-3/4}\ {\rm G}
\end{equation}
Finally, the number of accelerated electrons can be obtained from
Eq.(24) which, for $\delta=1$, gives
\begin{equation}
N_e(t)={2\,\zeta\,E_{\rm s}\over \Gamma_0\,m_p c^2}\left[1-1.19\,\left({t\over t_{\rm dec}}\right)^{-1/4}
\right]
\end{equation}
where $E_{\rm s}={1\over 2}\Gamma_0\,M_{\rm s} c^2$ is the total energy in the
slow material.
From Eq.(32), (33), (35) and (36) the expressions for
$F_{\rm max}$, $\nu_m$ and $\nu_c$ can be computed
\begin{equation}
\begin{array}{l}
F_{\rm max}=1.4\,10^8\,{(1+z)\over D_{28}^2}\,
{(\zeta E_{53})\left(\epsilon_B\,A_*\right)^{1/2}
\over \Gamma_2}\times {1\over t}\ \ {\rm mJ} \\
\nu_m=9.15\,10^{16}\,(\epsilon_B\,A_*)^{1/2}
\left({\epsilon_e\over \zeta}\right)^2\,\left({p-2\over p-1}\right)^2
\times {1\over t}\ \ {\rm Hz}\\
\nu_c=8.2\,10^8\,{t_{\rm dec}^{1/2}\Gamma_2^2\over (\epsilon_B\,A_*)^{3/2}}
\times t^{1/2}\ \ {\rm Hz}
\end{array}
\end{equation}
with $E_{53}=E_{\rm s}/10^{53}$ erg and where the expression for $F_{\rm max}$
has been written in the limit $t\gg t_{\rm dec}$.
Compared to the forward shock case, it can be seen that
$F_{\rm max}\propto t^{-1}$
and that $\nu_m$ decays less rapidly (as $t^{-1}$ instead of $t^{-3/2}$).
For a wind environment, the cooling frequency has the same power law
dependence, $\nu_c\propto t^{1/2}$.
From these expressions the flux can be computed for the different
possible radiative regimes
(Sari, Piran \& Narayan, 1998), the results being given in Appendix A.
Let us for example take the following values of the parameters:
$\Gamma_2=1$, $\epsilon_e=\epsilon_B=0.33$,
$\zeta=10^{-2}$, $p=2.5$, $A_*=0.5$ and
$t_{\rm dec}=100$ s. Then, the transition from fast to slow cooling occurs
at $t=1.3(1+z)$ day. Now adopting 1 keV and 2 eV
as typical energies for
the X-ray and visible bands (i.e. $\nu_X=2.4\,10^{17}$ Hz
and $\nu_V=4.8\,10^{14}$ Hz) and a redshift $z=2$, it appears that
after only a few seconds $\nu_X$ becomes larger than $\nu_m$ and
then remains larger than $\nu_c$ in the slow cooling regime.
The corresponding temporal slope is $\alpha_X=(2p+1)/4=1.5$.
At the visible frequency, we initially have $\nu_c<\nu_V<\nu_m$ and therefore
$\alpha_V=0.75$. The visible frequency crosses $\nu_m$ at $t=2.6$ h
(in the fast cooling regime) and then $\nu_c$ (in slow cooling) at very
late times. A break from $\alpha_V=0.75$ to 1.5 is expected at
$t=2.6$ h.
Since these predicted slopes are only valid in the asymptotic
regime where $t\gg t_{\rm dec}$ we have performed a numerical
simulation with the burst parameters given above except for the fraction
$\zeta$ of accelerated electrons which is varied between 0.003 and 0.03.
We assume in
addition that $E_{53}=1$ and adopt
a rest frame reddening $A_V=0.5$ in the burst host galaxy.
The resulting X-ray and visible light curves
are shown in Fig.4. For $\zeta=3\,10^{-3}$ and $10^{-2}$ they exhibit chromatic breaks.
The break in X-rays is a consequence of the dynamics of the reverse
shock (it is already present in the bolometric light curve) while the
break in the visible is a spectral break (when $\nu_V$ crosses $\nu_m$).
The cases with $\zeta=3\,10^{-3}$, $10^{-2}$ and $3\,10^{-2}$ are very
similar to the early afterglow light curves of respectively
GRB 050802, GRB 050922c and GRB 050801 (see Panaitescu et al, 2006
and Panaitescu, 2006).
The subsequent evolution of the afterglow
will depend on the behavior of $\epsilon_e$ and $\epsilon_B$ in the
forward shock. If they increase enough with time the forward shock
contribution will eventually dominate but the moment of the transition is difficult
to estimate in the absence of any reliable physical model for the possible
variations of the shock microphysics parameters. If the forward shock takes
over after about one day, the multiwavelength fits of GRB afterglows obtained
in the pre-SWIFT era will remain valid but the early afterglow
will be explained by the reverse shock. At the transition,
a change of slope or the presence of a bump may however be expected.
While such accidents have been observed in some bursts they do not seem to be
a generic feature of GRB afterglows.
A much more radical point of view can still be adopted: it would to
suppose that in some cases the forward shock never takes over so that the afterglow
is entirely produced by the reverse shock! The results shown in Fig.4
seem to indicate that this possibility should not be excluded a priori
even if, clearly, considerable work will be needed to confirm it.
As for the forward shock hypothesis it will have to be confronted to a large
amount of multiwavelength afterglow data and show that it can
provide a consistent picture for their interpretation.
\begin{figure*}{}
\begin{center}
\begin{tabular}{cc}
\resizebox{0.33\hsize}{!}{\includegraphics{ea1.eps}}&
\resizebox{0.33\hsize}{!}{\includegraphics{ea2.eps}}
\resizebox{0.33\hsize}{!}{\includegraphics{ea3.eps}}\\
\end{tabular}
\end{center}
\caption{Early afterglow light curves produced by the reverse shock for
$\epsilon_e=\epsilon_B=0.33$, $p=2.5$, $\Gamma_2=1$, $A_*=0.5$,
$t_{\rm dec}=100$ s, $A_V=0.5$ and, from left to right, $\zeta=3\,10^{-3}$,
$10^{-2}$ and $3\,10^{-2}$ (see text for details). The full (resp. dotted) line is the
X-ray (resp. the visible) afterglow. Compare these results to the
early afterglow light curves of respectively
GRB 050802, GRB 050922c and GRB 050801
as shown in Panaitescu (2006).}
\end{figure*}
\section{Conclusion}
We have developed a simplified model which enabled us to follow simultaneously the
dynamics of the internal, external and reverse shocks in GRBs. We were mainly interested
by dissipation in the reverse shock when the Lorentz factor in the material which
is ejected at late times by the source decreases to small values, $\Gamma_{\rm f} < 10$.
The propagation of the reverse shock then extends over quite a long time needed to
decelerate the fast moving part of the ejecta down to
$\Gamma\sim \Gamma_{\rm f}$. We have obtained
the dissipated power as a function of observed time for different
burst environments
(wind or constant density).
Its evolution shows a stri\-king resemblance
with the early afterglow light curves observed by SWIFT,
especially in the
wind case.
However the reverse shock contribution is normally expected
at low energy and to appear
in X-rays it requires a transfer of the dissipated power
to only a small fraction
($\zeta\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 10^{-2}$) of the electron population. If this is possible, SWIFT XRT
observations could be better explained by the reverse shock than by the standard afterglow
produced by the forward shock.
We have also computed the optical emission from the reverse
shock. The comparison with the X-ray light curve often reveals the
presence of chromatic breaks during the first hours. Such breaks
have been observed and are
difficult to explain with the standard model where the afterglow
comes from the forward shock.
We have finally even proposed that in some cases the entire afterglow
could be produced by the reverse shock. We fully understand that,
to be validated, this non standard view still has to show that it
can successfully explain multiwavelength observations of a reasonable
sample of GRB afterglows. We aim to perform these necessary tests in a work
in preparation.
|
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| 9,902
|
Scientists shed light on dehydrogenation
July 18, 2012 0 By Angie Bergenson
An understanding of dehydrogenation could pave the way for more efficient storage technologies
Scientists from the University of California have gained new insight on the kinectics of hydrogen release, also known as dehydrogenation. Dehydrogenation refers to the release of stored hydrogen molecules. The efficient release of hydrogen is a vital part of the effective storage of hydrogen fuel. For several years, problems with inefficient storage methods have held hydrogen fuel back. Researchers believe that a better understanding of dehydrogenation could unlock new technologies and methodologies to store hydrogen fuel more efficiently.
Computer simulation highlights the capabilities of aluminum hydride
Researchers from the university has created a computer simulation that models dehydrogenation from aluminum hydride, a material that has been gaining acclaim for its prospects in the realm of energy storage. The simulations mimic the basic mechanisms of the chemical processes that go into dehydrogenation. According to Professor Chris Van de Walle, the simulations show that aluminum hydride has the capability to quickly release hydrogen molecules quickly, but not too fast to cause any significant problems.
Hydrogen adoption could be hastened through more efficient storage
Hydrogen is becoming a more popular alternative to conventional fuels. The auto industry has, thus far, emerged as one of the largest advocates for hydrogen fuel in recent times. While other industries are beginning to warm to the notion of utilizing hydrogen as a source of energy, the issue of efficient storage has kept the adoption of the alternative energy relatively slow. If the issue of efficient storage can be rectified, hydrogen fuel could become one of the dominant forms of energy.
Further research of dehydrogenation and storage technologies requires funding
Researchers from the University of California believe that the insight provided through their computer simulations could further the development of efficient hydrogen fuel storage technologies. Time will tell whether this is the case, however, as the research and development necessary for new hydrogen storage systems is subject to adequate funding.
Related article(s) and resources:
http://www.sciencecodex.com/calculations_reveal_fine_line_for_hydrogen_release_from_storage_materials-95037
CategoryEnvironmental Science News Hydrogen Fuel Cell Hydrogen news Hydrogen Storage Renewable Energy News Research United States
Tagsaluminum hydride dehydrogenation efficient energy storage hydrogen fuel hydrogen fuel research hydrogen fuel storage professor chris van de walle University of California
Forze pits hydrogen-powered vehicle against fossil-fuels at Silverstone
Natural gas may be an ideal substitute for fossil-fuels
|
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| 872
|
{"url":"https:\/\/math.stackexchange.com\/questions\/2663453\/prove-that-every-symmetric-polynomial-can-be-written-in-terms-of-the-elementary","text":"Prove that every symmetric polynomial can be written in terms of the elementary symmetric polynomials\n\nHow do I prove that\n\nany symmetric polynomial P is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials.\n\nI have no clue of where to start, I just know the basic definition:\n\nThe polynomial $P(x_1,x_2,...,x_n)$ is symmetric if for any permutation $\\sigma$ of $\\{x_1,x_2,...,x_n\\}$, $$P(x_1,x_2,...,x_n)=P(x_{\\sigma_1},x_{\\sigma_2},...,x_{\\sigma_n})$$ The elementary symmetric polynomials for a polynomial consists of $n$ variables, $\\{x_1,x_2,...,x_n\\}$, and are defined as $\\{e_1,e_2,...,e_n\\}$: $$e_0=1\\\\ e_1=\\sum_{i}x_i\\\\ e_2=\\sum_{i,j}x_ix_j\\\\ e_3=\\sum_{i,j,k}x_ix_jx_k\\\\ ....................\\\\ e_n=x_1x_2...x_n$$\n\nCan I use induction, if possible where do I start ?","date":"2019-09-15 16:21:32","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9034357666969299, \"perplexity\": 112.83838618690727}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514571651.9\/warc\/CC-MAIN-20190915155225-20190915181225-00096.warc.gz\"}"}
| null | null |
Build Something
Bending the Sides
Steam bending is a very different procedure from other woodworking techniques I have used in the past, so it took some getting used to. The first side I bent took nearly an hour and a half. The second only took a half hour and turned out much better than the first side. The heat is the most important factor for a successful bend. If the iron is too hot, it will scorch the sides, turning them brown or even black (you can use a piece of paper to see if the iron is hot enough to scorch the wood.) If the iron is too cool, the wood will never bend effectively. The sides seemed to bend best when the pipe was at least hot enough to cause drops of water to instantly skip off the pipe on contact. As far as water goes, I kept the sides soaked through for this mandolin. On both sides I had some cracking on the outside edge of the larger bend (it was repairable, but still annoying.) Since I built this mandolin, I have changed my technique a bit. I try to use as little water as possible, wiping or drizzling on water with a rag in order to only wet the exterior on the side that will contact the iron. I also used thinner sides (around 0.1 inches or less compared with 0.125 inches thick on this mandolin.) This resulted in essentially no cracking on my later mandolins.
Planing the Sides
These are the planed sides ready for bending. I do not have access to a thickness sander so I used a planer to bring the rough cut sides to thickness. The sled in the second picture helps keep the very thin sides flat while the pass through the planer. I tried to cut out the sides as close to quartersawn as possible from the maple block. As far as width goes, they are about 1.75 inches across. Since the body is supossed to be about 1.625 inches tall, these sides just made the needed width considering the compound curves the side must make. I would go with at least 2 inches on my next attempt at this design so there is plenty of width for the angle bends of the sides.
Bending Iron Setup
My bending setup consists of a iron pipe supported by two pieces of angle iron and tied down to a 2x3 with pipe clamps. The propane tank is supported by a loose clamp. This setup worked quite well for my purposes. It is quick to heat up and holds a fairly steady temperature without too much fuss.
To bend the sides, I first get the section I want to work with wet. I then rock the section back and forth on the iron until it gets very hot through to the opposite side of the wood. At that point the wood becomes flexible and can bend surprisingly easily with gentle even presure. After a few seconds on the heat, the bottom side of the wood dries out, so I rewet the wood then quickly reapply heat to keep the wood bendable.
When the side was close enough to fit on the form, I used rubberbands to secure them and let them dry and cool. When they are cool they retain there shape surprisingly well. At this point, I also glued down and sanded the portions of the side which split. With some sanding, the sides on the finished project don't show the splits at all.
Next... make a neck.
Setup/ Fretting
Have a question? Want to make a comment?
Let everyone know what you think on this project's forum topic.
Mandolin No. 002
This project is a sequel to my first mandolin. Much like a movie sequel, I won�t need to use as much time on character development...
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 316
|
AnydayGuide
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Unofficial Holidays → USA
National Oatmeal Nut Waffles Day
March 11 is National Oatmeal Nut Waffles Day. It's hard to imagine heartier or more specific breakfast, than the oatmeal nut waffles. Even if you never tried these waffles, we're sure you will love them.
Festivals on March 11, 2019
Baklahorani Carnival in Istanbul, Turkey
Carnival of Basel in Basel, Switzerland
UK Asian Film Festival in London, United Kingdom
Canberra Balloon Spectacular in Canberra, Australia
WOMAD in Adelaide, Australia
South by Southwest (SXSW) in Austin, USA
Guadalajara International Film Festival in Guadalajara, Mexico
Geneva International Motor Show in Geneva, Switzerland
Sofia International Film Festival in Sofia, Bulgaria
Cartagena Film Festival in Cartagena, Colombia
Enlighten Canberra in Canberra, Australia
Emirates Airline Festival of Literature in Dubai, United Arab Emirates
New Zealand Fringe Festival in Wellington, New Zealand
Adelaide Festival in Adelaide, Australia
New York International Children's Film Festival in New York, USA
← More Festivals
2012 Died: Ian Turpie, Australian actor, best known for hosting the Australia version of game show The New Prince is Right.
2011 An earthquake measuring 9.0 in magnitude stroke east of Sendai, Japan. The earthquake triggered a tsunami, that killed thousands of people. This event also caused the large nuclear accident at the Fukushima I Nuclear Power Plant.
2009 16 students were killed and 11 injured at a secondary school in Winnenden, Germany, by a recent-graduated Tim Kretschmer, who later shot himself.
2006 Died: Slobodan Milošević, Serbian politician, the 3rd President of the Federal Republic of Yugoslavia.
2004 191 were killed and 1,800 injured when trains simultaneously exploded on rush hour in Madrid, Spain.
2002 Died: James Tobin, American economist, teacher at Harvard and Yale Universities and Nobel Memorial Prize in Economics laureate in 1981. Tobin pioneered the study of investment, monetary and fiscal policy, financial markets and proposed an economic model for censored endogenous variables.
1992 Died: Richard Brooks, American director, producer, and screenwriter, best known for outstanding works Blackboard Jungle, Can on a Hot Tin Roof, Elmer Gantry, In Cold Blood and Looking for Mr. Goodbar.
1978 At least 37 were killed and more than 70 were wounded when an Israel bus was hijacked by Al Fatah, a leading secular Palestinian political party. This incident prompted Israel's Operation Litani, when Israel Defense Forces invaded Lebanon up to the Litani River.
1970 Died: Erle Stanley Gardner, American author, best known for the detective stories Perry Mason. He is considered as one of the best-selling American authors of the 20th century at the time of death.
1967 Born: John Barrowman, Scottish-American actor, singer, and dancer, best known for the role as Captain Jack Harkness in Doctor Who and Torchwood.
1963 Born: Alex Kingston, English actress, best known for her roles as Dr. Elizabeth Corday in medical drama ER and as River Song in Doctor Who.
1958 Died: Ole Kirk Christiansen, Danish businessman, founder of The Lego Group, Lego toys manufacturer.
1957 Died: Richard E. Byrd, American admiral and explorer. Byrd claimed hat his expeditions were the first to reach the North Pole and the South Pole by air.
1955 Died: Alexander Fleming, Scottish biologist, pharmacologist, and botanist, Nobel Prize laureate for discovery of penicillin. Fleming also discovered the lysozyme, the enzyme that damages bacterial cell walls.
1952 Born: Douglas Adams, English-American author and playwright best known for The Hitchhiker's Guide to the Galaxy, that became the basis for TV series, stage plays, comics, computer game a a feature film.
1941 President Franklin D. Roosevelt signed the Lend-Lease Act, that allowed American-built war supplies to be shipped to the Allies on loan.
1937 Died: Joseph S. Cullinan, American businessman, co-founder of Texaco, the largest oil retail brand, known in the USA, Latin America, West Africa and the UK.
1927 Roxy Theater was opened in New York City by Samuel Roxy Rothafel.
1921 Born: Astor Piazzolla, Argentinian bandoneón player and tango composer. His oeuvre revolutionized the traditional tango into a new style nuevo tango, that incorporated elements of jazz and classic music.
1921 Born: Frank Harary, American mathematician, who specialized in graph theory and standardized the terminology of graphs.
1918 The first case of Spanish flue occurred, that marked the beginning of a devastating worldwide pandemic.
1916 Born: Ezra Jack Keats, American author and illustrator. The Snowy Day, that was written and illustrated by Keats, is considered as one of the most important American books of the 20th century. Keats is also known for introduction of multiculturalism into mainstream American children's literature and development of the use of collage as a medium for illustration.
1915 Died: Thomas Alexander Browne, Australian author. Browne is best known for this novel Robbery Under Arms, that was published under a pen name Rolf Boldrewood.
1907 Born: Jessie Matthews, English actress, singer, and dancer. Matthews was much-loved personality of the British theater and film audience for her warbling voice an round cheeks, that is why film exhibitors voted her among the top ten stars in Britain.
1888 More then 400 people were killed during the Great Blizzard, that hit the eastern seaboard of the United States.
1873 Born: David Horsley, English-American director and producer, co-founder of Universal Studios in Hollywood.
1864 238 people were killed during the Great Sheffield Flood in Sheffield, England.
1818 Born: Marius Petipa, French-Russian dancer and choreographer. Today Petipa is considered as the most influential ballet master and choreographer of ballet.
1811 Born: Urbain Le Verrier, French mathematician and astronomer, best known for predicting the existence of Neptune using only mathematics.
1784 The Treaty of Mangalore was signed, that brought the Second Anglo-Mysore War (1780-1784) to an end.
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{"url":"http:\/\/cantorsattic.info\/index.php?title=Feferman-Sch%C3%BCtte&diff=2575","text":"# Difference between revisions of \"Feferman-Sch\u00fctte\"\n\nThe Feferman-Sch\u00fctte ordinal, denoted $\\Gamma_0$ (\"gamma naught\"), is the first ordinal fixed point of the Veblen function. It figures prominently in the ordinal-analysis of the proof-theoretic strength of several mathematical theories.\n\n## Veblen hierarchy\n\nEvery increasing continuous ordinal function $f$ has an unbounded set of fixed points;\n\nSince the set of fixed points is an unbounded, well-ordered set, there is an ordinal function $\\varphi^{[f]}$ listing these fixedpoints; it is in turn increasing and continuous. The Veblen Hierarchy is the sequence of functions $\\varphi_\\alpha$ defined by\n\n\u2022 $\\varphi_0 x = \\omega^x$\n\u2022 $\\varphi_{\\alpha + 1} = \\varphi^{[\\varphi_\\alpha]}$\n\u2022 for $0 \\lt \\beta = \\cup \\beta$, $\\varphi_\\beta(x)$ enumerates the fixedpoints common to all $\\varphi_\\alpha$ for $\\alpha \\lt \\beta$\n\n(For $\\alpha \\lt \\beta$, the fixed point sets of $\\varphi_\\alpha$ are all closed sets, and so their intersection is closed; it is unbounded because $\\cup_\\alpha \\varphi_\\alpha(t+1)$ is a common fixed point greater than $t$)\n\nIn particular the function $$\\varphi_1$$ enumerates epsilon numbers i.e. $$\\varphi_1(\\alpha)=\\varepsilon_\\alpha$$\n\nThe Veblen functions have the following properties:\n\n\u2022 if $$\\beta<\\gamma$$ then $$\\varphi_\\alpha(\\beta)<\\varphi_\\alpha(\\gamma)$$\n\u2022 if $$\\alpha<\\beta$$ then $$\\varphi_\\alpha(0)<\\varphi_\\beta(0)$$\n\u2022 if $$\\alpha>\\gamma$$ then $$\\varphi_\\alpha(\\beta)=\\varphi_\\gamma(\\varphi_\\alpha(\\beta))$$\n\u2022 $$\\varphi_\\alpha(\\beta)$$ is an additive principal number.\n\nAn ordinal $$\\alpha$$ is an additive principal number if $$\\alpha>0$$ and $$\\alpha>\\delta+\\eta$$ for all $$\\delta, \\eta<\\alpha$$. Let $$P$$ denote the set of all additive principal numbers.\n\nWe define the normal form for ordinals $$\\alpha$$ such that $$0<\\alpha<\\Gamma_0=\\min\\{\\beta|\\varphi(\\beta,0)=\\beta\\}$$\n\n\u2022 $$\\alpha=_{NF}\\varphi_\\beta(\\gamma)$$ if and only if $$\\alpha=\\varphi_\\beta(\\gamma)$$ and $$\\beta,\\gamma<\\alpha$$\n\u2022 $$\\alpha=_{NF}\\alpha_1+\\alpha_2+\\cdots+\\alpha_n$$ if and only if $$\\alpha=\\alpha_1+\\alpha_2+\\cdots+\\alpha_n$$ and $$\\alpha>\\alpha_1\\geq\\alpha_2\\geq\\cdots\\geq\\alpha_n$$ and $$\\alpha_1,\\alpha_2,...,\\alpha_n\\in P$$\n\nLet $$T$$ denote the set of all ordinals which can be generated from the ordinal number 0 using the Veblen functions and the operation of addition\n\n\u2022 $$0 \\in T$$\n\u2022 if $$\\alpha=_{NF}\\varphi_\\beta(\\gamma)$$ and $$\\beta,\\gamma \\in T$$ then $$\\alpha\\in T$$\n\u2022 if $$\\alpha=_{NF}\\alpha_1+\\alpha_2+\\cdots+\\alpha_n$$ and $$\\alpha_1,\\alpha_2,...,\\alpha_n\\in T$$ then $$\\alpha\\in T$$\n\nFor each limit ordinal number $$\\alpha\\in T$$ we assign a fundamental sequence i.e. a strictly increasing sequence $$(\\alpha[n])_{n<\\omega}$$ such that the limit of the sequence is the ordinal number $$\\alpha$$\n\n\u2022 if $$\\alpha=\\alpha_1+\\alpha_2+\\cdots+\\alpha_k$$ then $$\\alpha[n]=\\alpha_1+\\alpha_2+\\cdots+(\\alpha_k[n])$$\n\u2022 if $$\\alpha=\\varphi_0(\\beta+1)$$ then $$\\alpha[n]=\\varphi_0(\\beta)\\times n$$\n\u2022 if $$\\alpha=\\varphi_{\\beta+1}(0)$$ then $$\\alpha[0]=0$$ and $$\\alpha[n+1]=\\varphi_\\beta(\\alpha[n])$$\n\u2022 if $$\\alpha=\\varphi_{\\beta+1}(\\gamma+1)$$ then $$\\alpha[0]=\\varphi_{\\beta+1}(\\gamma)+1$$ and $$\\alpha[n+1]=\\varphi_\\beta(\\alpha[n])$$\n\u2022 if $$\\alpha=\\varphi_{\\beta}(\\gamma)$$ and $$\\gamma$$ is a limit ordinal then $$\\alpha[n]=\\varphi_{\\beta}(\\gamma[n])$$\n\u2022 if $$\\alpha=\\varphi_{\\beta}(0)$$ and $$\\beta$$ is a limit ordinal then $$\\alpha[n]=\\varphi_{\\beta[n]}(0)$$\n\u2022 if $$\\alpha=\\varphi_{\\beta}(\\gamma+1)$$ and $$\\beta$$ is a limit ordinal then $$\\alpha[n]=\\varphi_{\\beta[n]}(\\varphi_{\\beta}(\\gamma)+1)$$\n\nThe Feferman-Sch\u00fctte ordinal, $$\\Gamma_0$$ is the least ordinal not in $$T$$.\n\n## Gamma function\n\nThe Gamma function is a function enumerating ordinal numbers $$\\alpha$$ such that $$\\varphi(\\alpha,0)=\\alpha$$\n\n\u2022 if $$\\alpha=\\Gamma_0$$ then $$\\alpha[0]=0$$ and $$\\alpha[n+1]=\\varphi(\\alpha[n],0)$$\n\u2022 if $$\\alpha=\\Gamma_{\\beta+1}$$ then $$\\alpha[0]=\\Gamma_{\\beta}+1$$ and $$\\alpha[n+1]=\\varphi(\\alpha[n],0)$$\n\u2022 if $$\\alpha=\\Gamma_{\\beta}$$ and $$\\beta$$ is a limit ordinal then $$\\alpha[n]=\\Gamma_{\\beta[n]}$$\n\n## References\n\nOswald Veblen. Continuous Increasing Functions of Finite and Transfinite Ordinals. Transactions of the American Mathematical Society (1908) Vol. 9, pp.280\u2013292","date":"2019-09-17 19:25:20","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\": 0.9913080930709839, \"perplexity\": 266.91324908748277}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514573105.1\/warc\/CC-MAIN-20190917181046-20190917203046-00251.warc.gz\"}"}
| null | null |
Good afternoon, and welcome to our latest Axios Deep Dive on the coronavirus pandemic.
To get all the most important health care news delivered straight to your inbox every morning, be sure to sign up for Vitals, our daily health care newsletter.
Today's Smart Brevity count: 1,531 words, 5.5 minutes.
1 big thing: The fog of coronavirus
We're fighting the greatest public health crisis in a century, and we barely understand our enemy, Axios' Felix Salmon writes.
We cannot afford to stay in lockdown until a cure or vaccine arrives — but anybody trying to reopen our cities needs information that is frustratingly difficult to find.
The big picture: The single biggest obstacle to reopening the economy is a lack of visibility — we don't know the scope of the pandemic itself, or its economic fallout, or how its trajectory will change as we embark upon an ad hoc effort to reopen the economy.
We don't know how many people the coronavirus has killed, or how many people have had it.
The official tally of over 37,000 deaths is too low, because it's based on people who died after testing positive for the coronavirus, but we don't know how low.
We don't know how many Americans have lost their jobs because of the coronavirus shutdown.
The official tally is 22 million new applications for unemployment benefits. But millions more haven't been able to make it through the application process, or haven't tried.
The biggest employers will be able to rehire their legions of workers, but the bigger concern is the businesses that will never be able to reopen. We don't know how many of them there will be.
We don't know when we'll have a treatment, whether summer will tame the spread, or whether the virus could return in the fall even stronger. We don't know whether we're immune once we've had it, or for how long.
We don't know whether tech will allow us to trace it, or whether enough Americans would sign up for that, even if it does.
We don't know when it'll be safe to fly, go to a game, or pack into a school or a church.
The bottom line: It's shocking and a bit scary how much we do not know, despite how much we now do know.
2. Reopening may pit governors against mayors
Governors and mayors are walking a political tightrope as they decide how quickly to reopen their economies, Axios' Alayna Treene and Stef Kight report.
The fear: Opening up too fast would only require a fresh set of lockdowns later.
"That would be like releasing someone from prison, and then saying, 'Sorry, you need to come back to serve more time,'" Paul TenHaken, the Republican mayor of Sioux Falls, South Dakota, tells Axios.
Yes, but: Republican governors in particular will likely face considerable political pressure to open up as quickly as possible — which could put them at odds with mayors in their state.
TenHaken issued a "Safer at Home" proclamation for his city, which has a high share of the state's rising number of COVID-19 cases. But South Dakota Gov. Kristi Noem has refused to enact a statewide order.
Between the lines: Although President Trump's formal guidelines leave most of these decisions up to local officials, he has also tweeted criticisms of Democratic governors' continuation of stay-at-home orders.
Protests against the restrictions are also cropping up.
"I'm very concerned about any type of pressure — political or otherwise — to make hasty decisions based on the economy as opposed to the science," Tampa Mayor Jane Castor tells Axios.
She pushed for a shelter-in-place order for her city long before Florida Gov. Ron DeSantis issued one statewide, and DeSantis has already loosened some of those restrictions in the days since the White House released its reopening plan.
3. It all comes back to testing
Detroit Mayor Mike Duggan announced yesterday that the city will start making diagnostic tests available to all employees of "essential" businesses, and to city employees performing essential services.
It's the first program of its kind, and a model for other cities to follow. And it's a reminder that testing remains the limiting factor in every facet of our response to the coronavirus, Axios' Sam Baker writes.
You may be tired of hearing about testing. You may think it has become an almost myopic focus in a multifaceted crisis. But there is simply no way to work around it, or to put that problem to the side and focus on something else. All roads lead back to testing.
Between the lines: Detroit's plan to test people who aren't experiencing symptoms, even if it's just a small group, is critical.
Many parts of the country, though, simply don't have the capacity to test anyone who isn't feeling sick. That will never be good enough.
Although more rapid tests are becoming available, turnaround times of up to a week increase the risk that people will spread the virus without knowing it.
If you test someone too soon after they've contracted COVID-19, they won't have enough of the virus in their system, resulting in a false negative — which means some infected workers may slip through the cracks even with widespread testing.
Our thought bubble: Every incremental step that brings more people out into the world creates some level of additional risk. We don't have a vaccine and we can't stay inside forever, so all we can do is try to find a level of risk that's manageable.
The only way to manage that risk is to stay on top of the virus' spread, and the only way to do that is with testing. And so the limitations in testing will always, necessarily, restrict everything else we try to do.
4. Isolation can be bad for mental health
Every treatment has some side effects — including social distancing.
Prolonged isolation, coupled with sustained job losses, could increase the risk of depression, as people are removed from the community support system that keeps them going, Sam writes.
That could in turn increase the risk of suicide, as a recent opinion piece in JAMA Psychiatry explains.
"A grim tradeoff is already being made between saving different lives: Saving the lives of those who are most vulnerable to COVID-19 versus saving the lives of those who are most vulnerable to suicide, substance abuse, and domestic violence," experts at Johns Hopkins write.
What to watch: Virtual check-ins are all the more important here — from simply keeping in touch with friends and family, to online support groups, to telehealth visits with mental health professionals.
None of this means social distancing was a bad idea, or that it needs to end quickly — it's definitely less deadly than the coronavirus. But even the most necessary interventions still need some careful management.
The National Suicide Prevention Lifeline (1-800-273-8255) provides 24/7, free and confidential support for anyone in distress, in addition to prevention and crisis resources. Also available for online chat.
Photo: Karen Ducey/Getty Images
Lab scientist Alicia Bui tests the blood of patients who have recovered from the coronavirus, looking for antibodies that prove exposure and may confer immunity to the virus.
These serology tests will likely be an important part of getting high-risk employees, especially health care workers, safely back to work.
6. Q&A: masks, lending books, self-isolating
Axios' Rashaan Ayesh is answering readers' questions about the coronavirus. Drop us a line at [email protected].
Q: What are the best practices for reusing face masks?
The FDA recommends using cloth masks, not N95s, and the CDC advises laundering them in a washing machine with detergent.
The University of Utah says cloth masks should be washed in 160°F water with soap or detergent, and recommends soaking them for five minutes in a bleach solution.
Q: What is the best way to keep up with appointments like physical therapy?
The American Physical Therapy Association recommends rescheduling non-urgent in-person care. Some specialists are using telehealth and videoconferencing to work with patients.
Q: Can I offer or accept home-cooked foods or share books?
Consult the CDC's social distancing guidelines in deciding how best to be a good neighbor.
Apps like NextDoor form digital groups that can connect you with neighbors.
Q: How long should I self-isolate after testing positive?
As always, ask your doctor. The CDC says people who've tested positive need to isolate at home for at least seven days after symptoms first appeared, wait at least 72 hours after fever is gone, and hold off until respiratory symptoms have improved.
Q: Can I microwave or freeze my food to kill the virus on the packages?
The FDA says there's no evidence of food packaging or food being associated with coronavirus transmission. There also isn't "direct data for a temperature-based cutoff for inactivation" for the virus, per the CDC.
7. 1 helpful thing: Art and culture come to you
Many renowned educational and cultural institutions are meeting Americans where they are — at home — with a slew of new intellectual pursuits even in a time of lockdowns and social distancing.
What's happening: Whether you're looking for an intensive 40-hour course or a short video on art history for your lunch break, you may have more options than ever before, Axios' Naomi Shavin writes.
Ivy League schools are offering a combined 450 free online courses, and many other universities around the world are also putting popular courses online for free. Some 1.5 million people have already registered for Yale's famous happiness course.
New York's Museum of Modern Art is offering free classes that include "What is contemporary art?" and "Fashion as design."
The Barnes in Philadelphia borrowed a page from restaurants' playbook to launch "Barnes Takeout," a series of short online lectures that focus on a single work of art.
From symphony halls to Broadway theaters, performance venues are streaming past performances and informational talks.
If the life of the mind gets to be too much, high-end gyms and exercise programs have made headlines for offering free online videos, too.
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\section{#1}}
\textwidth 160mm \textheight 220mm
\usepackage{amssymb,latexsym}
\begin{document}
\topmargin -10mm \topmargin 0pt \oddsidemargin 0mm
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\newcommand{\nonumber\\}{\nonumber\\}
\begin{titlepage}
\vspace*{10mm}
\begin{center}
{\Large \bf Viscous Cosmology and Thermodynamics of Apparent
Horizon}
\vspace*{20mm}
{\large M. Akbar~\footnote{Email address: ak64bar@yahoo.com}
\footnote{Email address: makbar@camp.edu.pk}}\\
\vspace{8mm} { \em Centre for Advanced Mathematics and Physics\\
National University of Sciences and Technology\\
Peshawar Road, Rawalpindi, Pakistan}
\end{center}
\vspace{20mm}
\centerline{{\bf{Abstract}}}
\vspace{5mm}
It is shown that the differential form of Friedmann equations of FRW
universe can be recast as a similar form of the first law
,$T_{A}dS_{A} = dE + WdV$, of thermodynamics at the apparent horizon
of FRW universe filled with the viscous fluid. It is also shown that
the generalized second law of thermodynamics holds at the apparent
horizon of FRW universe and preserves dominant energy condition.
PACS numbers: 04.70.Dy, 97.60.Lf
\end{titlepage}
\newpage
\renewcommand{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0} \setcounter{page}{2}
\paragraph{Introduction:}
In the cosmological setting, one can associate Hawking temperature
and entropy with the apparent horizon analogous to the hawking
temperature and entropy associated with the black hole horizon
\cite{a9,a2,a15}. In the case of de Sitter space, the event horizon
matches with the apparent horizon of FRW universe with $k = 0$
however for more general cosmological models, the event horizon may
not exist but the apparent horizon associated with the Hawking
temperature and entropy always exists. The thermodynamic properties
associated with the apparent horizon of FRW universe filled with the
perfect fluid has been studied by many authors (see for examples
\cite{a9, a2,jac, cai, hct, btz,a10}). The extension of this
connection between thermodynamics and gravity has also been carried
out in the braneworld cosmology \cite{b1,b2,b3,b4}. For a general
static spherically symmetric and stationary axisymmetric spacetimes,
It was shown that the Einstein field equations can be rewritten
\cite{ksp,Pad,PSP} as a first law of thermodynamics. More recently,
by considering a masslike function, it has been shown \cite{b5} that
the equilibrium thermodynamics should exist in the extended theories
of gravity and the Friedman equations in various theories of gravity
can be rewritten as a first law $TdS = dE$ at the apparent horizon
of FRW universe. More recently, Cai et al \cite{hct} has shown that
by employing Clausius relation, $\delta Q = TdS$, to the apparent
horizon of a FRW universe, they are able to derive the modified
Friedmann equation by using quantum corrected entropy-area
relation.\\
The cosmological models with perfect cosmic fluid has been studied
widely in literature however viscous cosmic fluid came much later in
the study of the universe \cite{min}. It has been revealed by Barrow
\cite{Barr}(also see \cite{Gio}) that the thermodynamic entropy
associated with bulk viscosity violates the dominant energy
condition and can decrease very well. This fact was resumed
\cite{Zim} in the framework of the solutions given in reference
\cite{Gio}. By taking into account the entropy of the sources only,
it has been deduced that the entropy of the bulk viscous sources
cannot decrease since it would violate the second law of
thermodynamics for large cosmic time values. However, in order to
state the generalized second law, one has to include both the
entropy connected with the sources and the entropy associated with
background geometry. The most of the study of the second law in the
cosmological context has been carried out at the event horizon (see
for examples \cite{Gio,Zim}) however the validity of second law
within the apparent horizon of FRW universe is of great important to
investigate. In this letter we will also investigate this issue. The
cosmological models with bulk viscosity have been studied in the
literature from various point of view (see for examples
\cite{Gio,bre,zimd}). \\The purpose of this letter is twofold. The
first is to show that the Friedmann equations of FRW universe
pertaining cosmic bulk viscosity can be rewritten as a similar form
of the first law of thermodynamics. The other is to discuss the
generalized second law within the apparent horizon of FRW
universe.\\
Let us start with a FRW universe of metric
\begin{equation}\label{1}
ds^{2}= -dt^{2}+ a^{2}(t)(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}),
\end{equation}
where $d\Omega^{2} = d\theta^{2} + sin^{2}\theta d\phi^{2}$ stands
for the line element of 2-dimensional unit sphere and the spatial
curvature constant $k = +1$, 0 and $-1$ represents a closed, flat
and open universe, respectively. The above metric (1) can be
rewritten in spherical form
\begin{equation}\label{2}
ds^{2}= h_{ab}dx^{a}dx^{b}+ \tilde{r}^{2}d\Omega^{2},
\end{equation}
where $\tilde{r} = a(t)r$, $x^{0} = t$, $x^{1}= r$ and $h_{ab} =
diag(-1, \frac{a^{2}}{1-kr^{2}})$. One can work out the dynamical
apparent horizon from the relation
$h^{ab}\partial_{a}\tilde{r}\partial_{b}\tilde{r} = 0$ which turns
out $\frac{1}{\tilde{r}^{2}_{A}} = H^{2}+k/a^{2}$, where $H \equiv
\frac{\dot{a}}{a}$ is the Hubble parameter and the dots denote
derivatives with respect to the cosmic time. let the universe be
filled with a viscous fluid and $u^{\mu}=(u^{0}, u^{i})$ is the
four-velocity of the fluid. In comoving coordinates $u^{0}=1$ and
$u^{i}=0$. Define $h_{\mu\nu} = g_{\mu\nu}+ u_{\mu}u_{\nu}$,
$w_{\mu\nu}=h^{\alpha}_{\mu}h^{\beta}_{\nu}u_{[\alpha;\beta]}$ and
$\theta_{\mu\nu}=h^{\alpha}_{\mu}h^{\beta}_{\nu}u_{(\alpha;\beta)}$
as a projection tensor, rotation tensor and expansion tensor,
respectively. The scalar expansion is $\theta \equiv \theta
^{\mu}_{\mu}= u^{\mu}_{;\mu}$ and
$\sigma_{\mu\nu}=\theta_{\mu\nu}-\frac{1}{3}h_{\mu\nu}\theta$ is the
shear tensor. Taking into consideration of metric (1),
$u^{\mu}_{;\nu} = h^{\mu}_{\nu}\frac{\dot{a}}{a}$ which implies that
the rotation and shear tensors vanish, i.e. $w_{\mu\nu} =
\sigma_{\mu\nu} = 0$. While the scalar expansion is $\theta =
3\frac{\dot{a}}{a} = 3H$. Hence the energy-momentum tensor of
viscous fluid in the background of metric (1) finally becomes
\cite{bre}
\begin{equation}\label{5}
T_{\mu\nu}=(\rho +P-\zeta \theta)u_{\mu}u_{\nu}+(P-\zeta
\theta)g_{\mu\nu},
\end{equation}
where $\rho$, $P$ and $\zeta$ are the energy density, thermodynamic
pressure and bulk viscosity respectively. The components of
energy-momentum tensor are $T_{00} = \rho $, $T_{0i}=0$ and $T_{ij}
= (P-\zeta \theta)g_{ij}$. Therefore, the total effect of the bulk
viscosity is to reduce the pressure $P$ of the perfect fluid by an
amount $\zeta \theta$, so that the effective pressure of the viscous
fluid turns out to be $\tilde{P} = P -\zeta \theta$. The energy
conservation $T^{\mu\nu}_{;\nu} = 0$, yields $\dot{\rho} + 3H(\rho +
\tilde{P}) = 0$. Now we turn to discuss the thermodynamic
interpretation of Friedman equations at the apparent horizon of FWR
universe. For this purpose, we first define the surface gravity
$\kappa =
\frac{1}{2\sqrt{-h}}\partial_{a}(\sqrt{-h}h^{ab}\partial_{b}\tilde{r})$
at the apparent of FRW universe. By utilizing above relation, one
can easily find that the surface gravity at the apparent horizon of
FRW universe yields
\begin{equation}
\kappa =
\frac{-1}{\tilde{r}_{A}}(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}).
\end{equation}
When $\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\leq 1$ implies the
surface gravity $\kappa \leq 0$ which leads to the temperature $T =
\kappa / 2\pi \leq 0 $. However, in reference \cite{a9}, an
approximation, $\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\ll 1$,
has been used while determining horizon temperature. They found $T =
\frac{1}{2\pi \tilde{r}_{A}}$ by approximating surface gravity
$|\kappa| = \frac{1}{\tilde{r}_{A}}$. Now we turn to define energy
of the universe within the apparent horizon. We take total matter
energy $E = V\rho$ inside a sphere of radius $\tilde{r}$ which is
also the Miser-Sharp energy \cite{sharp}, $E = \frac{\tilde{r}}{2
}(1-h^{ab}\partial_{a}\tilde{r}\partial_{b}\tilde{r})$, within the
apparent horizon. Since at the apparent horizon
$(h^{ab}\partial_{a}\tilde{r}\partial_{b}\tilde{r} = 0)$, so the
Mizner-Sharp energy is in fact total matter energy inside the sphere
of radius $\tilde{r}_{A}$ and is given by
\begin{equation}
E = V\rho.
\end{equation}
The entropy $S = A /4$ is the horizon entropy, where $A = 4\pi
\tilde{r}_{A}^{2}$ is the horizon area. Note that we use the units
$\hbar = c = G = \kappa_{B} = 1$. From the Einstein field equations
$G_{\mu\nu} = 8\pi T_{\mu\nu}$, the Friedman equations for the
viscous fluid of stress-energy tensor (3) can be written as
\begin{equation}
H^{2}+ \frac{k}{a^{2}} = \frac{8\pi }{3}\rho,
\end{equation}
\begin{equation}
\dot{H}-\frac{k}{a^{2}} = -4\pi (\rho + \tilde{P}).
\end{equation}
In terms of the apparent horizon, the Friedman equation (6) can be
rewritten as
\begin{equation}
\frac{1}{\tilde{r}^{2}_{A}} = \frac{8\pi }{3}\rho.
\end{equation}
Let the apparent horizon surface acts as the boundary of the thermal
system of the FRW universe. In general, the apparent horizon is not
constant but changes with time. Let $d\tilde{r}_{A}$ be an
infinitesimal change in radius of the apparent horizon in time
interval dt. This change in the apparent horizon will cause a small
change $dV$ in the volume $V$ of the universe. This constitutes two
spherical system of space-time with radii $\tilde{r}_{A}$ and
$\tilde{r}_{A} + d\tilde{r}_{A}$ with a common source of fluid with
non-zero pressure and energy density near horizon. Each space-time
constituting a thermal system and satisfying Einstein's equations,
differs infinitesimally in the extensive variables volume, energy
and entropy by $dV$, $dE$ and $dS$, respectively, while having the
same values of intensive variables temperature $T$ and pressure
$\bar{P}$. Thus for these two thermal states of the space-time,
there must exist a certain relation connecting these thermodynamic
quantities. To establish connection among the thermal quantities and
Friedman equations, we differentiate equation (8) which implies
\begin{equation}
\frac{1}{\tilde{r}_{A}^{3}}d\tilde{r}_{A} = 4 \pi (\rho +
\tilde{P})Hdt
\end{equation}
Multiplying both sides of this equation with a factor $
\tilde{r}_{A}^{3}(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}})$,
one can rewrite this equation in a form
\begin{equation}
\frac{\kappa}{2\pi}\frac {d(4\pi \tilde{r}_{A}^{2})}{4} = - 4\pi
\tilde{r}_{A}^{2}(\rho + P)H(1 -
\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}})dt
\end{equation}
One can recognize that the quantities $\frac{\kappa}{2\pi}$ and
$\frac{4\pi \tilde{r}_{A}^{2}}{4}$ are the temperature $T$ and
entropy $S$ respectively. Therefore the above equation can be
rewritten as
\begin{equation}
T dS = - 4\pi \tilde{r}_{A}^{3}(\rho + \tilde{P})H(1 -
\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}})dt
\end{equation}
Now we turn to the total matter energy (5) and taking differential
of it, one gets
\begin{equation}
dE = 4\pi \tilde{r}_{A}^{2}\rho d\tilde{r}_{A} - 4\pi
\tilde{r}_{A}^{3}(\rho + \tilde{P})Hdt.
\end{equation}
Using equations (11) and (12) one yields
\begin{equation}
dE = TdS + WdV,
\end{equation}
where $W = \frac{1}{2}(\rho - \tilde{P})$ is the work density. The
above equation is the unified first law \cite{Hay} of relativistic
thermodynamics. Note that the above thermal identity is obtained
from the Friedman equation with viscous fluid together with the
characteristics of the apparent horizon, while the author of
reference \cite{Hay} studied the thermodynamics of trapping horizon
of dynamical black hole. It has been found that the entropy
associated with apparent horizon is proportional to the horizon area
which is originally initiated from the black hole horizon entropy
that satisfies the so-called area formula \cite{wald}. In the case
of perfect fluid, $\zeta = 0$, one can compare the above thermal
identity with the standard form of the first law, $dE = TdS -PdV$,
of thermodynamics. In fact, the negative pressure term $-P$ in the
first law is taken the place of the work density $W$. Notice that
for pure de Sitter spacetime, $\rho = -P$, then one acquires the
standard form $dE = TdS-PdV$. Note that we are considering the
universe as a thermal object with apparent horizon as its boundary.
So it useful to define other thermodynamic quantities like Enthalpy
$H$ and Gibbs free energy $G$. The Enthalpy is defined by the
equation $H = U + PV$, where $U = E$ is the total internal energy
which is taken to be the total matter energy of the universe bounded
by the apparent horizon. So the Enthalpy of the universe bounded by
the apparent horizon can be written in terms of apparent radius $H =
\frac{4\pi}{3}(\rho + P)\tilde{r}_{A}^{3}$. So the heat capacity of
the universe enveloped by the apparent horizon at constant pressure
of the perfect is defined as
\begin{equation}
C_{P} = (\frac{\partial H}{\partial T})_{P} = (\frac{\partial
H}{\partial \tilde{r}_{A}}\frac{\partial \tilde{r}_{A}}{\partial
T})_{P},
\end{equation}
which yields $C_{P} = 4\pi \tilde{r}_{A}^{2}(\rho + P)
(\frac{\partial \tilde{r}_{A}}{\partial T})_{P}$. In general the
temperature, $T = \kappa / 2\pi \equiv \frac{-1}{2\pi
\tilde{r}_{A}}(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}})$.
However, if one considers the approximation,
$\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}} \ll 1$ (also see
\cite{a9, b1}), the temperature $T = \mid\kappa\mid / 2\pi \equiv
1/2\pi \tilde{r}_{A}$ which implies that the heat capacity of the
universe is negative provided the dominant energy condition holds.
On the other hand, if one consider general temperature $T =
\frac{-1}{2\pi
\tilde{r}_{A}}(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}})$
associated with the apparent horizon which implies the heat capacity
of the universe is positive definite provided
$\frac{\dot{\tilde{r}}_{A}}{H\tilde{r}_{A}}\leq 1$. It is also
interesting to note that the total matter energy $E = V \rho$,
entropy $S = A / 4 $ and temperature $T = 1/2\pi \tilde{r}_{A}$
satisfy the relation $E = TS$ .
\paragraph{Generalized Second Law:} Recently a lot of attention has
been granted to the generalized second law of thermodynamics in the
accelerating universe handled by the dark energy \cite{jia}. Using a
particular model for dark energy, the generalized second law has
been studied in reference \cite{bin} with the boundaries of universe
enveloped by the apparent as well event horizon. It is important to
investigate the generalized second law as defined in the region of
the universe bounded by the apparent horizon in more general
context.\\ Let us now consider a region of FRW universe bounded by
the apparent horizon filled by a perfect fluid of energy density
$\rho$ and pressure $P$. We assume that the region bounded by the
apparent horizon acts as a thermal system with boundary defined by
the apparent horizon. Since the apparent horizon is not constant but
varies with time. As the apparent radius changes , the volume
enveloped by the apparent horizon will also change, however the
thermal system bounded by the apparent horizon remains in
equilibrium when it moves from one state to another so that the
temperature of the system must be uniform and the same as the
temperature of its surroundings. This requires that the temperature
of total energy inside the apparent horizon should be in equilibrium
with the temperature associated with the apparent horizon because we
are not considering the flow of energy through the horizon. \\
In order to state that a given cosmological solution
satisfies/violates the second law of thermodynamics one should
define first what is the generalization of the second law to the
case of FRW universe. It should include both the entropy due to
sources and the entropy associated with the background geometry. So
the generalized second law can be expressed as
\begin{equation}
\dot{S}_{A} + \dot{S}_{m} \geq 0,
\end{equation}
where $S_{A} = A/4G$ is the Bekenstein-Hawking entropy associated
with the apparent horizon of FRW universe and $S_{m}$ is the entropy
due to the matter sources inside the apparent horizon. From the
Friedmann equations (6) and (7), it can easily be shown that
$\dot{\tilde{r}}_{A} = 4\pi \tilde{r}_{A}^{3}H(\rho + P)\geq 0$
provided the universe driven by the source of perfect fluid
preserves dominant energy condition. Since we are assuming the
region of a FRW universe enveloped by the apparent horizon as
thermal system so that the change of energy $dE$ of the universe
from one state to another must be connected through the Friedmann
equations together with the characteristics of the apparent horizon.
It has been shown \cite{b1,a15,b2} that the Friedmann equations at
apparent horizon of FRW universe filled with perfect fluid satisfied
the thermal identity $dE = T_{A}dS_{A} + WdV$ instead of satisfying
the standard first law $T_{A}dS_{A} = dE + PdV$, where $T_{A}$ is
the temperature associated with the apparent horizon. Since we are
assuming the local equilibrium so that the change of matter energy
$dE_{m}$ satisfies the first law of thermodynamics, $dE_{m} =
T_{A}dS_{m} - PdV$, where $T_{A} = |\kappa|/2\pi$ is the temperature
associated with the apparent horizon and is given by
\begin{equation}
T = \frac{1}{2\pi
\tilde{r}_{A}}(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}),
\end{equation}
where $\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\leq 1$ ensures
that the temperature is positive. From the first law, $dE_{m} =
T_{A}dS_{m} - PdV$, one can obtain
\begin{equation}
T_{A}\dot{S}_{m} = 4\pi \tilde{r}_{A}^{2}\dot{\tilde{r}}_{A}(\rho +
P)- 4\pi \tilde{r}_{A}^{3}H(\rho + P).
\end{equation}
Now we turn to find out the entropy change $\dot{S}_{H}$ associated
with the apparent horizon of FRW universe which yields
\begin{equation}
T_{A}\dot{S}_{A} = 4\pi \tilde{r}_{A}^{3}H(\rho + P)-2\pi
\tilde{r}_{A}^{2}(\rho + P)\dot{\tilde{r}}_{A}.
\end{equation}
Adding equations (17) and (18), one gets the expression for the
generalized second law
\begin{equation}
T_{A}(\dot{S}_{m}+\dot{S}_{H}) = 2 \pi
\tilde{r}_{A}^{2}\dot{\tilde{r}}_{A}(\rho + P).
\end{equation}
It is obvious from the above equation that
$\dot{S}_{m}+\dot{S}_{H}\geq 0$ implies that the generalized second
law holds provided $\rho + P \geq 0$.
\paragraph{Conclusion:}
It is shown that the differential form of Friedmann equations of FRW
universe filled with a viscous fluid can be rewritten as a similar
form of the first law ,$TdS = dE + WdV$, of thermodynamics at the
apparent horizon of FRW universe. It can easily be seen that the
similar identity also holds for the perfect fluid when bulk
viscosity of the fluid is zero. It is also shown that the heat
capacity of the universe filled with perfect fluid is negative if
one considers the approximate temperature $T = |\kappa| / 2\pi=
1/2\pi \tilde{r}_{A}$. However if one utilizes the general
expression $T = \frac{-1}{2\pi
\tilde{r}_{A}}(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}})$, the
heat capacity of the universe is positive definite provided
$\frac{\dot{\tilde{r}}_{A}}{H\tilde{r}_{A}} \leq 1$ which implies
that the universe within the apparent horizon of FRW universe is
thermodynamically stable. We also verify that the generalized second
law of thermodynamics at the apparent horizon of FRW universe holds
provided that the matter source satisfies the dominant energy
condition. It is interesting to investigate the generalized second
law for the extended theories of gravity in a region bounded by the
apparent horizon of FRW universe. The work in this respect is under
progress.
\section*{Acknowledgments} I would like to thank Rong-Gen Cai for
his useful comments. The work is supported by a grant from National
university of Sciences and Technology.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 330
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Q: How do I select a string after a radio button, with Jquery? Code sample:
<table>
<tr>
<td>
<input type="radio" name="SomeName" value="3" checked>
My String $10.00
</td>
</tr>
</table>
How do I select 'My String $10.00' to hide it from the client?
$('???').next().hide();
Edit:
The HTML above is auto generated..I have no control over it....
A: $('input[name=SomeName]').parent().text();
If you want to hide that text, you would be best off enclosing it in a paragraph, span or label tag, so you could do something like this:
$('input[name=SomeName]').parent().find('span').hide();
A: jQuery does not allow easy access to text nodes. Perhaps you could change the HTML to something like the following?
...
<input type="radio" id="myID"/>
<label for="myID">My String $10.00</label>
...
Then you can do:
$("input[type=radio]").next().hide();
or, if you want to have it work no matter the order of the elements or any other stuff in between:
$("input[type=radio]").closest("td").find("label").hide();
Since you edited the post to mention you have no control over the HTML, try something along these lines:
// Tiny jQuery extension to provide $().outerHTML();
jQuery.fn.outerHTML = function() {
return $($('<div></div>').html(this.clone())).html();
}
var radioButton = $("input[type=radio]");
var parentElement = radioButton.parent();
var radioHTML = radioButton.outerHTML();
var hiddenText = radioButton.parent().text();
// Hide the text:
parentElement.html(radioHTML);
// Display the text again:
parentElement.html(radioHTML + hiddenText);
A: Since you have no control of the HTML I would go with one of the following solutions.
First solution: I don't think you can use hide() on text nodes in jQuery since that function only operates on elements (I'm speculating). But here's how you can remove the text.
$(document).ready(function() {
$("input[type='radio'][name='SomeName']").parent().contents().filter(function() { return this.nodeType != 1; }).remove();
});
You can read more about contents() here.
Second solution: You can inject span elements around the text nodes and hide the spans.
var parent = $("input[type='radio'][name='SomeName']").parent();
parent.contents().filter(function() { return this.nodeType != 1; }).wrap("<span></span>");
parent.find("span").hide();
A: he says he can't control the HTML.
But it's true, you can't access the text node easily to use hide() on it.
Hack sol'n: try reading everything in the into a var, stripping out everything after the tag with a regexp and then re-loading the result.
var chunk = $('input[name="SomeName"]').parent().html();
$('input[name="SomeName"]').parent().empty().append( chunk.replace(/(<[^>]+>).*/, '$1') );
or something like that (not tested)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,072
|
Q: Redirect customer to recently acquired product in woocommerce I'm selling digital goods (videos) that are unlocked after you buy them on my WooCommerce shop.
What I'm trying to do is that after successful purchase completion, the customer is redirected to the recently acquired product. The shop is already configured for redirec them logged-in and knowing that they acquired the product.
There is no cart o quantity, so there is no chance you'll add more than one product.
How can this be possible to make? I want to skip the after checkout page and redirect the customer directly to product they just bought.
Thanks!
A: There you go:
/***
* Redirect to custom page when reaching thank you page
*/
add_action( 'woocommerce_thankyou', 'woocommerce_thankyou_action' );
function woocommerce_thankyou_action( int $order_id ) {
$order = wc_get_order( $order_id );
$first_order_item = $order->get_items()[0];
$first_product_id = $first_order_item->get_product_id();
if ( ! $order->has_status( 'failed' ) ) {
wp_safe_redirect( get_permalink( $first_product_id ) );
exit;
}
}
In this example, I use the first order item as reference since you're only talking about one product and not multiple products.
This hook goes inside the functions.php file of your child theme.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,465
|
Дата дебюта:
Himchan
Daehyun
Youngjae
Jongup
Yongguk
Zelo
Лейбл:
Members of B.A.P (비에이피): Yongguk, Himchan, Daehyun, Youngjae, Jongup and Zelo. On August 23, 2018, it was announced that Yongguk's contract with TS Ent. has expired and B.A.P will continue as a 5-member group until their contracts end. Zelo's contract ended on December 2, 2018 and he also decided not to renew his contract with the agency. B.A.P debuted on January 26, 2012 under TS Entertainment. As of February 18, 2019, all of the remaining B.A.P members' contracts have been terminated and they will not be renewing them.
Kpop goods B.A.P
Official Fandom B.A.P: Baby
Stage name: Himchan, 힘찬
Real name: Kim Him Chan, 김힘찬
In the group B.A.P: subvocalist, rapper, visual
Rabbit: pink
Facts about Himchan
1) Kim Himchan was born and raised in Seoul, South Korea.
2) Himchan's family: parents, older sister.
3) Kim Himchan's education: Korea National University of Arts (traditional arts), National High School of Korean Traditional Music (07′).
4) Himchan's love for music came from an early age. He was studying Western and traditional Korean (gukak) music when he received a call about an opportunity to enter the entertainment industry, which he was also interested in. Kim Himchan can play the janggu (traditional Korean musical instrument) and has also worked with artists who played the shamisen (Japanese traditional musical instrument) during the pre-debut.
5) Kim Himchan has been a trainee for 1.5 years.
6) Himchan was the second member of B.A.P to be officially announced in August 2011.
7) Kim Himchan was the sub-vocalist, rapper and visual of B.A.P from January 26, 2012 to February 18, 2019 when he signed with TS Ent. expired and he decided not to update it. All B.A.P members have hinted at resuming activities under a different name.
8) Himchan made his debut as an actor on January 13, 2018, playing the role of Lee Sun in the historical play 'Yodo'.
9) On July 24, 2018, a certain woman accused Kim Himchan of sexual harassment. The police became interested in this case, Himchan denied the allegations, arguing that everything was by mutual agreement.
10) Himchan's ideal type is a kind woman.
Stage name: Daehyun, 대현
Real name: Jung Dae Hyun, 정대현
In the group B.A.P: main vocalist, face of the group
Rabbit: white
Facts about Daehyun
1) Jung Daehyun is from Donnae-gu, Busan, South Korea.
2) Daehyun was born on the same day as Lee Teri.
3) Jung Daehyun has an older brother.
4) Daehyun wants to become an actor.
5) Jung Daehyun wrote many songs: How about you, Dark Light, Shadow and I Can't Fly.
6) Daehyun is close to Seungkwan (Seventeen), Kwon Hyunbin (former JBJ member) and Lee Su-Woong (Boys Republic).
7) Jung Daehyun loves to sing.
8) Daehyun's ideal type: Someone like Shin Saimdang.
Stage name: Youngjae, 영재
Real name: Yoo Young Jae, 유영재
B.A.P: lead vocalist
Rabbit: yellow
Youngjae facts
1) Yoo Youngjae was born in Bangbae-dong, Seoul, South Korea.
2) Youngjae family: parents and older brother.
3) Yoo Youngjae's education: Uijeongbu Technical High School.
4) Youngjae was a smart student, he scored 90% in science, history, Korean and English.
5) Yoo Youngjae developed a love for music before entering high school. He trained under JYP Entertainment for a year, where he auditioned with J-Hope (BTS) and Dino (Halo). Youngjae left JYP as his debut day was nowhere near. He was introduced by an acquaintance to an agent of TS Entertainment, with whom he subsequently signed a contract after passing an audition.
6) Youngjae has been training for a total of 4 years.
7) In early 2019, Yoo Youngjae founded his independent label JWORLD.
8) In March 2019, it was announced that Youngjae signed with a new agency and debuted as a solo artist on April 19, 2019 with 'Fancy'.
9) Youngjae's ideal type is someone who loves him very much.
Stage name: Jongup, 종업
Real name: Moon Jongup, 문종업
B.A.P: Main dancer, lead vocalist
Rabbit: green
Jongup Facts
1) Jongup Moon was born in Seongnam, Gyeonggi-do, a satellite city of Seoul, South Korea.
2) Jongup family: parents, two older brothers.
3) Education of Moon Jongup: Hanlim Multi Art School, Sunae Middle School.
4) Jongup has been interested in dancing since high school. Dancing gave him a sense of comfort and he began attending tournaments and competitions. On stage, Moon Jongup felt fascinated on stage and wanted to become a professional artist. Jongup then auditioned for TS Entertainment, where he was accepted and offered a position in a new boy group.
5) Moon Jongup starred in SECRET's 'Shy Boy', 'Starlight Moonlight' and Bang & Zelo's 'Never Give Up' videos a year before B.A.P's debut.
6) Jongup has been a trainee for 1.5 years.
7) The biggest influence on the musical taste of Moon Jongup was Chris Barun, he also became a model of behavior.
8) Jongup was the main dancer and lead vocalist of B.A.P from January 26, 2012 to February 18, 2019 when he signed with TS Ent. expired and he decided not to renew it. All of the B.A.P members have hinted that they may be resuming activities under a different name and under a different agency.
9) Jongup's ideal type: He prefers girls older than him with the same interest in anime.
Stage name: Yongguk, 용국
Real name: Bang Yong Guk, 방용국
In the group B.A.P: main rapper, leader
Rabbit: red
Yongguk Facts
1) Bang Yongguk was born in Incheon, South Korea. As a child, he briefly moved to the offshore Ijak Islands near Incheon.
2) Yongguk's family: parents, older sister (Natasha), older twin brother (Yongnam).
3) Bang Yongguk's education: Kyunghee Cyber University, Yuhan High School ('08), Gae Woong Middle School.
4) Yongguk speaks Korean, Japanese and English and Spanish.
5) Bang Yong Guk was the leader and main rapper of the k-pop group B.A.P from 2012 to 2018 until his contract with TS Entertainment expired on August 23, 2018 and he decided not to renew it. All of the B.A.P members have hinted at a possible future reunion under a different name.
6) Yongguk developed his songwriting skills after posting rap lyrics on an online forum when he was still in middle school.
7) Bang Yong Guk was a member of the underground hip hop group Soul Connection, his stage name was Jepp Blackman.
8) The hip-hop duo 'Untouchable' were the first to recommend Yongguk to their agency TS Ent. Yongguk signed with TS in 2010.
9) A year before B.A.P's debut, Bang Yong Guk and Zelo debuted as a rap subunit of Bang & Zelo.
10) On March 15, 2019, Yongguk released his first solo album "BANGYONGGUK", which ranked 9th on Billboard's World Albums Chart.
11) Yongguk's ideal type is a virtuous woman.
Stage name: Zelo, 젤로
Real name: Choi Jun Hong, 최준홍
B.A.P: Lead Rapper, Lead Dancer, Maknae
Rabbit: blue
Birthday: October 15, 1996
Zodiac Sign: Libra
Facts about Zelo
1) Zelo was born and raised in Mokpo, South Korea.
2) Choi Junhong's family: parents, older brother, Mochi (dog).
3) Education Zelo: School of Performing Arts Seoul (15′), Sungdae Middle School.
4) Nicknames of Choi Junhong: Noona "Killer" (xD).
5) Zelo speaks Korean and English.
6) As a child, Choi Junghong dreamed of becoming a football player, but subsequently changed his mind, his love for music grew in him and he began to think about a career as a singer. Zelo is also interested in dancing and beatboxing.
7) At the age of 10, Zelo started attending a well-known music academy in Gwangju where Big Bang's Seungri and BTS' J-Hope also studied. There, Choi Junhong began to study rap and dance. He also started attending auditions, but was unsuccessful due to being too young.
8) Choi Junghong ended up under TS Entertainment and made his debut in the hip-hop duo 'Bang & Zelo' along with fellow B.A.P Yongguk in 2011.
9) Zelo joined k-pop group B.A.P on January 26, 2012 under TS Entertainment, where he was the lead rapper, dance lead and "cute, innocent" maknae. The contract with TS ended on December 2nd, 2018 and he decided not to renew it. All the members of B.A.P have hinted that a reunion in the future under a different name is possible.
10) Zelo's Ideal Type: "A person with a sweet smile who is good at English."
посмотреть еще →
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,183
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Name mystery from Nancy's blog
Alright, Nancy's fantastic name blog sometimes draws attention to naming mysteries and usually she provides immensely satisfying resolutions, but in this particular case, I'm left totally intrigued and wanting to tap into the excellent hive mind we have here. It's been slow, right, and you are looking for a distraction in our post-Deneen era?
The name is Caster (http://www.nancy.cc/2017/04/10/mystery-monday-baby-name-caster/), which shows up with a very high debut in 1953 and then trails off.
yob1953.txt:Caster,M,21
yob1956.txt:Caster,M,5
Nancy adds:
"Caster doesn't seem to be a variant of some other name (like Casper, or Lancaster). So I'm assuming this usage corresponds to someone named Caster — either real or fictional — who was in the public eye for several years in a row.
The tricky thing is, of course, that any online search for the name "Caster" turns up all sorts of extraneous stuff — fishing, furniture, music (stratocaster), sports (sportscaster), and so forth.
Still, I was able to track down a few clues.
Records suggest that the majority of these 1950s Casters had middle names that started with D. Here's a Caster D. born in 1953, and another Caster D. born in 1957.
And every single D-middle I tracked down included the letter L and/or the letter R. Some examples: Dell, Derrell, Derrel, Derriel, Daryl, Deryl, Derald, Derra, Doria, and Doral. A handful of people even had combination names like Casterdale or Casterdell (b. 1953).
Finally, it looks like most of the people named Caster D. were born in the South.
Do you have any idea where the name Caster might have come from?"
The fact that there are combination names with D/l/r middles makes me particularly intrigued. Is it a heard-aloud phenomenon? Are the D/l/r middle names homaging a surname? Does anyone want to showcase their google skills or their excellent pop culture recall? Help a name puzzler out, please!
By lucubratrix
‹ Middle Name Suggestions Help with baby number 9... ›
By katjsh
When I first read this Castor Dell hit some memory trigger, it made me think of old Disney movies searching only turned up a small reference to a fantasia scene with a Centur named Castor and about 15 years too early. A little more searching turned up nothing that seems to fit but a few mentions in the right time frame.
USS Castor that was at Pearl Harbor and also used during Korea
a french rebuilding effort after WWII also called Castors
and Annie Glenn's maiden name was Castor. None of these seem to be the right fit but I'm enjoying the search and look forward to reading what everyone else finds.
By NotAGuestAnymore
I tried typing in only bits & pieces, so see what Google auto-suggested. Caste Del got me a hit on Castel del Monte in Italy. Wikipedia says that in the 1950s, a bright red compound in the soil around the castle was found to contain a strain of bacteria which was then used in chemotherapy. Interesting to my nerd-brain, but I don't imagine people in the southern U.S. were naming their kids because of a chemo drug.
I also got a hit on the Castor Delgado Perez residence in Sao Paolo, which was built in the 1950s. But again, I don't imagine there was much of a fandom for Brazilian architecture in the U.S. south.
And there is a town called Castor in Bienville Parish, Louisiana. Its only claim to fame in the 1950s was a deadly tornado that killed a family of 6. It was the Sullivan-Smith family, so nothing to explain on the DLR names. Also seems like a weird/morbid thing to name babies after the actual town. I'd think in this case, any namesakes would have been given honor names after the people who died.
Oh-and there is a user-submitted name on Behind the Name for Caster. It says it's a South African variant for the Greek name Castor. The only hits for South Africa + Caster were for 2016 Olympic runner Caster Semenya.
That's really interesting.
Most of the Casters that I'm finding are African American.
I just sent a message to a Casterdale and a Casterdaral (ETA: and a Casterdaryl and a Casterdarial and a Casterderria and a Casterderal and a Casterdara) on facebook -- I think that a person who is a part of the name trend might be the best way to get that information.
The more people I've turned up on social media with compound names, the more I think that we're looking at a two part homage of Caster and a second part (surname?) that sounds like Derrial (or Darryl), which in some more drawling accents would ellide into Dell or Dale. Messages sent; I'm hoping someone eventually likes checking their other folder for mail from nice-if-nosy name nerds and decides to take pity on them and spill their family name origin story. What's extra wild is that judging by obituaries of these Casterd*r*l names have siblings with completely boring (well, very normal, not-at-all-unexpected) names, sometimes even large sibsets of very boring names.
I think what really interests me is how many of these are First-Middle packages, something which I usually associate with explicit honors ("Horatio Nelson Surname for Horatio Nelson", not "Deneen Ivory Surname for the girl in the dish soap advertisement"). That suggests to me that there's a really good story here, rather than an incidental "I heard this nice name someplace random and it stuck".
By Karyn
I have absolutely nothing to add, but I'm really enjoying this thread.
In the past, I actually considered messaging people with first-last name combos that I was considering for my daughter. In the end, we ended up loving our first-choice name so much that it wasn't necessary, but I'm not ruling it out in the future :)
By MarCee
Lucubratrix, thanks for sharing this name mystery! I'm thoroughly enjoying this name-sleuthing thread.
I don't have much to add, but I did find a "Caster Dale" that was born in 1935 (before Caster showed up on the SSN list). His obituary lists a son named "Caster D. Jr" who plausibly could have been born between 1953 and 1956.
The similarity of Caster Sr's name to all of the 1950s babies could just be a coincidence, or it could suggest the Caster D_r_l combo goes back a few decades, with some spike in prominence in the early 1950s.
I found that guy, too! I tried to find different versions of his obituary that might list more details about why he was famous enough to spawn a series of namesakes, but I could only find that he was buried at Arlington due to military service... and if he had brought notoriety rather than fame to the name, then I wouldn't think that you'd have so many Caster D_r_l individuals. After all, you might use a name that you heard on the news because it sounded great even though it was attached to someone not particularly praiseworthy (see all the murder victim name spikes), but I don't think you'd deliberately use the middle name or make it a compound name.
I'm really curious about why the first part is mostly spelled Caster (not Kaster, not Castor, not Castar) but the second part is so hypermutable -- though I did find one Castordarryl. People have always been into creative spellings, so it's really more the contrast with the consistency of the first part that seems mysterious. I wonder if the namesake had the first part printed (on a jersey, in an advertisement, in the song name) but the second part was entirely a heard-aloud phenomenon? Perhaps it really is honoring an original Caster D. Surname, who only got referred to as Caster D in writing?
1935 Caster's obituary said he was a Command Sergeant Major in the Army--which is the highest enlisted rank. His job would have been to represent all of the enlisted soldiers in his command to the commanding officer. I briefly considered whether "Caster D" could have been a military leader/hero of some sort, and some soldiers coming home from the Korean War in 1953 might have chosen to name their sons after him. I suppose this is still possible, but the 1935 Caster would only have been 18 in 1953--definitely not a Command Sergeant Major yet and probably not influential enough to inspire a lot of namesakes unless he did something really heroic in Korea that saved a lot of lives.
The middle name is the most interesting part of this mystery, I think. I also looked for evidence that the original D_r_l was a surname adapted to form a middle name (like all of the Robert Lee Surnames floating around the South), but I haven't found any evidence yet of a Caster with a compelling "D" surname on any of the numerous geneology and gravefinding websites that I've checked. (I don't have an account with any of these websites, so my search capabilities are limited there.)
If the D_r_l name was originally a surname, then it's also reasonable that there might be a spike in the use of that name as a first name too. Most of the D* names listed on Nancy's blog are already trending up in 1953 and peak in the sixties. Dell and possibly Derald show a bump in 1953.
Dell is generally trending up before it spikes up in 1953, staying up until 1960 then declining dramatically in the sixties and seventies .
Dell 1948 59
Derald looks to be generally trending down from a high in 1929 before it spikes a little bit in 1953 and again in 1957 before petering out again:
Derald 1948 32
I'm not sure if either of these names are the key to the original inspiration for Caster. If so, Dell seems like the more likely candidate. It potentially explains the three or so people named Casterdell that I've found, but not the consistency in spelling for "Caster" and inconsistency in the D middle name.
MarCee, I really appreciate that you're also on board this mystery, and I think you make excellent points all around! Thank you!
Thanks also for your insight into the workings of the military. I also had contemplated that 1953-1935 = rather young to be a super-prominent namesake, but you're right that it could be possible given military service often makes heroes of such shockingly young men. I do think we should be able to turn up news clippings about whatever Caster D. Sr's military adventures were, though, right? Especially since you appear to have varsity level search skills at your disposal!
I agree that what makes this mystery so compelling is the middle name (or compound name) bundle. (I updated my above post with the many Casterd_l/r names I found on facebook; there were surprisingly many and I'm hopeful that someone might write back!) I'm intrigued by the suggestion that the D_l/r names on the whole also went up during that time prompted by the same person/thing/event. I guess the alternative explanation for the hypermutability is that the names Dale and Dell and Derald were very on-trend... so to homage an original Caster D., perhaps they were creative in filling in the sounds of the moment.
For example, check out the curve of this plot, especially if you require a,l,e,r endings to filter out the more distinct sounding Derrick, Darius, Dalton, Delbert: http://www.babynamewizard.com/namevoyager-expert#prefix=dar,der,dal,del&sw=m&exact=false
By Elizabeth T.
If the namesake were only known as Caster D., though, I would expect you to have found Casterdons, -daves, -davids, etc. I think this guy was moderately well known by his whole name.
Fair enough!
So, sorting through the SS death index for pre-1953 Casters yields Caster D. Giffin (b. 1922), but I cannot turn up anything else about him to suggest why he might have spawned so many namesakes.
He did, however, have grandparents named... Caster (grandpa) and Della (grandma).
Oh, now that's interesting. A cross-gender namesake could explain the variations-on-a-theme middle names. Any chance there was a bump in Della C___ names in 1953? Or perhaps Dolores or Darlene or similar.
Alas, no bump in Della. Darlene had already been rising.
Unfortunately, I'm not finding much of interest about this 1922 Caster D. to suggest he was a namesake, either... but I may not be looking in the right places.
Does anyone have twitter? There's a casterdarl who is fairly active. I feel like facebook hides messages from nonfriends so deep in the site that especially a casual user might not locate them, so switching platforms is appealing. After all, finding testimony from people who had the name is how we finally cracked Deneen...
Yeah, the problem with the death certificates is that our Caster may not have been deceased pre-1953. It's possible he was still alive (or could even be currently still alive) which is going to slow down our search.
This is a good point, but I am guessing that the namesake impetus for a 1953 spike would ilkely have been an adult at that time, and someone born before the 1930s would have to be blessed with longevity to still be alive today.
I don't know... Pre-1930 doesn't seem so terribly old that it's unreasonable to think that your mystery man is still alive.
Here are some data from the 2010 census to back that up (page 2 in particular): https://www.census.gov/prod/cen2010/briefs/c2010br-09.pdf
Any mention of military awards or honors he might have earned? That might give us a lead on something major that might have happend to inspire a bunch of namesakes while he was still quite young.
I keep running into dead ends. I haven't been able to find any record of military awards or ribbons for 1935 Caster, although I did find out a few other (unhelpful) things about him--like he went by Dale. Another Caster I found went by Cass, so apparently the namesake isn't so strong that it has a natural nickname.
I suspect part of the problem is that it's still reasonably likely that a Korean war vet is still alive. I know my husband encountered that problem when trying to research his father's Korean war service. Because of privacy issues, the military & sites like Ancestry.com avoid making that kind of information public access until the people are deceased. So unless the family has mentioned it in an obit or something, we may not be able to find out.
It's another thing entirely when trying to look up military service awards or military census data for earlier periods.
By HungarianNameGeek
I don't think it's likely to be related to the question, but a search for 1953 combined with Caster eventually turned up a Belgian-French comic book (Tintin: Destination Moon) that was published in 1953 by Editions Casterman.
I love Tintin (currently making a big resurgence in the peer group of my eldest, since it's one of the graphic novels at the school library, leading to the hilarious situation where his very obscure name is not perceived as that weird because there's a minor recurring Tintin character that shares his name)... but I think it's unlikely to be related to this phenomenon.
I finally sorted out the SSDI data. Here are the pre-spike Casters (at the ones who have died, which is a fair point). None of them have Daryl-tyep surnames, but few of them have middle initials, so it's possible that some of them were Caster D_r_l Surnames, besides Mr Giffin (1922).
Veteran?
DUKECASTERGrey 2/27/1887?GASSAWAYCASTERWister 10/23/1886GAYCASTER 09/18/1889BERRYCASTER 10/09/1889BANKSCASTER 11/18/1889SMITHCASTER 3/30/1893ELLISCASTER 4/11/1898TEMPLINCASTERWilliam 08/20/1898MITCHELLCASTER 7/18/1903SLAGLECASTERLee 9/10/1903SHELTONCASTERT 10/19/1903WESTCASTER 12/20/1904JOHNSONCASTER 1/1/1905ROBBINSCASTER 1/24/1906WILSONCASTER 5/23/1906CROFFORDCASTER 11/8/1908CAUSEYCASTERH 4/17/1909BROCKCASTER 9/9/1909LARSONCASTERBellman 2/10/1910HARGROVECASTERL 3/11/1910TUCKERCASTER 9/6/1910MCCOYCASTERBV12/13/1910JOHNSONCASTER P2/3/1912RADLECASTER 4/10/1912LESTERCASTER 7/9/1912WOODCASTER 12/25/1912BURCHFIELDCASTERAlvaydV5/4/1913SCOTTCASTERZV9/10/1913HORTONCASTER 12/11/1913JEFFERSONCASTER 5/30/1914BUCKCASTERC 9/5/1914STEIGERWALTCASTEREV11/11/1915ROSSCASTER 5/4/1916CLARKCASTERWV6/18/1916HARRISONCASTER V9/2/1916CHANCECASTER 10/2/1916HARRELLCASTERG 3/17/1917GRAYCASTER 9/12/1917MCKENZIECASTERA 2/17/1918DAVISCASTERCV1/17/1919PATTONCASTERDenton 9/10/1920BURNETTCASTERL 11/22/1920SELLERSCASTER 9/21/1921GIFFINCASTERDV8/30/1922HOWARDCASTER 10/23/1922BARNESCASTER 6/25/1923WOODENCASTER P6/30/1925DUKECASTER 9/2/1925UNDERWOODCASTERB 11/23/1925YOUNGERCASTERP 7/7/1927EDWARDSCASTERLV6/24/1928DAWKINSCASTER 10/21/1928BOYKINCASTER 11/23/1928YOUNGCASTERFV1/5/1933JACKSONCASTERLV1/29/1935WILLIAMSCASTERCV7/10/1935WESTCASTEREV7/23/1935NELSONCASTER V6/25/1936TOTTENCASTER 2/22/1939HERRINGCASTERL 10/12/1940THOMPSON SRCASTER 9/13/1948
Also of note is that the Caster D spike in 1953 starts in September.
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AkzoNobel has made an agreed offer to acquire BASF's Industrial Coatings business. The transaction would include technologies, patents and trademarks, as well as securing supply to customers worldwide.
Two manufacturing plants – one in the UK and one in South Africa – would also be transferred to AkzoNobel. The business supplies products for a number of end uses, including coil, furniture foil and panel coatings, wind energy and general industry and commercial transport.
The planned transaction is expected to be completed in the second half of 2016, subject to regular consultation with employee representatives and satisfaction of certain closing conditions, including receipt of required regulatory approval.
For more information, visit www.coatingsworld.com.
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| 1,923
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Litsea alveolata är en lagerväxtart som beskrevs av C. K. Allen. Litsea alveolata ingår i släktet Litsea och familjen lagerväxter. Inga underarter finns listade i Catalogue of Life.
Källor
Lagerväxter
alveolata
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| 4,934
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Le Troglodyte du Mérida (Cistothorus meridae) est une espèce d'oiseaux de la famille des Troglodytidae.
Cet oiseau peuple la cordillère de Mérida.
Liens externes
Troglodytidae
Oiseau endémique du Venezuela
Oiseau des Andes boréales
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| 2,129
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| null | null |
**Gotham Writers' Workshop ®**
**WRITING FICTION**
THE PRACTICAL GUIDE FROM NEW YORK'S ACCLAIMED CREATIVE WRITING SCHOOL
WRITTEN BY GOTHAM WRITERS' WORKSHOP FACULTY EDITED BY ALEXANDER STEELE
B L O O M S B U R Y
NEW YORK • BERLIN • LONDON
**FROM GOTHAM WRITERS' WORKSHOP'S FOUNDERS**
Gotham Writers' Workshop began with a single class taught in a living room on the Upper West Side of New York City. The class was free. After three hours, everyone had a choice. They could leave, or, if they felt they had learned something worthwhile, they could pay for the rest of the course. Everyone decided to stay, and the first semester at Gotham Writers' Workshop had begun.
Those original students spread the word. So we offered more classes. Word of mouth traveled wider. We hired teachers, rented an office. Soon we were teaching classes in various locations throughout New York City. Eventually we expanded into online classes, and we drew students from all over the world. Today we employ over a hundred instructors who teach more than six thousand students a year.
Despite our growth, we still think of ourselves as a grassroots organization. Class size remains small enough to fit into a New York living room. Our teachers continue to bring their passion for writing to every class. Our founding principles are unchanged.
Simply put, we believe anyone can write. We believe writing is a craft that can be taught. True, talent cannot be taught, only nurtured, but the craft of writing _can_ be taught. We're devoted to teaching the craft in a way that is so clear, direct, and applicable that our students begin growing as writers during their very first class.
There's no easy formula for creating great fiction, but a fundamental knowledge of writing craft is, more than anything, what will allow your talent to blossom on the page. Such knowledge is what we offered in that first class, and it's what our teachers—who are all writing _and_ teaching professionals—continue to offer every one of our students.
Now we've put the Gotham style of teaching into a book. The ability to write—to write with excellence—is in your hands.
Jeff Fligelman and David Grae February 2003
**HOW TO USE THIS BOOK**
You shouldn't just read your way through this book, but write your way through it as well. After all, you're reading this book because you want to write.
Sprinkled throughout every chapter, you'll find numerous writing exercises, indicated by the words _Your Turn._ Quite literally, this means it's your turn to apply the knowledge you've just learned to your own writing. You shouldn't worry about turning these exercises into brilliant works of fiction. Rather, you should simply focus on experimenting and having fun with the task at hand. If one of the exercises spawns a wonderful idea that you would like to expand into a longer piece of writing, a piece you hope to finish and perhaps publish, by all means help yourself. In fact, toward the end of the book you'll be advised to do just that.
You also might find it useful to keep your work on these exercises in a notebook, either the paper kind or the computer kind. If you do all or most of the exercises in this book—and you should—you'll have a wide resource of ideas and fragments from which to draw or be inspired the next time you're looking for a fiction project.
Not that we want to make things too easy for you, but at the back of this book you will find a "cheat sheet" that gives you a checklist of many of the key points on writing craft that appear in this book. You may want to keep the cheat sheet handy when writing your next work of fiction.
You will also find numerous passages from works of fiction cited throughout this book. If one or more of these works looks interesting to you, you should get your hands on a copy and read it. If the work is a novel or novella the title will appear in _italics,_ and if the work is a short story the title will appear in "quotes." A number of the short stories that appear in this book can be found in _The Vintage Book of Contemporary American Short Stories,_ edited by Tobias Wolff.
In particular, you should read the short story "Cathedral" by Raymond Carver, preferably in tandem with reading this book. "Cathedral" is referred to repeatedly throughout this book, and having read the story will enhance your understanding of these references. You will find "Cathedral" reprinted in its entirety in the Appendix.
Last, you'll find additional information on the art and business of writing at the Gotham Writers' Workshop Web site: www.writingclasses.com.
**CONTENTS**
[CHAPTER 1
**FICTION: THE WHAT, HOW, AND WHY OF IT**](Facu_9781596917910_epub_c5_r1.html#bb1)
BY ALEXANDER STEELE
[CHAPTER 2
**CHARACTER: CASTING SHADOWS**](Facu_9781596917910_epub_c6_r1.html#bb2)
BY BRANDI REISSENWEBER
[CHAPTER 3
**PLOT: A QUESTION OF FOCUS**](Facu_9781596917910_epub_c7_r1.html#bb3)
BY DAVID HARRIS EBENBACH
[CHAPTER 4
**POINT OF VIEW: THE COMPLETE MENU**](Facu_9781596917910_epub_c8_r1.html#bb4)
BY VALERIE VOGRIN
[CHAPTER 5
**DESCRIPTION: TO PICTURE IN WORDS**](Facu_9781596917910_epub_c9_r1.html#bb5)
BY CHRIS LOMBARDI
[CHAPTER 6
**DIALOGUE: TALKING IT UP**](Facu_9781596917910_epub_c10_r1.html#bb6)
BY ALLISON AMEND
[CHAPTER 7
**SETTING AND PACING: I'M HERE THEREFORE I AM**](Facu_9781596917910_epub_c11_r1.html#bb7)
BY CAREN GUSSOFF
[CHAPTER 8
**VOICE: THE SOUND OF A STORY**](Facu_9781596917910_epub_c12_r1.html#bb8)
BY HARDY GRIFFIN
[CHAPTER 9
**THEME: SO WHAT'S YOUR STORY REALLY ABOUT?**](Facu_9781596917910_epub_c13_r1.html#bb9)
BY TERRY BAIN
[CHAPTER 10
**REVISION: REAL WRITERS REVISE**](Facu_9781596917910_epub_c14_r1.html#bb10)
BY PETER SELGIN
[CHAPTER 11
**THE BUSINESS OF WRITING: DRIVING YOURSELF NUTS FOR FUN AND PROFIT**](Facu_9781596917910_epub_c15_r1.html#bb11)
BY CORENE LEMAITRE
CHEAT SHEET
APPENDIX: "CATHEDRAL"
BY RAYMOND CARVER
ACKNOWLEDGMENTS
CONTRIBUTORS
[CHAPTER 1
**FICTION: THE WHAT, HOW, AND WHY OF IT**](Facu_9781596917910_epub_c4_r1.html#aa1)
BY ALEXANDER STEELE
Hello, you look familiar.
As the dean of faculty at Gotham Writers' Workshop, I'm surrounded by people with a desire to create fiction. On a daily basis I work with our fiction teachers, folks so talented and intelligent they could have made a killing in most any field, but instead have opted to pursue the precarious life of a fiction writer. Frequently I observe our fiction classes, some in regular classrooms filled with students from New York City and the surrounding areas, some in cyberspace classrooms, filled with students from all over the United States, and from as far away as Africa, China, and Australia. I see the whole wide world in these classes—doctors, lawyers, accountants, janitors, policemen, undertakers, housewives, retirees, students, psychics, zookeepers, and everything else.
It's a fact: a staggering number of people out there harbor an intense desire to create fiction.
Why?
Though this chapter will cover more than a philosophical inquiry into why we write fiction—as the title promises—let me see if I can find an answer to this vexing question, preferably by the chapter's end.
**A BRIEF DEFINITION OF FICTION**
Let's start with a simpler question: what is fiction? In the broadest sense, fiction is simply a made-up story.
The business of making up stories has been going on for a long time. Somewhere in the shadowy past, our cave-dwelling ancestors began conjuring stories and telling them to each other. The tradition grew, and some of these stories eventually attained the "best-seller" status of myths, tales destined to be passed on through generations and to migrate across continents and to shape the way people thought. At some point, some of these stories started to get written down with the intention that they would be _read._ An enterprising Mesopotamian writer chiseled _The Epic of Gilgamesh_ into stone tablets some four thousand years ago, and if you think revision was difficult on a typewriter...
Anyway, this brings us to the more narrow definition of fiction: a made-up story told in prose with words alone.
Words alone.
That's the unique challenge and wonder of written fiction. There's no actor or storyteller using gesture and inflection. No painter or filmmaker showing settings or close-ups. Everything is done with those little symbols we call letters, which are melded into words, which multiply to form sentences and paragraphs. And by some alchemical process those words interact with the reader's imagination in such a way that readers are taken inside the reality of a story—like Alice stepping through the looking-glass—and once there they can experience and feel and care about this alternate reality as deeply as they do for the meanderings and heartbreaks of their own lives.
For us humans this process is strangely important. We seem to have a primal need for fiction, or really any kind of story, that is as deeply rooted as our need for food, shelter, and companionship. I see two reasons for this.
The first reason: entertainment. We crave entertainment, and stories are one of the key ways we satisfy this desire. The second reason: meaning. Our curiosity, and perhaps insecurity, compels us to explore continually the who, what, where, when, and why of our existence. Some call this lofty goal a search for _Truth._
A good piece of fiction will satisfy one or both of these needs extremely well and do so in a miraculously low-tech manner. All fiction ultimately requires is words interacting with the reader's imagination, a combination that provides, for many people, the most powerful form of storytelling possible, not to mention the most portable.
**A MATTER OF FORM**
Well come back to entertainment and meaning shortly, but now let's take a brief look at the basic forms of fiction.
First, the novel. Typically a novel runs at least eighty thousand words (about 320 pages of double-spaced typing). Some novels run a bit shorter than this and many run way longer. Novels are usually broken into chapters, which give the reader a much-needed mental break.
A novel is the literary equivalent of a symphony, the big, ambitious form of fiction. Novels aren't just longer than other forms of fiction. They generally have more of everything: more characters, more scenes, more developments, more _heft._ They may have a central story, but the story is usually surrounded by a whole swirling world of activity. Someone once told me she could tell if a work was a novel or short story simply by hearing the first sentence. Interpret that as you may.
Some novels are sprawling. Leo Tolstoy's _War and Peace_ is an ocean, bearing countless characters over numerous years and thousands of miles, immersing the reader in a span of history, encompassing all aspects of humanity. But J. D. Salinger's _The Catcher in the Rye_ covers only a few days and never leaves the side of that mixed-up teenager, Holden Caulfield. And then there's James Joyce's _Ulysses,_ which staggers on for almost eight hundred pages, weaving in and out of various minds and styles, but staying within the confines of a single Dublin day.
Writing a novel is a long haul that can swallow years of a life, a test of endurance for even the hardiest of souls. Nevertheless, for many an aspiring writer, the novel is the great white whale of fiction, and these people will not rest until they have spilled the blood of several hundred pages. Godspeed to them all.
Next, there's the short story. Short stories tend to run no longer than fifteen thousand words (about sixty pages of double-spaced typing), and most run shorter than this. The average short story is about the length of the chapters in the book you're holding, though recently flash fiction—stories that run only a page or two—has come into fashion. Short stories are the literary equivalent of songs. They are not necessarily less emotionally complex than novels, just as "Amazing Grace" is no less powerful than Beethoven's Ninth Symphony, but the scope of a short story is narrower. Often short stories focus on a single event, or at least a single aspect of a character's life.
"The Swimmer" by John Cheever stays focused on a man's dogged attempt one summer afternoon to travel home via the neighboring swimming pools. "A Bullet in the Brain" by Tobias Wolff stays focused on a few significant minutes while a book critic stands in line at the bank. "Carried Away" by Alice Munro stretches from World War I to World War II but stays focused on a librarian's strange relationship with a man who loses his head, literally. These stories dig deep yet never wander outside their tightly focused spotlights.
Sometimes, related short stories are written to be collected in a book, as in Sherwood Anderson's _Winesburg, Ohio,_ where the stories feature different characters, all of whom live in the same small town, or as in Denis Johnson's _Jesus' Son,_ where the same misfit character drifts through every story. The stories here can be enjoyed individually, but read together they have a cumulative effect.
Short stories are perhaps the best first step for the beginning fiction writer simply because they demand less time commitment than a novel does. But short fiction is an exacting form. Whereas a novel may be forgiven a bit of flabbiness, short stories must be kept on a strict diet. Every word _counts._ The best short stories employ a precision and economy reminiscent of poetry.
Then there's the novella, which hovers in between the novel and the short story. In length, novellas run from about fifteen thousand words to about eighty thousand words. Some novellas combine the broader scope of a novel with the lean telling of a short story, as with Joseph Conrad's _Heart of Darkness,_ which covers a long river journey by steamer through Africa. Other novellas combine the narrower scope of a short story with the leisurely unfolding of a novel, as with Franz Kafka's _The Metamorphosis,_ which covers a few weird days in the life of a man who wakes up to discover he's been turned into an insect.
To generalize any further about these forms would do them an injustice. They are elastic forms and can be many things to many different writers. The only indisputable difference is length, and no one really agrees on that either. Perhaps the only truly indisputable difference is that the titles of novels and novellas are italicized while short story titles are put in quotation marks.
Which form should you focus your efforts on? Well, a story should take the form it wants to take, the form in which it is most comfortably told. For example, you may start writing a short story and then discover that the characters and situations demand a much larger canvas. They won't stay within the small frame of short fiction. Then you'll either need to narrow your focus or cancel that summer trip and start working on a novel. Some writers choose one of these forms and stick with it, while others bounce back and forth between the forms.
**LITERARY AND GENRE FICTION**
Fiction can be further subdivided into two camps— _literary fiction_ and _genre fiction._ Literary fiction refers to stories with some aspiration of being considered "art." Most of the stories here appeal to a somewhat elite readership, especially in the case of short fiction. Genre fiction refers to stories that usually fall within the popular genres of mystery, thriller, horror, fantasy, science fiction, western, and romance. Here you'll find stories geared for a broader audience. (Sometimes you'll hear the term _mainstream fiction,_ which usually refers to literary fiction that has broad commercial appeal.)
The easy distinction is to say that genre fiction is fun, popular, and less important than literary fiction, which strives for layers of depth and artistic heights. There is some truth in this notion. Most genre writers will proudly admit that their chief motive is to keep their readers entertained. Most literary writers will readily affirm that they're trying to express something about the human condition. Both types of fiction are equally valid, with plenty of readers in both camps to prove it.
There is nothing terribly wrong with this division in the fiction house. _Vive la différence._ We now have several hundred channel choices on our TV sets. Why shouldn't we have as wide a range of options with our fiction?
Some prefer, say, the stylish prose of Amy Tan, a prize-winning literary author, while others prefer, say, the screaming terror of Stephen King, the big daddy of genre writers, while others still enjoy moving their tent between camps.
Actually, there is much common ground here. The literary writers need not view the genre writers as slackers and the genre writers need not view the lits as snobs. In fact, they can learn a great deal from each other. A literary work should keep readers entranced and turning pages well past bedtime, and an entertainment work will be all the more entertaining if it has some real insight and resonance.
This book will mostly focus on writing literary fiction, as is true of the GWW Fiction classes. We have separate classes for the various genres of fiction. However, while the genre classes deal with the specific needs of their genre, the majority of what is taught in these classes is exactly the same as what is taught in the Fiction classes. The same elements of craft apply. And, really, when it comes to fiction—good is good.
**AT ITS BEST**
If you look at the great works of fiction throughout the ages, you will notice just how brilliantly these stories have satisfied the dual need for entertainment and meaning. A few examples from the past two centuries:
_Pride and Prejudice_ by Jane Austen (1813)
With fairy-tale charm, the romance between Elizabeth and Darcy develops, enlivened by the push-pull of two magnetic personalities. Cold realities intrude too, in the form of gossip, suspicion, money, and meddling, and several couples in this English countryside fall prey to recklessly bad romances. Step by intricate step, the mating dance is revealed with such wit and precision, the book may almost serve as a relationship primer. Ultimately hope of true love shines through, and we take joy that Elizabeth and Darcy are still living happily ever after.
"The Tell-Tale Heart" by Edgar Allan Poe (1843)
Right away, we're sucked inside a nightmare. Trapped in a house with an old man who has a sickening vulture eye. Leaping against our will into murder. Chatting with policemen while the heart beats relentlessly louder, louder, LOUDER. We sweat through a terror made all the more terrible because the psychotic villain is inside our very own mind. It's one of the most frightening tales ever told, and no one loves it more than children.
_The Adventures of Huckleberry Finn_ by Mark Twain (1885) The most engaging (and revolutionary) thing about this book is the way it's told in the believable voice of an uneducated backwoods kid. It really wasn't okay to use bad grammar and slang until this book came down the pike. What a break-the-rules joy it is rafting up the Mississippi River with Huck and his companion, the escaped slave Jim, watching these two get themselves in and out of all sorts of trouble, some vaudevillian, some quite serious. We also gain miles of insight into the less noble sides of human nature by seeing it all through Huck's innocent eyes.
"The Lady with the Dog" by Anton Chekhov (1899)
A life-weary man from Moscow begins an affair with a young woman in a seaside resort. Both are married, neither expects the affair to last. Yet, for the first time, the man is caught in the undertow of genuine love. It's a tale of adultery told in shades of gray. No one wears a scarlet A or commits suicide with a rushing train. Instead we see the quiet yearning and uncertainty of the human heart, and while waiting to see where things will lead, the suspense overwhelms.
_The Great Gatsby_ by F. Scott Fitzgerald (1925)
First there's the vicarious pleasure of spending the summer among the elite of Long Island, attending the lavish parties, listening to jazz in the moonlight. Then, like Nick, the narrator, we become intrigued by this enigma of a man, Gatsby. Who is he? Where did he come from? What does he most desire—the elusive Daisy, the kind of class money can't buy, or simply the dream of life? Like a perfectly cut jewel, every facet shimmers and hypnotizes and reflects differently upon each observation. And it's nice to be reminded that rich people aren't always happy.
"A Good Man Is Hard to Find" by Flannery O'Connor (1955)
A tough old bird of a grandmother takes a car trip through the American South with her grumpy son, his snooty wife, and their two obnoxious children. And, oh, yes, Grandma has secretly smuggled her cat along. It's a hilarious ride, bound to conjure memories of your own hellish family trips. But when the family meets up with a dangerous escaped convict, things take a sharp turn down the darkest of roads. There lies both evil and salvation.
_One Hundred Years of Solitude_ by Gabriel Garcia Márquez (1967)
The saga focuses on a single family in a single town in a single century, but the whole of history seems to roll by in a biblical flow of passion, sadness, absurdity, and miracles. The Latin American town begins an Eden but is soon overrun with business, politics, war, and a quagmire of family scandals that would make a soap opera blush. It's nearly impossible to keep track of the time or the characters or to separate the magical from the mundane, but if you "listen" patiently at the storyteller's heels, you'll see all the butterflies and blood of our world.
"The Things They Carried" by Tim O'Brien (1990)
A platoon of soldiers trudges through Vietnam. Much of the story is simply a litany of "things" the soldiers carry with them, a list that evolves from equipment to personal items to emotion, told with a mesmerizing mix of documentary fact and poetic meter. Amid the baggage, a young lieutenant ponders his past and future back home in New Jersey with a girl he barely knows. After reading this story, in some sense, you have been to war.
These stories entwine entertainment and meaning so artfully that you could read them for either of these purposes without failing to let the other get under your skin. Don't pressure yourself to write a classic, a sure way to stifle creative freedom, but challenge yourself, as these writers have done, to keep your readers breathlessly turning pages and then give them something that lingers and reverberates even after they finish that last word. Isn't that what usually satisfies us most as readers?
YOUR TURN:
Choose a work of fiction that you cherish. In a single sentence, try to state the major reason why you love reading this work. Then list several ways with which you think the author achieved this effect. The reasons don't have to employ any fancy terms and they don't have to make sense to anyone but you. You're simply trying to tune in to the source of the magic.
**SEE THE SEEDS**
"As Gregor Samsa awoke one morning from a troubled dream he found himself transformed in his bed into a monstrous insect."
— _The Metamorphosis,_ Franz Kafka
In the beginning is an idea. Ideas are seeds from which the mimosa tree or watermelon or delphinium of a story will arise. There are no rules about what constitutes a proper seed. It can be a character, a name, a situation, structure, overheard dialogue, a setting, a theme, even a vague feeling.
While passing through an obscure nook of Notre Dame cathedral, Victor Hugo noticed the Greek word for _fate_ carved in the stone. He imagined a tormented soul driven to engrave this word. From this seed sprang his monumental novel _The Hunchback of Notre Dame._
Ideas are everywhere. The writer of fiction must learn to search the world for these seeds.
Probably the most fertile place to look for ideas is right inside the backyard of your own life. Herman Melville drew on his whaling adventures for _Moby-Dick_ and Philip Roth has drawn endless inspiration from his crazy Jewish family. You've got stuff to draw on too. If you don't think so, look a little harder. There are probably hundreds of things in your seemingly mundane existence that, if looked at with a little insight and whimsy, could be turned into good material. Your home life, relationships, work, hobbies, chance encounters. Sure, the eccentric and exotic make for good stories, but so does the ordinary, especially in contemporary fiction, where the ordinary flourishes like a spider plant in ample sunlight. (See, I drew that image from my very own window.)
Even the little things in your life can spark a story. Let's say you're having technical problems with your computer, so, horror of horrors, you have to call Tech Support. Telephone hell—pushing buttons, eternal waiting, trying to reason with computerized voices, trying to explain to computer people, contemplating throwing your computer out the window. But you know what? This very situation could prove useful in a story. Perhaps a character must send a life-or-death message that can be received only by e-mail but the tech problem is making this impossible. Or perhaps the frustration of dealing with Tech Support triggers all the other frustrations in a character's life, causing a major emotional crisis, perhaps poured out to the puzzled person on the other end of the line. You see, even the tiniest seed can sprout multiple story ideas.
Flannery O'Connor said, "Anyone who has lived to the age of eighteen has enough stories to last a lifetime." Zoom in for a close look at some of the events and people in your past, things that have haunted you, things you thought you had forgotten. Remember that girl from the other side of the tracks in your third-grade class whom you and your friends made fun of, until you caught a poignant glimpse of her eating alone in the lunchroom, after which you bought her a bracelet? Good seed.
Search your thoughts. Fyodor Dostoyevsky's changing philosophical views led him to write _Crime and Punishment_ (and he was lucky enough to have spent time in a Siberian prison to help with the last section of the book). What are the things you most love? What are the things you most hate? If you were to make a list of answers to either of these questions, you would have a collection of ideas that are of passionate interest to yourself.
But the fictional version of _You_ doesn't have to be the whole story, or even any part of the story. Indeed, if you're too egocentric with your ideas, your work may take on the yawning indulgence of that person who is always trying to give elaborate descriptions of her dreams (though dreams can certainly be a rich source for stories). A good writer must keenly observe things outside of himself or, as Henry James said, develop "the power to guess the unseen from the seen." Look around at other people and imagine who they really are and what it would be like to walk around in their shoes, whether the footwear is designer heels or clunky orthopedics.
One of the pleasures of reading fiction is the way it gives a secret peek into the lives of others—those people in the passing cars or at the cash registers or on the television screens—people we may never meet. This is a bit like stealing a glimpse of a person in the nude through a window or overhearing an argument between lovers in a restaurant. Whether these glimpses are enticing or unsettling, they usually provide a certain voyeuristic thrill. On a deeper level, it's actually very comforting to see that other people are just as lost and flawed as we are. In a way, fiction is firm affirmation that We Are Not Alone.
Learn to see, and then reveal, those secret peeks, be they about someone like yourself or someone entirely different. This is another fringe benefit of being a writer. As you search for ideas, your powers of observation (and other senses as well) will intensify. The world around you will become more alive, vibrant, multidimensional, entertaining, meaningful.
Feel free to search for seeds far from home too. You can look in a newspaper any day of the week and chances are you'll find a multitude of seeds for stories. I'll do it myself, right now. Granted my local paper happens to be _The New York Times,_ but I'll bet you could do this with most any newspaper.
Let's see, on the front page there's an article about the fellow who painted those dog pictures, the most famous of which shows a poker-playing pug slipping a pawed ace to a pal. This man has won very little respect or reputation, but his art is probably better known to many people than that of Cezanne or Van Gogh. Certainly there's a story in there somewhere.
Here's an article in the sports section about a pitcher who is expected by his team to bean (hit) a batter in an upcoming game because this player beaned a player from the pitcher's team two seasons ago. But the pitcher seems reluctant. To bean or not to bean? That's a story.
Elsewhere. A faulty carnival ride left seventeen people hanging upside down for a period of time. A substitute teacher attacked his class with a broom. The obituaries report the passing of a gentleman who belonged to five country clubs. Story, story, story.
You may have heard the old maxim _Write what you know._ The advice has merit, but it's not the whole truth. If you want to write about a famous fashion model who befriends a lame penguin while on a magazine shoot in Antarctica, go for it, even if it has absolutely zero relationship to your own life. You may have to do some research on models and penguins and Antarctica, but it may be intriguing. How could it not be? And I would guess that even if you write about things totally alien to your existence, you will still be writing, in some way, about what you know. The tone or emotions or perspective will be your own. The truest maxim in this respect might be _Write what ignites your interest._
Don't shy away from surprising seeds. GWW teacher Jess Row somehow got himself stuck on the idea of echolocation, the sound-detection technique used by bats to navigate. Though he didn't really know much about echo-location, he liked the idea and began imagining a girl who believed she had the power of echolocation, which she used to pursue the spirit of her departed mother. Following this bizarre seed, Jess created the much-acclaimed story "The Secrets of Bats."
History is an incredibly rich source of story ideas. Toni Morrison heard tell of a slave woman who murdered her child to prevent the child from being a slave herself. Out of this haunting incident grew Morrison's _Beloved._ Salman Rushdie's _Midnight's Children_ was largely inspired by the contemporary history of India, the protagonist even being born at the moment of India's independence from colonialism.
So, in addition to absorbing the world that is immediately around you, feel free to absorb the world that is around you in a broader sense, even if that takes you to other time periods or the far reaches of the universe.
Ideas are everywhere, and there is literally no limit to what you can write about.
YOUR TURN:
Write down ten things that might possibly serve as story ideas, drawing from things that happened to you over the past week—people, emotions, thoughts, situations. Nothing is too big or small, cosmic or microscopic. Then review your list and pick the idea that looks the most promising for a story. The right idea will probably give you a buzz when you see it. Then list several ways in which this idea might be turned into a fictional story. Will your idea result in a brilliant story? Maybe, maybe not. But you'll probably discover how plentiful ideas can be.
Once you start absorbing the world as a writer, your problem will quickly shift from _I don't have any good ideas_ to _I have so many great ideas I can't possibly live long enough to get all of them down._ This is a wonderful problem for a writer to have.
So how do you know when you have the right idea, the one that's truly worth pursuing? Well, you'll know. People who live in New York City know that if you're riding the subway and across the aisle from you sits a wild-eyed person wearing garbage-can couture who is clearly on his way to nowhere, you should avoid making eye contact with that person because the second you do the person will lock those wild eyes onto your soul and start jabbering about apocalypse or Porky Pig or who knows what and the conversation will continue with or without your consent for an uncomfortably long period of time. Ideas are like that person. When the right idea enters your head it will loudly and persistently announce its presence. You may acknowledge its presence right away or it may take you a few days, but you'll know when the right idea has arrived.
Bear in mind, however, that a single big idea won't give you a whole story. A fictional work is really an accumulation of many ideas. A single word may have inspired _The Hunchback of Notre Dame,_ but, somehow or other, Hugo managed to cultivate some five hundred pages from this seed. He also got the idea for a noble hunchback, a gypsy girl with a dancing goat, the Festival of Fools, a mob storming the cathedral walls, and a host of other things that he skillfully wove into a multicolored, yet unified, tapestry.
Before I forget, let me remind you to write your ideas down. Most writers don't go far without a notebook handy. The "eureka" ideas may stay with you, but once you get going ideas will start popping in your head like popcorn and you can't possibly remember them all. The jottings in your notebook don't have to make any sense and you won't use every single one of them, but sooner or later some of those notes will flower into something very useful.
Now, a word of caution. The seeds you pick up from the world are just that, seeds. Once you plant your seeds in the soil of a story, let those seeds grow into fiction, not fact. Don't confine your story to the way things really were, in a literal sense. Fiction demands better storytelling than real life, even if the fiction seems perfectly "real." Often beginning fiction writers stick too rigidly to the facts and inevitably their stories feel a little flat or indulgent. (True, many memoirs are quite compelling, but one reads a memoir with a different set of expectations than one reads fiction with.)
Lorrie Moore's "People Like That Are the Only People Here" is a somewhat autobiographical story about a mother seeing her baby through a terrible illness. The real-life seed idea was certainly emotional enough. But Moore chose to fictionalize the story (even though the protagonist is a writer) so she could further deepen the story's impact. If she had not done so, the story probably wouldn't have the shape, suspense, clarity, irony, and humor (yes, humor) that make it so unforgettable.
The fiction writer must water the soil with imagination until the story yields the maximum amount of entertainment and/or meaning. The goal is to write a story that a total stranger will enjoy (though some writers find it helpful to picture someone they know as the total stranger). The stranger reading your fiction won't care about you or your life or your observations. Not a bit. All the stranger ultimately cares about is a great story well told.
Will you be sacrificing honesty along with the facts? No. By bending reality into fantasy you are not lessening the Truth inherent in your idea. Rather, you are increasing it. Life is a blur in which it is difficult to see anything clearly because a zillion things are going on all at once. Art is all about sharp focus.
_Attempted Theory #1:_ Hey, back to our initial question. Perhaps this is the big reason why we write fiction—as a way of understanding ourselves and the world around us. The fiction writer takes a fragment of reality and examines it from several angles until it starts to make some damn sense. By focusing life through the lens of fiction, truths are revealed and magnified and understood. Order is made from chaos. It's like therapy but cheaper and more fun, and perhaps even more effective.
Yes, a good answer. But not the complete answer. If fiction were such an effective method of making sense of things, why am I generally more confused than, say, my blissfully satisfied sister who has never written a word of fiction in her life? I sense there is a more definitive reason out there, perhaps something like Descartes's sweeping proof of human existence: "I think, therefore I am." So... let's continue examining the creative process of a fiction writer.
**SHOW UP FOR WORK**
"If I'd 'a' knowed what a trouble it was to make a book I wouldn't 'a' tackled it, and ain't a-going to no more."
— _The Adventures of Huckleberry Finn,_ Mark Twain
At some time or another, most everyone fancies they have a story to tell. As you may have just learned, ideas are rather cheap and easy to come by. But few of these people actually manage to get the story on paper, fewer still stick with it long enough to go through several drafts, and fewer still move on to complete numerous stories.
For a work of fiction to exist it must get written down. For it to be any good, a lot of work must be done. If you want to be a great writer and you have a choice between being brilliant and lazy or being a little clueless but motivated, choose the latter. You stand a far better chance. Sure, such intangibles as creativity, talent, and inspiration play a role, but work is where the real action is.
The best way to get good at writing fiction is to write and write and write. Do it enough and you can't help but get better. Watch a kid playing a quick-reflex computer game. You will never be that adept at the game because you will never put in that many hours practicing. Go to a lake and try skipping stones across the water for a few hours. I guarantee your skipping skills will improve. Frequent practice is how I made the passage from the worst writer who ever lived to someone who occasionally turns out something worth reading, and most of the fine writers I have the pleasure of knowing will tell you the same thing.
Much of your improvement will be so incremental you won't even notice it. But then one day, after x number of months or years, voilà, there is bound to be a moment where it seems a switch was flicked, instantly changing you from Bad to Good. You'll cruise through, instinctively knowing where to turn and how to hug the curves and when to downshift and accelerate. It's an exhilarating ride and well worth the long wait.
If you're serious about creating fiction, you should set aside designated writing times, preferably most days of the week. If you leave it catch-as-catch-can, it will become all too easy to catch nothing. Some writers prefer the first blush of morning, others opt for the graveyard shift. Find the time at which you feel the most free and stimulated.
Force yourself (and the other people in your life) to stick to the schedule. It's actually more important to stick to the schedule than it is to write something wonderful at these sessions. Indeed, if you work for five hours and end up with zilch, you've still done your work for the day. (Hey, creative writing isn't like most jobs.) If you simply show up for work every time, you will have developed a discipline that will become your greatest strength. Eventually progress will be made. Some writers measure their time in pages, not hours, but unless you've got plenty of spare time, you may not want to put this pressure on yourself.
Find a place where you feel comfortable creating. Don't worry if you don't have a study with bay windows on the coast of Maine. A section of your den or a stretch of your bed can work just fine. Most writers need solitude, but some prefer the stimulation of public places. I know a professional writer who uses the corner table at a little French café as his office. He works in the cafe every day all day long, and even takes meetings there. He doesn't even order very much.
A regular time and place for writing seems to be the key for most writers. Then again, Joyce Carol Oates claims to have no regular writing habits and she turns out more fiction than, well, just about anyone. So the real trick is to find what works best for you.
YOUR TURN:
Create a week-long writing schedule for yourself, encompassing at least five hours of writing time, with the installments lasting at least an hour each. Work on a piece of fiction using this schedule for a full week. (If you have a story in progress, use that. If you need a story idea, you'll find plenty of "triggers" in the exercises throughout this book.) The idea is to utilize a writing schedule, not write a masterpiece, so don't worry about the result. When the week is done, analyze how well the schedule worked. If the schedule needs adjusting, do so. If your discipline needs adjusting, do so.
There are really two kinds of writing time, what I call _hard time_ and _soft time._ Hard time refers to what is normally thought of as "writing"—at the computer screen or typewriter or pad of paper. Obviously, you'll need to put in plenty of hard time since this is where the words get written. Soft time refers to the time when you are not actually writing but pondering your work. This can happen anywhere—walking the dog, buying groceries, losing money at the casino. Pondering is a major part of the job, which is one of the cool things about being a writer.
When I started writing, I used to stare at the blank page until my forehead bled (to borrow an oft-used metaphor). I thought that's what writers did. Yes, it was torturous, but I took a certain masochistic pleasure in it. As I began to write professionally and get better, my technique shifted. I began doing more soft time in the early stages of a project, letting my mind wander in a leisurely manner. Perhaps I would do relevant research or have conversations with people about my ideas. Perhaps I would just ponder. I would take notes and maybe even write fragments here and there. After a while, I had an abundance of ideas about my story. Then... I took the story into hard time. The work flowed with relative ease. And it was better. I seldom needed to bandage my poor forehead.
I also recommend soft time as a tool, like a crowbar, to help out when you're stuck. If you absolutely can't figure out the resolution of your story or how to describe the mysterious stranger in the alley, don't wear yourself out with worry. Go do something else while keeping the problem somewhere in your mind. Or do some really soft time where you take a complete break from your work. Instead of desperately searching high and low for that elusive solution, let the solution come to you. Like a runaway cat that grows hungry, that solution will return home when it's ready.
Perhaps the very best time for soft time is in bed at night. As you're lying there, easing into sleep, let your mind play over some aspect of your story, maybe something you're having trouble with. You'll be amazed at what brilliant ideas sneak in at this relaxed moment. You should keep a notepad and pen by your bedside and you should make yourself jot down these ideas or they may easily disappear into the mist of your dreams. Sometimes the idea is so inspiring you simply must spring to full alert and get some writing done.
Also be aware that there are two types of writer inside every person. There's Free Spirit, who wears flouncy attire and meditates at ashrams and never stops her kids from crayoning the walls. She writes whatever and whenever she wants and doesn't give a fig what anyone thinks of it. Then there's Stern Editor, who wears button-down shirts that peek out exactly one inch from the sleeves of his double-breasted suits. He won't allow a single word that isn't necessary and he can be a stickler for logic and grammar. (He's a lot like GWW Fiction teacher Peter Selgin, who you'll meet in the Revision chapter.) Both of these people are crucial to a story's success. But they seldom see eye to eye so it's best to keep them separated.
In the early stages of a work, banish Stern Editor from the room and let Free Spirit reign in chaos. Much of the work will be fragmentary, rambling, and incoherent, but you will be tapping into that deep well where your thoughts are both wise and childlike. Pump that well until you sweat. Many writers swear by the virtues of freewriting, which means you can write whatever you want just as long as you don't lift your pen from the page or your fingers from the keyboard. Free Spirit loves freewriting. At the very least, it's a great kick-start for those times when you're stuck in the mud. Sooner or later, something interesting will emerge.
YOUR TURN:
Take this opening phrase: _Sam wasn't sure if it was a wonderful sign or a sign of disaster but Sam knew._ . . Write down that fictional opener, then keep going. Freewrite, meaning write without stopping or even thinking too much, just scribble away however things come out. You should write for at least five minutes but feel free to go as long as you like. No one will see this but you, and you have permission for this to be nothing but gibberish. Just feel what it's like to write in a white heat.
At some point, perhaps not until after you've dashed off a complete first draft, you'll send Free Spirit out for some herbal tea and invite Stern Editor in with his set of finely sharpened pencils. Oh, he'll make you cut and correct and shape and answer a bunch of difficult questions, but pay attention, because your future readers will be every bit as demanding as he is. Then you'll probably alternate these two helpful sides of your psyche for a while, letting each have his or her say at what you deem to be the proper times. Toward the end of the process, submit yourself to Stern Editor's iron law while Free Spirit is off prancing in the meadow, hopefully conjuring up your next big idea.
_Attempted Theory #2:_ Ah, perhaps the answer lies here. Maybe writing fiction is akin to those personal challenges we call recreation, such as playing golf or climbing mountains or doing crossword puzzles or building ships inside bottles. These things are rewarding because they're not so easy. They awaken us by making us feel the vibrations of our inner potential, regardless of the outcome. Writing is one of the best possible personal challenges because the room for growth is as limitless as outer space and you're never too young or old to give it a go.
Could this be the big reason we want to write fiction? Uh, no, wait a second... I know many writers who say they enjoy having written something much more than the actual writing of it and will clip their toenails twice in one day to procrastinate. How would they fit into this explanation? Sorry, let's keep going.
**DON'T BE A CHIMP**
"If I could have reached my rod I would have blown his guts out."
— _The Big Kill,_ Mickey Spillane
So let's see where we are in the creative process. Promising ideas + hard work = good fiction. Well, not quite. Something is still missing.
To tell a story effectively, you will need some mastery of _craft._ By craft we mean the time-tested practices that have proven helpful to the construction of good fiction.
Good writing comes down to craft far more than most people realize. True, anyone can write a story without training, which separates fiction writing from such activities as performing heart surgery or piloting a helicopter. But a working knowledge of craft is almost always necessary to make a story really good, worthy of being read by all those strangers. You could build a chair without any knowledge of woodworking because you have a good idea of what a chair is like. You would cut the wood and hammer the pieces together, and sure enough you would have a chair. But it would probably be wobbly, unsightly, and destined to break. It certainly wouldn't sell. The same is true of fiction.
You should learn craft because it works. The "rules" of fiction craft weren't created by any one person in particular. They simply emerged over time as guiding principles that made fiction writing stronger, in much the same way the mortise-and-tenon joint emerged as a good way to join parts of a chair.
Let's say you learn that it's better to _show_ a character trait than to _tell_ about it. _{Show, don't tell_ is something of a fiction mantra, like the carpenter's _Measure twice, cut once.)_ So you go back to your story in progress, cross out the line "Kathy was a dishonest woman," and insert a moment where you show Kathy doing something dishonest. Perhaps Kathy realizes the teenage cashier has given her ten dollars too much in change but Kathy slips the bill in her purse without a word. Most likely the dishonesty trait will be illustrated more dramatically, more memorably. We'll gain a more dimensional sense of Kathy as a "real" person. If dishonesty comes into play later in the story, we'll be better prepared for it. You haven't grown any wiser or more intrinsically talented. You've just picked up some _craft_ And craft makes all the difference.
In addition to making fiction better, knowledge of craft can actually make the writing easier. There is a theory that if you put a bunch of chimpanzees in a room with a bunch of typewriters, eventually one of them will tap out _Hamlet._ I have some doubts about this theory but I will say this: if those chimps know something about craft, they will get there faster. When you work with craft, you're not floundering so much, waiting to stumble accidentally into something good. Once you have some craft at your fingertips, you'll look a lot less like one of those chimps, showing teeth and screeching as you maniacally play with that toy of a keyboard.
Now, I can hear what some of you are thinking. You don't want to trod the familiar paths. You're a rebel, busting to blaze some new ground, perhaps with purple ink. Good for you. But get this: you can actually break the rules better if you know a little something about them in the first place. Take Frank Lloyd Wright. He reenvisioned architecture, creating buildings and spaces that seemed to emerge out of their natural surroundings as if they were always there. (He also served as the model for the maverick architect in Ayn Rand's _The Fountainhead.)_ Yet Wright knew the structural principles of architecture through and through, and this is why his Imperial Hotel in Tokyo was one of the few buildings that withstood the Great Kanto earthquake of 1923.
Rules are made to be broken. If you follow the rules of craft too scrupulously, you'll likely end up with a story that's more of an A+ assignment than a vibrant work of art. The great writers usually break a few rules, and the greater the writer the larger the transgressions. James Joyce opened the floodgates of the human mind, letting the prose gush higher and farther and weirder than ever before. Ernest Hemingway kicked all of what he called the "bullshit" out of prose (even when he wrote about bulls), making prose simpler than was ever thought acceptable.
GWW Fiction teacher Brian Dillon likes to give each of his students five pieces of different colored candy on the first day of class. Each color represents an element of craft—red/character, black/plot, green/dialogue, orange/point of view, yellow/theme. After explaining how each color corresponds to its craft element, Brian tells his students to eat the pieces of candy. There is a point to this fiction-class version of the Eucharist. The teacher wants his students to digest the elements of craft, merge them into their systems, which is very different from slavishly following the rules.
This book will mostly focus on craft. After reading this book once or twice (and doing the nifty exercises), you will have some familiarity with all the major craft elements that go into fiction writing. They are easy enough to pick up, though you can spend a lifetime learning to manipulate them to the satisfaction of yourself or the world at large. And, believe it or not, this book is not the only place you will learn about writing craft.
As late as 1920, there weren't many, if any, good books on writing craft and there were virtually no educational programs devoted to creative writing. So where did James Joyce, Thomas Mann, Willa Cather, F. Scott Fitzgerald, Ernest Hemingway, and Gertrude Stein, all of whom wrote at that time, learn their craft? They read. They read a lot. And they analyzed the hell out of what they read. They also discussed things with other writers, often in cool expatriate locations, and that's also a good thing to do.
Painters often learn their craft by studying the masters, and writers should do the same. If you want to see how to imbue a story with the dirt and essence of its setting, read William Faulkner's _The Sound and the Fury._ If you want to see how to manipulate perception through point of view, read Henry James's _The Turn of the Screw._ If you want to see how to turn everyday life into the rising action of a plot, read Raymond Carver's "Cathedral."
Read widely and adventurously. There is no telling where you might pick up something useful. You should read some of the so-called great writers, but, if you're so inclined, don't hesitate to study those lesser lights who might make a literature professor sneer. Perhaps Mickey Spillane, creator of the _Mike Hammer_ detective series, didn't quite have Edith Wharton's craftsmanship, but he probably knew a few tricks.
If you read a story that leaves you a little cold, ask yourself why. What's missing? Is the plot not quite plausible? Does the story feel pointless? Are the sentences too show-offy? Sometimes you can learn volumes from a story you don't like. And it's fine, by the way, if you find yourself bored by some work that everyone else seems to think is The Greatest Story Ever Written.
Trust your own taste. As Duke Ellington said about music: "If it sounds good, it _is_ good." At the end of the day, you should be writing the kind of thing that you enjoy reading. Figure out why you like what you like, then try to utilize some of the techniques that will help you get there. Following your own taste is a good way to let your true self emerge, as long as you manage to draw a line between emulation and imitation.
YOUR TURN:
Return to the work of fiction that you chose as a favorite. Get your hands on a copy of this story, then pick a passage that you especially like. Write out a page or so of this section, word for word, just to let yourself feel what it might have been like to create that particular arrangement of words. You may gain some insight into _how the_ author did what he did. At the very least, you'll see that everyone does it the same way—one word at a time.
Once you start mastering your craft you'll be on your way to accomplishing what has been achieved by all the writers you've admired through your own reading, on your way to holding strangers in thrall with your fiction. Out of nothing—literally nothing but the invisible vapor of imagination—you will create stories that tickle, torment, intrigue, inform, entertain, and maybe even _change_ your readers. All with words alone. Your words.
_Attempted Theory #3:_ Aha! Perhaps that is the real reason why we write fiction. The satisfaction, nay, the intoxication, of creating something we sense will soon captivate legions of readers, making us like Scheherazade, who (over the course of a thousand and one nights) transformed her murderer into her bridegroom through the sheer hypnosis of a tale brilliantly told. Who knows what earthly delights may follow—prestige, adulation, fame, money, sex, travel, the respect of our parents. And now that we're getting egotistical about it, let's face it, great fiction could give us our one real shot at the impossible—immortality.
**THE BIG ANSWER**
Forgive me if I got carried away back there. Sometimes we scribblers do indeed become drunk with power. But now I see, with some humility, that I haven't really succeeded at my task. The flaw in my last theory is that some people write fiction with no real intention of ever showing it to anyone, and often they're deliriously happy doing so. Alas, I've come up with several possible and plausible reasons why we write fiction, good ones at that, but it seems the "smoking gun" is still missing.
So I asked a bunch of GWW students and teachers why they write fiction, in the hope of finding an obviously dominant answer. Here are some of the responses:
I like blank paper.
To meet people I find interesting.
Writing puts me into a world that has not been written yet.
I spend much of my time contemplating love and death.
When I am writing a surge of complete happiness takes over.
To make readers hear the sound of their own heartbeats, that sound that whispers up to us: you are alive.
When I manage to turn pages and pages of crap into a little bit of art, I feel like that girl in the _Diamonds Are Forever_ ad.
Writing gives me permission to be a child and to play with words the way that children play with blocks or twigs or mud.
Writing makes me a god, each new page enabling me to create and destroy as many worlds as I please.
It allows me to spy on my neighbors.
It's the only socially acceptable way to be a compulsive liar.
I want to cleanse the past.
To discover, to express, to celebrate, to acknowledge, to witness, to remember who I am.
I find out what might have been, what should have happened, and what
I fear will happen.
It's a means of asking questions, though the answers may be as puzzling as a rune.
This question drives me crazy.
There is nothing else I want to do more.
My soul will not be still until the words are written on paper.
Because I can.
Because I must.
I can't not.
If I don't I will explode.
I want to be good at something and I've tried everything else.
Oh, well. The question took me on such an interesting journey, I suppose I no longer care about reaching the desired destination. Suffice it to say there is no ultimate answer for why we want to write fiction. It's as mysterious as everything else about human nature, and if human nature weren't so mysterious we probably wouldn't need to write (or read) fiction in the first place.
[CHAPTER 2
**CHARACTER: CASTING SHADOWS**](Facu_9781596917910_epub_c4_r1.html#aa2)
BY BRANDI REISSENWEBER
When I taught creative writing on a pediatrics ward at a hospital I met a long-term patient, a thirteen-year-old girl who had been in and out of the hospital since she was two years old. She was sharp and witty but rarely ever wanted to write with me, no matter how enticing the writing project. She eyed me from the corner of the hospital playroom as I wrote with other young people, but every time I'd approach her she would send me away, telling me that, after all, the hospital wasn't school.
One day, I found her reading a book in her room. I sat down and asked if she would read to me, which she did. That afternoon, I learned that she loved to read books, so we talked about some of our favorite stories. I asked, thinking it was a simple question: "Why do you enjoy reading?"
She looked at me, scratched her shortly cropped hair, and then opened her book again. I thought she was through with me as her eyes began to follow the lines on the pages. After a few minutes she looked up at me and said: "Because I get to meet lots of different people."
We eventually wrote a story together. It was fantastical and full of the kinds of people she wanted to be around: those who could fly, aliens who would befriend her, people who were outrageous, graceful, and courageous, just like her. But what stuck with me most was her response to my question—that she read to meet people. That answer to what I thought was a simple inquiry lies at the heart of good storytelling.
When you read fiction, you are, first and foremost, meeting people. Characters are the core of a story and interact with or influence every other element of fiction. Characters are what drive a story, carrying the reader from the first to the last page, making readers care. How exciting would Ken Kesey's _One Flew Over the Cuckoo's Nest_ be without Randle McMurphy, the rabble-rouser asylum patient who shakes up the system? Without Miss Amelia, the self-reliant and cross store owner who is unlucky in love, Carson McCullers's "The Ballad of the Sad Café" would be about a dull, dusty town. And without the mysterious and glamorous Jay Gatsby, F. Scott Fitzgerald's _The Great Gatsby_ would be far from great.
Good writers create a sense that their characters are people—physical, emotional, living, breathing, thinking people. The more you manage to make your characters feel real, to create the illusion of an actual person on the page, the more likely your reader is to fall into the story, past the language and the words, letting the real world recede and be replaced by the fictional world you have created. As a writer, you want your reader to feel that your characters are substantial, authentic, dimensional. Real enough to cast shadows. Creating characters that seem dimensional and lifelike requires some artistry, to be sure, but with a little knowledge such a miraculous feat is entirely possible. Let's examine the process.
**THE BEAT OF DESIRE**
Desire beats in the heart of every dimensional character. A character should want something. Desire is a driving force of human nature and, applied to characters, it creates a steam of momentum to drive a story forward. You may create a character with quirky habits and high intellect and vague tendencies toward adventure, but if all he does is sit on the couch and snack on lemon squares, the reader is going to find more excitement in thumb twiddling. Give that same character a desire to travel from Florida to Maine in a hot-air balloon and that begins to propel the story into motion, especially if the character doesn't know how to acquire or pilot a hot-air balloon.
A character's desire can be huge, looming, and intoxicating, like the desire to ease loneliness, to seek the revenge of a son's death, or to climb to the peak of Mount Everest. Or the desire can be smaller and simpler: to find a wedge of stellar Brie, to escape the complaining of an ailing wife, or to coax the orchids into finally blooming in the backyard garden.
The grandness or simplicity of the desire is not important as long as the character wants it badly. In Katherine Anne Porter's short story "Theft," the main character simply wants to retrieve her empty purse. However, this desire is rendered important to her and, therefore, it is important to the reader. A strong desire helps the reader identify and sympathize with the character, whereas a character without a strong desire will bore your readers, a great way to get them to abandon your story for good. After all, why should the reader care about a character retrieving a purse if she only _kind of_ wants it back?
One of the benefits of spending time drawing a main character who has a strong desire is that the story line will grow organically from the character's need. In Vladimir Nabokov's _Lolita,_ for example, the main character, Humbert Humbert, desires nymphets (his word for beautiful preteen girls), a category in which the youthful Lolita reigns queen. The story grows out of Humbert Humbert's attempts to possess Lolita's body and affections. If he didn't crave Lolita with such fierceness, there would be no story.
What happens when characters don't have desire? I once had a student who wrote a piece about two boys exploring their grandmother's vast mansion. The reader followed the children up creaking stairs, into the dank attic, behind a false wall, and past steamer trunks filled with old photographs.
The description was wonderful, but the story fizzled out quickly. Why? Because the characters didn't want anything. They were content with their adventures. Description, no matter how brilliantly crafted, cannot carry a story. What the piece needed was a driving force, a momentum to thrust the reader forward. If the boys found themselves trapped in the attic, the summer sun quickly heating up the small space to an unbearable degree, creating a desire for escape, that would have added some tension and interest. The gears of the story machine don't start swirling into action until characters have a desire.
Desires, however, are not always as straightforward as they are in Porter's "Theft" and Nabokov's _Lolita._ For instance, in Raymond Carver's "So Much Water So Close to Home," Claire attempts to come to terms with her husband's decision to continue with a fishing trip even after he and his buddies find a young woman's dead body floating in the river. They secure the woman's wrist to a tree trunk so her body won't float away, and it is not until they are on their way home, when they can conveniently reach a phone, that they report the body to the police.
The story revolves around Claire, who wants to understand what has happened on this weekend, and why her husband and his friends dealt with the body so insensitively. Out of Claire's intense desire to seek understanding, she questions her husband, which leads to arguments between them. She scours the newspapers for information and travels to attend the young woman's funeral. She begins to sleep in the guest bedroom and is awakened one night to her husband breaking the lock on the bedroom door, simply to prove that he can do so. Claire's desire to understand and come to terms with the decision her husband made is complex and circuitous, but no less compelling.
YOUR TURN:
Think of a character. If that's too vague, make this character some kind of performer—actor, singer, magician—who has hit middle age and is finding that his or her career is now mostly faded glory. Or use a parent or child who is having difficulty with his or her own parent or child. Then think of a specific desire for this character. One driving desire. Make the desire something concrete—money, a career break, the touch of a certain person—instead of an abstract desire like love or personal growth. Once you find the character and desire, jot them down. We'll be coming back to this character shortly.
HUMAN COMPLEXITY
Nothing is less compelling in a story than a character who acts like a million other characters you've encountered, exhibiting only one facet: the kindly grandma, the sinister janitor, the heroic patient. It's easy enough to fall into this trap because it's so easy to see people as types, at first. For example, the well-situated investment banker—are you picturing formal suits and a furrowed forehead? Long hours, lots of technical gadgets like handheld palm pilots and cellular phones, and a whole lot of excess cash? That's a good place to start with this character, but your specific character should transcend this type. Richard, an investment banker, might have a lot of excess cash and work long hours, but maybe he also makes anonymous donations to the local Humane Society. On evenings when he's had too much imported beer, he calls his sister's last known number, even though she hasn't lived there for at least a year.
Such distinctiveness makes Richard different from any other person who might fall into his type. When you create characters, explore the specific and unique details that will make them complex; not a type but a real person. We carry with us our histories, our experiences, our memories, each of our bundles distinctly different from anyone else's. Craft characters in the same fashion.
In Joyce Carol Oates's "Where Are You Going, Where Have You Been?" Connie, the main character, is a fifteen-year-old girl who is self-absorbed and insecure. Yet Oates didn't stop there. She gave Connie qualities that raise her beyond this type. Connie has a "high, breathless, amused voice which made everything she said sound a little forced, whether it was sincere or not." She easily abandons her friend when a popular guy comes along. She has disdain for her sister for still living at home at twenty-four years old and for not being as beautiful as Connie. She is convinced that her mother likes her better because of her beauty and that their arguments are just a front, "a pretense of exasperation, a sense that they were tugging and struggling over something of little value to either one of them." In the end, when her family is away at a barbecue and the sinister Arnold Friend pulls up and threatens to harm her family if she doesn't submit to him, Connie eventually relents, sacrificing herself by walking toward Arnold, toward what she "did not recognize except to know that she was going to it."
From the small details, like the sound of her voice, to the larger details of the ultimate decision she makes, Connie is revealed as a complex character. Not every fifteen-year-old teenager possesses this combination of traits. Not every fifteen-year-old would make the decision to walk out that door at the end of the story. Connie is not a type. She is a dimensional character, substantial enough to cast a shadow.
Writers are sometimes drawn specifically to the allure of the all-good or all-evil character, which is another version of the typecast character. Unless you're writing a fairy tale, you'll want to avoid these extremes. Aristotle once wrote that a character should be one "whose misfortune is brought about by some error or frailty." Whether your characters meet misfortune or not, flaws will make them more interesting and authentic.
Frankie Machine, the main character in Nelson Algren's _The Man with the Golden Arm,_ a card dealer in the nighttime circuit of Chicago's bars, is a predominantly good character but certainly not without flaws. The most obvious flaw is his frantic struggle with drug addiction. Also, he seeks solace in an affair with Molly-O, avoiding his wife, who is cooped up at home in a wheelchair. And the crushing guilt Frankie feels over his wife's immobility eventually causes him to leave her.
Those characters who are _not_ fundamentally good should also be rendered with multiple facets. Bad guys aren't bad every single second of the day. Sometimes, they're just hanging out eating their take-out Chinese food, or waiting in line with their car at the car wash, or even doing something kindly, like helping an old lady pick up apples that have fallen from her grocery bag.
_Lolita_ stirred a lot of controversy when it was published and Nabokov spent quite a bit of time insisting that his own knowledge of nymphets was purely scholarly, unlike the fictional Humbert Humbert, who molested young girls. In _Lolita,_ Nabokov committed one of the toughest acts of the fiction writer: staying true to the humanness of a reprehensible character. Humbert Humbert is as disgusting and deplorable a character as any ever written and it would be easy to cast him in a light that shows him as only horrid. Yet Nabokov allows him some appealing traits: decided charm, dazzling intelligence, a sense of shame for his weakness, and, ultimately, a genuine love for Lolita.
Literature is filled with great villains. Part of what makes them so compelling is the tiny bit of ourselves we can see in them. Usually there is something, however small, that a reader can relate to. In _Lolita,_ the reader can relate to Humbert Humbert's inability to resist a desire he knows is wrong. Although Humbert's desire is extreme, that basic idea of wanting and indulging in what you shouldn't—be it greasy foods, cigarettes, or too much mind-numbing television—is a very human trait. In showing Humbert Humbert as something more than an inhuman monster, the impact of his misdeeds is much more powerful.
YOUR TURN:
Recall the worst person you've ever met. A psychotic boss, a back-stabbing friend, a playground bully. Or make someone up. Next, assign one redeeming quality to this character—kindness, courtesy, sympathy, a fondness for animals. Then write a passage with this person in action. Perhaps you show a sadistic ex-spouse helping a homeless person find shelter, or a bank robber arranging a baby-sitter on behalf of a woman he's just tied up. The result? A fully dimensional villain.
CONTRASTING TRAITS
A fascinating element of human nature is that we all possess contrasting traits, sometimes subtle, other times greatly conflicting. These contrasts provide endless opportunity to make your characters complex.
We see this very clearly in Oates's "Where Are You Going, Where Have You Been?" where Connie behaves one way with her family and another way when she's out with her friends:
_Everything about her had two sides to it, one for home and one for anywhere that was not home: her walk that could be childlike and bobbing, or languid enough to make anyone think she was hearing music in her head, her mouth which was pale and smirking most of the time, but bright pink on these evenings out, her laugh which was cynical and drawling at home_ —" _Ha, ha, very funny"_ — _but high-pitched and nervous anywhere else, like the jingling of the charms on her bracelet._
Details such as her girlish walk at home and her fluid saunter outside of home show the reader Connie's dual personality, which hints at both her childlike innocence and her devious, secretive side. This contrast in Connie's actions helps to reveal the complexity of her character, to show the struggle with her own identities—who she was as a young girl and who she will become as a woman.
You might think of contrasting qualities as places where the characterization is unexpected or not quite matching up. A college football hero bravely battling leukemia need not be cowardly in order to exhibit contrasting qualities. A more subtle option would be preferable. His pride could be wounded by a bad call in the game, or he could drive recklessly while his younger brother is in the passenger seat, or betray a friend's secret. Any of these would add a level of complexity to the character without being too predictable.
Regardless of what kind of contrasting traits you give your characters, keep in mind that contrasts do not leap forward and say, _Here I am, a contrast! Revel in my humanity!_ The best contrasts are so seamlessly sewn with the characterization that they're not easy to spot; they seep into the characterization. The reader should experience the tension, not be spotting contrasts like stop signs along the road.
YOUR TURN:
Return to the character for whom you created a desire. Now give this person two contrasting traits. Let's say you chose an actress hoping to win an audition. Maybe she's overly considerate to people but turns into a witch if she feels slighted by someone. Jot down the contrasting traits. We'll be coming back to this character again soon.
CONSISTENCY
Unless your character swallows some kind of Jekyll/Hyde potion, you don't want to have him or her behave one way for most of the story and then change with just a snap at the end. All actions and behaviors should seem authentic and true to the character based on what you have established. Contrasting qualities are important, but your characterizations should still be consistent. Hmm... That seems like a contradiction: our characters should have contrasting qualities, but should be consistent.
Here's the difference. Contrasting qualities are moments of humanness. Keeping a character consistent is a general issue of good characterization. Rod may be a cocky, angry guy, defensive when someone asks the time, a person who cuts to the front of a long line at the convenience store when he has only one item. But perhaps he also stops to look, with interest, at a poster for a carnival. Something is going on there. Maybe he's putting on airs with his defensiveness and he really does like the fun, carefree things in life. Or maybe the poster reminds him of something he did with his own son before he divorced his wife and they moved to another state. Either way, it's a contrast in action: he projects an angry persona but shows interest in a carnival, a happy and rather playful event. It's believable that he could have these conflicting traits. However, if we see Rod as self-assured, gruff, and dismissive for the first nine pages of a story and then, on page ten, he starts wandering around a carnival, excited to ride the Ferris wheel _for no reason at all,_ then the character has not stayed consistent.
Characters can do something "out of character" as long as you show the reader a glimmer of that tendency ahead of time. If a shy character who usually plays it safe does something courageous or risky, the reader needs to see where this is in his realm of capability before it happens.
Nothing is worse than walking away from a story thinking there is no way _that_ character would have done _that_ action. Even if readers are very surprised by what your character does (which is a good thing), the characterization should still be consistent.
In "Queen Devil" by Kathy Hepinstall, Nick, the narrator's brother, does something utterly surprising—he shoots and (presumably) kills his wife. Quite a shock. Nick, up to that point, had been crafted as a man who deeply missed his wife and children, who recently left him. He's the kind of guy who, seeing his daughter's pink hair comb in his fishing tackle box, has this reaction: '"My baby,' he says. 'My sweet girl.'"
What makes Nick's character consistent and believable? Despite Nick's genuine affection for his family, the intensity of his anger is apparent throughout the story. He talks about his wife and his resentment toward her. He drinks a great deal. Both of these indications of instability make Nick a volatile character and show the angry energy that simmers below the surface.
Some of his dialogue with his sister, Jill, betrays what he is capable of. For example, when his wife leaves, Nick calls Jill: "When Nick's wife told him she was leaving him, he called me up and was not my brother, suddenly. He said, 'Remember, I'm a hunter. Remember, I have a closet full of guns.'"
Later, he says of his wife: "She won't get away with this." He then quickly backtracks when Jill questions him: '"All I mean,' he says, the sweet tone in his voice back, 'is that she'll miss old Nick someday.'"
So these desperate ideas have been, in one way or another, roaming around in Nick's mind and the writer makes sure that the reader senses that. His motivations for his final, violent action are clear.
You do want a sense of surprise in your characterizations; it's part of what creates a satisfying sense of journey and discovery in a story. We see this kind of thing often in fiction: the humble and submissive Chief Bromden escapes from the oppressive ward at the end of _One Flew Over the Cuckoo's_
_Nest;_ the hardened, bitter Joy in Flannery O'Connor's "Good Country People" removes her wooden leg, making herself feel completely vulnerable to the Bible salesman she at first disdains. You want your characters to be consistent, but you don't want characters who are completely straightforward and predictable, so much so that they are incapable of discrepancy or change.
**THE ABILITY TO CHANGE**
Characters should possess the ability to change, and the reader should see this potential. Change is particularly important for a story's main character. Just as the desire of a main character drives the story, the character's change is often the story's culmination. While a main character usually does change to some degree, either dramatically or in the more gentle form of a realization, this does not mean your character actually has to make a change at the end of the story or that the change has to be whole and complete. However, the reader should see that the character is capable of doing so throughout the story—the choice should be there. If you don't create the potential for change, the character will feel predictable and the reader will quickly lose interest. That puts you back on the couch with the character who has no desire, just hanging out and eating lemon squares.
In Anton Chekhov's "The Lady with the Dog," the main character, Dmitry, changes drastically. At the beginning of the story, his attitude toward women, who he deems "the inferior race," is dismal, although he recognizes his inability to spend more than two days without them. He finds his wife of "limited intelligence" but is fearful of her and has many fleeting affairs. The story centers around Dmitry's desire for Anna, a desire that starts out calmly, when he first hears about the buzz of the new woman in Yalta, "the lady with the pet dog," and builds in intensity through their affair. By the end of the story, Dmitry feels he is truly in love for the first time in his life. The story ends with Dmitry holding his head in frustration, wondering how he and Anna will be together. His attitude changes from women being expendable to finding this one woman absolutely necessary.
Going back to _Lolita,_ the reader knows that Humbert Humbert is capable of not pursuing nymphets—not that it would be easy or desirable to him to change his ways. However, the reader sees that he is capable. He holds back and resists during times when he must, such as when Lolita's mother is around. He laments his own desire for nymphets and wishes that he could resist more often. In the end, he returns to Lolita when she is seventeen and pregnant, nowhere close to her former nymphet self, and he finds he is still in love with her. He's changed in that he loves Lolita beyond her nymphet status, but there's little doubt that, left alone with another young nymphet, he would likely try to quench his desire. So, Humbert Humbert does change, but not entirely.
YOUR TURN:
Return to the character for whom you have created a desire and contrasting traits. Time to bring this person to life. Write a passage where this character is pursuing his or her desire in some way. For example, perhaps the actress is traveling to an audition to which she was not invited. (Oh, yes, it'll help tremendously if you put some obstacles in the character's path.) You don't have to bring this "quest" to a conclusion, but have something happen that allows both contrasting traits to emerge and also try to include some hint that the character is capable of change. That's a lot of juggling, so don't worry if it comes out a little clumsy. Dimension doesn't always happen overnight.
**WHERE CHARACTERS COME FROM**
To craft fascinating and memorable characters, you will need a starting point. So where will your characters come from? Look around you, and into your memory and mind. Your characters will emerge from people you know and see and even imagine. Inspiration for characters is everywhere.
Writers often construct characters by beginning with interesting people or characteristics of people they know. Some writers even start with their own personality as a basis for a character and build from there. Nelson Algren immersed himself in the downtrodden of the Chicago he wrote about in _The Man with the Golden Arm._ He likely based the main character, Frankie Machine, on a man named Doc who dealt at the poker games Algren played in on Division Street. Parts of Algren went into Frankie too.
Like Frankie, Algren served in the army during World War II, and when he was discharged he came home with very little in his pockets. Starting with people you know, including yourself, lets you create characters that spring from a strong foundation of knowledge and intimacy.
However, keep in mind that when you deal with real people, you have to leave room for creative invention. I've worked with writers who say their character is based on someone they know, and then they spend all their time making sure the character's actions stick to fact. Instead of wondering _What would my character do in this situation?_ they're wondering _What would my real-life brother (or aunt, or best friend) do in this situation?_ You can drive yourself crazy trying to figure out if you're writing the story "right" or if the actual person would do what you have them doing. Much better to let yourself fictionalize these people, transform them into characters that suit the needs of your story.
You can also draw inspiration from people you don't know. People watching is a great activity for developing characters. Observe people in action and then take it a step further by imagining what kind of situation they are in. See that young girl on the bench at the bus stop wiping her eyes and trying to hold back her tears? Why might she be doing that? See the group at the bar, flight attendants on a layover? How do they get along? What is going on between the redheaded man who seems so self-conscious and the woman who's avoiding talking to him?
You can take this all a step further and introduce the question _What if?_ What if someone tried to sit down by the weeping girl and console her? What would the girl do? What if one of the other men put his hand on the knee of the woman avoiding conversation with the redheaded man? How would each react?
Of course, you might find that characters people your imagination constantly. You don't need to investigate where they come from. Just pluck one out who seems particularly evocative to you and get to the task of getting acquainted.
**GETTING TO KNOW THEM**
Take the time to get to know your characters as if they were good friends, even the unpleasant characters whom you would probably not befriend in real life. Investing time during the developmental stages helps you understand your characters more intimately, which allows you to put them on the page with more authenticity.
Put your characters in different scenarios and imagine how they would get themselves out of or even deeper into those situations. For instance, what would your character do after accidentally walking out of a department store with an item she had tried on, like a bracelet or a hat, but didn't pay for? Characters do not exist in a vacuum, so imagine how they act and react in the world. Chris, born and raised in Brooklyn, is likely to react differently on a subway in New York City than on a bus in Atlanta, Georgia. In New York City, he's on his home turf, knows the stops by memory. In Atlanta, he might be watching out the window with anticipation, looking for street signs to see if he's reached his stop. He might be more inclined to put his bag on the seat next to him, something he'd never consider doing in the busy subways of the city.
While fleshing out your characters, you should consider the following categories:
Appearance: You can't judge a book by its cover, but a cover can be informative and set up expectations. How your character holds herself, what she chooses to wear, or what kind of expression she has when she walks down the street are meaningful. Think of your character in three dimensions, taking up space. The style and presence with which characters inhabit the world reveal a great deal about their attitude and personality.
Background: A woman who grew up in a family with seventeen kids is going to have a much different experience from a woman who grew up as an only child. While we certainly can't make sweeping value judgments about characters' backgrounds, the characters will undoubtedly be impacted by their previous experiences. How they grew up, how they loved, lost, learned... all these things will help to shape them.
Personality: This is shaped largely by the previous two categories, the end result of everything the person is and has been. What is your character really like? How does her mind work? What are his inclinations? Disposition? Outlook? Hopes? Fears? A character's personality contains the larger truths of the person, which will indicate how she will act and react in a story.
Primary Identity: What is your character's primary definition of herself? Ask several people to respond to this question: _Who are you?_ Some will answer by occupation or ethnicity, others by gender or age. The answer to that question is usually what the person identifies with most strongly, how he defines himself. A person who answers "I am a lawyer" has a much different primary identity than the one who answers "I am Hindu." They may both be lawyers, they may both be Hindu, but they identify more strongly with different parts of their identity.
QUESTIONS
Ask yourself all sorts of questions about your characters to get a better idea of who they are. Many writers like to make lists of questions which they answer in writing. This type of "homework" helps you gain a wellspring of knowledge about your characters.
You might start with questions that address the basics about a character:
* What is your character's name? Does the character have a nickname?
* What is your character's hair color? Eye color?
* What kind of distinguishing facial features does your character have? Does your character have a birthmark? Where is it? What about scars? How did he get them?
* Who are your character's friends and family? Who does she surround herself with? Who are the people your character is closest to? Who does he wish he were closest to?
* Where was your character born? Where has she lived since then? Where does she call home?
* Where does your character go when he's angry?
* What is her biggest fear? Who has she told this to? Who would she never tell this to? Why?
* Does she have a secret?
* What makes your character laugh out loud?
* When has your character been in love? Had a broken heart?
Then dig deeper by asking more unconventional questions:
* What is in your character's refrigerator right now? On her bedroom floor? On her nightstand? In her garbage can?
* Look at your character's feet. Describe what you see there. Does he wear dress shoes, gym shoes, or none at all? Is he in socks that are ratty and full of holes? Or is he wearing a pair of blue-and-gold slippers knitted by his grandmother?
* When your character thinks of her childhood kitchen, what smell does she associate with it? Sauerkraut? Oatmeal cookies? Paint? Why is that smell so resonate for her?
* Your character is doing intense spring cleaning. What is easy for her to throw out? What is difficult for her to part with? Why?
* It's Saturday at noon. What is your character doing? Give details. If he's eating breakfast, what exactly does he eat? If she's stretching out in her backyard to sun, what kind of blanket or towel does she lie on?
* What is one strong memory that has stuck with your character from childhood? Why is it so powerful and lasting?
* Your character is getting ready for a night out. Where is she going? What does she wear? Who will she be with?
For a downloadable version of these and many other "character questions," go to the Writer's Toolbox area of the Gotham Writers' Workshop Web site—www.writingclasses.com.
You certainly won't end up using all the information you gather from these question—as a matter of fact, you shouldn't—but the more you know your characters, the better you will be able to draw them on the page in a believable way. The novelist E. M. Forster, in his classic book _Aspects of the Novel,_ wrote that a character is real when the writer knows everything about him: "He may not choose to tell all he knows—many of the facts, even of the kind we call obvious, may be hidden. But he will give us the feeling that though the character has not been explained, it is explicable..."
YOUR TURN:
Go out into the world and find a character.
Observe someone you don't know, like a fellow diner in a restaurant, or someone you know only a little, like the bank cashier you see once a week. Talk to him or her, if you like, though you don't have to. Make some notes, mental or written. Then fill in the unknown blanks of this person by answering all or most of the suggested questions on the preceding pages. You'll be making up most of the details, but that's okay. This _is_ fiction.
**KINDS OF CHARACTERS**
Not all characters must be developed with the same depth. Your main concern with characterization falls on the most prominent characters in your story. They, of course, should be developed the most fully.
This is particularly vital for the main character of the story, often referred to as the protagonist. All of the dimensional aspects we've discussed—desire, complexity, contrast, consistency, change—invariably come into play with a story's protagonist. Stories tend to have one protagonist, although you will find that some novels have two or more protagonists. Jay Gatsby is the protagonist of _The Great Gatsby,_ a man shrouded in mystery who devotes his life to winning back the elusive Daisy. The protagonist of Raymond Carver's "Cathedral" is an unnamed fellow who unexpectedly sees in a new way with the help of a blind man.
Stories sometimes include an antagonist, a person who poses a formidable obstacle to the protagonist's desire. Tom Buchanan, Daisy's husband, is the antagonist of _The Great Gatsby_ because he actively stands in the way of Gatsby winning Daisy. Tom may be considered a "bad guy" because he is a philanderer and is sometimes rough with Daisy, but antagonists don't have to be "bad." Robert, the blind man who comes to visit in "Cathedral," is the story's antagonist but he's a perfectly nice fellow. Of course, there are often numerous major characters, aside from the protagonist and antagonist, especially in novels. In _The Great Gatsby,_ for example, Daisy and Nick are certainly major and fully dimensional characters.
Secondary characters are the supporting cast. Some of the secondary characters will go through a bit of development, but not of the same intensity as the main characters. The trick with the lesser players in your story is to find a few defining details that really capture their essence. Jordan is a secondary character in _The Great Gatsby,_ defined by her stature as a golf pro and her gossipy tendencies, as is Meyer Wolfsheim, who is defined as a shadowy figure of the business underworld. In "Cathedral," the wife is a secondary character, kind to Robert, irritated by her husband, but mostly in the background.
Extras are the characters who populate the fictional world but don't have a significant impact on the story. They appear but don't achieve any dimension beyond their limited role. The waitress who makes an appearance for one scene need not be thoroughly examined, nor the ex-husband who doesn't play a central role but occasionally takes the kids for a weekend. In _The Great Gatsby,_ Gatsby's servants are extras, as are many of the lively attendees at his parties. "Cathedral," on the other hand, does not have any extras.
Often you'll hear the terms _round_ and _flat_ characters. Round characters are fully developed and lifelike, possessing the qualities we've discussed in this chapter. E. M. Forster notes that a round character is capable of surprising a reader in a convincing way, which echoes the need for contrast and consistency in characterization. Flat characters are those who are characterized only by their role or a minor action. Flat characters are not necessarily a bad thing; it's important to let very small characters be flat. Fleshing them out too much gives them an emotional weight that will mislead the readers or steal focus from the stars of the story.
**SHOWING AND TELLING**
A key question remains: how do we put seemingly real people on the page, conveying a sense of their total humanity using not blood, flesh, and muscle but merely words?
There are two basic methods of revealing a character in fiction: showing and telling. Sometimes it is most efficient for the narrator just to _tell_ the reader about a character. For example, in _The Man with the Golden Arm,_ the narrator simply tells the reader about Frankie Machine and his friend, Sparrow, as they sit in jail:
_The tranquil, square-faced, shagheaded little buffalo-eyed blond called Frankie Machine and the ruffled, jittery punk called Sparrow felt they were about as sharp as the next pair of hustlers. These walls, that had held them both before, had never held either long._
The characters aren't shown with specific actions that reveal their physical traits or their hustling. The writer just tells us what he wants us to know. Here telling is appropriate because it makes a quick distinction between Frankie and Sparrow and it introduces their racket as hustlers, in and out of jail.
But do not overrely on telling. Writing instructors frequently proclaim _Show don't tell_ in the margins of student manuscripts, often adorning the advice with an exclamation point. For very good reason. Revealing information through showing is generally more interesting than telling about it, because showing gives the reader more with which to engage actively. The bulk of characterization should come through showing characters to the reader.
For example:
_Greta is a twenty-three-year-old artist and interior designer who dislikes having a roommate._
Now the basic information about Greta is out in the open and you can get on with things. In this _telling_ example about Greta the reader learns her name, her age, her occupations, and her dislike of having a roommate. However, in most cases it would be stronger to _show_ Greta's character rather than just tell about it.
The trick in showing instead of telling is to find the specific details that will convey the necessary information while the reader's attention stays on the character's emotion and actions—the interesting stuff. For example:
_After a stressful week at Mr. Feίnmen's, experimenting with materials that might transform their front foyer into a low-ceilinged cave, Greta sat at a secluded corner of the café, sipping tea. Maybe once her roommate left for the night, she could have a little time to experiement with molding the wire mesh into skeleton marionettes._
The basic facts are still in this version—you get an idea of Greta's age, and learn about her job as an interior designer and her dislike of having a roommate. But in this version there's even more information. You get a sense of Greta's eccentric, even macabre, artistic taste. Also the nature of her roommate difficulty begins to appear: Greta doesn't feel like she has her own space to do her art at home. You also see how Greta handles stress: she chooses to hide in a caf _é_ instead of confronting the problem. Interestingly, she seems to favor cavelike surroundings in both her artistic and her personal life. She drinks tea, which tells the reader something different than if she were drinking a beer. A martini would say something else.
And, best of all, the second version is much more interesting to read because it gives the reader an opportunity to interact more with the story. The reader's attention stays focused on the action and on Greta's desire, which creates momentum and tension while also conveying bits of characterization.
Showing also allows you to slow down and reveal the character's intricacies gradually. In real life you don't sit down and lay out all the beautiful and ugly things about yourself at once. You don't fess up to everything right away. Instead, those things about you that make you human and individual reveal themselves gradually over time to the people around you. This same thing happens in fiction, only it doesn't take a lifetime.
There are four ways to show a character's traits:
* Action
* Speech
* Appearance
* Thought
These four methods allow you to reveal characters in all their dimensional glory.
ACTION
You know the clichés: _Actions speak louder than words and I'll believe it when I see it._ Action is in demand in stories because it reveals so much to the reader. A character's personality comes through in the way she handles the next-door neighbor who leaves his garbage outside his apartment until someone else takes it out, the ways she spends her Tuesday nights, the way she copes with the screaming man on the train. Action is usually the strongest method of revealing a character.
In _The Great Gatsby,_ Daisy is revealed very distinctly by her actions when she sees Nick, the narrator and Daisy's cousin, for the first time in years:
_She laughed again, as if she said something very witty, and held my hand for a moment, looking up into my face, promising that there was no one in the world she so much wanted to see. That was the way she had._
Daisy treats Nick with earnest attention. She grabs hold of his hand and looks into his face, as if he's the only person in the world at that moment. This action helps to characterize Daisy's charm and her background as a Southern belle.
While all actions are revealing, the actions a character takes in the time of crisis often cut to the core of the character's true self and intentions. After Daisy admits to her husband, Tom, that she loves Gatsby and will be leaving Tom for him—showing her romantic longing—Daisy and Gatsby drive back from the city. Daisy, at the wheel, strikes and accidentally kills a woman (Tom's lover, no less) then drives away—showing her nervousness under pressure. Gatsby offers to say he was driving to protect Daisy and drops her off at home. He waits outside, thinking she's locked up in her room, as they had agreed to a code she could make with the lights should Tom give her any trouble. Nick, however, sees her downstairs with Tom:
_Daisy and Tom were sitting opposite each other at the kitchen table with a plate of cold fried chicken between them and two bottles of ale. He was talking intently across the table at her and in his earnestness his hand had fallen upon and covered her own. Once in a while she looked up at him and nodded in agreement._
_They weren't happy, and neither of them had touched the chicken or the ale_ — _and yet they weren't unhappy either. There was an unmistakable air of natural intimacy about the picture and anybody would have said that they were conspiring together._
Here Daisy's true nature is revealed. In this moment, when she's pledged her love for Gatsby, then killed a woman with her reckless driving, she follows her husband and stays in the safety of her marriage, leaving Gatsby to take the fall. This reveals a great deal about Daisy: her duplicity, submissiveness, cowardice, and inability to withstand social pressure. She's come a long way from the charming belle we first meet in the beginning.
Flannery O'Connor, in her book _Mystery and Manners,_ describes how she once gave a few of her early stories to a lady who lived down the street. The woman returned them saying, "Them stories just gone and shown you how some folks would do." To this, O'Connor commented: "I thought to myself that that was right; when you write stories, you have to be content to start exactly there—showing how some specific folks will do, will do in spite of everything." Place fourteen characters in the exact same circumstances and you should get fourteen very different courses of action and approaches to the situation—fourteen different illustrations of what each character _will do._
**SPEECH**
Characters are also revealed through their speech. What people say, how they say it, and what they don't say are all very illuminating. If you want to get to know someone, what do you do? You talk to them.
In _The Great Gatsby,_ Daisy talks with a childlike giddiness, as seen in these three snippets of dialogue:
_"Do you always watch for the longest day of the year and then miss it? I always watch for the longest day in the year and then miss it."_
_"I'll tell you a family secret," she whispered enthusiastically. "It's about the butler's nose. Do you want to hear about the butler's nose?"_
_"You see I think everything's terrible anyhow," she went on in a convinced way. "Everybody thinks so_ — _the most advanced people. And I_ know. _I've been everywhere and seen everything and done everything."_
Daisy's questions and confidences and blanket statements show the reader her wide-eyed wonder and naïve nature. As the story progresses, however, we begin to glean that there is tension beneath the gaiety.
**APPEARANCE**
A glimpse of someone can give you a lot of information about his personality. You can draw conclusions from physical looks, clothing style, gait, and facial expression. The way a character appears gives the reader information about how this person presents himself and occupies space in the world.
In _The Great Gatsby,_ the reader first meets Tom Buchanan, Daisy's husband, this way:
_He had changed since his New Haven years. Now he was a sturdy, straw-haired man of thirty with a rather hard mouth and a supercilious_ _manner. Two shining, arrogant eyes had established dominance over his face and gave him the appearance of always leaning aggressively forward. Not even the effeminate swank of his riding clothes could hide the enormous power of that body—he seemed to fill those glistening hoots until he strained the top lacing and you could see a great pack of muscle shifting when his shoulder moved under his thin coat. It was a body capable of enormous leverage_ — _a cruel body._
Tom's aggression and confidence are apparent in the way he stands with his legs apart on the front porch. The description of his mouth, his eyes, and the way he inhabits his clothing gives the impression of a strong and unrelenting man.
Don't just go for the obvious when focusing on your character's appearance. Sometimes oddball details, like the description of Meyer Wolfsheim's cufflinks made of human teeth and the "two fine growths of hair which luxuriated in either nostril," can be particularly revealing.
THOUGHT
Fiction has the pliancy to get inside characters' minds, often with more grace and depth than other forms of storytelling. In movies and plays thought is not as easily conveyed, but in fiction the character's thoughts can be bared directly to the reader.
In _The Great Gatsby,_ the reader has direct access to Nick's thoughts since the story is told from his point of view. Here is a glimpse inside his mind:
_I liked to walk up Fifth Avenue and pick out romantic women from the crowd and imagine that in a few minutes I was going to enter into their lives, and no one would ever know or disapprove. Sometimes, in my mind, I followed them to their apartments on the corners of hidden streets, and they turned and smiled back at me before they faded through a door into the warm darkness. At the enchanted metropolitan twilight, I felt a haunting loneliness sometimes, and felt it in the others_ — _poor young clerks who loitered in front of windows waiting until it was time for a solitary restaurant dinner—young clerks in the dusk, wasting the poignant moments of night and life._
This moment of thought shows Nick's loneliness in the way he fantasizes about entering into women's lives, getting smiles from them before they move on, a moment of acknowledgment. The reader sees a very secret side of Nick, something he likely wouldn't share with anyone.
A SYMPHONY OF METHODS
Use these four methods—action, speech, appearance, thought—in concert to create a sense of depth in the moment-to-moment experience of the story. In real life, we experience people in a variety of ways, often simultaneously, and mixing the methods re-creates this sense of reality.
In "Cathedral," the narrator's wife is expecting a visitor: a blind man, Robert, whom she once worked for and continued to keep in touch with over the years. The narrator isn't thrilled about his visit and is uncomfortable about the fact that the man is blind. Here's what happens when the narrator first meets Robert:
_This blind man, feature this, he was wearing a full heard! A beard on a blind man! Too much, I say. The blind man reached into the back seat and dragged out a suitcase. My wife took his arm, shut the car door, and, talking all the way, moved him down the drive and then up the steps to the front porch. I turned off the TV. I finished my drink, rinsed the glass, dried my hands. Then I went to the door._
_My wife said, "I want you to meet Robert. Robert, this is my husband. I've told you all about him." She was beaming. She had this blind man by his coat sleeve._
_The blind man let go of his suitcase and up came his hand._
_I took it. He squeezed hard, held my hand, and then he let it go._
_"I feel like we've already met," he boomed._
_"Likewise," I said. I didn't know what else to say. Then I said, "Welcome. I've heard a lot about you."_
Notice how the different methods of showing are used in combination to capture these three characters in this particular moment. With the narrator, we get his thoughts that the beard on the blind man is "too much" and his somewhat nervous actions of finishing up his drink and preparing to answer the door as well as his dialogue, which is mostly pleasantries he should say rather than genuine welcoming. The moment isn't limited to characterizing the narrator, either. The reader sees Robert's appearance, his speech, which is confident, and his actions, which are genial. The narrator's wife is also characterized through her attentive actions, her beaming face, and the excitement with which she introduces Robert to her husband. A tremendous amount of character information is portrayed in just this brief passage.
The four methods of showing can also work in opposition with one another to interesting effect. Do you always say exactly what you're thinking in an argument? Can you act like you're having a good time at your wife's important but stuffy work banquet when you would rather be out sailing on your cousin's keelboat? Might your body language give you away?
Often a truth is revealed about a person when there is a discrepancy between two or more of the four methods of showing: action, speech, appearance, and thought. For example, George may tell his sister over the phone that the Ye Haw Singles Sock Hop she's urging him to go to sounds ridiculous and desperate, but at the same time he might be writing down the date, time, and location of that very sock hop. George's speech _("Are you crazy? What a stupid ideal")_ says something much different than his actions _("Well, maybe I'll want to go")_
In _The Great Gatsby,_ Daisy shows a discrepancy shortly after we first meet her. Tom leaves the dinner table to take a phone call from his mistress and Daisy suddenly excuses herself and follows Tom. The dinner guests can hear their rising voices as they argue. When Daisy returns to the room with Tom, she says about the phone call, "It couldn't be helped," then she begins babbling blithely about a bird outside. However, her actions—abruptly leaving the dining room and having an argument with Tom—contradict her carefree speech. Even though Daisy acts as if everything is fine, her actions tell the reader, and the guests, otherwise.
Effective fiction makes use of all four methods of showing characters, whether the methods are supporting or contradicting each other. The idea is to blend them, rather like the balance of instruments in a symphony. Think of these four methods as the strings, winds, brass, and percussion of an orchestra, and yourself as the composer who must unite these sounds into a harmonious—or purposefully discordant—whole.
YOUR TURN:
Return to the character for whom you filled out the questionnaire. You're going to put this (now fictional) person into the world and let him reveal himself. Imagine this person is entering the waiting room of a therapist's office for the first time. The type of therapy is up to you (it may even involve bringing a spouse or a pet), but chances are this person will be feeling a little stressed. Keeping the character in the waiting room, write a passage where he is revealed through all four of these "showing" methods—action, speech, appearance, and thought. For a bonus round, you can put this character in an even more stressful situation, like observing someone being held at gunpoint. What will your character do _then?_
**ONLY RELEVANT DETAILS**
Once you create fully realized characters, there is really no limit to how much you can know or show or tell about them. But when you shape your characters into the context of a story, make sure to pick and choose carefully what details you include. In other words, resist the urge to stuff in everything.
The fact that your character, Lance, spent a year and a half at Harvard might not be relevant in a story about Lance and Addie, a couple married for forty-three years who are now dealing with the recent death of their longtime pet, a Doberman pinscher named Eugene. However, the detail about Harvard might become important if, instead of being about the dog's death, the story is about how Lance is now regretting the decisions he made in the past to drop out of school and go to Paris with a woman he met at the student union. But do you need these details in the story about the dog's death? No, the Harvard education (or lack of it) likely isn't necessary. Every character detail included in your fiction should work to advance or enhance the story you are telling.
Don't let extraneous details sit around cluttering up your characterization. In "Everything That Rises Must Converge" by Flannery O'Connor, Julian and his mother are about to leave for her Wednesday night weight-loss class at the Y:
_She was almost ready to go, standing before the hall mirror, putting on her hat, while he, his hands behind him, appeared pinned to the door frame, waiting like Saint Sebastian for the arrow to begin piercing him. The hat was new and had cost her seven dollars and a half She kept saying, "Maybe I shouldn't have paid that for it. No, I shouldn't have. I'll take it off and return it tomorrow. I shouldn't have bought it."_
_Julian raised his eyes to heaven. "Yes, you should have bought it," he said. "Put it on and let's go." It was a hideous hat. A purple velvet flap came down on one side of it and stood up on the other; the rest of it was green and looked like a cushion with the stuffing out. He decided it was less comical than jaunty and pathetic. Everything that gave her pleasure was small and depressed to him._
_She lifted the hat one more time and set it down slowly on top of her head._
Julian's mother frets over this elaborate-looking hat—deciding to wear it, then deciding not to—regretting her decision to buy it and needing to hear from her son that it was fine for her to do so. These details show her insecurity, vanity, and slow pace. And perhaps most importantly we see that her behavior is a constant source of aggravation to her son. At first glance, this scene with the hat might not seem particularly important, but the hat, as well as the traits illustrated, plays an important role in the story's unfolding when they meet up with a woman on the bus who has the same hat!
**WHAT'S IN A NAME?**
Last, names are not trivial; they should feel right for the character. Granted, your parents likely named you before they even met you, and certainly before you developed a significant personality, but as an author, you have the opportunity to let the name of a character play a role in the characterization.
So avoid wishy-washy names that don't say much about the character, such as Joe Smith or Jane Jones. And avoid giving all of your characters similar names, like Mike, Mark, Mick, and Mary, as that only serves to confuse the reader. Instead, look for ways to reveal something about your characters through their names.
Some writers favor names that seem to make a literal statement about the character. Dickens used such pointed and colorful names, as in _Nicholas Nickleby_ (a story largely about money), where you'll find such characters as Wackford Squeers, Sir Mulberry Hawk, and Miss Snevellicci. _Lolita's_ narrator takes on the appropriately lumbering name of Humbert Humbert.
But names can also be revealing in more subtle ways. In Carson McCullers's _The Heart Is a Lonely Hunter,_ the name Antonapoulos is appropriate for that character's fussy and persnickety nature, while his friend's name, Singer, echoes his more toned-down simplicity. And don't neglect nicknames. Frankie Machine's nickname comes from the sound of his real last name—Majcinek—and his ability to deal cards so flawlessly (his arm seemed like a machine).
YOUR TURN:
Go to the phone book. Open it up and point to a name, any name. That person is your character. Think about who that person might really be. Or what character would live well with that particular name. Let a picture form in your mind. if you like, jot down random details about this character. If you're so inspired, you may apply this character to any of the previous exercises in this chapter. And there's nothing stopping you from doing this exercise anytime you feel like finding a character. There are plenty of names in the phone book!
Sometimes characters go nameless, as is the case with the unnamed protagonist of "Cathedral." Some characters are known only by such names as "the man" or "the girl," as in Ernest Hemingway's "Hills Like White Elephants." In these instances the writers may have wanted the characters to have a certain anonymity, but you should be sparing with this device as it can run the risk of seeming pretentious and, worse, it can deprive you of a great way to characterize.
Indeed, in _The Great Gatsby,_ Fitzgerald managed to characterize a whole group of people mostly by their names alone, as Nick recalls some of the guests at Gatsby's parties:
... _I can remember that Faustina O'Brien came there at least once and the Baedecker girls and young Brewer who had his nose shot off in the way and Mr. Albrucksburger and Miss Haag, his fiancée, and Ardita Fitz-Peters, and Mr. P. Jewett, once head of the American Legion, and Miss Claudia Hip with a man reputed to be her chauffeur, and a prince of something whom we called Duke, and whose name, if I ever knew it, I have forgotten._
Names are like the wrapping on a present, offering just a hint of what may be inside the person.
[CHAPTER 3
**PLOT: A QUESTION OF FOCUS**](Facu_9781596917910_epub_c4_r1.html#aa3)
BY DAVID HARRIS EBENBACH
When I landed at Vermont College to enter a graduate program in writing, I had more than a hundred pages of my first novel in hand. I had been working on the thing for more than a few months already. Yet I'd also hit a wall—I wasn't exactly sure what to make of these pages I'd written. I handed off the partial draft to my adviser, Ellen Lesser. My secret hope, of course, was that she would declare it a work of true genius, giving me all the motivation I'd need to keep at it and write the rest of the book. What she actually said in response was less blindly positive, but much more helpful. What Ellen told me, in short form, was to cut fully half the pages I'd already written.
The first thing I decided, naturally, was that the novel was completely worthless garbage, and that I had to abandon it immediately, possibly shredding or burning the draft I had. But after I took a little time to calm down, I looked more closely at Ellen's comments, and more closely at the work itself. That's when I really began to learn.
She had actually liked many parts of the book—but she also pointed out the huge glut of exposition that was clogging up the first chapter while little was happening in the story. She highlighted a conversation between my main characters that looked like a real enough conversation, but wasn't accomplishing anything significant. She focused on scenes where characters engaged in the normal activities of life—brushing teeth, getting dressed, walking from here to there—even though those activities had little to do with the concerns at the heart of the novel.
In particular I remember a scene showcasing one of my two main characters pacing the length of the apartment with his cat draped over his shoulder. He had been reading and sleeping, on and off; now he fixed himself a peanut butter sandwich; now he looked out this window; now he looked out that one; it was hot. Nothing much was going on. As he paced he was thinking through the background of his life, details I was sure the reader would need and want to know. Meanwhile, the reader was falling asleep.
My adviser suggested that I had actually written these pages for myself—that I'd written them to get to know the people and places and situations in the book, so that I could write authentically about them. And that was well and good, probably even essential for me. But the reader didn't need to see all of that prep work. They needed the story, right then, right there.
In other words, I needed to think more about plot.
**PLOT VERSUS REAL LIFE**
Life may be interesting. Life is often moving and eventful. But rarely does life actually contain plot. I eat a sandwich because I'm vaguely hungry; I go to work and come home again, feeling pretty much the same as I did the day before; all sorts of people enter and exit the stage around me as I go through my day—but a lot of it seems more random and anticlimactic than what happens in good short stories and novels. Even death, probably the only true ending we have available to us as human beings, often happens right in the middle of things, leaving many things unfinished.
Now, I suppose you might say plot _is_ like real life, if, as Elmore Leonard advises, you leave out all the boring parts. But that brings us back to my premise that plot is _not_ exactly like real life. And so plot is one of the elements of craft that clearly separates the real world from the world of fiction. When people ask _What's it about?_ in regard to their own existence, they're not sure they should even expect an answer. Yet they also, on some level, ask that question every time they approach a short story or novel—and this time, there had better be an answer. Successful fiction does have a point, does have a fascinating and meaningful sequence of events. The purpose of this chapter is to show how life can be brought to the page, by bringing plot to life.
Think for a moment about a story you really enjoyed—one of those stories that you just couldn't put down, that you had to keep reading because of all the suspense. Chances are this story had a magnificent plot. As you read, one event set into motion another one, more captivating than the last; the situation became more and more tense, with you wondering all the while how everything would turn out.
This is the most tangible benefit that plot brings to fiction. Although in some high-minded literary circles the term _page-turner_ is less than a compliment, the fact is that some of our most celebrated novels—Toni Morrison's _Beloved,_ for example, or Dickens's _David Copperfield,_ or Richard Wright's _Native Son_ —work as well as they do partly because they have plots that pull the reader irresistibly along, from beginning to end. We care about what's happening, and we're concerned about what might happen next, And if we don't care about these things, the stories in our hands may end up abandoned before we've even finished them, left to gather dust on an untouched shelf somewhere.
At the heart of most great fiction is the excitement created when we really feel that the work is after something specific—when it has plot.
The way all this happens is pretty simple, at least in theory. Plot makes fiction coherent by drawing together all the characters, settings, voice, and everything else around a single organizing force. That's right, _one_ organizing force. After all, a short story, even though it might have a big impact on a reader, is actually a very small and focused world, and the same is surprisingly true of a novel. Works of fiction are not, and cannot be, about a million things—they are usually about just one thing. And that thing, the force that draws everything together in a successful piece of fiction, is a single, pressing question.
**THE MAJOR DRAMATIC QUESTION**
This question—often known as the major dramatic question—is generally a straightforward yes/no question, one that can be answered by the end of the story. _Will Brian find a job? Will Jamie and Ana move to separate apartments? Will Shira finally stop ignoring her inner child?_
If you stop to look around, you'll find major dramatic questions organizing things throughout the entire body of literature. Consider the following short stories. Bernard Malamud's "The Magic Barrel" is a busy piece, populated by fascinating characters and a memorable storytelling style, but the bottom-line concern is whether rabbinical student Leo Finkle will find himself a wife. In Peter Cameron's "Memorial Day," we wonder whether the little boy narrating the story can somehow get back his old life—with his parents still married. In "Sonny's Blues," by James Baldwin, our narrator asks whether Sonny can rise above his difficult life and his suffering, and, in the echoes of that, whether we all can.
The single major dramatic question remains the central organizing force even in the relatively complex world of novels. In _Pride and Prejudice,_ by Jane Austen, the question is whether Elizabeth Bennet will end up with Mr. Darcy. The question in Ernest Hemingway's _For Whom the Bell Tolls_ is whether Robert Jordan will escape his apparent fate by surviving his military mission. In John Steinbeck's _East of Eden_ the question seems to be whether Cal is to be forgiven for who he's turned out to be. Hundreds of pages, but in all these cases, one dominant question is at the forefront. One of the major reasons why we keep reading is because of the suspense the major dramatic question creates. We need to find out what the answer will turn out to be.
Which naturally leads me to those all-important answers. It's obvious enough that you'll have to provide an answer, given that you've raised a question, but it's worth noting that your choice of possible answers is diverse— _yes, no,_ and _maybe_ are each fair game. That's right, it's okay if Brian finds a job, and it's okay if he doesn't, assuming that the events of the story and the nature of Brian's character justify the outcome. You're even allowed to say _maybe,_ as long as you've convinced the reader that you've tried your best and that, in the end, neither _no_ nor _yes_ would really be honest.
One thing to bear in mind, though, is that readers tend to be unsatisfied if this answer, whatever it is, comes through some kind of _deus ex machina,_ a Greek term that refers to a random act of God or luck that resolves everything. This is one reason why some readers shy away from Dickens now—the miracles that resolve many of his novels can seem a little too easy to the modern reader. Instead, the answer should come directly from the actions or thoughts of the protagonist.
The most important rule, however, is that your answer has to match your question. If you spend the whole story asking whether Brian will find a job, and you build tension around his repeated and increasingly urgent attempts to get a job, you can't end by concluding _Yes, he will learn to love again._ You must conclude with a decision about his employment future. This mismatch between question and answer is going to provoke the reader to say, with some irritation, _I can't tell what this story is about!_ And that's the last thing you want.
The major dramatic question can seem strange to some writers, as though it's a foreign element that has to be forced into a piece where it might not belong, but it's an organic part of your fictional universe, connected to all the other elements of the piece. Most fundamentally, the question arises from the relationship among three elements: the protagonist, his or her goal, and the conflict blocking that goal.
YOUR TURN:
Think about one of your favorite works of fiction— _Gone with the Wind,_ "Hills Like White Elephants," _Charlotte's Web_... Try to figure out what the major dramatic question might be. Remember, this is a question that can usually be answered with a _yes, no,_ or _maybe._ Sometimes this question takes a little fishing around to find. You may even want to leaf through or reread the story you've picked.
PROTAGONIST
The protagonist is, simply enough, the main character in your work. He or she will be the most complex and dimensional character in the piece, the one illuminated most fully and followed most closely. The major dramatic question always centers around the protagonist, focusing above all on what will happen in his or her life.
Because of all this attention the protagonist gets, there's usually just one protagonist in a given work of fiction. Certainly there are novels with multiple protagonists—William Faulkner's _The Sound and the Fury,_ Gabriel Garcia Márquez's _One Hundred Years of Solitude,_ and Rick Moody's _Purple America_ are just a few examples—but for the most part even novels follow just one protagonist. Writers often use the added length in a novel not to explore a bunch of characters but to even more thoroughly explore _one._
A classic example is J. D. Salinger's _The Catcher in the Rye,_ which offers the reader the one and only Holden Caulfield. Holden, the tough-talking but ultrasensitive teenager, is smart and inept and world-weary and inexperienced and hopeful and miserable and lonely and young—all at once. He has become one of literature's favorite characters, in part because we get so deep into his mind that at times we almost become him.
Of course, as much as we might be drawn in by Holden's engaging and sympathetic personality, that personality alone won't be enough to keep us reading for the entire length of a book. In order to keep us turning those pages, Holden's got to have a story.
GOAL
Chapter 2 on Character suggests that desire lies at the heart of all characters. Here I suggest that the protagonist's desire is the key to the story's plot. Just as the story will be driven by _one_ question, that question will come into play in the first place because of the _one thing_ that the protagonist wants most. Call it the goal. Specifically, the protagonist's goal is going to be about getting a _yes_ or _no_ answer to the major dramatic question.
This goal may be conscious, with the protagonist knowing what he or she is after, or it may be unconscious, driving the protagonist's actions without ever making itself explicitly known to him or her. Either way, it pushes the question into the open for the reader.
Further, this goal may either be concrete, such as a job the character wants, or abstract, such as a desire for a sense of self-worth. It's important to note, however, that in practice abstract goals are associated with concrete ones, and vice versa; food can provide a feeling of comfort, power is often tied to money, and, of course, a job can provide a sense of self-worth. Generally, abstract goals need to be represented by something concrete in a story, or else the whole story is going to feel, well, abstract, and it will be harder to formulate the events of the plot.
In _The Catcher in the Rye_ Holden's goal is to find a place where he belongs. He pursues this abstract idea in a concrete way, by going to New York City on a hunt for a person, any person, who is going to understand him and not turn out to be a "phony." From understanding our protagonist and his or her goal, we uncover the major dramatic question. The major dramatic question of _The Catcher in the Rye: Will Holden find a place where he belongs?_
Whether the character will get what he or she wants is another issue altogether.
YOUR TURN:
Imagine a protagonist who seems to have it all—a home, financial security, a loving spouse. Give this person a name and flesh out some details. Then figure out an abstract goal for this person. Then attach that abstract goal to a concrete goal that might work in a story. For example, if this person abstractly desires adventure, then perhaps make his or her concrete goal to sail around the world. Hint: the goal should probably stem from something that's missing in this person's seemingly perfect life.
CONFLICT
The protagonist's goal may fly in the face of what other characters want, and the goal may even fly in the face of physical and social reality. In other words, there are obstacles in our main character's path. These obstacles create conflict. No, we can't make it easy for our characters, no matter how much we'd like to; making it easy makes for bad fiction. Plot depends on conflict.
And to keep things really interesting, the conflict should keep escalating. These forces—the ones pulling our main character toward the desired outcome and the ones pushing in the opposite direction—increase in equal measure across the fiction, like a pair of well-matched arm wrestlers, until one of them has to give out. The more fierce the fight, the better the story.
Conflict can take many forms. Some obstacles are external to the character, and can be found in other people (such as antagonists, characters who actively try to prevent the protagonist from meeting goals), or in societal structures, in nature, in acts of God, or in any number of other external possibilities. A woman seeking shelter may not have the money to get it, for example, or a man seeking love may not encounter anyone interested in providing it.
Other obstacles are internal. The same woman may not _feel worthy_ of shelter, and the man may not _know how_ to seek love. In these cases the struggle at hand takes place largely within the minds of our characters, between desires on the one hand and fears and personal inadequacies on the other. Stories that truly move us, stories with real depth, generally require that at least some of the conflict be internal.
Often there are many sources of conflict, both internal and external, within a given piece. Yet while all these obstacles are painful for the protagonist, they're good news for the story, and for the reader. After all, _A starving family goes out to beg for food but on their way out the front door they find food on their doorstep_ isn't much of a story.
_The Catcher in the Rye_ is rife with conflict. Holden Caulfield repeatedly attempts to find situations where he might find himself safe and understood—with authority figures like Mr. Spencer and Mr. Antolini, with peers like Stradlater or Carl Luce, with potential girlfriend-material Sally Hayes, with family, and even with a prostitute. But the attempts fail again and again; everywhere he goes he encounters external obstacles, meeting "morons" and "phonies" and people who abuse his trust. He also runs up against internal obstacles when he fails to express himself adequately to the people around himself, when he fails to be clear even with himself; he probably doesn't know he _has_ a goal, and certainly doesn't know how to attain it. Yet he pursues this goal more and more intensely, by the end even considering a vague and desperate plan to run off to Colorado.
Does Holden ultimately find what he's looking for? Well, he does decide to abandon his Colorado plans and stay with the one person he really trusts—his sister Phoebe—but this positive move also gives his parents the chance to put him in a psychiatric institution, and then ship him off to yet another phony school. While Holden has not achieved his goal, he has not given up the hope of getting there one day. _The Catcher in the Rye_ is one of those stories that ends with a _maybe._
YOUR TURN:
Return to the protagonist you created for the previous exercise, the one with a concrete goal. Make a list of obstacles—internal and external—that might block the achievement of the goal. List as many obstacles as you can think of, more than you could possibly use in a story. Last, in one sentence, create a major dramatic question that you could employ for a story about this character. Remember, this is a question that can be answered with a _yes, no,_ or _maybe._
**THE STRUCTURE OF PLOT**
Now well take a close look at the structure that grows out of—and supports—the major dramatic question, and, thus, the plot. Structure, in fact, is inseparable from plot. Plot is the sequence of events in a piece that drives toward answering the major dramatic question; structure is the overarching shape that keeps the sequence of events in good order. Luckily, we do not have to reinvent structure every time we start writing a new fictional work—we have a model readily available to us.
The model is certainly a tried-and-true one, having existed for more than twenty-three hundred years, coming from Aristotle's _Poetics,_ a discourse on how drama works. In fact, much of our thinking about storytelling derives from _Poetics,_ which itself was born from the thriving Greek theater and from the surrounding mythology, both of which were rich with story and plot. What Aristotle codified still plays itself out in story after story and novel after novel written in our time.
The accepted model of plot offered by _Poetics_ is more complex than it at first seems. It says that fictional works have a beginning, a middle, and an end. Well, maybe that's a _duh_ —but more importantly it tells us that those sections have distinct roles in the successful telling of a story.
Now well walk through these three sections, discussing the function of each one, and illustrating those functions by looking closely at Raymond Carver's "Cathedral," which is, at its simplest, about a man whose wife's old friend—a blind man—has come to dinner.
THE BEGINNING
The beginning of a story has to get three things done: it has to drop the reader right into the middle of the action, it has to provide all the _necessary_ background information to get the reader up to speed, and it has to establish the major dramatic question. Let's discuss each of these in turn.
The first job of a story's beginning is to start at the right time. It should not start when things are quiet, when nothing's happening, when things are much the same as they have always been. Think about how boring it would be if your friend told a story about something really exciting but began the anecdote two days before anything important actually got started. After all, the whole reason we tell the story is because something about life is new and different, something's happening that stands out—and your responsibility, as the writer, is to begin the work at that point of change. The first sentence of "Cathedral," for example, reads,
_This blind man, an old friend of my wife's, he was on his way to spend the night._
We do not begin the story a few weeks earlier, when nobody was coming to visit. We begin on the day that's the focus of the whole piece.
Yet starting at a point of change means that the reader will by necessity lack crucial background information that explains both the events at hand and the characters reacting to those events. The beginning of the story therefore also has to provide sufficient exposition so that readers will know what's going on and why. The most important issue here is to strike the right balance. The reader does not have to know everything, and certainly not right away; too much background slows the fiction down and can lead to boredom or confusion if the background contains the irrelevant, and some information should just be saved for later.
Exposition will certainly keep coming throughout the work. The key is to supply all the background needed for the time being—and not any more. Finding this balance, of course, is a matter of trial and error at first, but in the long run it will become a matter of instinct. With experience, you learn how to feed the reader information on a need-to-know basis only, and how to keep that information interesting and relevant by integrating it with ongoing action.
"Cathedral" is unusual in that it has a long beginning and does provide a lot of exposition at the outset. However, all of the exposition is useful. Primarily, we learn about the protagonist, who is also the narrator, and who is, interestingly enough, never named. We discover, for example, that the narrator has strange perceptions of the blind, derived from the movies, and that "A blind man in my house was not something I looked forward to." We are told that the blind man is coming to visit because his wife has died and he's going to be in the area, visiting in-laws. We also learn that the narrator's wife worked for the blind man (as the narrator continues to call him, even after learning that his guest's name is Robert) ten years ago, and that they've kept up a correspondence ever since, through her divorce to a first husband and through her marriage to the narrator.
More important, we learn that the protagonist isn't very interested in any of this. His wife, some time ago, showed him one of the poems she wrote about the blind man, his response being he "didn't think much of the poem." She played him one of the tapes sent by the blind man as part of their correspondence, and even though the narrator was mentioned on the tape (or perhaps because of that), he wasn't much curious about that, either. In fact, he seems to have little investment in life altogether. It seems he has no friends of his own. His disinterest in his wife's poetry is a disinterest in poetry in general. He has a job he doesn't like, but he has no plans to do anything about that. He describes drinking as a "pastime." The narrator is a lump.
This protagonist's goal is a strange one, especially if we're used to considering _goal_ and _ambition_ synonymous (which we shouldn't, in fiction), and it's one that he may not even consciously know about. Yet it's still a goal. He wants, more than anything, to give in to his own inertia, to avoid new experiences that would cause him to look too deeply at his own life. In the context of this story, his goal is to be unaffected, unchanged, by the arrival of his unwelcome guest.
The story begins with conflict, just because the narrator is so dreading the blind man's arrival, and that resistance becomes a serious issue as the time of his arrival draws nearer and husband and wife fight about it:
_"If you love me," she said, "you can do this for me. If you don't love me, okay. But if you had a friend, any friend, and the friend came to visit, I'd make him feel comfortable."_
Already this visitor is disrupting our protagonist's stable, if unenviable, existence. And early on, we sense what the major dramatic question of the story will be—and it's essential that the story establishes this as close to the beginning as possible. The question is nicely illustrated in some of the protagonist's musings about the blind man:
_[The blind man and his wife had] married, lived and worked together, slept together_ — _had sex, sure_ — _and then the blind man had to bury her. All this without his having ever seen what the goddamned woman looked like. It was beyond my understanding. Hearing this, I felt sorry for the blind man for a little bit. And then I found myself thinking what a pitiful life this woman must have led_... A _woman who could go on day after day and never receive the smallest compliment from her beloved. A woman whose husband could never read the expression on her face, be it misery or something better._
In this passage, the narrator could easily have been describing his relationship with his own wife, whom he doesn't understand, and the reader begins to see a parallel between the blind man and the narrator. This awareness, along with all the other background we've encountered, gives us our question: _Will the narrator, in the course of this story, come to change, so that he can truly "see" (or understand) himself and his life?_ The reader probably hopes so; the narrator certainly hopes not. In any case, by this point we have our beginning, and we understand a great deal.
Now, despite these several responsibilities—starting with action, giving background, establishing the major dramatic question—typically the beginning is very short, often quite a bit shorter than the beginning of "Cathedral." The reader simply doesn't want to spend much time getting caught up, and so the exposition has to be limited. The reader wants to get to the interesting stuff, to the action—in other words, to the middle of the work.
THE MIDDLE
Although in practice the beginning and middle of a piece of fiction may overlap a bit, as they do in "Cathedral," they are quite different entities. First of all, the middle typically takes up the vast majority of space in the piece, more pages by far than the beginning or the end. No surprise—it has the most work to do. The story's middle usually contains additional exposition, further developing the characters and situations we've learned about in the beginning. It's also where the core action of the story happens, where everything but the opening and closing events takes place.
The middle section is, most importantly, where the protagonist's path toward his or her goal is blocked again and again by increasingly daunting obstacles, and where the forces arrayed against him or her become ever more powerful. In the middle, conflict increases and increases until it can increase no further. Some people picture a sort of "plot graph," where tension is represented by a line that rises as we move from left to right and then suddenly drops off at the end. However you visualize it, the bottom line is that tightly linked events must build in tension and conflict toward a crisis.
Of course, the events of the middle section are not by any means random happenings. It's worth emphasizing here that the fictional world is a cause-and-effect one; E. M. Forster once observed that _"The Queen died and then the King died"_ is not a plot, but _"The Queen died and then the King died of grief'_ is, because it contains cause and effect. In the fictional universe, things happen as a result of the actions of characters, and the actions of characters are a response to things that happen. Your story's middle shouldn't contain a jumbled pile of events in some arbitrary order—it should contain a chain of events, each one tightly linked to the event before.
In "Cathedral," the boundary between beginning and middle may be a bit blurry, but we can probably safely locate it at the point where the blind man arrives at our narrator's home. What follows is clearly a _chain_ of events—every awkward move by the protagonist provokes a countermove on the part of the blind man, who is the antagonist (albeit a friendly one) of this story.
The tension and conflict increase when the guest actually shows up. From the very outset the narrator doesn't know what to say to this blind man, asking him what side of the train he sat on while traveling there because, irrelevantly, the scenery is better on one side of the train than the other. Because of his preconceived notions of the blind, the narrator is also caught off guard by just about every aspect of his guest—the fact that he doesn't wear dark glasses, that he's worked many jobs, that he smokes, even that he has a beard. If this protagonist wants more than anything to avoid new and eye-opening experiences, he's already in trouble.
All through drinks and dinner, the narrator is exposed to more and more information about this guest, as his wife focuses all conversation on the blind man and ignores the protagonist almost entirely. "I waited in vain to hear my name on my wife's sweet lips," he grumbles to us at one point. In one sense, he's glad to be left out, because he's able to keep to himself during the meal, but in another sense the scene is pretty awkward. The protagonist is even more clearly painted as a person who is quarantined in his own limited world. And he begins to grow a little jealous of all the attention the blind man is getting, calling him "a regular blind jack-of-all-trades" after hearing endless stories about the man's past.
The conflict further increases when the narrator's wife abandons them by falling asleep on the couch while the two men sit in front of the television. Now the protagonist is the guest's only company. He in fact pushes the blind man to go off to bed, but the guest would rather stay up and get to know the protagonist. The narrator quickly turns to methods to escape intimacy; he smokes a little pot with the blind man, and they sit in front of the television.
But then something interesting starts happening. Robert's relentless charm, finally, begins to affect the narrator just the least little bit. The narrator tries to fight it, struggling to find the right channel on TV, but, surprisingly, he also admits to himself that he's glad for the change of pace. Most evenings he watches TV and smokes pot by himself. After failing to find something better, they settle on a show about churches and cathedrals.
Around this time internal conflict is developing in the narrator. Although he's trying to fight it, our narrator is starting to recognize the emptiness of his own life as he feels growing curiosity and sympathy toward Robert. He asks the blind man,
_"Do you have any idea what a cathedral is?_... _If somebody says cathedral to you, do you have any notion what they're talking about?"_
The blind man admits he knows very little about the subject, in fact, and asks our narrator to try to describe what he's seeing on TV. Our narrator does his best. Part of him now almost wants to do a good job, even though it flies in the face of his goal to stay mired in his own inertia. His own limitations, however, hold him back; he doesn't explain cathedrals very well.
After some prodding on the subject of religion by the blind man, the narrator goes further:
_"I guess I don't believe in it In anything. Sometimes it's hard. You know what I'm saying?"_
And:
_"You'll have to forgive me," I said. "But I can't tell you what a cathedral looks like. It just isn't in me to do it. I can't do any more than I've done."_
The struggle inside the narrator is now growing to an almost titanic level. He is truly torn, in that he's still clinging to the status quo (his original goal), but in this moment part of him would actually like to become reconnected to the world and the higher meaning in life.
But this development doesn't serve as an answer to our major dramatic question. The question was not whether he would come to _want_ that recon-nection, that open-minded vision, but whether he would _get_ it. Remember that it's crucial that your story answers not just any question but the question it's been pursuing all along. So, while our protagonist has changed a little, the story isn't over. In fact, he is now squarely facing the biggest conflict possible in his life, between his shut-down inertia on the one hand and his rusty interest in a meaningful life on the other.
The blind man forces him to face that conflict directly, by suggesting that the narrator get a pen and paper so that they can draw a cathedral together. This will force the narrator to take his guest's perspective, to feel the world through him, raising the tension to its highest point, and all this will lead to one of the more moving endings to be found in short fiction.
THE END
The end of the work is likely to be the shortest part of the piece, particularly in contemporary fiction. There tends not to be an extended denouement, where we fully play out all the ultimate ramifications of what's taken place. Yet the end of a story does have significant responsibilities. This section of the story may be the shortest, but it's also the place where everything comes together.
The end generally follows a pattern that could be called "the three C's"—crisis, climax, and consequences. The crisis is the point where tension hits its maximum, and the climax is where the tension breaks, and where we get the answer to our major dramatic question. Then, the consequences, however briefly handled, are alluded to at the very end of the piece. "Cathedral," like most stories, follows this pattern.
We return to the narrator as he tries to draw a cathedral, with the blind man's hand over his own, so the blind man can follow the movement. This event produces the crisis, where the conflict between the protagonist's original goal and all the internal and external obstacles becomes too much to stand. He starts out simply:
_First I drew a box that looked like a house. It could have been the house I lived in. Then I put a roof on it. At either end of the roof, I drew spires. Crazy._
He's trying, but he hasn't yet broken through his limitations, and there's a real possibility that he'll drop the effort and sink back to his regular life at any moment. But the blind man is urging him on all the time, and he's gaining momentum.
_I kept at it. I'm no artist. But I kept drawing just the same._
The narrator's wife wakes up to see this taking place, and recognizes it for the amazing event it is. "What's going on?" she asks desperately. The tension continues to hold at maximum while we keenly hope that our protagonist will really come to life, despite everything, right in front of us. And then the blind man pushes the story toward its climax:
_"Close your eyes now," the blind man said to me._
This is asking everything of our narrator. If he can close his eyes to experience this moment in all earnestness, he will have actually _opened_ his eyes, opened himself to the larger world around him. He closes his eyes—but what's happening inside him?
_So we kept on with it. His fingers rode my fingers as my hand went over the paper. It was like nothing else in my life up to now._
And then the climax finally comes, the moment where we get the answer we've been waiting for. Remember that the major dramatic question was not whether our narrator will have a moment of insight—and he already has—but whether he will change, whether he will now be a person who truly _sees._ We get our answer in the story's final lines, when the blind man tells our narrator to open his eyes again:
_But I had my eyes closed. I thought I'd keep them that way for a little longer. I thought it was something I ought to do._
_"Well?" he said. "Are you looking?"_
_My eyes were still closed. I was in my house. I knew that. But I didn't feel like I was inside anything._
_"It's really something," I said._
An ending like this can send a reader straight into tears—what an enormous release, getting our _yes_ answer in the way that we get it. Our narrator is, for apparently the first time in a long while, completely open, seeing himself and the world in a whole new way, and we know it, ironically, because he's keeping his eyes _closed_ here. He says he's no longer in his house, and we understand him to mean that he's no longer trapped in himself in the same way. In answer to the blind man's final question, our protagonist is, at last, looking and "seeing."
This climax, of course, contains the consequences, though they play out more in the reader's head than on the page itself. We don't know that we've got a turnaround as dramatic and permanent as with Ebenezer Scrooge in Dickens's _A Christmas Carol,_ say, but we suspect that things will never be quite the same. All the things we noted at the beginning—the narrator's dread of new experiences, his preconceptions about people he hasn't met, his disinterest in his wife, in himself, and in his own existence—all these attitudes are now endangered species. Whether they'll outright disappear we can't be sure, but we know that, at least for the time being, he has risen above them, and we have reason to hope that the effects may be long-lasting. Of course, we are told none of these things directly. As in most contemporary stories, we are more likely to feel the consequences than to read about them.
It's been said that an ending should feel inevitable but unexpected—that, looking back, it is the only ending that really would have made sense, but that it still felt striking and surprising when it happened. Think of a good murder mystery. At the end, we realize all the clues were there, but we just hadn't put them together right. Now that we know what we know, in other words, it seems obvious.
In "Cathedral," for example, the ending is certainly unexpected—how could someone so stubbornly shut down ever open up again? But if we look back from the end, we see that there are also clues scattered throughout showing that he's more interested in changing than we might have first thought. There's his jealousy about the blind man's range of experience; his sympathy for the blind man's wife, who'd never even been seen by her husband; his shy and ambivalent pleasure at staying up alone with his guest. If we look back, we see how things have been leading, somewhat inevitably, to this unexpected conclusion.
YOUR TURN:
Return to the character you created in the previous exercises, the one with the seemingly perfect life. Now write an entire story centering around the major dramatic question you created. Your story should have a beginning, a middle with escalating conflict, and an ending that includes crisis, climax, and consequences. One more thing. This story can be no longer than five hundred words. Not five hundred pages. Five hundred _words._ Afterward, if you're so inclined, you may turn your idea into a longer work.
**APPLYING STRUCTURE TO NOVELS**
For the most part, what we've discussed in regards to "Cathedral" applies just as well to novels as it does to short stories. Novels too need a beginning, a middle, and an end, and the three sections serve the same functions here as they do in shorter work. In Jane Austen's _Pride and Prejudice,_ for example, we begin in the midst of an exciting development—a wealthy and single Mr. Bingley has moved into the area near the Bennets, and Mrs. Bennet already has ideas about marrying off one of her daughters—and so we start to get a sense of the major dramatic question: _Will Elizabeth get married?_ Then the novel moves through a middle, where her protagonist's desires become more intense while being frustrated in ever more intense ways, and then in the end we are treated to a happy ending where all these complications are removed, and our central question is answered: Elizabeth gets her man.
Therefore, nothing radical changes about the basic structure of plot in moving between the short and long forms of fiction. Yet there are, or at least there can be, some differences worth talking about.
First of all, although the novel follows the same plot structure as a short story, the great increase in size with the novel can bring some change to the relative size and content of the three sections—beginning, middle, and end. For example, the beginning of a novel might take up the entire opening chapter, or possibly a bit longer, giving us an opportunity to take in more background information, as long as it's integrated with important action. Ultimately, in a novel, we'll know a lot more about our characters and their world, and although we'll continue to encounter exposition throughout the entire work, we can certainly get a good chunk of information at the start. The reader, who already signed up for a long ride, will allow you the room for more exposition up front.
The same size increase can hold true for the ending. In some cases, and _Pride and Prejudice_ is a good example, the end will actually allow the consequences to play out for quite a while, telling us in a relaxed way how things have really ultimately turned out. This is less common in the contemporary novel— _East of Eden_ isn't decided until the very last sentence—but is still possible. In Russell Banks's _Rule of the Bone,_ for example, the final chapter is mostly devoted to consequences. The point, however, is that the extra room to be found in a novel allows extra room for the end to breathe, if needed.
But the main difference in the proportions allotted to these three sections is that where in a typical short story the middle takes up the vast majority of the work, in a typical novel the middle takes up the vast, vast majority of the work. That leaves hundreds of pages to develop your characters, raise and intensify your obstacles, relate numerous events. As with a short story, none of those pages should be wasted. Each page should move us farther along a linked chain of events, farther along that rising arc of conflict and tension toward the book's climax. If the release offered by a climax after ten pages can be powerful, imagine how much impact we can experience when it comes after 350 pages or so.
SUBPLOTS
Another difference between short stories and novels is that the length of a novel allows the plot to be more complex, containing more twists and turns and the like. This opens the door for the possibility of subplot. Subplot, when it exists, is a plotline that runs alongside the main plotline of the book. It may concern a character other than the main character, or it may focus on an issue that is not quite the main issue at hand, but it's not unrelated to the main plot. In fact, the subplot exists only because it's relevant to the main plot, commenting on it, exploring it. The connection between the two threads may not always be obvious, but it must always be there.
Sometimes the subplot serves as a lesser parallel to the main plot. In Chaim Potok's _The Chosen,_ while the main character, Danny Saunders, struggles to come to terms with his religion and the future before him, his friend Reuven, the narrator, engages in a very similar struggle. We also see paralleling in Anne Lamott's _Rosie,_ where both the title character and her mother learn to engage themselves with the real world, a world that is frequently intimidating and sometimes truly dangerous. In _Pride and Prejudice,_ the marriage between Elizabeth and Mr. Darcy is not the only match to be made.
In other cases, the subplot is designed as a contrast with the main plot. In Salman Rushdie's _Midnight's Children,_ for example, the focus is on the rise and fall of its main character, Saleem Sinai, but in the background is the fall and rise (and fall again) of the book's antagonist, Shiva. As the fortune of one character goes, so goes the other—but in the opposite direction. Although the fates of Saleem and Shiva are different, they are not unrelated; as opposites, they are bound together as tightly as possible. We see a similar contrast in Tolstoy's _Anna Karenina_ between the destiny of Anna, on the one hand, and Levin, on the other. One is damned, and the other is saved. In Charles Dickens's _David Copperfield,_ the downfall of the noxious Uriah Heep must be ensured before we can move forward to David's happy ending.
One of the natural consequences of subplot is that novels often have multiple climaxes rather than just one. For example, Uriah Heep loses before David Copperfield wins out, and both moments are climactic, though the second one is much more so. While the single major dramatic question must wait until the very end of the book for its ultimate answer, in novels lesser questions will be raised and answered along the way, whether there is subplot or not. In Harper Lee's _To Kill a Mockingbird,_ Tom Robinson's guilty verdict is certainly climactic, and it answers the pressing question of his fate, but it does not give us the final answer we need from this book. In _East of Eden,_ we experience a number of resolutions, some painful and some a relief: the villainous Kate dies, and then the naive Aron runs away, and _then_ the patriarch Adam suffers a stroke, and _only then_ do we get the answer to the book's major dramatic question. Each climax resolves one concern but allows the main one to keep growing in intensity until it too finally breaks wide open.
Subplot is not, of course, necessary to a novel. In a book like _The Catcher in the Rye,_ a subplot would disrupt the intensity of Salinger's focus on Holden Caulfìeld. The added length of the novel, though, does leave open the choice—how do you want to explore the main plotline and protagonist? By focusing tightly and directly on them, or by creating echoes for the main plot in other characters and their struggles?
**HOW PLOT EMERGES**
Perhaps by now you are in a state of panic. _Do you honestly expect,_ you may be asking, _that I'll be able to figure out my major dramatic question, satisfy all the requirements of Aristotle's beginning-middle-end model, and work out all the myriad details of my plot before I even start writing?_
Luckily for all of us, the answer is _no._
Sure, it's possible that you'll have firm ideas about all these issues before you begin. You may find, for example, that you're the kind of writer who likes to outline a piece thoroughly before starting to write it. However, if you're not that kind of writer, relax. You're in good company. For most writers, the first draft is the one where the inspiration runs wild, venting itself in thrilling and undisciplined fashion, and for most of us, there's no room for such careful plotting in that first explosion. Plot is instead something that emerges over time, over the course of a number of drafts. The issue of exactly when it emerges isn't so important. The only rule is that it must, at some point.
I'll return for a second to that overlong partial draft of a novel I brought with me to graduate school. Yes, I did need to cut out half the pages I'd written, but remember that my adviser didn't dismiss those pages as having been a waste of time. As she suggested, I'd needed to write all those go-nowhere scenes—to see how my characters acted in a variety of situations, and above all to see what they were like when they had the leisure to quietly be themselves. And of course it worked. A hundred pages deep into my novel, I showed every sign of knowing my characters well. Even when not writing, I thought about them constantly, empathized with their troubles enormously, and sometimes accidentally fell into their speech patterns when talking to my friends. Those extra pages had been valuable.
After all, plot is almost inseparable from character. In fact, they are so tightly intertwined that it often raises the chicken-or-the-egg question. Much of the time it's hard to say which of the two comes first in writing the work. But whichever comes first, it usually contains the other as well.
Maybe you have a fascinating protagonist in mind. Well, whatever fascinates you about that protagonist probably involves her or his big goal and therefore the major dramatic question (and therefore plot). On the other hand, you might have a really great question—but it can be good only if it's relevant to someone in particular. _Will some unspecified person have a religious experience?_ is a much less interesting question than _Will this disaffected Buddhist monk from Muncie have a religious experience?_
Well, whether plot or character comes first, the point is that they both need to be compelling, and they should both inform one another. And if you're the kind of person who writes brilliant characters but can't seem to get plot jump-started, take a closer look at those characters of yours; they may know what the story is before you do.
And so, instead of imposing some abstract idea of plot from above, we turn to the story, to see what it has to say. After several rereads, you might decide that the primary issue in this piece seems to be whether the protagonist can assert his own needs in his love life. Or you might find that it's about whether your protagonist can regain a sense of innocence, or leave home, or get around to retiring. Whatever your major dramatic question is, it's probably already there to be found in your first draft, even if it's hidden so deep that you didn't notice writing it into the story.
YOUR TURN:
Take one of the characters you worked with in the exercises from chapter 2. Find a major dramatic question for that person. For clues, study whatever it is you wrote about that person. He or she is probably secretly holding the answer... or, we should say, the question.
One likely spot to look for the major dramatic question is the climax, the place where you'll often find your question is answered, even if you didn't know you had one. It's fairly common to base a story around a good climax, whether it's one you design in advance or one you produce while wandering through a first draft. But read the work as many times as it takes until you can see the question. It's the key to everything.
YOUR TURN:
Imagine this as a story climax: a person is rushing through a chaotic place—Times Square, Pamplona while the bulls are running, Mecca during a pilgrimage... Decide where this character is going and why, bearing in mind that this is the story's climactic moment. Now start writing a story that is headed toward this climax. Feel free to steal a character from one of the previous exercises. Write as little or as much of the story as you like, but even if you write only one sentence, make sure it is approaching this peak.
Once you've unearthed a workable major dramatic question, you might want to think about the possibility of an outline. Yes, an outline. As much as it might seem like a potential killjoy, many writers at some point—before a first draft or after a draft or two—find outlines absolutely indispensable. This is especially true with novels, which are unwieldy creatures under the best circumstances, and can be downright unmanageable if you don't get a good grip on them early enough. Outlines work because they allow writers to distill their amorphous creations into their crucial parts, to find the places where tension will need to be increased and the place, or places, where crisis will set in and climax will result.
If the idea appeals to you, you might start by taking down some notes on how your story or novel could divide into a beginning, middle, and end, and think in detail about each section. As far as the beginning, where should it start, what exposition will you provide, and what's your major dramatic question? As for the middle, what additional exposition will be necessary and when will you reveal it? What conflict will the protagonist encounter—and how does the conflict _increase?_ In considering the end, what's your crisis, your climax, and what are your consequences? Are they in the right order?
Another way to start is to write down all the events you want to include in your work, making sure each one offers some conflict between the protagonist and his or her goal, and then outline an order for those events, one that ensures that the conflict will, with each step, _increase._ In any case, the point is to boil the work down to its basics and see whether those basics are the right ones, and whether they're in the right order. You don't have to stick with the outline—your ideas will almost certainly change as you keep writing—but if you look at it as a tentative guideline, it can be helpful.
Whatever tools you use, be reassured that you don't have to have your plot nailed down before you begin to write. The story needs plot in the long run, but just like it took me so many pages to get to know my characters, it will take you time and writing and revising to get to know what your story is actually about. Have faith—you probably wouldn't have started writing the piece if it wasn't about something.
YOUR TURN:
Create an outline for a short story, novella, or novel that is structured as follows: the story should open with the protagonist setting out on a trip. The destination could be as close as the corner store or as far away as the other side of the universe, but the story should end when the character either arrives at the destination or returns to the starting point. Your outline should have a clear beginning, middle, and end, as well as crisis, climax, and consequences. Don't feel the need to make everything perfect, though. You can make different choices if and when you write the piece. On a journey, we don't always stick to the planned route.
**FORM VERSUS FORMULA**
Talking about plot like this really puts some writers off. Discussing structural models and plot graphs and outlines and the like can make the work of writing fiction seem like plugging numbers into a formula. And of course, although writing fiction is nothing whatsoever like plugging numbers into a formula, there is a great deal to be said for understanding the basic requirements of form. In the end, there is no real conflict between the requirements of form and the boundless nature of the creative impulse.
Take the analysis of E. M. Forster, who once said that there are only two plots in all of fiction: somebody goes on a journey, or a stranger comes to town. This (arguable) theory breaks down the vastness of literature to help us understand that stories are driven by the actions of characters, and that they must center around a time when something changes in the life of those characters.
Yet isn't it an oversimplification? Doesn't it imply that there are only two stories in all of literature? Hardly. In this chapter alone we've discussed many wildly different versions of the "somebody goes on a journey" plot—from "Sonny's Blues" to _David Copperfield, For Whom the Bell Tolls,_ and _The Catcher in the Rye_ —and we could certainly go back as far as _The Odyssey_ or the biblical Exodus for other examples. As for "a stranger comes to town," we've mentioned _Beloved, Pride and Prejudice,_ "The Magic Barrel," and, of course, "Cathedral."
The basic forms may be few, but the variations are infinite. Similarly, understanding that plot requires a protagonist, a central question, conflict, and a beginning, middle, and end doesn't limit us at all. Rather, it gives us something concrete to work with as we follow the urges of our imagination.
The possibilities are countless, and there are many examples where standard storytelling conventions have been successfully flouted. Nontraditional works are one of the great traditions of literature and from them we learn that there are many ways to tell a good story.
First of all, we don't have to present the events of a story in chronological order. Morrison's _Beloved,_ for example, moves in two time frames that unfold side by side, following the protagonist both as a slave and in her life afterward, during Reconstruction. In _The Sound and the Fury,_ Faulkner uses linear time only when it suits him. In the first chapter, where Benjy's mind is a continuous stream of free associations, we experience events in the present time as essentially simultaneous with events from weeks or months or years ago.
Writers are continually experimenting with story structure. For example, Tim O'Brien's story "The Things They Carried" is structured around long lists of things carried by a group of soldiers in Vietnam. The novel _The Mezzanine_ by Nicholson Baker is similarly irreverent, the plot covering the length of time it takes to ride a single escalator between two floors, and the text is regularly broken up with footnotes!
Yet while all of these are nontraditional in terms of specific form, all of them follow the general demands of structure, plot, and beginnings, middles, and ends. All of them succeed partly because they pay attention to plot; even Jean-Paul Sartre, whose existentialist philosophy seems to reject any kind of conventional form, can't resist having the protagonist of _Nausea_ struggle and then change, by book's end. In regard to plot, while the specifics change, the general rules, for the most part, hold. Even if we try to _avoid_ plot, in telling a story we often end up creating it anyway. And that's probably a very good thing.
[CHAPTER 4
**POINT OF VIEW: THE COMPLETE MENU**](Facu_9781596917910_epub_c4_r1.html#aa4)
BY VALERIE VOGRIN
When I consider a photograph of myself taken from several feet away I see a caricature—comically high eyebrows and a crooked chin. When my mother looks at me she sees herself as a younger woman, sort of. When my husband gazes at me he sees a big smile and bright eyes and a mop of tousle-ready hair and twelve years' worth of complicated history. From a traffic helicopter I am one of many toy-size drivers inching my way up Interstate 5. What makes me in turn humorous, poignant, beautiful, and insignificant is point of view. Point of view (also referred to as POV) is equally influential in fiction writing.
Consider the story of a lovers' triangle. Imagine how you might respond to that story presented primarily from the point of view of the husband, left at home with his young son over Thanksgiving weekend while his wife slips away for a ski trip with her lover in Vermont. Now how would you respond if the same story was presented from the point of view of the unfaithful wife, whose husband hasn't made love to her, or spoken a kind word to her, in four years, since she was six months' pregnant; or from the point of view of the lover himself, recently flunked out of law school, adrift in the world, desperate for someone to tell him what to do next? Or what if this story's events are observed by some fourth party, such as the young son or a private detective hired by the husband to spy on his wife?
There you have it—more than anything else, the point of view you choose for your story or novel will affect the way readers respond emotionally to your characters and their actions. Your choice of point of view will also influence other elements of your piece, such as tone and theme. Depending on who is narrating the lovers' triangle story, the tone of the story could be repentant, cruel, caustically funny, wistful, or bitter. The story's theme could be the improbability of marriage, the slippery slope of fidelity, the sacred nature of vows, the tenuousness of love, the fickleness of women, the perfidy of men, etc. And all of this depends on the point of view the writer chooses for the story. As I was saying, pretty darn powerful, this POV business.
Point of view is my favorite topic to teach. I like that there are so many variations to consider. Yet I also appreciate that POV is based on a very basic concept: things look different depending on who is doing the looking and what their vantage point is. Point of view, like microscopes and telescopes, can reveal things ordinarily unseen. But point of view is something of an underdog, in that many writers, inexperienced and experienced, don't give it a second thought. Poor, misguided fools.
When it comes right down to it, POV deals with the following issues:
* Who is speaking: a narrator or a character?
* Whose eyes are seeing the events of the story unfold?
* Whose thoughts does the reader have access to?
* From what distance are the events being viewed?
There are a multitude of ways to handle these issues, and this makes POV a rather complex topic. It's like you've just walked into a restaurant for the first time. You're hungry, unfamiliar with the cuisine, and the host hands you a three-pound menu, twenty laminated pages of culinary possibilities. To a novice fiction writer, the array of point-of-view choices may appear just as overwhelming. There, under entrees, is something called "third-person multiple-vision point of view." _Huh? Does that come in a red sauce?_
This is one reason why some writers just shrug and point to something on the menu that sounds familiar. A better bet might be to spend some time with someone who has your best interests at heart, such as me. Thus I will perform the duties of a shrewd, seasoned waiter, walking you through the menu, helping you make informed choices.
FIRST PERSON
A story told from the first-person POV is narrated by a character in the story, usually the story's protagonist. The narrator tells the story of what _I_ did. If the story is about a crime, the narrator is at the scene of the crime. As the police cars pull up, lights flashing, the narrator might be standing in a pool of blood holding a switchblade, or watching from the backseat of the getaway car, or peering out the window of a second-story apartment across the street. The narrator is the story's eyewitness, the reader's means of perception. The reader experiences the fictional world through the narrator's eyes and ears and nose and skin.
_I saw my wife laughing as she parked the car. I saw her get out of the car and shut the door. She was still wearing a smile. Just amazing. She went around to the other side of the car to where the blind man was already starting to get out. This blind man, feature this, he was wearing a full beard! A beard on a blind man! Too much, I say._
As demonstrated in this example from "Cathedral," the narrator does more than simply observe. Here we are getting the narrator's take on what he sees.
Now watch how the first-person narrator in Margaret Atwood's story "Weight" tells us what she thinks and how she feels.
_I am gaining weight. I'm not getting bigger, only heavier. This doesn't show up on the scales: technically, I'm the same. My clothes still fit, so it isn't size, whatever they tell you about fat taking up more space than muscle. The heaviness I feel is the energy I burn up getting myself around: along the sidewalks, up the stairs, through the day. It's the pressure on my feet. It's a density of the cells, as if I've been drinking heavy metals._
There's nothing standing between this character's consciousness and the reader.
When writing in the first person, you are also writing in the voice—the words and tone—of the character. Writers often create memorable voices for their first-person narrators. Here's Richard, the narrator of Thorn Jones's "Cold Snap":
_Son of a bitch, there's a cold snap and I do this number where I leave all the faucets running because my house, and most houses out here on the West Coast, aren't "real"_ — _they don't have windows that go up and down, or basements (which protect the pipes in a way that a crawl space can't), or sidewalks out in the front with a nice pair of towering oak trees or a couple of elms, which a real house will have, one of those good old Midwest houses. Out here the windows go side to side. You get no basement. No sidewalk and no real trees, just evergreens, and when it gets cold and snows, nobody knows what to do._
Jones convinces us that we're hearing Richard's voice for ourselves. Richard might be sitting two stools down from yours at the local tavern, close enough that you can smell his whiskey breath.
The first-person narrator may even use the reader as a confidante, perhaps addressing the reader directly. I chose to do this in my story "Who Can Say Otherwise?" in which a teenage girl narrates the unlikely story of her love affair with a middle-aged rock star:
_I'm telling you this so you know, so I can try to explain how it is. You'll see the pictures in the tabloids and say, "But she's no one special!" and of him, looking a little less dissipated, but with that same famous, wasted face, you might ask, "What's wrong with that man?" I'm writing against all that._
Sometimes a narrator addresses a specific someone. For example, Philip Roth's narrator in _Portnoy's Complaint,_ Alexander Portnoy, relates his story to his psychoanalyst. The narrator may tell her story to herself in the form of a diary, as in Helen Fielding's _Bridget Jones's Diary._ J. M. Coetzee's _Age of Iron_ consists of a single book-length letter written by a mother to her daughter. Though this kind of thing isn't necessary, you might find it helpful in choosing your narrator's words if you imagine what kind of person the narrator is speaking to.
The main advantage of first person is intimacy. The writer can eliminate almost all distance between the reader and the story by placing the reader into the narrator's skin. Also, the narrator's voice can reveal a lot to the reader about the kind of person he is. When we hear Richard speak in "Cold Snap" a picture forms in our minds. We would be surprised to see this guy show up wearing Italian loafers or a pinkie ring.
But the first-person POV does offer some challenges. The writer is stuck in the narrator's skin, along with the reader. All you have to work with is that one character's observations and thoughts. You're not free to wander anywhere, physically or mentally, unless your narrator comes along.
You're also limited by the intelligence and vocabulary of the first-person narrator. Say your story is about an eleven-year-old girl who wants to spend the summer with her ballet-teacher mother in New Orleans despite the fact that her father, an uptight D.C. lawyer, refuses even to consider it. Is the girl up to the job of telling her own story? How mature is she? Does she understand enough about her divorced parents' relationship to make it clear to the reader? Will her observations be interesting enough to keep the reader involved in the story? After all, some eleven-year-olds would make splendid, entertaining companions on a long train trip and others you'd want to shove off the train at the first stop.
YOUR TURN:
Get inside someone's skin. Write a passage from the first-person POV of a person walking to a mailbox to send a difficult letter—breaking up with someone, confessing something unpleasant... Then pick another character also walking to a mailbox to deliver a difficult letter and write from that character's first-person POV. These characters can be anyone you like, but make them the opposite sex from each other and quite different in age. Remember, this is first person, so you should inhabit these characters and tell things the way they would tell them.
FIRST PERSON: MULTIPLE VISION
Most often first person uses just one first-person narrator, but occasionally there are multiple narrators. A short story writer is confined by space, and more than one narrator will usually play havoc with the writer's ability to create a tight, coherent story. But a novelist, working with plenty of elbow room, may decide that a story will be strongest if more than one witness describes the story's events.
In the novel _The Sweet Hereafter,_ Russell Banks uses four first-person narrators to tell and retell the same basic story of a tragic school bus crash: the bus driver, a man whose two small children were killed in the crash, a New York City negligence attorney hoping to make some big bucks off the bereaved, and a teenage girl who will never walk again as a result of the accident. Each character gets a shot at telling his or her own story. The book is divided into discrete sections in which each character presents his or her own version of the truth. The reader hears from Billy Ansel, the young father:
_And then there were the folks who wanted to believe that the accident was not really an accident, that it was somehow_ caused, _and that therefore, someone was to_ blame... _Naturally, the lawyers fed off this need and cultivated among people who should have known better. They swam north like sharks from Albany and New York City ._ . . _slipping their cards into pockets of mourners as they departed from the graveyard, and before long that segment of the story had begun_ — _the lawsuits and all the anger and nastiness and greed that people at their worst are capable of_
The reader may consider Mitchell Stephens, Esq., to be a vulture, but perhaps there is more to him than that:
_People immediately assume we're greedy, that it's money we're after, people call us ambulance chasers and so on, like we're the proctologists of the profession, and, yes, there's lots of those. But the truth is, the good ones, we'd make the same moves for a single shekel as for a ten-million-dollar settlement. Because it's anger that drives and delivers us._
But Nichole Burnell, the girl who is now paralyzed, someone who can reasonably be assumed to have reason to be angry, has a different take on it:
_It just wasn't right_ — _to be alive, to have what people assured you was a close call, and then go out and hire a lawyer, it wasn't right... Not if I was, like they said, truly lucky... There was no stopping Mom and Daddy, though. They had their minds made up. This Mr. Stephens had convinced them that they were going to get a million dollars from the State of New York and maybe another million from the town of Sam Dent. Daddy said they all have insurance for this sort of thing; it won't come out of anybody's pocket, he kept saying; but even so, it made me nervous._
Banks actually _forces_ the reader to participate in making the story's meaning by deciphering the similarities and differences among the versions. Who is telling the truth about the accident? Whose motives are noble and whose are not?
One variation on the multiple first-person narrator POV uses the epistolary technique—the story is presented as a series of letters exchanged between characters. Though this technique was more common in centuries past, appearing in novels such as _Pamela_ and _Les Liaisons Dangereuses,_ it still turns up in contemporary fiction. An extremely popular recent example is the _Griffin & Sabine_ trilogy by Nick Bantock.
On rare occasions, such as in William Faulkner's "A Rose for Emily," you'll see the first-person plural, where _we_ is used instead of _I_ even though one person is usually speaking for the _we._
One of the chief strengths of first-person multiple-vision POV is the reader's intellectual involvement in the story. It doesn't allow the reader to sit back and be told what to think and feel. The reader must piece things together for himself, which can make for an interesting reading experience.
You might choose this point of view for your own novel if your characters have strikingly different perspectives and you want readers to hear each character's voice directly and to draw their own conclusions. Of course, many writers find it's not that easy to create a single strong convincing voice, much less a handful of them. And you will almost certainly lose some of the focus of first-person singular as the reader slips in and out of several characters' skins.
**FIRST PERSON: PERIPHERAL**
Although the first-person narrator is usually the protagonist, you may choose to have your first-person narrator be another character in the story. A famous example of a peripheral narrator is found in F. Scott Fitzgerald's _The Great Gatsby._ Most of the events narrator Nick Carraway describes in the novel concern the misadventures of the protagonist, Jay Gatsby. Nick's primary job is to observe and relate the story, as he does here:
_And as I sat there, brooding on the old unknown world, I thought of Gatsby's wonder when he first picked out the green light at the end of Daisy's dock. He had come a long way to this blue lawn and his dreams must have seemed so close that he could hardly fail to grasp it._
The peripheral point of view is effective when the story's protagonist is blind to his or her own actions and when that blindness or its consequences are significant enough to strongly affect someone who stands outside the action, as in _The Great Gatsby._
Say you want to write the story of a failing marriage in which both spouses believe themselves to be the injured party. The point of the story is that they're trapped, blind to the wider sense of the truth. So a better choice for a narrator than either the husband or the wife might be someone who is capable of observing things fully, such as the couple's adolescent son. There's nothing he can do to help his parents. He can only watch and learn—that's what makes him peripheral.
But there's a real challenge with this POV as the narrator must report on the protagonist while stuck in the body of a bystander. Nick isn't Gatsby's shadow. He has to go home sometimes. And he's only just met Gatsby. A writer often has to get creative to work around a problem like this, as when Fitzgerald has Nick's (sort of) girlfriend, Jordan, relate the history of the romance between Daisy and Gatsby.
**THE UNRELIABLE FIRST PERSON**
In a sense, all first-person narrators are somewhat unreliable. Even the most scrupulous characters may, unconsciously perhaps, shade the truth or emphasize one fact over another to make themselves look ever-so-slightly better. A boy telling the story of his sister running away from home might not want to own up to his role in her unhappiness. Even an honest fellow like Nick Carraway may distort the truth a bit.
However, if the answer to the question _Who is speaking,_ for example, is an autistic person, a very young child, a psychopath, a cat, a jealous lover, or a habitual liar, the reader understands that the ordinary skepticism does not apply. This narrator has extraordinary limitations and her version of the facts is not to be trusted.
The reader understands after only a few sentences that the narrator of Edgar Allan Poe's "The Tell-Tale Heart" is insane, although he tries to convince us otherwise:
_True!_ — _nervous_ — _very, very dreadfully nervous I had been and am; but why will you say that I am mad? The disease has sharpened my senses_ — _not destroyed_ — _not dulled them. Above all was the sense of hearing acute. I heard all things in the heaven and of the earth. I heard many things in hell._
In Poe's story the narrator's madness leaves the reader off balance, unable to distinguish between the narrator's delusions and reality. The narrator's unreliability adds to the story's unsettling effect. Contemporary writers have used unreliable narrators to underscore my point—that there's no such thing as a reliable narrator. The unreliable narrator emphasizes the philosophical view that there is no such thing as a single, static, knowable reality.
Using an unreliable narrator forces the writer to create two versions of the truth, a steep challenge. But if the POV is handled well, the results can be quite intriguing.
YOUR TURN:
Write a passage from the POV of an unreliable narrator who skews the facts, intentionally or unintentionally. For example, what might a child at a boisterous cocktail party hosted by her parents confide about the guests? What erroneous conclusions might she draw from their behavior and their jokes? Might her false equations add up to cold, hard truths? Of course, if you favor a deluded or deceitful character, go for it. Whomever you pick, see if you can make a reader understand the narrator's unreliability.
**THIRD PERSON: SINGLE VISION**
With the third-person point of view the narrator is _not_ a character in the story. The narrator is a voice created by the author to tell the story. The narrator tells the story of what _he_ did or what _she_ said. Third person has numerous variations with unwieldy names. Never fear. These variations are quite manageable when broken down into their parts.
Perhaps the most prevalent version of the third person is the third-person single vision. With this POV, the narrator has access to only one character's mind. Thus, single vision refers to the way the narrator views a story's events—through the eyes of a single character. The story is told _by_ the narrator, _from_ the perspective of a single participant in the action. The character whose point of view is being recognized by the author is called the point-of-view character. (This term really applies in any type of POV.) The entire story is filtered through the point-of-view character's consciousness.
In the short story "Earth to Molly," Elizabeth Tallent intends for the reader to understand that the opinions the narrator expresses are Molly's:
_At the hotel, really a shabby bed-and-breakfast, the landlady, pinching her upper lip in displeasure at having to hoist herself from her chair, let Molly into her room and left her with the key. The landlady was a long time retreating down the hall. The dolor of her tread, with its brooding pauses, was not eavesdropping but arthritis. Molly was sorry for having needed her to climb the stairs, but of course the old woman complained her stiff-legged way up them all the time, showing lodgers to their rooms. Why, oh why, would anyone spend the night here? A prickly gray carpet ran tightly from wall to wall. It was the color of static, and seemed as hateful._
Notice that it's Molly who is thinking, "Why, oh why, would anyone spend the night here?" just as the details concerning the carpet are filtered through Molly's consciousness.
While this narrator seems to stand just behind Molly's shoulder, or perhaps even lurk in her mind, the third-person narrator may also stand back at a little distance. This may create an ironic or comic effect, as in this description in Kingsley Amis's _Lucky Jim_ of an aspiring academic reacting to a tiresome joke by his superior:
_[Dixon] tried to flail his features into some sort of response to humour. Mentally, however, he was making a different face and promising himself he'd make it actually when next alone. He'd draw his lower lip in under his top teeth and by degrees retract his chin as far as possible, all this while dilating his eyes and nostrils. By these means he would, he was confident, cause a deep and dangerous flush to suffuse his face._
With many of the advantages of first person (the reader empathizes with the point-of-view character much as she does with a first-person narrator), employing an "outside" narrator allows the writer to craft the language in ways that may be implausible coming from the mouth of a first-person narrator. Also, if your narrator is a fictionalized version of yourself, allowing the third-person narrator to tell the story avoids the appearance of self-indulgence.
The third-person single vision is an excellent POV if your point-of-view character is someone with limited intellectual powers or verbal skills. For example, what if the girl who wants to spend her summer vacation with her mother in New Orleans is autistic? Even though she may be quite perceptive, she can't tell her own story if she doesn't have the words to describe what her world is like. The situation would be similar with a character who has little formal education. No matter that he's the smartest person in the story, you'll probably find that a narrator with a greater facility with language will be more successful in conveying the character's shrewdness than the character himself.
The disadvantage of this POV—perhaps the only one—is that the point-of-view character must be present for everything that takes place in the story, just as with a first-person narrator. If your point-of-view character overhears a conversation, she may report that to the reader. However, if the conversation takes place in a health-food store across town, the discussion is off-limits.
YOUR TURN:
Imagine an incident in a department store in which a salesperson and a customer clash over something—shoplifting, rudeness, racial misunderstanding... Using the third-person single-vision POV, write a passage detailing this clash through the eyes of the customer. As is customary with third-person single vision, include the character's thoughts.
**THIRD PERSON: MULTIPLE VISION**
As with first-person POV, a writer using the third-person POV may decide that two or more heads are better than one. The multiple-vision POV allows the writer to show a story's events from different angles.
This point of view is most often used in longer pieces of fiction—novellas and novels. In a shorter piece you might find that you don't have the room to develop numerous point-of-view characters. After all, the reader needs to know fully who these people are in order to make sense of what they're thinking and feeling. Indeed, writers often arrange the perspectives of the point-of-view characters to emphasize their differences.
In _The Watch,_ a novella by Rick Bass, the three point-of-view characters are Hollingsworth, the middle-aged proprietor of a country story in Mississippi, his seventy-seven-year-old father, Buzbee, who has run away to live in a tree in the woods, and Jesse, a cyclist-in-training. The following paragraphs represent two of these characters:
_Hollingsworth would sit on his heels on the steps and tremble whenever Jesse and the others rode past, and on the times when Jesse turned in and came up to the store, so great was Hollingsworth's hurry to light his cigarette... that he spilled two cigarettes, and had barely gotten the third lit and drawn one puff when Jesse finished his Coke and then stood back up, and put the wet empty bottle back on the wire rack, waved, and rode off, the great backs of his calves and hamstrings working up and down in swallowing shapes, like things trapped in a sack._
_The first thing Jesse did in the mornings when he woke up was to check the sky, and then, stepping out onto the back porch, naked, the wind. If there wasn't any, he would be relaxed and happy with his life. If it was windy_ — _even the faintest stir against his shaved ankles, up and over his round legs_ — _he would scowl, a grimace of concentration, and go in and fix his coffee._
Both characters have active inner lives, but Hollingsworth directs his thoughts outward. He craves companionship; in an earlier section we see that he invents names for the other riders in Jesse's group. Jesse is self-absorbed. The wind has no existence for him except as a condition affecting his ride each day. These are my conclusions as a reader, conclusions Bass intends for me to draw.
As a general rule you should make distinct transitions between point-of-view characters. You do not want your reader to be unsure of whose eyes are witnessing the events of the story. Rick Bass never switches point-of-view characters mid-paragraph, and he uses white space on the page to mark the transition for the reader. Novelists often make this switch at a chapter break; that is, each chapter belongs to a single point-of-view character.
Many fine novels use this technique of alternating points of view. Sometimes each point-of-view character gets equal time. But a book may be dominated by one point-of-view character, with only occasional switches to a secondary character's viewpoint. Sometimes the points of view alternate in a pattern and the reader becomes less conscious of the switches as she reads on. In _Happenstance: Two Novels in One About a Marriage in Transition,_ Carol Shields's switcheroo between the husband's and wife's point of view is impossible to ignore—the reader must turn the book over to begin the second half. And in another POV twist, Shields gives no instructions or even hints as to which half should be read first. The reader's experience of the book is literally in his own hands.
No matter what variation or twist you choose, if you elect to use more than one point-of-view character make sure you have good reason. There's no sense in it if the characters view the world in nearly identical terms. As with first-person multiple-vision stories, much of the interest is generated by the disparities and similarities that emerge between the points of view.
Third-person multiple vision also provides a wider view, often creating an effect like a collage. In _The End of Vandalism,_ Tom Drury, the writer, has different episodes seen through the eyes of multiple characters, with the result being a picture of an entire made-up world—Grouse County. The focus is on the community.
With access to more than one character's thoughts the writer gains flexibility. Your story may seem roomier once you leave the confines of a single character's head. In a third-person multiple-vision POV, each character's experience is interesting, but the writer highlights what's most interesting by juxtaposing the various viewpoints.
As occurs with first person, the flexibility you gain with multiple viewpoints costs you focus. The reader's attention and concern are spread more thinly. But this can just as easily be seen as an advantage. With the addition of just one more consciousness the reader is immediately engaged in a more complex way. The reader must observe and draw conclusions based on how the different characters' beliefs contradict or confirm each other. The reader's divided sympathy may be the point of the story.
You might show, for instance, how disastrous it would be for one sympathetic character if another sympathetic character's desires are fulfilled. First you introduce Lily, a young widow, as a point-of-view character, showing the reader how desperate she is and how hard she's trying to be a good mother, and what awful luck has brought her to this place, alone and broke and applying for a job as an attendant at a Laundromat, a job that would allow her to keep her toddler son with her while she worked. (Did I mention that the boy has a rare degenerative disease for which the only promising treatment centers are two thousand miles away?) Then in the next chapter you introduce a second point-of-view character, Jack, who recently lost his hand in a farming accident. He's behind on his car payments and his thoughts are on his beloved dogs, four retired seeing-eye dogs, and the few pieces of dog food rattling in the bottom of the twenty-pound bag. And inevitably, in the way of stories, Jack applies for the same job as Lily because he wants to stay in town and try to get his bearings while he decides what to do next. This is a job he can perform one-handed while he learns to use his prosthetic hand. Now you've engaged the reader by putting him in a very thorny spot. Who should he root for? Which of these wretched souls is more deserving? Multiple-vision points of view can add the desirable kind of complexity to a story, the kind that honors the way our lives are entwined and our sympathies are divided.
YOUR TURN:
Return to the previous exercise, the one with the clash at the department store. Write a passage about the exact same incident through the POV of the salesperson. Then write again about the same incident, this time from the POV of an innocent bystander. You will then have viewed this department-store clash through the eyes of three different characters. Who has the most interesting point of view on this incident?
**THIRD PERSON: OMNISCIENT**
Think "god's-eye view." Think Zeus enthroned at the top of Mount Olympus, the archetypal deity peering down from heaven. Omniscient means all knowing, and thus the writer is always omniscient; the writer should always know everything there is to know about each character and the setting, and every event related to the story, past, present, and future. What distinguishes the omniscient point of view is that the writer who employs it is free to share directly some, or all, of this vast amount of information with the reader.
In each of the points of view I covered earlier, essential information is filtered through the consciousness of one or more of the characters. In the third-person omniscient point of view, the story's information is filtered through the narrator's all-knowing consciousness. Through the omniscient narrator you have the ability to do any of the following: enter the mind of any or all of the characters, interpret the story's events, describe incidents unobserved by any of the story's characters, provide historical context for the story, and inform the reader of future events.
Prior to the twentieth century, most fiction employed omniscient narrators, including many of the big names in literature such as Fielding, Dickens, Tolstoy, Flaubert, and Austen. Their omniscient narrators often had authoritative, opinionated voices, like the one we hear in Washington Irving's _Rip Van Winkle:_
_Whoever has made a voyage up the Hudson must remember the Kaatskill Mountains. They are a dismembered branch of the great Appalachian family, and are seen away to the west of the river, swelling up to a noble height and lording it over the surrounding country. Every change of season, every change of weather, indeed every hour of the day, produces some change in the magical hues and shapes of these mountains, and they are regarded by all the good wives, far and near, as perfect barometers._
Soon thereafter a variety of social changes occurred related to the rise of democracy (and the decline of empires), Freud, religious skepticism, feminism, and so forth which over time resulted in the (now seemingly paternal, heavy-handed, one-sided, bigheaded) omniscient point of view falling out of favor with contemporary writers. Once enough writers traded in omniscience for more limited points of view, this kind of omniscience seemed old-fashioned and fell out of favor. Yet it is still an effective device.
Here's a twentieth-century example of omniscience from Eudora Welty's "No Place for You, My Love":
_They were strangers to each other, both fairly well strangers to the place, now seated side by side at luncheon_ — _a party combined in a free-and-easy way when the friends he and she were with recognized each other across Galatoire's. The time was a Sunday in summer_ — _those hours of afternoon that seem Time Out in New Orleans._
_The moment he saw her little blunt, fair face, he thought that here was a woman who was having an affair. It was one of those odd meetings when such an impact is felt that it has to be translated at once into some sort of speculation._
_With a married man, most likely, he supposed, slipping quickly into a groove_ — _he was long married_ — _and feeling more conventional, then, in his curiosity as she sat there, leaning her cheek on her hand, looking no further before her than the flowers on the table, and wearing that hat._
_He did not like her hat, any more than he liked tropical flowers. It was the wrong hat for her, thought this Eastern businessman who had no interest whatsoever in women's clothes and no eye for them; he thought the unaccustomed thing crossly._
_It must stick out all over me, she thought, so people think they can love me or hate me just by looking at me. How did it leave us_ — _the old, safe, slow way people used to know of learning how one another feels, and the privilege that went with it of shying away if it seemed best? People in love like me, I suppose, give away the short cuts to everybody's secrets._
Welty's narrator enters the minds of both characters. She also interprets the characters' reactions and draws definite conclusions. Yet she is gentle, not insistent. There's room for the reader to draw his own conclusions.
As some contemporary writers have discovered, omniscience doesn't require adopting a biblical tone or throwing the literary equivalent of thunderbolts. A modern and more subtle form of omniscience appears in works by Andre Dubus, Michael Ondaatje, Nicola Barker, Ellen Gilchrist, and Alice Munro, to name a few.
Postmodern writers, such as Milan Kundera, have adopted a conspicuous form of omniscience, eschewing verisimilitude. They call attention to the novel as a _made_ thing. They flaunt their godlike powers. For example, in _The Unbearable Lightness of Being,_ Kundera often interrupts the flow of the story to comment on the story, its themes, or fiction itself:
_It would be senseless for the author to try to convince the reader that his characters once lived. They were not born of a mother's womb; they were born of a stimulating phrase or two from a basic situation. Tomas was born of the saying_ "Einmal ist keinmal." _Tereza was born of a rumbling of a stomach._
_The first time she went to Tomas's flat, her insides began to rumble. And no wonder: she had had nothing to eat since breakfast but a quick sandwich on the platform before boarding the train . ._ .
Once Kundera has made his point, reminding us of the fictional nature of his characters, he returns to their story.
When you think about it, omniscience represents freedom. And freedom is good, yes? Instead of being limited by the intelligence and maturity and sanity of your characters, omniscience provides you with a way to take charge, to make sense of your characters' bizarre behavior or the customs of the planet you just invented.
You can use omniscience to create suspense by supplying the reader with information unknown to the characters. The narrator might inform the reader that even as Sue and Harry prepare for their wedding a giant wave is approaching their town and that when the tsunami strikes in one hour only one of them will survive. Or the narrator might tip off the reader that Miss Harriet Wood, beloved by her first-grade students and their parents and school administrators and stray dogs everywhere, is entertaining a dangerous criminal in her home, and not a representative from an educational book company as she believes. The narrator can make Miss Harriet's story downright painful for the reader, as he anxiously turns the page, hoping against hope that Harriet will get a clue before this villain can harm her.
However, there are reasons why omniscience is used rarely now. Omniscience usually calls attention to the presence of the writer—an undesirable thing for writers who want their readers to suspend their disbelief willingly. Omniscience may seem impersonal to the reader, who is used to being asked to care for a particular character or characters. Omniscience is not for the faint of heart; most writers find it's easier to manage POV when they're limited to revealing the thoughts of just one or two characters. Too much freedom makes them dizzy, like riding a unicycle across the high wire without a net.
YOUR TURN:
Using the omniscient POV, write a scene in which something gets broken at a wedding reception. A gift, a bottle of champagne, somebody's heart... Demonstrate at least three of the five omniscient powers—entering the mind of any character, interpreting events, describing unobserved incidents, providing historical context, revealing future events. There is plenty of opportunity here, as there are bound to be many people in attendance. Relish your godlike ability to know and see everything.
**THIRD PERSON: OBJECTIVE**
This is the ultimate POV challenge, a real test of your abilities to reveal information in scenes. The narrator in the third-person objective point of view is denied access to even a single character's mind. The writer must reveal _everything_ about the story (background, characterization, conflict, theme, etc.) through dialogue and action. The effect is a bit like reading a journalist's account of events, getting only the hard facts.
In his story "Little Things," master short story writer Raymond Carver demonstrates that he's up to the challenge of this point of view:
_He was in the bedroom pushing clothes into a suitcase when she came to the door._
_I'm glad you're leaving! I'm glad you're leaving! She began to cry. You can't even look me in the face, can you?_
_Then she noticed the baby's picture on the bed and picked it up._
_He looked at her and she wiped her eyes and stared at him before turning and going back to the living room._
_Bring that back, he said._
_Just get your things and get out, she said._
_He did not answer. He fastened the suitcase, put on his coat, looked around the bedroom before turning off the light. Then he went out to the living room._
_She stood in the doorway of the little kitchen, holding the baby._
_I want the baby, he said._
_Are you crazy?_
_No, but I want the baby. I'll get someone to come by for his things._
_You're not touching this baby, she said._
This unnamed couple goes back and forth, arguing about who gets the baby. Ultimately, gruesomely, each grabs hold of the child and, Carver deadpans, "in this way the matter was decided." This isn't the kind of thing most people want to believe they're capable of. How could Carver ever convince us of the thoughts a mother or father might have that would allow them to act so atrociously? The objective POV solves the problem for Carver. By reporting the events rather than trying to explain them, he makes what occurs credible.
The primary strength of the objective point of view is that it offers a sense of integrity and impartiality. Objective POV prevents a writer from overexplaining because the writer can't really _explain_ anything at all.
Here's the downside. One of fiction's major attractions is that we, as readers, are allowed insight into the murky minds of others—unlike in real life, where we are left guessing at what's behind a boss's costly dentistry, a child's smirk, a lover's raised eyebrow. The opacity of the objective POV denies us these insights. A story told in objective POV is like a flower minus its scent and vivid colors, a vaguely interesting oddity perhaps, but not likely to attract much attention.
YOUR TURN:
Take the wedding reception passage from your omniscient POV exercise, and revise it using the objective POV. Employ your powers of observation and describe what takes place, as though you are a journalist writing a news account. Remember, this time you can't enter anyone's head. But, what does the behavior of the characters reveal about their thoughts?
SECOND PERSON
As with the third-person points of view, second-person POV stories are told in the voice of a narrator. In second person, however, the narrator tells what _you_ did or said.
When Jay Mclnerney published his novel _Bright Lights, Big City_ in 1987, his use of the second-person point of view created quite a stir in literary circles. His choice of POV was denigrated by some critics as a gimmick. Readers couldn't remember seeing this done before, and they certainly weren't accustomed to being addressed like this by a narrator:
_You are at a nightclub talking to a girl with a shaved head. The club is either Heartbreak or the Lizard Lounge. All might come clear if you could just slip into the bathroom and do a little more Bolivian Marching Powder_... _Your brain at this moment is composed of brigades of tiny Bolivian soldiers. They are tired and muddy from their long march through the night. There are holes in their boots and they are hungry. They need to be fed. They need the Bolivian Marching Powder._
The novel's POV caused many readers to get the sensation that they were the protagonist, the barhopping, late-night snorter of cocaine. Though the book was a best-seller, other novelists didn't rush to follow suit. The POV choice felt like a novelty act—a how-many-times-do-you-need-to-watch-a-man-eat-a-car type thing.
Lorrie Moore employed the second-person POV quite differently in her collection of short stories _Self-Help._ The narrators of many of the book's stories (i.e., "How to Be an Other Woman" and "How to Become a Writer") mimic the advice-giving voice of self-help writers. This is from the story "How":
_Begin by meeting him in a class, in a bar, at a rummage sale. Maybe he teaches sixth grade. Manages a hardware store. Foreman at a carton factory. He will be a good dancer. He will have perfectly cut hair. He will laugh at your jokes._
_A week, a month, a year. Feel discovered, comforted, needed, loved, and start sometimes, somehow, to feel bored. When sad or confused, walk uptown to the movies. Buy popcorn. These things come and go. A week, a month, a year._
_Make attempts at a less restrictive arrangement. Watch them sputter and deflate like balloons. He will ask you to move in. Do so hesitantly, with ambivalence. Clarify: rents are high, nothing long-range, love and all that, hon, but it's footloose. Lay out the rules with much elocution. Stress openness, non-exclusivity. Make room in his closet, but don't rearrange the furniture._
Imagine what you could do with this. What would be the title of your "self-help" short story?
I must tell you, however, that the second person is closely associated with Mclnerney and Moore. Second person has possibilities and it's quite fun to use, but if their goal is publication, it's up to other writers to make it their own—to make it fresh—by using it to create a different effect.
One writer who has done so recently is Helen Dunmore in her novel _With Your Crooked Heart:_
_You lie down on the warm stone, and wriggle your body until it fits. Then you relax, and the terrace bears you up as if you are floating out to sea. Sun has been pouring onto it since seven o'clock, and every grain of stone is packed with heat. Sun pours on to the glistening mound of your belly, on to your parted thighs, your arms, your fingers, your face. No part of you resists, no part does not shine. The moist lips of your vulva are caught in a shining tangle of hair._
Though I do feel the odd sensation, at first, of being addressed, I don't think Dunmore's motives for using second person have much to do with directly addressing the reader. Instead, I hear the voice to be narrating a particular character's experience, putting words to a set of feelings and sensations that would otherwise be unexpressed. I find the voice to be quite intimate, as if the narrator is whispering in the character's ear as she lies in the hot sun. If you can find a compelling way to use the second-person POV, go forth and conquer. Otherwise, proceed with caution.
YOUR TURN:
Rewrite one of your first-person POV passages using the second-person POV. Though you may do little more than switch the pronoun / to _you,_ the effect may be profound. And feel free to change anything you like to fit this new POV. Compare the two versions and consider the different emotional impact of each.
DISTANCE
Early on I stated that one of the questions POV answers is _From what distance are the events being viewed?_ And yes, it's one more thing to factor into your point-of-view decision-making.
EMOTIONAL DISTANCE
This is the distance that we sense between the narrator and the characters, a distance that affects how close the reader feels to the characters. We usually think of emotional distance as an abstract idea, like asking someone, _How close do you feel to your sister?_ But with POV this distance can actually be measured, if we think about it in terms of camera distance:
Long shot: _The man hurried through the cold night._
Medium shot: _The man hurried through the night, squinting against the cold._
Close-up: _As the man hurried through the night, he felt the bitter cold air on his lips._
If the narrator is close enough to feel the cold on the character's lips, we presume the narrator's empathy for the character—very little emotional distance.
Often a writer will pick a camera distance, so to speak, and stick with it for the entire story. But sometimes the camera distance will change during the course of the story.
Jane Smiley opens her novel _A Thousand Acres_ with a panoramic shot:
_At sixty miles per hour, you could pass our farm in a minute, on County Road 686, which ran due north into the T intersection at Cabot Street Road_... _Because the intersection was on this tiny rise, you could see our buildings, a mile distant, at the southern edge of the farm. A mile to the east, you could see three silos that marked the northeastern corner, and if you raked your gaze from the silos to the house and barn, then back again, you would take in the immensity of the piece of land my father owned, six hundred forty acres, a whole section, paid for, no encumbrances, as flat and fertile, black, friable, and exposed as any piece of land on the face of the earth._
Because this land is at the center of the novel's conflict, it is important that Smiley establishes it as a physical, known presence. Yet we wouldn't choose this distance for an entire story or book if we wanted the reader to care about the characters. At this distance the characters would be specks. That distance changes in the next chapter. Smiley zooms the camera in on her narrator, so close the lens could touch her skin:
_Linda was just born when I had my first miscarriage, and for a while, six months maybe, the sight of those two babies [her nieces], whom I had loved and cared for with real interest and satisfaction, affected me like a poison. All my tissues hurt when I saw them, when I saw Rose with them, as if my capillaries were carrying acid into the furthest reaches of my system._
TIME DISTANCE
If it isn't specified, we normally presume that the events in the stories we read occurred relatively recently. In this case we might say there is very little distance in time between the narrator and the story. Though the story is written in past tense (as most are) a writer often creates the effect of immediacy—of the story occurring just now, as we read:
_A young man said he wanted to go to bed with Alexandra because she had an interesting mind. He was a cabdriver and she_ had _admired the curly back of his head. Still, she was surprised. He said he would pick her up again in about an hour and a half. Because she was fair and a reasonable person, she placed between them a barrier of truthful information._
In this, the opening passage of Grace Paley's story "Enormous Changes at the Last Minute," we are meant to be drawn into the present of the story—Alexandra's thinking about how to react to the cabdriver and the hospital visit to her father that the cabbie is driving her to.
Writers occasionally try to narrow this time distance by telling the story in the present tense, as Margaret Atwood did in her novel _Surfacing:_
_He feels me watching him and lets go of my hand. Then he takes his gum out, bundling it in the silver wrapper, and sticks it in the ashtray and crosses his arms. This means I'm not supposed to observe him; I face front._
Does this sound more immediate to you than the previous passage? Well, yes, and that's the intent. But because past tense has been a convention of fiction for so long, most readers now find its use invisible. Formerly quite unusual, present tense doesn't upset many soup bowls anymore.
Sometimes a writer specifies that the events of the story took place long ago, creating a substantial time distance. When this occurs, the reader may feel that the story is tinged with nostalgia or that the account may be in some other way suspect, memory having been eroded over time. The distance is striking in George Eliot's story "The Lifted Veil," which begins:
_The time of my end approaches. I have lately been subject to attacks of angina pectoris; and in the ordinary course of things, my physician tells me, I may fairly hope that my life will not be protracted many months._
The narrator goes on to reflect on the span of his lifetime. He tells us, "my childhood perhaps seems happier to me than it really was, in contrast with all the after-years." Several pages later he's leaped ahead to describe his young adulthood: "At Basle we were joined by my brother Alfred, now a handsome self-confident man of six-and-twenty, a thorough contrast to my fragile nervous ineffectual self."
In an instance like this, when the reader is aware that the story's events occurred long ago, the emotional urgency and suspense of the story may be diminished. In Eliot's story we know the narrator is about to die—there's no hope of a different outcome to the story's events. But this kind of time distance allows the narrator to tell the story with an interesting perspective, often fusing emotions of both the past and the present.
**THE POV CONTRACT**
What you must never forget is that point of view establishes a contract with the reader. POV tells the reader what kind of story he is reading. Break this contract and you risk losing the reader's trust in you. Thereafter the story will never feel quite "real" to the reader. You will distract the reader from the smooth red-carpet-like unfolding of your story.
Novice writers sometimes break the POV contract with a careless slip. One of my fiction students chose a third-person limited single-vision POV for her story, the point-of-view character being Barbara, the CEO of a cosmetics company. Thus, the contract is this: all the events should be filtered through Barbara's consciousness.
My student's story was moving right along, with competent handling of character development and conflict. Then, _bang!_ The narrator entered the head of a minor character and the story stumbled, like so:
_Barbara spent the next ten minutes listening to her accountant. She couldn't focus on the papers Ted waved in front of her or his words. All she could think of was of the message John had left on her answering machine._ I'm heading to Montana. I know this is sudden and I feel like a jackass, a cliché, but I've really got to get away and figure some things out.
_She pushed back from her desk and grabbed her jacket from the chair. She thanked Ted for his attention to detail. "And as soon as I finish with today's meetings I'll look over these ledger sheets."_
_Once again feeling dismissed and belittled, and wondering why he kept working for such an ungrateful boss, Ted began to gather up the papers from her desk._
_"You can leave those papers where they are, Ted. I said I'd get to them later."_
No offense to administrative assistants, but in this story Ted has a walk-on role. His only reason for existence in the story is to inform Barbara that one of her trusted employees has been embezzling. My student confessed she hadn't even noticed the lapse. To adhere to the POV contract while still revealing Ted's attitude, my student can simply describe Ted's actions. She might rewrite the offending paragraph to read:
_Ted sighed and started to gather up the papers from her desk. "Fine, just fine," he muttered, "I'il just go sit in my corner and wait meekly for the madam to summon me."_
As a safeguard against POV abuses, you might write down your point-of-view rules regarding omniscience, reliability, and distance. When you finish a draft you can check every paragraph against these rules.
But as many a recording artist knows, contracts are made to be broken. Occasionally a very daring writer will break the POV contract deliberately, to achieve a special effect—when that is what the story needs.
In Richard Russo's _Empire Falls,_ the preface is presented by an omniscient narrator. Then a third-person narrator follows the protagonist, Miles Roby, for two chapters; we get a chapter told from the POV of Miles's estranged wife and one (suddenly switching from past to present tense) from the POV of Miles's daughter. Thereafter Russo establishes a pattern of third-person multiple vision, the vision shifting with chapter breaks. But, wait, in chapter 7, we're in a tavern and the POV suddenly starts shifting back and forth between the tavern's owner and the only two customers in a way that looks suspiciously like omniscience. However, none of this feels careless. The author wants us to view the inhabitants of this town in a slightly unpredictable manner and we trust that he knows what he's doing.
**HOW TO CHOOSE**
As I warned you, the POV menu is complicated, but I hope you're beginning to feel comfortable with it. And I'm betting that by now you're convinced that your choice of POV is one of the biggest choices you will make with a piece of fiction. As stated earlier, it affects everything. As with all significant and complex decisions, your task will be easier if you're able to narrow your choices.
I suggest asking yourself, _Whose story is this?_ Many times you'll be able to answer immediately. Stories written in the first person usually do belong to the narrator, just as stories written in third-person single-vision POV usually belong to the point-of-view character. That character most often has the most at stake. Whether the reader thinks the story has a happy ending depends on whether the point-of-view character thinks it does. If your story clearly belongs to a single dominant character, then first person or third-person limited is the obvious choice. Then it's just a matter of deciding if you want the story told in the character's voice or not.
Stories populated by extended families, blended families, marriages, soccer teams, submarine crews, and people linked by a situation (as in _The Sweet Hereafter)_ often have more than one protagonist. Many times each and every character has a goal, and frequently the goals conflict with one another. In this case you must ask yourself, _What's most interesting to me about this story?_ Say your story is about a prestigious jazz quintet in which the trumpet player, after a dozen years with the group, wants out. As you consider this scenario, are you most interested in how the trumpet player struggles with his feelings of obligation and the obstacles the other four musicians create? If so, use one of the single-vision POVs. Maybe, however, you are most intrigued by what happens to the group as a whole. Does a leader emerge? Do any of the other members secretly work to help the trumpet player? Will the group be able to stay together if the trumpet player succeeds in leaving? Your interest in the group's story doesn't necessitate a multiple-vision or omniscient point of view; perhaps there is one character who can act as the spokesperson, providing the eyes and ears for the whole group—but you will at least want to consider them because of the flexibility they offer.
Another question you can ask yourself is as basic as it gets. _What kinds of stories do I like to read?_ If you had a whole glorious weekend to read—if by some miracle you weren't going to be allowed to do anything but read—what would you pick up first, a multigenerational epic or an intense story with a memorable protagonist? Are you most interested in psychological dynamics—what happens to one person as she faces adversity or social dynamics—or in how individuals react and play off one another as they struggle to achieve their individual goals? Which movie better suits your tastes, _Rocky_ or _A League of Their Own?_
You can't, however, be confident you've found the best POV for your story when it's just an abstract idea. You need to taste that POV. Just as the wine connoisseur holds a sip of wine in her mouth and swirls it across her tongue and cheeks to get the full effect, you need to discover how the POV feels on the page. Does it sound like you expected it to? Does it have the complexity of flavor your story needs?
You may be able to learn all you need with only a page or two of a draft. You may reach a bull's-eye certainty after considering only a couple of points of view. Then again, maybe not.
I wrote half a novel in first person because my inspiration came in the form of a sentence landing in my head on my morning walk. The voice said, "My name is Eleanor Sweetleaf and I've lived in this house since I was three days old." Who was I to argue with inspiration? But a year or so later I was stuck. Eleanor Sweetleaf was a good soul, but not quite the right voice for the story after all. She took her story a little too seriously. After grinding my teeth for a while, I began rewriting in the third-person single-vision POV—a better choice, I quickly realized. The revised story was funnier and I felt freer to play with language. Once I let go of my death grip on first person I found I was also willing to change other aspects of my story. My husband was horrified at first; he didn't understand how I had failed to realize sooner that I was writing in the wrong point of view. He felt bad about all that "wasted" work. But what he didn't know (yet) was that writing is all about trial and error. And he also didn't know that it's the writer's duty to fully exercise the enormous power of point of view.
[CHAPTER 5
**DESCRIPTION: TO PICTURE IN WORDS**](Facu_9781596917910_epub_c4_r1.html#aa5)
BY CHRIS LOMBARDI
About twelve years ago, my best friend was reading a draft of a story I'd written about a woman recently returned from years in a far-off country, grieving the lover she had left behind. In one scene Ruth, the protagonist, is insomniac and considers calling her lover four time zones away. Or rather, she gets up at two a.m. and stares at the telephone, an old, black instrument with a battered dial, even though it's the late 1980s:
_The light from the street made the phone a ghost._
My friend, reading this, looked up from the page and cried out: "Where do you get such descriptions? You never notice anything!"
And she's right. When walking down the street together, she was always the one to point things out, while I remained absorbed in my thoughts. But it seems that even back then I noticed details from my peripheral vision and filed them for later use. I learned, in the back of my brain somewhere, the heft of a hammer in the palm, the way a set of keys feels like home. Somewhere I noticed the weird shadows cast by city lights that turned familiar objects eerie. When I was imagining Ruth contemplating the telephone, she (and I) saw a ghost.
My first teacher of fiction was the novelist and much-heralded writing teacher John Gardner, who taught that any good writer is creating, with words on paper, "a vivid and continuous dream." By _vivid_ he meant a dream that feels as sharp and focused as real life. By _continuous_ he meant a dream that remains vivid, not allowing the reader's mind to wander out of the fictional world.
When I think of description, I think of film, which is quite similar to a dream state. Think about it. You enter a darkened theater and for a couple of hours you are enveloped by an alternate reality that leaves you blinking as you emerge. With the movie, it's the filmmaking that keeps you engrossed. With fiction, more than anything else perhaps, it's the description that envelops you because really everything in a work of fiction, except for the dialogue, is a description of some sort. When writing this description you want to make sure the reader experiences the story as vividly and continuously as if he or she is watching a spellbinding film. You don't want the story fading out in the middle like an old Super 8 home movie shown on a bad projector. You want to ensure that your movie is written in full color, even if the colors are gentle, muted, not blazing at all.
_Webster's New World Dictionary_ offers two definitions for the verb _describe:_
1.to tell or write about; give a detailed account of
2. to picture in words
To give a detailed account. To picture in words. That particular _Webster's_ scribe is a poet.
For the purposes of storytelling, description is anything that creates a picture in a reader's mind. If the descriptions are good enough, the reader will forget about the rain outside his window, the fact that her chair is a little uncomfortable, the fact that the rent is due. The reader will be swept along by the words, believing every moment of the story, as if it's a dream or a movie, or as if it were actually happening.
**THE FIVE SENSES**
You write and read with your brain, but you live your life most defìnably in your body. To convey that experience, you need the _physicality_ of it. Your morning trek to work consists of a series of aggravations, or so my writing students have told me repeatedly. But at bottom, it consists of your feet on the carpet, the feel of your jacket on your skin, the noise of the street, and so on. That's how we learn the world.
To bring a reader into your fictional world, you need to offer data for all the senses. You want to make your readers see the rain's shadow, taste the bitterness of bad soup, feel the roughness of unshaved skin, smell the spoiled pizza after an all-night party, hear the tires screech during the accident. Note that I've referred to all five senses. Don't be tempted to focus only on sight, as many beginning writers do. It may be the sound after the party that your character really remembers. You may find that the feel of the fabric of a character's dress tells more about her upbringing than her hairstyle does.
In Anna Quindlen's spellbinding novel _Black and Blue,_ the protagonist—a battered woman fleeing her attacker—meets her first new friend in a suburban Florida town:
_She was wearing pink linen shorts and a matching blouse, white sunglasses, and pink nail polish. She sounded like an actress playing Blanche du Bois in summer stock, and looked and smelled as if she'd groomed herself as painstakingly for that morning as I had the morning I got married. A drawl and Diorissimo, or something that smelled a whole lot like it._
We get a strong sense of this character because we are experiencing her through our senses, in this case sight (her clothing), sound (the way she talks), and smell (her perfume).
You need these kinds of sensory details to support more general statements or abstract descriptive phrases. You may write poetic, sweeping statements, in sentences whose music makes the reader smile. But giving too many of those without sensory detail is kind of like serving the aperitif without the meal. I might get drunk, but I'll fall asleep during the movie.
William Faulkner, in his story "Barn Burning," begins with smell and expands to include other senses:
_The store in which the Justice of the Peace's court was sitting smelled of cheese. The boy, crouched on his nail keg at the back of the crowded room, knew he smelled cheese and more: from where he sat he could see the ranked shelves close-packed with the solid, squat, dynamic shapes of tin cans whose labels his stomach read, not from the lettering which meant nothing to his mind but from the scarlet devils and the silver curve of fish_ — _this, the cheese which he believed he smelled and the hermetic meat which his intestines believed he smelled coming in intermittent gusts momentary and brief beneath the other constant one, the smell and sense just a little of fear because mostly of despair and grief, the old fierce pull of blood._
Here Faulkner weaves in smell, sight, taste (even if vicarious), and the physical sensations attached to emotion—all pretty compact within this rather robust passage.
Or how about this brief passage from Amy Tan's "Rules of the Game" that utilizes all five senses:
_We lived on Waverly Place, in a warm, clean, two-bedroom flat that sat above a small Chinese bakery, specializing in steamed pastries and dim sum. In the early morning, when the alley was still quiet, I could smell fragrant red beans as they were cooked down to a pasty sweetness. By daybreak, our flat was heavy with the odor of fried sesame balls and sweet curried chicken crescents. From my bed, I would listen as my father got ready for work, then locked the door behind him, one-two-three clicks._
We see, hear, smell, feel, and even taste this world. We are physically _there._ The most powerful method for luring readers into the fictional world is through sensory experience.
YOUR TURN:
Pick a character and imagine he or she has gone spelunking (cave exploring) with a group of friends. Unfortunately, your character has become separated from the group and now he or she is groping through a pitch-dark passage (without a flashlight), searching for either a way out or the missing companions. Write a passage bringing this scene to life through sensory description. Since vision is limited, you'll have to rely on hearing, smell, touch, and taste. Let the reader physically experience this place through these senses.
SPECIFICITY
Your descriptions can't just offer sensory details, though; the details also have to be specific. The cumulative effect of specific sensory details is verisimilitude—the sense that these events have really happened.
Many years ago a high school writer friend of mine, who's now a professor, asked about my use of _his intense gray eyes._ "What does that mean?" he asked. I've never forgotten it. Vagueness is often our first impulse when we're getting something down. When I wrote _intense gray eyes,_ what did I mean? I meant, first, that the eyes were slate gray, and second, that they glittered a little, like he had extra tear ducts. But that is not what I conveyed with my vague description.
Specificity also prevents a sort of writer's laziness. _She was a beautiful blonde._ That's vague enough not to give us a picture at all, and it smells like it was easy to write. Give us specific details about this blond beauty, like so:
_Her nose was dusted ever so lightly with freckles, as softly colored as the skin below._
Paint a picture with your words. For example, Jeannette Winterson offers this sweeping description in her novel _The Passion,_ a fable of eighteenth-century Venice:
_There are exiles too. Men and women driven out of their gleaming palaces that open so elegantly to shining canals._
It sounds powerful, with adjectives like _gleaming_ and _shining_ suggesting the glamour of what's been lost. But we're not actually in the picture until the author follows it up with:
_One woman who kept a fleet of boats and a string of cats and dealt in spices is here now, in the silent city. I cannot tell how old she may be, her hair is green with slime from the walls of the nook she lives in. She feeds on vegetable matter that snags against the stones when the tide is sluggish. She has no teeth. She has no need of teeth. She still wears the curtains that she dragged from her drawing-room window as she left._
Note the stones, the green hair, the lack of teeth, the curtains. With these specific details, Winterson brings this mythical woman alive as a macabre figure in her near-noir romance. Specifics can make the reader believe anything, including that an aristocrat fleeing the French Revolution ended up feeding in Venice's canals while she played in its casinos. Or that all Venetian boatmen, like Villanelle, the book's narrator, have webbed feet. The specific details weave a world, and the reader is willing to stay in it—to watch Villanelle fall in love with an aristocrat's wife, and later to watch her pair up with one of Napoleon's cooks.
Specific descriptions make true more homespun locations as well. Louis B. Jones's _Ordinary Money_ shows the reader its location in working-class northern California by simply directing the reader there:
_There is a stop sign at the 7-Eleven, and you go left onto Robin Song Lane, then right onto Sparrow Court, and Wayne and Laura Paschke's house is the third on the left, the same model as the neighbor's, but painted an out-of-date sherbert green, with a big chicken-wire thing on the side, left there by the previous tenant_ — _and the hard lawn and the oil-stained driveway which always provide a landlord with a reason for keeping the damage deposit._
The author is so specific in conjuring this place that it's almost impossible not to believe it truly exists.
If, say, a character drives a car, consider telling us what kind of car. Earl, the car thief in Richard Ford's "Rock Springs," drives a cranberry-colored Mercedes. Not only can we picture that particular car, but it also tells us a few things about Earl and his taste in stolen vehicles.
Think of yourself as a collector—of sensations, of objects, of names. Especially names. Don't be like one of my favorite poets, John Berryman, who famously said: "I don't know one damned butterfly from another." I'm as guilty as many in this; urban chick that I am, the names of trees and such send me scurrying to books. But I go to those books to learn the names of trees and colors and everything else because I know those names will notch up the clarity of my fiction.
You should do this too. Name exact colors, for example—not that you should rattle off every gradation in the Crayola 64 box, but learn and use the names of some: ocher, cornflower, or even something like "pale pink shading to white." Name fabrics, tastes, musical instruments. Even brand names can be useful, though if overused they come off as a cheap thrill, and distract the reader. (Of course, Bret Easton Ellis disagrees with me, as readers of his novel _American Psycho_ can attest. But there the brand names support the theme of American greed.)
And sometimes a list of names itself becomes accomplished description. Students of Homer call them "heroic catalogs," after those breathless recitations of a hero's armor, a goddess's boudoir, an army's food supply, that march through Homer's _Iliad._
Watch how Barbara Kingsolver, in _The Poisonwood Bible,_ paints the Congo with little more than this list:
_All God's creatures have names, whether they slither across our path or show up for sale at our front stoop: bushbuck, mongoose, tarantula, cobra, the red-and-black monkey called_ ngonndo, _geckos scurrying up the walls. Nile perch and_ nkyende _and electric eel dragged from the river._ Akala, nkento, a-ana: _man, woman, and child. And everything that grows: frangipani, jacaranda,_ mangwansi _beans, sugarcane, breadfruit, bird of paradise._
YOUR TURN:
Think of a place well known to you from your youth—a street, park, school... Write a passage where you describe this place with great specificity. What color were the bricks? Was the slide straight or curving? How far was the pond from the house? If you can't remember key details, fill them in with your imagination. For a bonus round, do the same for a person you knew from this place.
**THE BEST WORDS**
What is description made of? Words, of course. If you're bringing the movie in your head to the page, words are the strands of light that determine the colors, and shadows, and clear shapes.
Mark Twain once noted that the difference between the right word and the almost right word is the difference between lightning and a lightning bug. Always challenge yourself to find the best possible word to convey the picture in your mind. Quite often the perfect word comes to you instinctually and, no, you shouldn't agonize over every word as you fly through a first draft. But at some point, find the words that best sustain the magical illusion of your story.
Let's return to this line from _The Passion:_
_One woman who kept a fleet of boats and a string of cats and dealt in spices is here now, in the silent city._
Everything is pretty straightforward in that sentence except for the phrase _a string of cats._ Why did the author choose _string?_ She could have used any number of other words— _collection, group, family, pack, litter, entourage, coterie,_ to name just a few. But obviously she felt there was a particular meaning in the word _string_ that made it feel just right. Perhaps she liked the sense of the cats following in single file or the sense that the cats were somehow attached to the woman. Regardless of whether the author found this word instantly or spent half a day worrying over it, the word _string_ makes a strong and specific impact.
How big is your vocabulary? Though you don't want to show off by using elaborate words all the time, you should always seek to widen your choice of word possibilities. Keep a dictionary around. An old, old language, English has absorbed words from Latin, French, Spanish, Asian languages, and many others, giving us a range of choices that rivals the spectrum of the rainbow. If you're at a loss for a word, the dictionary and its cousin, the thesaurus, could be your best friends.
Just watch out for adjectives and adverbs. Like sirens, they can lure you into the perilous waters of weak description.
When many people think description, they often think adjectives and adverbs. As you know, adjectives describe nouns, as in _her light hair,_ and adverbs describe verbs, as in _she walked lightly._ Think of the pattern of speech: _The word I'd use to describe Alan is fulsome._ But the truth is that adjectives and adverbs can be very lazy words. They deceive you into thinking they're doing their job when really they're not doing much at all. Remember my _intense gray eyes._ That's two adjectives pretending to really describe those eyes. But they haven't done much at all—a hint of sensory, a hint of specificity, but nothing that brings those eyes, or their owner, to life.
And a sentence with too many adjectives and adverbs is like an unpicked apple tree, the boughs sagging from the weight. Like so:
_She walked gracefully into the spacious room, swiftly removing a letter from her designer-label purse and regarding us all with her intense gray eyes._
Despite all those adjectives and adverbs, we're getting little more than the bare facts. This tree needs picking.
If you look carefully at good description, you'll notice that writers are often quite sparing in their use of adjectives and adverbs. In "Cathedral," the narrator relates his first impression of the blind man's eyes:
_At first glance, his eyes looked like anyone else's eyes. But if you looked close, there was something different about them. Too much white in the iris, for one thing, and the pupils seemed to move around in the sockets without his knowing it or being able to stop it. Creepy. As I stared at his face, I saw the left pupil turn in toward his nose while the other made an effort to keep in one place. But it was only an effort, for that eye was on the roam without his knowing it or wanting it to be._
How many adjectives do you pick out of that passage? Three: _different, creepy,_ and _left (white_ is being used as a noun). And Carver isn't depending on those adjectives to do the real work.
However, when used sparingly and well, adjectives and adverbs can be quite effective. Let's return once more to that sentence (with the cats) from _The Passion:_
_One woman who kept a fleet of boats and a string of cats and dealt in spices is here now, in the silent city._
This sentence contains one well-placed adjective— _silent_ —and it works magnificently, adding a perfect and necessary final touch to this sentence.
Adjectives and adverbs are helper words, what the grammarians call "modifiers." They help refine the impression cast by your true building blocks: nouns and verbs. At a writers' conference a few years ago, a supposedly clever expression was circulating: _Are your verbs working hard enough?_ Granted, the expression isn't all that clever, but it points to a truth. The stronger your nouns and verbs are, the better they can support your carefully chosen modifiers.
Look at this passage from F. Scott Fitzgerald's _The Great Gatsby._ As one of Gatsby's famous parties begins:
_Suddenly one of these gypsies in trembling opal seizes a cocktail out of the air, dumps it down for courage, and moving her hands like Frisco dances out alone on the canvas platform._
Look carefully at this sentence. It features only two adjectives (or three, if you count _alone),_ but its nouns and verbs carry maximum impact. Not _a woman_ but _one of these gypsies,_ not _takes_ but _seizes._ Notice how the strong verb phrases alleviate the need for modifiers, as in _dumps it down for courage_ and _moving her hands like Frisco._ (The last phrase refers to a jazz dancer of the 1920s.) I'm fascinated by the fact that I can draw such sparing use of modifiers from one of our more florid writers.
For a more contemporary example, let's look at the following portrait of the narrator's mother from Melanie Rae Thon's story "Nobody's Daughters":
_Past noon, Adele still fogged. I knew everything from the sound of her voice, too low, knew she must be on night shift again: nursing home or bar, bringing bedpans or beers_ — _it didn't matter which. I saw the stumps of cigarettes in the ashtray beside her bed. I saw her red hair matted flat, creases on her cheek, the way she'd slept. I smelled her, smelted the smoke in her clothes, the smoke on her breath._
You'll find very few modifiers in here. But notice the strength of the nouns: _stumps of cigarettes, creases on her cheek._ And notice such strong verbs as _fogged_ and _matted._ The nouns and verbs paint a picture.
As previously noted, strong verbs can even alleviate the need for adverbs. For example, _she walked lightly_ can be effectively transformed into _she glided_ or _she floated,_ each more evocative than the version leaning on the adverb.
Look at this example from Arundhati Roy's _The God of Small Things:_
_By early June the southwest monsoon breaks and there are three months of wind and water with short spells of sharp, glittering sunshine that thrilled children snatch to play with. The countryside turns an immodest green. Boundaries blur as tapioca fences take root and bloom. Brick walls turn mossgreen. Pepper vines snake up electric poles. Wild creepers burst through laterite banks and spill across the flooded roads. Boats ply in the bazaars. And small fish appear in the puddles that fill the PWD potholes on the highways._
Note how vibrant this place is made through such dynamic verbs as: _breaks, snatch, blur, root, bloom, snake, burst, spill, ply._ No adverbs needed. Though a few adjectives are sprinkled in, they are invariably linked to strong nouns that don't get overshadowed by them.
I'm not telling you to avoid adjectives and adverbs entirely. But first focus on the best possible nouns and verbs, then find the modifiers that enhance these words, adding subtle touches to the foundation.
YOUR TURN:
Pick a person you know. Fictionalize the name, which will also give you license to alter other characteristics, if you so desire. Now describe this person as vividly as you can. Here's the catch: you _cannot_ use a single adjective or adverb. This will force you to use strong nouns and verbs and employ some of the other techniques you've picked up in this chapter. Though challenging, you will probably end up with a very well-drawn picture of this person.
TRICKS OF THE TRADE
Now that we've covered some of the brass tacks of good description, it's time to look at some ways to further expand your palette of descriptive options.
First, learn to embrace figurative language, a fancy expression for figures of speech, as in similes and metaphors. These are scary-sounding words out of an English class, but they're really shorthand for the way we think, the way we process information and emotions.
A _simile_ is defined (by the _American Heritage Dictionary)_ as "A figure of speech in which two essentially unlike things are compared, the comparison being made explicit typically by the use of the introductory 'like' or 'as'..." A _metaphor_ (according to _Chambers's Twentieth Century Dictionary)_ is "a figure of speech by which a thing is spoken of as being that which it only resembles, as when a ferocious man is called a 'tiger.'"
We use these every day. When you tell a friend, _I was like a house on fire!_ or _He's such a wet blanket!_ you're doing it—taking an image or idea from the universe of common memory and yoking it to a person or experience. Of course, the examples I've given are hackneyed—that's one reason why they work well on the phone or on the street. Everyone understands them.
But in fiction, your task is to use similes and metaphors that are too fresh, too surprising, to be something you've heard on the phone. Why should you bother? Because figures of speech are a stealthy way of reaching into your reader's subconscious. You're pulling up visual images, remembered experiences, bits of their own dreams, and showing them anew. Your descriptions now have double the power.
Here are two arresting similes from Mary Gaitskill's "A Romantic Weekend":
_She felt like an object unraveling in every direction._
_His gaze penetrated her so thoroughly, it was as though he had thrust his hand into her chest and begun feeling her ribs one by one._
In Calvin Baker's novel _Naming the New World,_ a metaphor appears when a man sees the rising sun as
_a beautiful almond with honey edges._
Now, you would never say on a street corner, _Wow, look at that sunrise! A beautiful almond. Yeah, with honey edges._ But this jazzy metaphor used in description feels just right, especially as it takes a little bow toward the Deep South, where the novel takes place.
My partner, a poet, told me when we met, "I hate similes; I like metaphors better." I agree that metaphors feel more powerful, but I think similes are a far suppler instrument. You can do anything with them—stick them in dialogue, give them to a first-person narrator, embed them in news headlines or gossip. Metaphors lend themselves to a heavier narrative style, which may or may not work for your story, depending on its tone. And an extended metaphor can dominate a story entirely, as when the protagonist of Franz Kafka's _The Metamorphosis_ finds himself transformed in his bed into a giant insect. That's when you find yourself at the level of allegory where a whole story stands for something else.
Are you one of those people whose writing prompts comments like _That's very poetic_ or _Wow, it's almost poetry?_ If so, count yourself among the lucky few who already know a few things about lyricism. By lyricism, I mean prose that plays with sound and rhythm in the way that poetry does.
Feel the lyricism in the final line of James Joyce's "The Dead":
_His soul swooned slowly as he heard the snow falling faintly through the universe and faintly falling, like the descent of their last end, upon all the living and the dead._
How do you know if you've got any lyricism? It helps to read your work aloud and hear the ebb and flow of the rhythm and hear how the words slide and sing. You'll also hear where things start to _clunk._
What does all this lyrical effect do for fiction? Just like figures of speech, lyricism sinks your story deeper inside the subconscious of the reader. If music says things words can't express, text that feels like music also carries those nonverbal meanings, immersing the reader in the experience in a rather primal way. And just because I say _lyrical,_ this doesn't mean you must use long, elaborate sentences.
Ernest Hemingway knew how to make beautiful music of simple words and short sentences, as in the following descriptive passage from "True at First Light":
_Then I looked through the trees at the Mountain showing very big and near this morning with the new snow shining in the first sunlight._
Notice the almost iconic power of the image, rendered through the chantlike rhythm.
In "Cathedral," the narrator, who doesn't even _like_ poetry, manages a simple lyricism when he asks the reader to
_Imagine a woman who could never see herself as she was seen in the eyes of her loved one. A woman who could go on day after day and never receive the smallest compliment from her beloved. A woman whose husband could never read the expression on her face, be it misery or something better. Someone who could wear make-up or not_ — _what difference to him?_
To further deepen your descriptions, consider onomatopoeia, achieved when words sound like what they are. I just did it earlier, when I mentioned prose going _clunk._ If people in your stories _murmur,_ if crowds _buzz,_ if the tea kettle _hisses,_ you're employing onomatopoeia.
In this passage from Barry Hannah's "Testimony of a Pilot," check out the effectiveness of the onomatopoeia _whistling_ at the very end:
_It was a grand cannon, set up on a stack of bricks at the back of my dad's property, which was the free place to play. When it shot, it would back up violently with thick smoke and you could hear the flashlight battery whistling off._
Also consider alliteration, where two or more words have a common initial sound. Alliterations comes naturally to us; it's a game we've played since we were three. _Meet Bobby Bumblebee!_ Alliteration can be overused, but when used judiciously it introduces a wonderful grace note to a description.
Notice how alliteration helps conjure the sense of quietly falling snow in that passage from "The Dead":
_His soul swooned slowly as he heard the snow falling faintly through the universe and faintly falling, like the descent of their last end, upon all the living and the dead._
When done well, these creative elements can blend together in an effortless flow. Note Bharati Mukherjee's description, in _Leave It to Me,_ of the place where her narrator was born:
_I have no clear memory of my birthplace, only of the whiteness of its sun, the harshness of its hills, the raspy moan of its desert winds, the desperate suddenness of its twilight: these I see like the pattern of veins on the insides of my eyelids._
In addition to alliteration ("harshness of its hills"), note how Mukherjee also uses simile (those veins) and onomatopoeia ("raspy moan"). Her rhythm's not bad, either. Try reading that paragraph aloud. Hear the music.
Finally, I'm going to pass on one of my own trade secrets, a way of conjuring fresh images that's often got me out of a description jam: use an image or adjective usually associated with one sense unexpectedly with another. It's a poet's trick, known as synesthesia. John Keats used it here:
_Taste the music of the vision pale_...
A couple that I've used in my own work:
_the sound that washed your senses his dark chocolate voice_
Try it. At the very least, synesthesia is fun to play with; at best, your description will jump to life in a startling way.
YOUR TURN:
Take one of the previous exercises from this chapter and revise it by leaning on such devices as simile, metaphor, lyricism, alliteration, onomatopoeia, and perhaps even synesthesia. Run wild, using as many of these devices as you can. The results may be a bit overripe, but you will have cultivated your inner poet.
TELLING DETAILS
You've picked up a lot of techniques to energize and excite your descriptions. You may be tempted to run free with them, alliterating here and bursting with high-flung metaphors there, layering on smells and tastes and sounds, until your readers feel gorged with sensations.
But it's important, ultimately, to choose your descriptive details. As readers, we know what it's like to slog through a thicket of description—to lose track of a story in the avalanche of detail about the lush tropical stream, the cold Manhattan apartment, the overview of a village at the top of a mountain. All I wanted to know, you want to say to the author, is what she looks like, and you gave me three rambling pages describing her every detail.
There's a fine line between lush description and the kind that chokes the reader. Such description is easy to fall into when you're describing a place you think may be foreign to your readers, or even working to get the details of someone's clothing or gestures. Be particularly careful of language that's so beautiful you notice it just for that. Always ask yourself: _Does the description interrupt the flow of the story?_
Anton Chekhov, one of the pioneers of the contemporary short story, gave us the classic definition of what a story does: "the casual telling of a nuclear experience in an ordinary life, rendered with immediate and telling detail." What did he mean by the _telling_ detail?
A telling detail does what it says: it tells the essence of what it's describing. Telling details are the Scotch tape holding up Susie's hemline in the back, the tiny piece of ice that never seemed to melt in the bottom of Mom's martini, the street sign on the corner that still says, to this day, school crossing, though the school is long gone. A telling detail can speak volumes in a very short amount of time. They help you achieve a golden mean—enough description to paint the picture, but not so much as to weigh it down.
Look at the opening of Anna Quindlen's _Black and Blue:_
_That butterscotch-syrup voice that made goose bumps rise on my arms when I was young, that turned all of my skin warm and alive with a sibilant_ S, _the drawling vowels, its shocking fricatives. It always sounded like a whisper, the way he talked, the intimacy of it, the way the words seemed to go into your guts, your head, your heart_
The telling detail of the character being described is his voice. The author gets the most out of her description of the voice by using synesthesia ("butterscotch-syrup" to evoke the voice's smoothness and sweetness), simile ("like a whisper"), and a precise catalog of detail: "sibilant _S,_ the drawling vowels, its shocking fricatives." For the character's response to the voice we get a quick hit that tells all—"made goose bumps rise on my arms when I was young" and "the words seemed to go into your guts, your head, your heart."
Soon enough this man will be described visually, his actions named. But the voice is how he is introduced and his voice is what we'll remember. Just as many people, if they remember nothing else about _The Great Gatsby,_ remember that Jay Gatsby felt the voice of his beloved, Daisy, was "full of money."
In Toni Morrison's _Beloved,_ the eyes of Sethe, the protagonist, are certainly a telling detail. (Handled so much better than my _intense gray eyes.)_ Here is how Sethe's eyes are seen by her old friend Paul D:
_irises the same color of her skin, which, in that still face, made him think of mercifully punched-out eyes._
Later, another character sees Sethe's eyes this way:
_Since the whites in them had disappeared and since they were as black as her skin, she looked blind_...
Such telling details stick with us and define the place, character, or atmosphere. And they stay in the reader's mind with an almost hypnotic force.
You may not know which of your details, at first, are the telling ones. It's only when all of them have made it out of your head and onto the page, only when you've gotten to the end of your first or second draft, that you'll notice which have borne repeating. What does your protagonist remember about his childhood home years later? What feature of Vietnam's spectacular sunsets represents the whole, years later? I ended up having to answer the latter question in one of my novels, when too much detail about Asia threatened to choke. What remains now are colors associated with tastes, "watermelon colors," "rose ice cream skies," repeated through three Asian countries and my character's dreams. The combination of color and taste and sunsets seemed to be the telling detail that most reflected the emotional response to the place. You'll know when you've found the telling detail: it's the detail that sticks with you the most.
Until you find that telling detail, however, be generous. As the story in your head starts to move and your hands follow it, try to write it all down, everything that comes to you, especially any sensory detail.
I can't tell you how many times I've looked at a student's work and asked for more detail about this or that—a place, a person—only to be told,
"I didn't want to overdo it." Novice writers, just getting their chops, need to worry more about saying _enough._ You're so familiar with the scene in your head that you may think just a few words are needed to bring it alive. And it's possible that you're right—but it's unlikely that you know, right away, which few words those are. Get it all on the page first, and then cut back as needed. Even if your preferred style is on the minimalist side, if you like Raymond Carver more than Arundhati Roy, I encourage you to be generous on the page.
A brilliant young writer, whom I knew in graduate school, favors a stripped-down, economical delivery. His stories, from draft to draft, undergo constant unfolding and compression, compression and unfolding. One draft may be four pages, the next eleven, the next five, and so on, as he fills out the scene and then pares the excess.
For right now, give it all you've got. Eventually you'll find the right time to pick and choose the most telling details.
YOUR TURN:
Return to the previous exercise, where you let your poetic impulses run wild. Pick a telling detail—one particular thing that most embodies the thing you described. Revise the passage, this time focusing only on that one telling detail. And while you're in there, this time try to keep the description from being too long or overwrought. You should end up with a description that is both economical and effective.
DESCRIPTION TRAPS
I've spent this whole chapter encouraging you to utilize many different descriptive methods, to bring your movie ever more vividly and continuously to the reader's mind. Now I need to point my usher's flashlight at some examples of what you _don't_ want to do with description. Bad description stops readers cold, yanking them from the spell of your story, the last thing you want to do.
First and foremost, avoid clichés. I know there's nothing new under the sun. But anything you can do to loosen the grip of overly familiar language is a plus:
Bone-chilling cold
He smiled daggers
Her cascading hair
Sleeping like the dead
Turning on one's heel
Feet planted firm on the ground
Such expressions have been used so many times that they're meaningless now. They leave the reader unengaged, painting almost nothing in the mind's eye.
A student in my class a few summers ago was a sweet and voluble retiree with white hair and a big laugh. I was surprised, therefore, when he stood up to protest during my lecture on description. "What if she really did have 'legs that don't quit'?" he demanded. He raised his chin and looked at me, his lips pursed, either a defiant schoolboy or a guy calling for his lawyer.
I told him that the phrase had meaning to him because of the layers of movies, books, and TV shows that used the phrase. And that the same things that made the phrase work for him have dimmed it for the purposes of improving and strengthening any story he might tell.
Also watch out for being imprecise or even sloppy with your description. Take this example:
_He felt like a punching bag without air._
We'll give some credit here for using a simile. But not much. Punching bags don't have air, and anyone who knows this will immediately stop believing in this story and this writer. Make it _a balloon without air_ or _a punching bag without stuffing_ and we're back inside the fictional illusion.
How about this one:
_She tossed her head at me._
Here we assume the writer means something like _she tossed her hair_ or _she tilted her head,_ rather than that she actually took off her head and tossed it. But with writing this sloppy, it's hard to be sure.
Also problematic are mixed metaphors. You can't have Joanne metaphorically swimming against a tide in one sentence and climbing a tall mountain a few lines later, or, worse, in the same sentence. The reader doesn't know if she's on land or water, and the power of either image is lost. If you want your mother to be a fish, fine, just don't turn her into an elephant three chapters later.
Sometimes, of course, we just need to get the story down, that first mad time, and we put down bad description—clichés, imprecise phrases, and such. That's quite all right. Think of those phrases as markers, as _blah blah blah_ written down. You can then tinker in your revision phase, replacing the bad descriptions with specific, precise, and interesting language. It's part of the fun of revision, even if you find yourself going _ouch!_ when you notice the cliché or ridiculously mixed metaphor.
**DESCRIPTION OF INNER LIFE**
Most of what we've been discussing has dealt with the externals: what places and people look like, how they sound, how they make themselves available to the senses. All of which is central to how we use description. But description is also used to portray the inner life of characters—their thoughts and emotions.
Essentially, the same rules of description apply to emotions and thoughts as to anything else. For example, you could write:
_Susanna was angry that Max didn't understand._
This sentence does the job, I suppose, but it actually conveys very little. _Angry_ is an idea, an abstract concept, a pointer to an emotion. Emotions are physical. They're expressed and felt in sensation or action or both. As with any kind of description, emotions are rendered more vividly when dealt with specifically, through the senses.
If you want Susanna to be angry, there are many good ways to get this across. Perhaps her chest feels tight and hollow or she can't breathe or her jaw tightened or she speaks in a gutteral voice. Any of these things will convey her emotion more descriptively than simply saying she was angry. For example:
_The second Max said the words, Susanna felt her skin flush hot. Rage closed her throat._
The poet T. S. Eliot said, when discussing Shakespeare's _Hamlet:_
_The only way of expressing emotion in the form of art is by finding an "objective correlative"; in other words, a set of objects, a situation, a chain of events which shall be the formula of that particular emotion; such that when the external facts, which must terminate in sensory experience, are given, the emotion is immediately evoked._
Eliot is asking you, in other words, to make the reader feel the same emotion as the person you're describing, by naming enough familiar details to evoke empathy.
Lynne Sharon Schwartz's narrator in her novel _Disturbances in the Field_ doesn't write:
_I was depressed after Althea was born._
Instead she writes:
_When she sucked at my breasts she was sucking the life out of me, and when she was done I swayed on my feet... I was cut off from the subtleties of common language and, like a non-native speaker, from idioms._
We are made to actually feel the emotion alongside her.
On a related note, the emotions and thoughts of characters may actually color all of the description in a work of fiction. As you learned in chapter 4, often the narration is filtered through the consciousness of a character, or perhaps several characters. Bear in mind that anything from a character's viewpoint will be somewhat subjective, and that this subjectivity will affect the way something is described.
John Gardner liked to have his students write a description of a barn from the point of view of a man who had just murdered someone. The idea was that the description of the barn would somehow take on the man's feelings or thoughts about the murder. Perhaps the claustrophobia of the enclosed barn would remind him of his emotions while killing, or perhaps the red color of the barn's door would remind him of blood. To some degree, this effect should occur anytime you're writing through the filter of a character's consciousness.
For example, Mary Gordon's novel _Men and Angels_ is told partly from the point of view of a young, disturbed live-in baby-sitter named Laura, who becomes infatuated with her employer's best friend. Here's what she thinks of him:
_She knew Adrian really liked her. He said she was a good listener. He was the handsomest man she had ever seen, with his thick gray curly hair, his open shirts, his shoulders. But really she wanted to be in the room with him without Anne there. If she went on and listened to Adrian, looked into his eyes when he told her things, praised whatever he said, someday he would like her more than he liked Anne._
Is Adrian really handsome? Maybe, maybe not. But he is to this character. Will Adrian someday like her? Perhaps not, but Laura thinks so. This third-person narrator is giving us Laura's perceptions, not objective fact.
In Frederic Tuten's _Tallien: A Romance,_ a first-person narrator reflects on his father, the charismatic union organizer:
_Nobleman that he was, riding down the fields of wrath, his terrible swift sword cutting a swath of fat pinky-ringed capitalists, defunct leases and eviction notices still clutched in their pudgy fists, Rex, the radical prince of the Confederacy, under whose ceaseless guard none would suffer except his periodically abandoned family, unpaid bills rolling up like waves against the door, his decade-old son staring up at the light bulbs, waiting for them, like stars blinking off into cold cinders, to go dead for failure of payment. ._ .
This man's memories of his father are certainly tinged with rage, and the depiction may or may not be objectively true.
YOUR TURN:
Describe a character who is going about the mundane job of cleaning his or her home. Write from the POV of this character (either first, second, or third person), which means the character's consciousness will inform the description. Here's the twist: the character has just recently fallen in love, and you should let this emotion color the description without being directly stated. Then rewrite the passage, but this time the character has just had a painful romantic breakup. You'll see how different the world looks depending on how people feel.
With his groundbreaking _Ulysses,_ James Joyce attempted to merge his descriptions as completely as possible with the minds of his characters. In the following passage, notice how the description follows a young woman's free flow of thought, where a sight of the sea unleashes barely related memories of chalk drawings and church incense:
_She gazed out towards the distant sea. It was like the paintings that man used to do on the pavement with the coloured chalks and such a pity too leaving them there to be all blotted out, the evening and the clouds coming out and the Bailey light on the Howth and to hear the music like that and the perfume of those incense they burned in the church like a kind_ _of waft._
There really is no limit as to how deep inward description may reach.
[CHAPTER 6
**DIALOGUE: TALKING IT UP**](Facu_9781596917910_epub_c4_r1.html#aa6)
BY ALLISON AMEND
I've been on a lot of bad dates. A lot. Some were blind dates; some I wish I had been blind for. But what amazes me is that the more I learn about fiction, and the more I learn about dating, the more they seem eerily parallel. Why? Because dialogue is the key to a successful date, and, I would argue, to successful fiction. There is nothing worse than sitting over a plate of cooling penne with nothing to say, and there is no substitute for the heady feeling you leave with when you just seem to "get each other." That connection hinges on dialogue.
Fiction can go without dialogue, and I've certainly been on a couple of great dates that didn't involve a lot of conversation, but, in general, dialogue is what keeps you coming back for more. The characters' interactions provide the scintillation that brings the reader to the story, and more often than not dialogue is a key part of this interaction. What makes the _War_ part of _War and Peace_ so boring (sorry, Tolstoy) is the fact that it's just the author droning on and on like a college lecture on geology. What's exciting is hearing Natasha speak with Andrei _(Peace),_ not reading dry re-creations of military maneuvers _(War)._
The characters are the ones in the story interacting with each other, so they are the people, not the author, who have the power to affect other characters. Perhaps the best way to let the reader really see the characters interacting with each other is to let them talk to each other. If it's done well, the readers will forget that the people they are reading about are the writer's creation. The characters will assume a life of their own. And isn't that the real purpose of fiction?
**DIALOGUE EXPLAINED**
Dialogue is everything in fiction that isn't narration. In other words, it's the stuff between the quotation marks—what the characters "say."
There is no official rule for how much dialogue to use in fiction. Some stories are dialogue-heavy, others dialogue-light. For example, Ernest Hemingway's "Hills Like White Elephants" is almost all dialogue, while _The Metamorphosis_ by Franz Kafka has practically no dialogue. Most stories find a balance between dialogue and narration. Switching between the two gives a work of fiction a nice diversity. Narration tends to have a dense feel, whereas dialogue—which reads quickly and offers lots of white space—has a zippier feel, making it like a cleansing dish of sherbet between courses. Again, I reference the theoretical "perfect date." You neither talk too much nor have to prod the conversation. It should be an exchange, a give-and-take.
There are two fundamental ways a writer can reveal any moment in a story—summary or scene. Summary is where the action is summarized, or "told." In contrast, a scene depicts the moment in real time, showing us exactly what transpires. Scenes are where dialogue makes its appearance. The effect is similar to that of watching a scene in a play or film, where the actors are speaking to and interacting with each other. Both scene and summary are frequently used techniques, and both have their place in fiction. But just as showing is more powerful than telling, scene is more powerful than summary. In fact, scene is the primary means by which a fiction writer "shows."
Save summary for moments when you want to relay information quickly and efficiently or when you want the narrator to revel in the pure telling of something. For the most important moments in your story, you'll want to switch to scene.
In Lorrie Moore's "People Like That Are the Only People Here," a mother brings her seriously ill baby to the doctor for some tests. Moore could have given the reader a summarized account, something like this:
_As the doctor explained that the baby had a tumor, the baby practiced his new pastime by switching the light on and off, on and off, increasing the Mother's nervousness and fear. When the doctor pronounced the words_ Wilm's tumor, _the room went dark._
This summary is fine. But notice how much more lifelike and dramatic the moment is when translated into scene. Here is what actually appears in the story:
_The baby wants to get up and play with the light switch. He fidgets, fusses, and points._
_"He's big on lights these days," explains the Mother._
_"That's okay," says the surgeon, nodding toward the light switch. "Let him play with it." The Mother goes and stands by it, and the Baby begins turning the lights off and on, off and on._
_"What we have here is a Wilm's tumor," says the Surgeon, suddenly plunged into darkness. He says "tumor" as if it were the most normal thing in the world._
_"Wilms?" repeats the Mother. The room is quickly on fire again with light, then wiped dark again. Among the three of them here, there is a long silence, as if it were suddenly the middle of the night. "Is that apostrophe_ s _or_ s _apostrophe?"_
We see and hear the scene with enough detail that it feels as if we're really there in the hospital witnessing it. Notice how effectively the on/off of the light plays against the dialogue. Also pay attention to the contrast between the surgeon's calm and the mother's nervousness, beautifully illustrated with her irrelevant question about the apostrophe. The summarized version gets the point across; the scene immerses us in the moment.
It's entirely possible to have a scene with no dialogue, where the thrust is conveyed just through physical action, but more often than not, dialogue will play a central role in a scene.
How do you know if a moment should be translated into dialogue or not? Well, dialogue tends to draw lots of attention to itself so you want to make sure you are dialoguing moments of real significance, be it character development, plot advancement, or a moment of extreme drama. A six-page scene of dialogue in which characters discuss carpool arrangements followed by a six-page scene in which the same characters reveal past infidelities serves the purpose of both inflating the importance of the first scene and diminishing the power of the second—if the reader even got to the second scene after slogging through the first six pages about the traffic on Main Street. The wise writer would relate only what was necessary about the carpool, perhaps not even using dialogue, then save the dialogue for the good part.
Key moments in a story lend themselves to being portrayed in dialogue.
If a moment is of real significance, the reader likes to be there, sitting front and center, watching and hearing. Often, authors choose dialogue to portray a confrontation scene, for example, when Patricia accuses her sister of stealing her boyfriend, or when Richard finally summons the nerve to ask his father if he lied about his military record. The dialogue doesn't have to show a cataclysmic moment for the characters, but the reader should come away from the dialogue scene with an increased understanding of the story.
In Charles Baxter's "Gryphon," a boy's boring suburban existence is exponentially expanded by the arrival of a mysterious substitute teacher who awakens his imagination. To get a sense of the teacher's unorthodox views, Baxter lets us hear her speak:
_"Did you know,"she asked, walking to the side of the room so that she was standing by the coat closet, "that George Washington had Egyptian blood from his grandmother? Certain features of the Constitution of the United States are notable for their Egyptian ideas."_
One of the kids at school is skeptical about the teacher, so it's important that we hear his reaction to her:
_"I didn't believe that stuff about the bird," Carl said, "and what she told us about the pyramids? I didn't believe that either. She didn't know what she was talking about."_
When the teacher reads the class's tarot cards and foretells a death, one of the students reports her to the principal, and she is dismissed. The climax of the story is when the protagonist explodes in anger at the snitch. This important moment is portrayed, of course, through dialogue.
_"You told," I shouted at him. "She was just kidding."_
_"She shouldn't have," he shouted back. "We were supposed to be doing arithmetic."_
_"She just scared you," I said. "You're a chicken. You're a chicken, Wayne. You are. Scared of a little card," I sing-songed._
_Wayne fell at me, his two fists hammering down on my nose. I gave him a good one in the stomach and then I tried for his head. Aiming my fist, I saw that he was crying. I slugged him._
Throughout the story Baxter alternates scene with summary, using ample portions of each, but he knows exactly which moments are worth letting the characters speak for themselves.
**THE ILLUSION OF REALITY**
Everybody talks. Well, practically everybody. One would think that dialogue would be one of the easiest aspects of fiction to pull off. After all, we use it every day. But good dialogue is deceptively difficult to write.
Your first task is to ensure that your dialogue sounds real. In past centuries, fictional dialogue had a certain theatricality, as in this line from Emily Bronte's _Wuthering Heights:_
_"Why, Master Heathcliff, you are not fit for enjoying a ramble, this morning. How ill you do look!"_
But nowadays dialogue tends to sound like actual people conversing with one another. What they say shouldn't seem rehearsed or robotic. And yet it's all too easy to write something along these lines:
_Upon spying the Grand Canyon for the first time, Jeannie-Lynn and Billy-Joe exclaimed, "What a splendid vista!"_
_"See?" Their mother pointed. "The scrub brush creates a harmonious palate of green-tinted lushness in the vastness of the canyon."_
_"I'll have to relate this to my fourth-grade class!" Jeannie-Lynn said._
Few people talk like that. They talk more like this:
_When they finally reached the edge of the Grand Canyon, Jeannie-Lynn and Billy-Joe opened their eyes wide in amazement. "Wow," said Billy-Joe._
_"That's so awesome," Jeannie-Lynn whispered._
_"See the scrub brush like we saw in Grandma's backyard?" Their mother pointed. The children nodded._
_"I'm going to talk about this in show-and-tell," Jeannie-Lynn said. "Can we take a picture?"_
The best way to get a feel for realistic dialogue is by listening to people talk. Listen to people on the bus, in the elevator, on the radio; pay attention to their speech patterns and the content of their conversations. Imagine writing their words down. Maybe even try writing them down. This will help develop your ear for dialogue. Being able to listen and mimic is the best preparation for writing realistic dialogue.
Two little tips for realistic dialogue. Contractions are good. Only a very formal person will say: _I do not think this is the best idea._ Most folks would say: _I don't think this is the best idea._ And though writers are instructed to avoid clichés, characters often use hackneyed phrasing. As a description, _hot as hell_ doesn't do very much. But it would be perfectly acceptable for certain characters to use this phrase in dialogue.
But simply capturing the sound of lifelike dialogue isn't enough. Actually, the realism of good dialogue is something of an illusion. Readers of fiction have a higher expectation for dialogue than the conversations of real life. Fictional dialogue needs to have more impact, focus, relevance, than ordinary conversation. The truth is most real-life conversations are dull, or at least they would come off as dull on paper. Try transcribing a conversation that you overhear. Or tape one and then type it onto the computer. It probably won't make any sense. If it does, it will most likely be tedious. The dialogue will probably take a long time to get to the point.
Let's look at a clip of lifelike conversation:
_"Hey. Um, hey."_
_"Oh, hey."_
_"Hey, Dana. It's Gina."_
_"Oh, hi. Wait, can you hold on? Okay, hi."_
_"Hey. What's up?"_
_"Good. I mean, nothing. How're you doing?"_
_"Good. Where are you?"_
_"On my cell."_
_"I mean, where."_
_"Oh, on my way after work, like, in the street."_
_"Yeah?"_
_"Um, yeah."_
The above selection is dull and would do absolutely nothing for a story, because it mimics real speech too closely. Now, were it fictionalized, it might sound more like this:
_"Hey, Dana. It's Gina."_
_"Hi. What's up?"_
_"Good. I mean, nothing. How're you doing?"_
Here we get to the point much more quickly. But this still isn't quality dialogue because there's no real significance to the conversation. Take a look at what happens when the dialogue is transformed to this:
_"Hey, Dana, it's Gina."_
_"Hi. Was I supposed to call you?"_
_"Yeah, it's Wednesday. Are you still up for seeing a movie?"_
_"I have to wait to see what Matt is doing."_
In this dialogue, you get a real sense of the characters and the tension between them. Gina's tone is a little challenging, as if she's used to Dana blowing her off. And we see that Dana has an avoidance of making concrete plans, due to her reliance on Matt. With just a few lines, this dialogue gives us a wealth of valuable information.
So, you see, dialogue has to seem real and yet not be too real and also do something important.
YOUR TURN:
Recall a dialogue exchange you had in the past few days. Do your best to write it down being faithful to what was actually said. Don't airbrush out the boring parts or make the dialogue snappy. Pretend you're transcribing a conversation from a tape recorder. Just write each character's name, then put the dialogue beside the name. (Jack: Hey, what's going on, man?) Then rewrite the dialogue exchange, this time making it concise and dramatically interesting. Why don't you fictionalize the names this time and feel free to embellish a bit.
DIALOGUE CONVENTION
Convention is the fulfillment of an established expectation. There are certain things our society has grown to expect because that's the way certain things are usually done. We expect a bride to wear white; we expect to be given a speech about wearing seat belts low and tight across our laps on airplanes. Dialogue too follows convention. The reader is used to dialogue looking and performing a certain way. Let's examine some of the conventions of dialogue.
Double quotation marks signal to the reader that someone is speaking:
_"Dude, you seen my left shoe?" "Dude, check your right foot."_
Occasionally, authors break with convention, foregoing double quotation marks for single ones, dashes, brackets, or even nothing:
— _Dude, you seen my left shoe?_ — _Dude, check your right foot._
But unless you have a compelling reason to do otherwise, stick with double quotes.
Usually dialogue dedicates one paragraph per speaker, no matter how short the speech:
_I sidled up to the bars of the drunk tank, resting my forehead on the cool steel. The keys to the jail dangled from the guard's belt._
_"Hot enough for you, mate?" I asked the guard._
_"Shut up," he said._
_"Okay, okay." I sat back down on the wooden bench and tried to close my eyes._
The fact that each line of dialogue gets its own paragraph highlights the importance and makes it easier to follow the flow of the conversation. Sometimes writers put different speakers in the same paragraph. While it's not wrong do to so, it can look confusing or intimidate the reader.
One of the major conventions of dialogue is the use of tags. In dialogue, most writers add speech tags, also known as attributions, so that the reader can follow who's saying what. _Said_ is the most frequently used tag. In fact, you could use nothing but _said_ and probably no one would notice. It may feel tedious to you to keep writing _he said, she said, the dog said,_ but the reader is trained to look at speech tags only to gather his bearings, the way commas signal pauses. Readers don't even notice that you've used the word _said_ 507 times. _Said_ becomes invisible:
_"You gonna drink that?" she said._
_"Yes, I am," he said._
You can use verbs other than _said,_ but you want to make sure they don't seem forced or get distracting.
_"You gonna drink that?" she asked._
_"Yes, I am," he replied._
However, it can be dangerous to veer too far from the _said_ paradigm. It's tempting to get out the thesaurus and have your characters _utter, express, state, announce, articulate, voice,_ etc., but overuse will provide a trampoline effect, making it seem as though all of your characters are springing five feet in the air when they speak:
_"You gonna drink that?" she sputtered._
_"Yes, I am," he proclaimed._
These tags are a little strong for such a banal statement. You _proclaim_ emancipation from tyranny, but you _say_ that you had a good night's sleep.
Similarly, adverbs in speech tags tend to make the author seem amateurish. Let what the character is saying tell the reader the tone of voice; don't have your characters speak _coquettishly_ or _snidely_ or _sarcastically._ Occasionally, adverbs are useful, but use them sparingly; they can draw attention to themselves in the wrong way, like an eighties hairdo. And more often than not they are just plain unnecessary, as seen here:
_"DON'T YELL AT ME!!!!" she screamed stridently._
Also, exclamation points in dialogue tend to make statements sound like lovesick teenage e-mail. Try at all costs to avoid using them!
When you attribute speech, make sure you place the tags in a logical or effective place. The following is jerky and awkward:
_"I don't," she said, "love you anymore."_
While this is appropriately devastating:
_"I don't love you anymore," she said._
But if the phrase is long, you might want to put the tag in the middle, so that the reader knows who is speaking. Tags in the middle should follow a natural "breath," or break, in the sentence. Like so:
_"I don't love you anymore," she said, "even though you still write me poems every day and shower me with gifts and tell me that I'm the most beautiful woman alive."_
Tags aren't the only way of indicating who is speaking. You can let the reader know who is talking to whom by having a character say someone's name, like so:
_"Hey, Pete, you got a light?"_
Be aware, though, that people don't usually call others by their names when they speak to them. Use this technique sparingly, as it can sound forced:
_"Bonnie Marie McGee, please pass the carrots."_
_"I'd be glad to, Aunt Fiona."_
Another effective way to attribute speech is to link an action with the dialogue, like so:
_"I don't think I believe in God." Bert put down his coffee cup to stare out the window._
Or a thought:
_"Get me a half-pound of that salami." Marsha wondered if she'd been a little harsh. "Please," she added._
Attribution for every single line of dialogue is not strictly necessary, as long as it's perfectly clear who is speaking when. Here's a passage from Hemingway's "Hills Like White Elephants," where a man and a girl are sitting in a bar in Spain:
_The girl looked at the bead curtain. "They've painted something on it," she said. "What does it say?"_
_"Anis del Toro. It's a drink."_
_"Could we try it?"_
_The man called "Listen" through the curtain. The woman came out from the bar._
_"Four reales."_
_"We want two Anis del Toro."_
_"With water?"_
_"Do you want it with water?"_
_"I don't know," the girl said. "Is it good with water?"_
_"It's all right."_
_"You want them with water?" asked the woman._
_"Yes, with water."_
With a sparing but skillful use of tags, we have no trouble knowing who is speaking when, despite the fact that there are three characters and none of these characters seems to have a name.
**STAGE DIRECTIONS**
Adding physical action to dialogue can help bring a scene to life. Take a look around you the next time you're at a party. You can tell someone's personality by how they interact with others. Gregarious people talk with their hands; seductive people run their fingers through their hair. Anal people gather all the toothpicks from the ashtrays and throw them away. Nervous people laugh too loud; attention seekers act outrageously to try to get others to notice them. All of these gestures, interspersed with dialogue, give a much more subtle and imaginative idea of the character than just _She tried to seduce him,_ or _Aiden was nervous._
If the author gives the reader no clue as to whether the characters are sitting or standing, eating lunch or driving a car, the scene can sound like floating heads reciting words. By mixing in narration details with your dialogue, you can shed light on your characters and give the scene a real physical presence. In a play or film, we have the actors to interpret the dialogue, through their gestures, movements, expressions, and tone of voice. This extra dimension can be achieved in fiction too with the skillful use of "stage directions."
Notice how physical action enlivens this scene from Denis Johnson's "Emergency." The drifter main character is talking to his friend and fellow orderly from the hospital, Georgie, who "often stole pills from the cabinets."
_He was running over the tiled floor of the operating room with a mop. "Are you still doing that?" I said._
_"Jesus, there's a lot of blood here," he complained._
_"Where?" The floor looked clean enough to me._
_"What the hell were they doing in here?" he asked me._
_"They were performing surgery, Georgie," I told him._
_"There's so much goop inside of us, man," he said, "and it all wants to get out." He leaned his mop against a cabinet._
_"What are you crying for?" I didn't understand._
_He stood still, raised both arms slowly behind his head, and tightened his ponytail. Then he grabbed the mop and started making broad random_ arcs _with it, trembling and weeping and moving all around the place really fast. "What am I_ crying _for?" he said. "Jesus. Wow, oh boy, perfect."_
The actions are important in that they show the drug-induced insanity of the characters, an integral element of the book's bizarre tone.
Similarly, thoughts can be used in stage directions, giving us an extra dimension, as in this clip from "Emergency":
_Georgie opened his arms and cried out, "It's the drive-in, man!" "The drive-in_ . . _." I wasn't sure what these words meant._
Stage directions are especially useful when there is a conflict between what a character says and what a character feels or thinks. If a character says she's not hurt and yet starts to cry, the reader knows that really she does feel injured. Like so:
_"Nothing you say can hurt me," she said, fighting back tears._
Also add action or thought if the tone of the words spoken needs to be explained. _"I hate you," she said slamming the front door_ is a very different sentence from _"I hate you," she said, hitting him playfully on the arm._
On the flip side, there is such a thing as too many stage directions. The reader doesn't really need to know every single time the character shifts his weight or scratches behind his ears or thinks about doing the laundry. The scene can sound "overacted." Like so:
_She took the Brita pitcher out of the refrigerator._ Did I drink that much, _she wondered,_ or did I forget to refill it? _She tilted the pitcher, pouring the clear, cold liquid into the glass. The condensation immediately began to bead down the sides. She returned the pitcher to the refrigerator, placing it next to the kiwi and an unopened jar of olives. "Oh," she said, "did you want a glass of water, too, Mom?"_
In this instance, the reader probably wouldn't even _get_ to the dialogue.
YOUR TURN:
Take the second version of the dialogue exchange you did for the previous exercise (the fictionalized one). Using the same dialogue that you wrote, rewrite the exchange, this time adding in tags and stage directions. Your tags should make it clear who is speaking and your stage directions should offer an added dash of nuance or meaning. Hint: it may help if one or both of the characters are engaged in a physical action. Then marvel at how well you've transformed real life into an interesting clip of fictional dialogue.
**INDIRECT DIALOGUE**
So far, we've been discussing direct dialogue—where the actual lines spoken are given. But there's another option and that's indirect dialogue—where the dialogue is summarized rather than quoted, appearing in summary rather than scene. Indirect dialogue can come in handy when the gist of what was said is more important than the actual dialogue.
Look at this passage from Tobias Wolff's "Smokers." Here the narrator is accosted by an annoying boy traveling by train to the same boarding school:
_He started to talk almost the moment he sat down, and he didn't stop until we reached Wallingford. Was I going to Choate? What a coincidence_ — _so was he. My first year? His too. Where was I from? Oregon? No shit? Way the hell and gone up in the boondocks, eh? He was from Indiana_ — _Gary, Indiana. I knew the song, didn't I? I did, but he sang it for me anyway, all the way through, including the tricky ending._
By just summarizing the annoying boy's questions, we are spared a monotonous conversation but get the most important information—the boy's desperate appeal for friendship and the narrator's annoyance at him—and we get it in a most economical fashion. In this instance, the reader doesn't really need the back-and-forth of the actual conversation.
Let's look at another example from "Smokers," where Wolff mixes both direct and indirect dialogue:
_As it happened, the courts were full. Talbot and I sat on the grass and I asked him questions I already knew the answers to, like where was he from and where had he been going to school the year before and who did he have for English. At this question he came to life. "English? Parker, the bald one. I got A's all through school and now Parker tells me I can't write."_
Here we get the gist of the conversation, but then, on an especially significant line, we get the actual quote.
So, in addition to asking yourself if a moment should be dialogued or not, you can also ask yourself if direct or indirect dialogue is the best choice for that particular moment.
YOUR TURN:
Return to the dialogue exchange you wrote in the previous exercise. This time convey the gist of it with just a few sentences of _summarized_ dialogue. In addition to conveying the facts, hint at the character personalities and/or tension in the exchange. If you wish to include a line or two of the actual words spoken, do so. The determine if this particular exchange would be better served by dialogue or summarized dialogue in a work of fiction—a choice you will always have.
**DIALOGUE AND CHARACTER**
Perhaps the best thing about dialogue is that it allows characters to speak for themselves. You don't really know someone in real life until you've talked to them and heard them talk to you; the same principle applies to fiction. Unless you're superficial, a person's outside appearance doesn't matter nearly as much as what he has to say. It's on the basis of what comes out of his mouth that you decide whether you like the person and want to spend more time with him. Okay, sometimes on dates we get superficial at first, but you know what I mean.
For example, rather than being told that Mr. Jackson is a highly educated and rather stuffy man who has an interest in German opera, let's hear him speak:
_"I am emotionally attracted to Ms. Mason. She has a Wagnerian formality that begs to be breeched."_
We get it, quite effectively.
Every person in life speaks in a somewhat unique fashion, and the same should be true for fictional characters. Just as you look for unique traits in your characters, look for the uniqueness in how they speak. Avoid having all your characters talk exactly the same way, or even having all of them talk just like you. Seek out the distinctive ways that characters express themselves when they open their mouths.
You should think about the plentitude of a character's speech. You might create characters who speak in never-ending segments that travel over hill and dale and try the patience of everyone in the room or characters who only grunt monosyllabic responses, which could be equally trying to those in the conversation. If you find yourself on a date with either of these habits, you're in trouble, but, fortunately, there are many variations in between.
People often have pet expressions that they use over and over again in dialogue. _Oh, my head!_ instead of _Oh, my God!_ or _Don't piss in my Cheerios_ instead of _Don't rain on my parade._ Jay Gatsby, for example, is fond of the term "ol' sport," and Bartleby in Herman Melville's "Bartleby, the Scrivener" would utter the phrase "I prefer not to" in response to just about anything, even when his employer insists he leave his job.
Ask yourself questions about how your characters might talk. Do they use incorrect grammar and colloquialisms, or do they speak "perfectly"?Does their background and social status affect the way they talk? Do they tend to beat around the bush or get to the point quickly?
Here's an example from J. D. Salinger's _The Catcher in the Rye,_ where Holden Caulfield is conversing with a New York City cab driver:
_"Hey, Horwitz," I said. "You ever pass by the lagoon in Central Park? Down by Central Park South?"_
_"The_ what?"
_"The lagoon. That little lake, like, there. Where the ducks are. You know."_
_"Yeah, what about it?"_
_"Well, you know the ducks that swim around in it? In the springtime and all? Do you happen to know where they go in the wintertime, by any chance?"_
_"Where_ who _goes?"_
These two characters clearly have different backgrounds and we hear it in the way they speak. We also hear the difference in their personalities. Holden is chatty, curious, even a bit nervous. The cabbie just wants to drive in peace without worrying about the damn ducks.
In Ethan Canin's "The Accountant," an accountant is at an adult baseball fantasy camp chatting with the legendary Willie Mays:
_Willie Mays said, "Shoot, you hit the ball, brother."
I ventured, "Shoot, yes."
Willie Mays said, "You creamed that sucker."
I said, "Say, I bet they sock you at tax time."_
Here it's humorous watching the accountant trying to sound loose and cool talking to Willie Mays. Though the two men are speaking a similar lingo, it's clear that the accountant is much less at home with it. Perhaps it's because he says "Shoot, yes," instead of the more natural "Shoot, yeah." When it comes to dialogue, such minute nuances make a world of difference.
Here's an example from Philip Roth's _Portnoy's Complaint,_ where a Jewish mother is conversing with her adolescent son:
_"I don't believe in God."_
_"Get out of those dungarees, Alex, and put on some decent clothes."_
_"They're not dungarees, they're Levi's."_
_"It's Rosh Hashanah, Alex, and to me you're wearing overalls! Get in there and put a tie on and a jacket on and a pair of trousers and a clean shirt, and come out looking like a human being. And shoes, Mister, hard shoes."_
_"My shirt is clean."_
_"Oh, you're riding for a fall, Mr. Big. You're fourteen years old, and believe me, you don't know everything there is to know. Get out of those moccasins! What the hell are you supposed to be, some kind of Indian?"_
Here both characters are from the same family but they clearly have different methods of expressing themselves. Notice that no tags are used, although the mother calls the boy by his name a few times. When characters speak this distinctively, tags become superfluous.
Now notice how different all the characters above sound from these two servants in an English manor in Kazuo Ishiguro's _Remains of the Day:_
_Miss Kenton had entered and said from the door:_
_"Mr. Stevens, I have just noticed something outside which puzzles me."_
_"What is that, Miss Kenton?"_
_"Was it his lordship's wish that the Chinaman on the upstairs landing should be exchanged with the one outside this door?"_
_"The Chinaman, Miss Kenton?"_
_"Yes, Mr. Stevens. The Chinaman normally on the landing you will now find outside this door."_
_"I fear, Miss Kenton, that you are a little confused."_
_"I do not believe I am confused at all, Mr. Stevens. I make it my business to acquaint myself with where the objects properly belong in a house. The Chinamen, I would suppose, were polished by someone then replaced incorrectly. If you are skeptical, Mr. Stevens, perhaps you will care to step out here and observe for yourself."_
_"Miss Kenton, I am occupied at present."_
_"But, Mr. Stevens, you do not appear to believe what I am saying. I am thus asking you to step outside this door and see for yourself"_
_"Miss Kenton, I am busy just now and will attend to the matter shortly. It is hardly one of urgency."_
_"You accept then, Mr. Stevens, that I am not in error on this point."_
_"I will accept nothing of the sort, Miss Kenton, until I have had a chance to deal with the matter. However, I am occupied at present."_
Not only do these characters speak in a formalized manner, but they become disturbed in a formalized manner. And though Mr. Stevens and Miss Kenton both strive for the propriety demanded by their setting, Miss Kenton has more trouble than Mr. Stevens with keeping her emotions in check. The interesting thing about Mr. Stevens is that, in the manner of the perfect English butler, he almost blends into the polished woodwork, and this is certainly reflected in his dialogue.
In addition to getting a sense of who characters are, dialogue can also convey a strong sense of the interaction between characters. Let's return to the man and girl in Hemingway's "Hills Like White Elephants." Here's what they say right after they get the Anis del Toro that they ordered:
_"It tastes like licorice," the girl said and put the glass down._
_"That's the way with everything."_
_"Yes," said the girl. "Everything tastes of licorice. Especially all the things you've waited so long for, like absinthe."_
_"Oh, cut it out."_
_"You started it," the girl said. "I was being amused. I was having a fine time."_
_"Well, let's try and have a fine time."_
We don't have to be told there's tension between this couple. We can _hear_ it. Almost as though we are sitting at the neighboring table. Here's a tip: tension between characters will almost always notch up the interest level of your dialogue.
In "Cathedral," look at the dialogue shortly after the blind man arrives.
_"Did you have a good train ride?" I said. "Which side of the train did you sit on, by the way?"_
_"What a question, which side!" my wife said. "What's it matter which side?" she said._
_"I just asked," I said._
_"Right side," the blind man said. "I hadn't been on a train in nearly forty years. Not since I was a kid. With my folks. That's been a long time. Yd nearly forgotten the sensation. I have winter in my beard now," he said. "So I've been told, anyway. Do I look distinguished, my dear?" the blind man said to my wife._
Here we clearly see the tension between the narrator and his wife, a continuation of an earlier argument about the narrator's unwillingness to welcome the blind man into their home. Despite the marital unrest before him, Robert, the blind man, seems remarkably at ease. We also see each character thrown into sharp relief through the words spoken. The narrator is being flip, even ornery, by asking the blind man on which side of the train he sat. The wife, poor woman, is trying to curtail her errant husband. And right off the bat Robert is demonstrating his relaxed friendliness and social ease. He even shows a touch of the poet by referring to "winter in my beard," a significant point considering poetry is something shared by the wife and blind man and disliked by the narrator. A rather complex character triangle is revealed quite specifically in just this short passage.
YOUR TURN:
Jessica, a somewhat stuffy university professor (you pick her field), stops at a gas station in some backwater place. As she fills her tank, Aivin, the attendant, approaches her. He is an uneducated sort (though not necessarily dim) and, being both bored and friendly, he wants to chat. Jessica would rather not chat but she also doesn't want to alienate Alvin because she would like directions to a nearby restaurant that won't be too greasy or ghastly. Write a scene between Jessica and Alvin, using dialogue, tags, and stage directions. Your main goal is to capture the flavor of these two people through how they speak.
**SUBTEXT**
Check out the relationship section of a bookstore. There are thousands of books on communication between partners. Our society may be excellent at talking, but we have trouble communicating. People often don't say what they mean. Sometimes they say the opposite of what they mean. They hide insults in sugary language (or sugary feelings in insults). They don't listen. They mishear. They don't answer. They remain silent.
Capitalizing on miscommunication improves fictional dialogue because it makes it more true to life. Misunderstandings can also add tension to the dialogue exchange. This tension results from the gap between what's being said and the subtext—the meaning beneath the surface meaning. Dialogue with subtext has two levels of meaning.
A great illustration of subtext occurs in the film _Annie Hall._ Woody Allen and Diane Keaton, who have recently met, are standing on a terrace carrying on a nervous conversation. She says: "Well, I-I-I would—I would like to take a serious photography course soon." But what she's really thinking appears in a subtitle: "He probably thinks I'm a yo-yo." Then he says: "Photography's interesting, 'cause, you know, it's-it's a new art form, and a, uh, a set of aesthetic criteria have not emerged yet." But his subtitle says: "I wonder what she looks like naked?" How much does this mirror real-life conversation? Probably more than any of us would like to admit.
Though you are unlikely to use subtitles with your dialogue, subtext can be enormously effective in fiction.
In _The Good People of New York_ by Thisbe Nissen, Edwin, long divorced, asks his daughter about her mother's live-in boyfriend.
_Edwin is silent for a time. "You didn't like Steven much, did you?" he finally asks._
_Miranda shrugs. "He was my orthodontist."_
Miranda's evasive answer contains much meaning. In the sullen manner of a teenager, she's explaining how embarrassing it is to have her mother date the man who tightens her braces, and her further embarrassment at discussing it with her father. But her response wouldn't be nearly as interesting or relevatory (or concise) if she were able to articulate her complicated feelings.
In _The Great Gatsby,_ when Gatsby shows Daisy his exquisite collection of monogrammed shirts, this is how she reacts:
_"They're such beautiful shirts," she sobbed, her voice muffled in the thick folds. "It makes me sad because I've never seen such_ — _such beautiful shirts before."_
Daisy is sobbing for many painful reasons but the shirts aren't one of them. Her inability to express her feelings adds tremendous poignancy to the moment.
Finally, let's look at another dialogue exchange from "Cathedral." The narrator and his wife are arguing as they prepare for the blind man's visit. As you read this, try to determine if the characters are speaking in subtext or not:
_"Maybe I could take him bowling," I said to my wife. She was at the draining board doing scalloped potatoes. She put down the knife she was using and turned around._
_"If you love me," she said, "you can do this for me. If you don't love me, okay. But if you had a friend, any friend, and the friend came to visit, Yd make him feel comfortable." She wiped her hands with the dish towel._
_"I don't have any blind friends," I said._
_"You don't have_ any _friends," she said. "Period. Besides," she said, "goddamn it, his wife's just died! Don't you understand that? The man's lost his wife!"_
If you answered _yes,_ you're right. If you answered _no,_ you're also right. The narrator _is_ speaking in subtext. When he says, "I could take him bowling," he's really saying how ridiculous it is to be entertaining a blind man. When he says, "I don't have any blind friends," he's really saying that a blind friend is worse than no friend. The wife, on the other hand, is saying exactly what's on her mind. This is a very lifelike situation, where one person is more inclined to speak sideways than another.
Having thus emphasized that dialogue should not just be a representation of everyday speech, I should add that there is real pleasure to be had in the way people actually talk, their confusion, their circumlocution, their mistakes, misunderstandings, repetitions, and their small talk. There exists a fine line between actual and fictionalized dialogue. But in fictional dialogue you want to make sure the poor communication serves a dramatic purpose that is understood by the reader. When you achieve this, your dialogue will take on layers of realism and depth.
As the old Yiddish proverb goes, "A man hears one word but understands two." Good advice for writing dialogue. A layered conversation is the difference between what seems to be a stage-set version of a house and a genuine lived-in home.
YOUR TURN:
Envision a husband and wife or any other kind of romantic pair. Give them names and think about who they are. One of these characters suspects the other of being unfaithful (in some way), and let's say the other character is (in some way) guilty. Write a dialogue exchange between these two where the sore topic is never referred to directly but instead simmers beneath the words spoken. Don't enter the thoughts of either character. And keep the conversation focused on tuna steak, which they are having for dinner at the moment. If you exhaust tuna steak, you can move on to politics or movies. Silly as this sounds, see if the finished product doesn't have a ring of truth to it.
BAD DIALOGUE
Bad dialogue makes characters seem like puppets, mere creations of the author. Of course they are, but the reader will conveniently forget that if the dialogue is well rendered. Bad dialogue exposes the author, in much the same way that Toto exposes the Great and Powerful Oz when he tugs back the curtain in _The Wizard of Oz._ When you can see the machinations behind the writing, the entire illusion is lost.
You remember Mulder and Scully from the TV show _The X-Files?_ They worked together for nine years and still, each week, Scully patronizingly explained simple medical terms to poor, brilliant Mulder. Why? So that the audience could figure it out. Lesson: dialogue is often _not_ a good venue for exposition:
Scully: _He's exsanguinating from a laceration in his jugular._
Mulder: _You mean he's bleeding to death from his neck?_
There's no way that Mulder, expert in all things paranormal and disgusting, wouldn't know what _exsanguinating_ means. Find some other way to impart the information (the dying man's wife wants to know what's going on; or show us the gaping, bloody wound). I made up that exchange, but it could very well have been in one of the episodes. While these exchanges had a certain charm on the TV show, they would ring phony in fiction.
Though you're not likely to be using the word _exsanguinating,_ you may find yourself trying to sneak in some exposition through your dialogue, like so:
_"Troy, you were six years old when your mother left you and your sister to join the circus as a high-wire acrobat."_
Presumably Troy hasn't forgotten this odd fact, so why is he being told? Read your dialogue over. If it sounds forced in places, or unrealistic, see if you can turn the dialogue around so it's not so obvious:
_"Troy, grow up. It's been twenty years since your mother left, and you're still harping about how much you hate aerialists. Don't you think it's time you let go?"_
Now there is a credible reason why the information is introduced. The speaker is making a point about "letting go," and the exposition just happens to be included. If you can't find a plausible way to get the exposition in the dialogue, then you're better off just putting the exposition in the narration:
_At the age of six, Troy and his sister had the misfortune to be abandoned by their mother, who fled to fulfill her lifelong ambition of performing as a circus acrobat._
Another thing to watch out for is preaching in dialogue. Some writers, once they gain your attention, use their stories as political platforms. Their characters expound the writer's views on various social issues or prejudices. Don't give your characters a podium from which to harangue the reader. If you want to write about the evils of corporate greed, draft a letter to the editor, but don't make poor fictional Johnny argue at length with his basketball buddy about the pitfalls of capitalism:
_"You may think the raffle is a good idea but I'm telling you it's merely a capitalistic ploy to get rich. Capitalism, my friend, is the root of all evil. Today it's a raffle. Tomorrow you'll be paying Guatamalan families a penny a week to produce your goods so you can travel first-class and keep a summer house in East Hampton."_
If the reader feels that the author is making the character's voice the author's own opinion, the reader might feel manipulated and bored. That said, there are some authors who made a career of exactly this—Ayn Rand, Jean-Paul Sartre, George Orwell, to name a few—but it's extremely hard to pull off.
One last thing on bad dialogue. For reasons as yet undetermined by modern science, profanity on the page is much more alarming and vulgar than spoken profanity. Even foul-mouthed characters appear to overuse swear words when they are written down. If you don't believe me, write a dialogue between rampant cussers, truck drivers or socialites, and you'll see what I mean. A couple of well-chosen profanities work much better than a string of four-letter wonders, bringing all the flavor of X-rated speech without overdoing it.
**DIALECT**
Dialect is like walking on eggshells—tread carefully. It's tough to do well, and even if it is done well it can be distracting.
If rendered carelessly, dialect runs the risk of sounding hackneyed, exaggerated, or even offensive, as in:
_Moishe tripped over a piece of gefilte fish that was lying on the kitchen floor._
_"Oy gevalt!" cried Sadie. "Bubbela, the scare you gave me."_
_"What, are you meshugana, leaving this fish on the floor?"_
We're being clobbered on the head with Yiddishisms. Less is more, as in:
_Moishe tripped over a piece of gefilte fish that was lying on the kitchen floor._
_"Oy gevalt!" Sadie said. "The scare you gave me."_
_"Well, I didn't expect there should be fish on the floor," Moishe said._
Here we get the flavor of the ethnicity without having it shoved down our throat.
Mark Twain is considered a master of dialect, but it's important to remember how difficult it is to read the character of Jim, the escaped slave, in _The Adventures of Huckleberry Finn_ because the dialect is executed so faithfully:
_"Well, you see, it 'uz dis way. Ole missus_ — _dat's Miss Watson_ — _she pecks on me all de time, en treats me potty rough, but she awluz said she wouldn't sell me down to Orleans."_
But it is possible to make dialect work smoothly if you focus on just giving a flavor of it—with key words and speech patterns and rhythms. In _Beloved_ —which deals with black characters living around the time of the Civil War, contemporaries of Twain's Jim—Toni Morrison manages to capture the essence of her characters' dialect without throwing it in the reader's face:
_"Wait here. Somebody be here directly. Don't move. They'll find you."_
_"Thank you," she said. "I wish I knew your name so I could remember you right."_
_"Name's Stamp," he said. "Stamp Paid. Watch out for that there baby, you hear?"_
_"I hear, I hear," she said, but she didn't._
Nothing about the dialogue looks terribly foreign to the reader, but we get the sense of dialect through the little touches like "Somebody be here directly" and "Watch out for that there baby, you hear?" Morrison has found the proper balance.
Another alternative to dialect is simply to state that a character has an accent or a dialect, or have another character comment on this fact, then write dialogue as you normally would. This method gets your point across without confusing the reader.
A semirelated matter: make sure that you're not making the reader experience the same speech difficulties as your characters. If you have a character who stutters, avoid showing it in this distracting fashion:
_"I_ — _I_ — _I'm not sh-sure," Joe said._
Go for the simpler:
_"I'm not sure," Joe stammered._
You also want to be fairly sparing with such circumlocutions as "uhm," "uh," "well," and "you know."
And, uh, well, I guess that's what I've got to say on dialogue.
[CHAPTER 7
**SETTING AND PACING** : **I'M HERE THEREFORE I AM**](Facu_9781596917910_epub_c4_r1.html#aa7)
BY CAREN GUSSOFF
I've lived in seven different states in fifteen years. I grew up in the shadow of the Yonkers Raceway in Yonkers, New York, my neighborhood hemmed in by squat WPA houses and freeway overpasses. I became accustomed to waking up to the smell of rotten apples and wet sheep in the fairylike mists of early-morning Marlboro, Vermont. In Boulder, Colorado, I wore layers—from shorts to earmuffs—as the whole world could seem to change in an instant as the strange weather patterns bounced off the walls of the valley, walled in by granite flatirons on two sides. I've settled, for now, in Seattle, Washington, where the meteorologists have a hundred words for rain, understandably so. I've begun to appreciate the various forms it can take, from an imperceptible mist that nonetheless soaks you through to your skivvies to a vigorous outpouring that makes you swear someone is dumping a bottomless pitcher right overhead. Yet with all the rain, there is almost never any thunder or lightning, very few true storms. Living here in the constant sop affects every aspect of my life—where I go, what I wear, how I get there.
Strangely enough, though, my earliest works of fiction blithely ignored setting, concentrating instead almost solely on character. I thought I was doing okay. All of my favorite books had memorable characters—Holden Caulfield, Leopold Bloom, Jay Gatsby—so memorable, in fact, that most of them are irrevocably etched into our cultural consciousness. I intuitively understood the importance of characters, and what they do in a piece of fiction. They not only give a reader a focal point, but they also act, interact, react, have relationships, offer judgments, tell us about other characters, speak with other characters, describe, ruminate, think... and, most importantly, they change.
In a way, I was right to concentrate on character. After all, without strong characters, a story will fall flat on its face. But my stories fell flat on their face anyway. My characters paced aimlessly through my stories like caged tigers, and, worse, I often couldn't even write beyond a few pages. Things just stalled. I noticed that people who read my work labeled my pieces _vignettes, anecdotes._ How frustrating. That was not what I wanted. I wanted to write great stories, full stories, stories like _The Catcher in the Rye, Ulysses, The Great Gatsby._
Here's what I failed to realize: Jay Gatsby is who he is _because_ of the Jazz Age. Leopold Bloom is who he is _because_ of Dublin. Holden Caulfield is who he is _because_ of Pencey Prep School and New York City. I was missing the world surrounding my characters and the enormous impact it had on them, not to mention the impact on the overall story.
When readers read a piece of fiction, they expect it to feel real, even if it's a life they don't and will never know. They want to enter into it, to live there, with the characters. Setting—which refers simply to time and place—grounds the reader in the story in the most physical sense.
Traveling to various fictional places and times can also be one of the most entertaining aspects of reading, be it a journey of escapism or straight into the familiar. Think about some of your favorite works of fiction. They've all taken you to some kind of setting—haven't they? Charles Dickens moved his characters through the sooty streets of nineteenth-century England; J.R.R. Tolkien's Hobbits in the _Lord of the Rings_ trilogy are from comfortable holes in Middle-Earth; Jeffrey Eugenides's _The Virgin Suicides_ uses the façade of a calm Vietnam-era American suburb to show the driving desperation hidden just beneath the surface.
It's easy to dismiss setting because, frankly, it can be easy to miss. Look carefully at effective stories and you'll find the setting is so deeply combined with the characters and the action that it's almost unnoticeable. Like a master woodworker whose joinery is invisible, the writer has embedded the setting into the story. But if you really attune your eyes, you'll see that this implanted setting has an immeasurable impact on the characters and plot and, well, everything else.
YOUR TURN:
Pick up one of your favorite works of fiction and read the first few pages. Pay attention to how quickly you get some indication of the setting. Notice how much or how little the setting is layered into the action and description. If you want to have some fun, try revising the opening of the story using a drastically different place or time. For example, put Scarlett O'Hara in contemporary Los Angeles and see how she does. (Probably very well.)
**PLACE**
When writers talk about place, they mean the specific and definite location of a story, on a large and small level. What planet, continent, country, state, city, neighborhood, street are the characters in? What does the office or building or river or cabana or castle or room look like?
The idea is to ground the reader in the place, whether it is the middle-of-nowhere Wyoming, as in Richard Ford's "Rock Springs":
_Where the car went bad there wasn't a town in sight or even a house, just some low mountains maybe fifty miles away, or maybe a hundred, a barbed wire fence in both directions, hardpan prairie, and some hawks sailing through the evening air seizing insects._
Or the toughest section of the Bronx from Tom Wolfe's _Bonfire of the Vanities:_
_It was as if he had fallen into a junkyard. He seemed to be underneath the expressway. In the blackness, he could make out a cyclone fence over on the left... something caught in it... A woman's head!... No, it was a chair with three legs and a burnt seat with the charred stuffing hanging out in great wads, rammed halfway through a cyclone fence..._
Or looking out at Dublin Bay in James Joyce's _Ulysses:_
_Woodshadows floated silently by through the morning peace from the stairhead seaward where he gazed. Inshore and farther out the mirror of water whitened, spurned by lightshod hurrying feet. White breast of the dim sea._
Or a room as bleak as this one from William Faulkner's _The Sound and the Fury:_
_We entered a bare room smelling of stale tobacco. There was a sheet iron stove in the center of a wooden frame filled with sand, and a faded map on the wall and the dingy plat of a township. Behind a scarred littered table a man with a fierce roach of iron gray hair peered at us over steel spectacles._
Or a room as luxurious as this one from F. Scott Fitzgerald's _The Great Gatsby:_
_The windows were ajar and gleaming white against the fresh grass outside that seemed to grow a little way into the house. A breeze blew through the room, blew curtains in at one end and out the other like pale flags, twisting them up toward the frosted wedding cake of the ceiling—and then rippled over the wine-colored rug, making a shadow on it as wind does on sea._
When dealing with place, don't neglect the possibility of including weather. As I mentioned earlier, my whole existence is directly and fundamentally affected by Seattle's weather. I know my corner of the world wouldn't be the same without the continual precipitation. Chances are there's something about the weather where you live that affects you dramatically, even though you may have ceased to think about it.
It may be like this sensual sunniness from Joyce Carol Oates's "Where Are You Going, Where Have You Been?":
_Connie sat with her eyes closed in the sun, dreaming and dazed with the warmth about her as if this were a kind of love, the caresses of love..._
Or this snowstorm from Leo Tolstoy's _Anna Karenina:_
_The terrible storm tore and shrieked between the wheels of the train and round the scaffolding at the corner of the station. The railway carriages, the pillars, the people, and everything that could be seen were covered on one side with snow getting thicker and thicker. Now and then the storm would abate for an instant, and then blow with such gusts that it seemed impossible to stand up against it._
Weather immediately deepens the visceral sensation of "being" in a fictional place.
When writing your fiction, you should always be asking yourself where the characters are. And you need to give the readers some indication of where the characters are, be it one setting or numerous settings. Some places you may paint in great detail, others not, but you and the reader need this awareness of location at all times.
Place also affects the action of a piece. What would be possible in midwinter Siberia differs tremendously from what is possible in Miami in May. Likewise, what's considered normal activity in the Wild West might be downright scandalous in Victorian England. (You'll notice that time and place are irrevocably intertwined.)
Consider the impact of place on the plot of Cormac McCarthy's _All the Pretty Horses._ The main character, sixteen-year-old John Grady Cole, is the last in a long line of west Texas ranchers. The conflict at the center of the story is about the disappearance of the cowboy and rancher way of life, which eventually forces Grady's mother to sell off the family ranch. Grady then travels to Mexico, hoping to find a land free of the progress and population that have started to choke the Texas cowboys. This novel depends heavily upon the arid and desolate Southwest borderlands. In fact, it just wouldn't make any sense if it were set anywhere else.
Setting is also a driving force in Flannery O'Connor's "A Good Man Is Hard to Find." While on a car trip with her family, a grandmother believes she recognizes the area they are passing through and urges the family to take a side trip:
_They turned onto the dirt road and the car raced roughly along in a swirl of pink dust. The grandmother recalled the times when there were no paved roads and thirty miles was a day's journey. The dirt road was hilly and there were sudden washes in it and sharp curves on dangerous embankments. All at once they would be on a hill, looking down over the blue tops of trees for miles around, then the next minute, they would be in a red depression with the dust-coated trees looking down on them._
_"This place had better turn up in a minute," Bailey said, "or I'm going to turn around."_
_The road looked as if no one had traveled on it in months._
Chaos ensues from taking the road less traveled. When the grandmother gets upset, the cat she has smuggled along gets loose, causing a car accident that places the family face-to-face with a murderer. Could the story have taken place during a jaunt down a suburban cul-de-sac? Perhaps not. And we wouldn't want it to.
How does the place, or places, in which your story is set affect the action? If the answer is _not at all,_ then you should probably look for ways to make the place play some kind of a role in what happens. Otherwise your characters just might be drifting through a vacuum.
YOUR TURN:
Return to something you have written, perhaps using one of the previous exercises. If you haven't dealt with the place of the piece fully enough, revise with an eye toward doing this. Don't choke the passage with too much setting, but ground the reader in the place and let place have some impact on the action. If you have already dealt with place fully, then revise the piece drastically _altering_ the place. Whichever route you choose, you should end up with a visibly different piece.
**TIME**
The notion of time is as integral to setting as place. Time can give us a sense of the backdrop of the story in the big sense—the era, century, year—and in the small sense—the season, day of the week, and time of day.
If we're in the nineteenth century, people may be traveling as they do in Charles Dickens's _Great Expectations:_
_The journey from our town to the metropolis was a journey of about five hours. It was a little past midday when the four-horse stage-coach by which I was a passenger got into the ravel of traffic frayed out about the Cross Keys, Wood Street, Cheapside, London._
And a century later they may be traveling as they do in Thorn Jones's "A White Horse":
_A faded, light-green Mercedes with a broken rear spring came bouncing too fast across the beach and skidded, sliding sideways as it stopped near the carousel._
We may be in springtime, as in D. H. Lawrence's _Lady Chatterley's Lover:_
_An English spring! Why not an Irish one? Or Jewish? The chair moved slowly ahead, past tufts of sturdy bluebells that stood up like wheat and over grey burdock leaves. When they came to the open place where the trees had been felled, the light flooded in rather stark. And the bluebells made sheets of bright blue colour, here and there, sheering off into lilac and purple._
Or autumn, as in Ernest Hemingway's "In Another Country":
_In the fall the war was always there, but we did not go to it any more. It was cold in the fall in Milan and the dark came very early. Then the electric lights came on, and it was pleasant along the streets looking in the windows. There was much game hanging outside the shops, and the snow powdered in the fur of the foxes and the wind blew their tails._
Or in a busy city during the day, as in J. D. Salinger's _The Catcher in_
_the Rye: Broadway was mobbed and messy. It was Sunday and only about twelve o'clock, but it was mobbed anyway. Everybody was on their way to the movies_ — _the Paramount or the Astor or the Strand or the Capitol or one of those crazy places. Everybody was all dressed up, because it was Sunday, and that made it worse._
Which is different from the same city in the same book late at night:
_I didn't see hardly anybody on the street. Now and then you just saw a man and a girl crossing a street, with their arms around each other's waists and all, or a bunch of hoodlumy-looking guys and their dates, all of them laughing like hyenas at something you could bet wasn't funny. New York's terrible when somebody laughs on the street very late at night. You can hear it for miles. It makes you feel so lonely and depressed._
As with place, you always want to locate your characters in time, whether or not you spend much space describing it. Be aware of the time, in both the big and the small sense, and give the reader whatever clues they may need to stay oriented. You may have heard the newscaster expression: _"It's ten p.m. Do you know where your children are?"_ At any given time, you should know where your characters are and what they are doing.
YOUR TURN:
Return to something you have written. Just don't use the piece you used for the previous exercise on place. But do the same thing as in that exercise, this time either dealing with the time more fully or drastically altering the time. See if your tampering with time affects this piece as deeply as the tampering of place did with the previous exercise.
SETTING THE MOOD
In addition to grounding the reader in a physical place and time, setting can actually enhance the emotional landscape of a piece, affecting the atmosphere and mood.
Edgar Allan Poe was a master of using setting to maximize the mood of his stories, as seen in the opening of "The Fall of the House of Usher":
_During the whole of a dull, dark, and soundless day in the autumn of the year, when the clouds hung oppressively low in the heavens, I had been passing alone, on horseback, through a singularly dreary tract of country, and at length found myself as the shades of the evening drew on, within view of the melancholy House of Usher._
As much as painting the time and place, perhaps more, Poe is evoking a mood, an emotional state—one of bleakness and danger and melancholy. Of course it's autumn, of course night is falling, of course the clouds hang oppressively. Practically every word in this passage tolls like a solemn bell. Poe underscores the tension before you even know what the tension is... or could be. Descriptively, Poe may be a little over the top, but dramatically, he's right where he should be, setting the stage for the dark tale about to unfold.
In a far more contemporary example, Lorrie Moore uses the setting of a hospital to convey the emotional state of her protagonist in "People Like That Are the Only People Here":
_The Mother studies the trees and fish along the ceiling's edge in the Save the Planet wallpaper border. Save the Planet. Yes! But the windows in this very building don't open and diesel fumes are leaking into the ventilating system, near which, outside, a delivery truck is parked. The air is nauseous and stale._
The setting seems even more nightmarish later with:
_Red cellophane garlands festoon the doorways. She has totally forgotten it is as close to Christmas as this. A pianist in the corner is playing "Carol of the Bells," and it sounds not only unfestive but scary, like the theme from_ The Exorcist.
Notice that in the above examples, the setting is being conveyed through the consciousness of the POV character. Perhaps some people would have found the "Carol of the Bells" charming, but this character found it harrowing.
YOUR TURN:
Imagine a character who is contemplating a major change in his or her life—dropping out of school, having a child, entering a risky business venture... Once you have fleshed out the character a bit, write a passage where this character is dealing with the change. You may or may not choose to have other characters involved. Here's the interesting part: let weather underscore the drama of the passage, be it the first gusts of autumn or a torrential downpour or any other act of the elements.
**SETTING AND CHARACTER**
While we're on the subject of characters, let me point out that setting plays a great role in who your characters are—how they dress, talk, socialize, work, travel, eat, and so forth. Much like animals, people behave a certain way in their natural habitat, and you want to pay attention to how your characters are shaped by their setting.
At this pool party in a ritzy suburb from John Cheever's "The Swimmer," the characters act in a way that is quite native to their environment:
_As soon as Enid Bunker saw him she began to scream; "Oh, look who's here! What a marvelous surprise! When Lucinda said that you couldn't come I thought I'd_ die." _She made her way to him through the crowd, and when they had finished kissing she led him to the bar, a progress that was slowed by the fact that he stopped to kiss eight or ten other women and shake the hands of as many men. A smiling bartender he had seen at a hundred parties gave him a gin and tonic and he stood by the bar for a moment, anxious not to get stuck in any conversation that would delay his voyage._
How are the characters in your fiction shaped by their setting? If you can easily take your characters out of their current setting and plop them elsewhere with no notable difference in who they are, then perhaps your characters are not affected deeply enough by the time and place in which they live. Don't pound setting into your characters, like forcing your Texas businessman to wear cowboy boots and exclaim _y'all_ every time he speaks, but look for the subtle ways that Texas may have seeped into his being.
Stories often contain characters who are forced to go outside their natural environment, which creates interesting dynamics and situations. In such cases, you'll need to be aware of how a character acts and reacts in a setting that is somewhat foreign. Imagine a young punk girl getting fitted for an evening dress at Neiman Marcus, or a socialite in a tattoo parlor. Literature is filled with examples of this, such as middle-class midwestern Nick entering Jay Gatsby's lavish world in _The Great Gatsby,_ or Earl, the car thief, being stranded for a few days in Wyoming in "Rock Springs."
At its most extreme, this becomes a "fish out of water" story, where the main conflict is between a character and a wildly unfamiliar surrounding. You may recognize the fish-out-of-water scenario from such TV shows as _Green Acres_ or _Northern Exposure,_ but it's also proven popular in fiction. An obvious example is Lewis Carroll's _Alice in Wonderland,_ where a well-bred Victorian girl falls through a rabbit hole, landing in a topsy-turvy world where turtles sing and cats disappear into nothing but a smile. Examples abound, such as _Great Expectations,_ where a boy from the English marsh country mingles with the dandies of London, or Thorn Jones's "A White Horse," where an American advertising man with amnesia finds himself wandering through the squalor of Bombay without a clue as to how he got there. A fish-out-of-water story can be enormously fun, but if you tackle one by ready to deal with your setting extensively.
_The Metamorphosis,_ by Franz Kafka, puts an interesting twist on the fish-out-of-water story, as Gregor Samsa, a young traveling salesman, wakes up one morning to discover that during the night he has been transformed into an insect. The world that was once so familiar and comfortable to Gregor has become an unnavigable and strange landscape. Here he is trying to manage a once simple task:
_He thought that he might get out of bed with the lower part of his body first, but this lower part, which he had not yet seen and of which he could form no clear conception, proved too difficult to move: it shifted so slowly; and when finally, almost wild with annoyance, he gathered his forces together and struck out recklessly, he had miscalculated the direction and bumped heavily against the lower end of the bed._
Throughout the story, Gregor is forced to see his familiar world from a whole new perspective. Offering a new perspective is one of the advantages of placing characters in an alien environment, because it forces the characters and the reader to see things with fresh, and often wary, eyes.
YOUR TURN:
Think up a character who is very much the opposite of yourself. Choose some of the following differences: sex, age, occupation, background, temperament... Now write a passage where this character must live for a while in an environment very similar to your own. Let the setting cause as much conflict as possible for the character. For example, if the character is a freewheeling bachelor, perhaps let him struggle tending to your houseful of kids. If the character is a spoiled rich kid, perhaps let her hold down your job for a day. Have fun letting someone else struggle with your setting!
In some cases, a setting becomes so overwhelmingly important that it actually performs as a character in its own right—it can act and change, and even become one of the most dominant features in your story. This is certainly true of most fish-out-of-water stories, where the setting is the major conflict, but it also applies to such diverse examples as the moors of _Wuthering Heights,_ the Jazz Age of _The Great Gatsby,_ the Depression of _The Grapes of Wrath,_ the Vietnam War of "The Things They Carried," the hospital of "People Like That Are the Only People Here," and the affluent suburbia of "The Swimmer."
Virginia Woolf used this technique in her novel _The Waves,_ where she traces the lives of a group of six friends from childhood to middle age. They each speak in a series of soliloquies, with the sea and roaring waves as the backdrop. _The Waves_ is punctuated by descriptive interludes that show a changing view of the sea as day evolves from dawn to nightfall. The waves themselves recur and change, creating a sense of unity in the novel and adding a level of meaning. The characters' speeches seem to be fluid stream-of-consciousness pieces about what it is to be, as the waves are. When the novel opens, we see the early waves of the morning:
_The wave paused, and then drew out again, sighing like a sleeper whose breath comes and goes unconsciously._
As the book progresses, and the characters age, so the day progresses and the waves become bolder:
_The waves massed themselves, curved their backs and crashed. Up spurted stones and shingle._
And at the end of the book, in which the characters face impending age and death, the waves do as well. The novel ends with the simple line:
_The waves broke on the shore._
**SETTING THE DETAILS**
If the purpose of setting is to ground the characters, and the readers, in the physical world of the story, and perhaps reflect the appropriate dramatic mood, then you, the writer, are going to need to create that physical world. With what? With the only thing you've got—words. If you've read chapter 5 in this book, you should already have a pretty good idea of how to bring your settings to life. Largely you'll be painting your fictional settings through the artful use of sensory and specific description.
Here's a section from Henry James's _The Portrait of a Lady._ Pay attention to the sensory and specific details in this depiction of a house:
_It stood upon a low hill, above the river_ — _the river being the Thames, at some forty miles from London. A long gabled front of red brick, with the complexion of which time and the weather had played all sorts of picturesque tricks, only, however, to improve and refine it, presented itself to the lawn, with its patches of ivy, its clustered chimneys, its windows smothered in creepers._
This is a setting you can see unmistakably, can't you? Not only does James give you a highly detailed visual picture of the house itself but he also gives you a sense of the house's history and he pinpoints its exact location. There is a lot of fuel for your imagination in such a small amount of space.
Such conciseness is actually important. You don't want your setting description to hit the pause button on the action too often or readers will go out to have a drink or do some shopping while you're there working carefully at the easel with your brush and paint. Readers depend on the forward movement in a story, so your story is best served by scattering your setting description throughout a piece rather than dropping it in giant globs here and there. You would also do well to mix a little action into your descriptions of setting.
Take a look at this selection from Marguerite Duras's novella _The Lover:_
_I get off the bus. I go over to the rails. I look at the river. My mother sometimes tells me that never in my life shall I ever again see rivers as beautiful and big and wild as these, the Mekong and its tributaries going down to the sea, the great regions of water soon to disappear into the caves of ocean. In the surrounding flatness stretching as far as the eye can see, the rivers flow as fast as if the earth sloped downward._
_I always get off the bus when we reach the ferry, even at night, because I'm always afraid, afraid the cables might break and we might be swept out to sea. In the terrible current I watch my last moments. The current is so strong it could carry everything away_ — _rocks, a cathedral, a city. There's a storm blowing inside the water. A wind raging._
Duras's details are sensuous, but she weaves the setting into the story by relating an anecdote about what the character's mother tells her to notice, thereby providing valuable character insight alongside the description. The flow of the story is not lost at all.
Remember the importance of telling details, those tidbits of information that carry so much power in their little shells? Skillful use of telling details will allow you to convey your settings quickly, yet effectively. In Richard Russo's _Empire Falls,_ Miles, the protagonist, runs a somewhat rundown diner. Instead of giving us every shabby detail of the diner's kitchen, Russo focuses on the antiquated dishwasher:
_Only one tub of dirty dishes remained, but it was a big one, so Miles lugged it into the kitchen and set it on the drainboard, stopping there to listen to the Hobart chug and whir, steam leaking from inside its stainless steel frame. They'd had this dishwasher for, what, twenty years? Twenty-five? He was pretty sure it was there when Roger Sperry first hired him back in high school._
With this one detail, the reader feels as if he is standing there in the kitchen beside Miles, and the story is ready to move onward.
As a rule of thumb, ask yourself how important a particular time or place is to your story and this will help you determine how much "space" to spend describing any given setting. "The Fall of the House of Usher" takes place entirely in the House of Usher, and so a good amount of description on the house is warranted. But you don't need to spend page after beautiful page describing a drugstore if your main character simply stops in to buy a bottle of aspirin.
Do all works of fiction deal with setting in at least some detail? Well, no, actually. Raymond Carver, for example, noted for the minimalism of his prose, often doesn't put in much setting. In "Cathedral," the entire story takes place in a house, and though Carver refers to the kitchen and the sofa and the window and the TV, he doesn't give any specifics about anything there, except that his wife bought a new sofa and the narrator liked the old one. But the lack of setting detail makes sense here because this story is told in the first person and the narrator isn't really taking note of his house or especially interested in it. We can also assume from the lack of detail that this is a fairly ordinary home, neither squalid nor lavish. If it were either, then the writer should have told us so. If you have a compelling reason _not_ to describe your setting, then you have permission to do so.
**THE REALITY OF SETTING**
Most fictional stories deal with authentic settings, portraying real places and times or at least seemingly real ones. Though, say, Thomas Hardy's Wessex County, home to several of his novels, cannot be found on any map, it's very similar to the genuine article in that particular part of the English countryside. When writing about these real or seemingly real settings, do your best to paint them accurately and vividly.
If you're writing about a setting that you know well, you shouldn't have a problem verifying the details. If you're writing about a setting you don't know well, you should do your best to gather as much information as you can, by visiting or just doing some old-fashioned research, made all the easier by the un-old-fashioned Internet. However, if you're writing about a setting with which you are not intimately familiar, you may want to give it a fictional name so as not to do something like anger the residents of Cleveland by mixing up the names of the cross-streets downtown. If you call your city, say, Leveland, you're off the hook.
Fictionalizing your location also gives you some dramatic license. In Beth Nugent's novel _Live Girls,_ the main character, Catherine, in her early twenties, drops out of college to live in a seedy motel and work as a ticket taker in a rundown adult theater. The city is unnamed, and is probably an amalgamation of several places:
_It is just another grim Eastern seaport collapsed in on itself with its own inertia, caving slowly into the dark heart at its center. It is a city full of rooms that rent by the week or the day or even the hour, and it is populated by the kind of people who would rent them; there are no families here, and no houses, and every day what respectable people there are move farther and farther away, where there are houses and families and respectable jobs and hobbies; they live in little developments built just for them, spreading outward from the city, like spores cast away from a plant._
Though this place feels authentic, Nugent could choose her details, calibrating them to the feel of the story, without limiting herself to a literal place.
Perhaps you're using a place and time that doesn't exist, or hasn't... yet. Look at this passage from George Orwell's speculative novel _1984:_
_The black moustachio'd face gazed down from every commanding corner. There was one on the house-front immediately opposite, BIG BBOTHER IS WATCHING YOU, the caption said, while the dark eyes looked deep into Winston's own. Down at street level another poster, torn at one corner, flapped fitfully in the wind, alternately covering and uncovering the single word INGSOC. In the far distance a helicopter skimmed down between the roofs, hovered for an instant like a bluebottle, and darted away again with a curving flight. It was the police patrol, snooping into people's windows._
The novel centers around Winston Smith, an average man who lives in Oceania, a totalitarian empire led by Big Brother. The setting isn't real, but the details certainly feel real. Here we have familiar sights—the flapping posters, the hovering helicopters—placed in an unfamiliar framework, helping to maintain an internal accuracy that rings true. If you find yourself creating settings that are only semirealistic, look for ways to blend the familiar with the fabricated to give your setting a sense of verisimilitude.
Even if you're creating a world that is wholly fantastical, as found in many works of science fiction and fantasy, you will still want the setting to _seem_ real. Believe it or not, these kinds of settings usually demand more homework than any other kind of setting, if you really want your time and place to be convincing. While working on the great trilogy _The Lord of the Rings,_ J.R.R Tolkien spent many years developing a complicated system of mythology and history, a detailed geography and a full set of maps, and the entirety of several languages (including Elvish) in order to build his imaginary world. The Hobbits didn't just live off in the ether of a fantasyland. They dwelled in a very specific place and time:
_Forty leagues it stretched from the Far Downs to the Brandywine Bridge, and fifty from the northern moors to the marshes in the south. The Hobbits named it the Shire, as the region of the authority of their Thain and a district of well-ordered business; and there in that pleasant corner of the world they plied their well-ordered business of living, and they heeded less and less the world outside where dark things moved, until they came to think that peace and plenty were the rule in Middle-Earth and the right of all sensible folk._
No wonder so many millions of readers seem to think this world really exists.
**THE PACE OF TIME**
The term _pacing_ is often used to mean a number of different things in fiction, but here we're going to refer to it as the manipulation of time. The fiction writer becomes a sort of Tolkienesque wizard, able to manipulate time with the wave of a staff (or tap of the keyboard), and pacing is one of the great tools in the fiction writer's bag of tricks.
The most prevalent way you manipulate time is by compressing and expanding it to fit the needs of your story. Time passes for our characters, but the writer controls how quickly or slowly it flows. Writers do not show every moment of a plot, every instance in a character's life from birth until death, but instead speed through or skip over sections of time that are irrelevant to the story, while slowing down and expanding the sections that are most important.
If you move too quickly through an important section, the reader can feel disappointed or even confused. Likewise, if you move too slowly or dwell on irrelevant events, you can bore your reader. I bet you can remember skimming or even skipping over entire pages of a book, to "get to the good part." That was most likely because the writer slowed down the pace in a section where nothing of particular interest was happening. So pay attention to how important your scene is to the overall story, and pace it accordingly.
Do you want to see how time can be magically slowed down? Take a look at this section from Toni Morrison's _Sula,_ in which Nell has just walked into her bedroom to find her husband, Jude, naked on the floor with her best friend:
_I am just standing here. They are not doing that. I am just standing here and seeing it, but they are not really doing it. But then they did look up. Or you did. You did, Jude. And if only you had not looked at me the way the soldiers did on the train, the way you look at the children when they come in while you are listening to Gabriel Heatter and break your train of thought_ — _not focusing exactly but giving them an instant, a piece of time, to remember what they are doing, what they are interrupting, and to go on back to wherever they were and let you listen to Gabriel Heatter. And I did not know how to move my feet or fix my eyes or what. I just stood there..._
This passage (which continues a while longer) covers only a few seconds of real time, but Morrison has slowed time to an excruciatingly slow pace to adequately reflect the horror of the moment in the narrator's mind. If you think about it, times of crisis do indeed seem to pass very slowly, each second exploding into an eternity. And as you can see, this is far more effective than telling readers about a character's emotions. _She felt shocked, betrayed, angry, and embarrassed_ just doesn't give the reader the same kick to the heart. But by giving us every awful thought, Morrison makes us understand just how shocked and torn her protagonist is. Obviously not every moment in the story is handled with such depth and breadth, but this one deserved the attention it got.
YOUR TURN:
Recall the most frightening moment of your life. If it's too scary or recent, go for the second most frightening moment. Using yourself as a first-person narrator, write a passage about this moment. Chances are time slowed down for you as you were living that moment, so slow time way down as you describe this experience. Include minute details and the panoply of your thoughts. You may end up writing several pages about several seconds. Feel free to embellish, though you probably won't need to.
Let's look at how Raymond Carver handles pacing in "Cathedral." After some initial exposition and a scene in the kitchen where the narrator and his wife argue, Carver jumps ahead to the blind man's arrival with:
_So when the time rolled around, my wife went to the depot to pick him up. With nothing to do but wait_ — _sure, I blamed him for that_ — _I was having a drink and watching the TV when I heard the car pull into the drive. I got up from the sofa with my drink and went to the window to have a look._
Very quickly, we've cut to the important moment of Robert's arrival. Then we get a scene that covers the narrator meeting Robert and everyone sitting around having a drink (clearly not the narrator's first of the day). In the scene we fully observe the first part of the conversation, but Carver skillfully moves us through the cocktail hour (which probably included several rounds) with:
_This blind man filled his ashtray and my wife emptied it._
It takes a while for even a chain smoker to fill an ashtray, so we get the sense that some time has passed, and then we cut to:
_When we sat down at the table for dinner, we had another drink._
Then we get a longish summary of the dinner and corresponding conversation. Note that one of the chief ways pacing is achieved is by alternating scene and summary. Here the author has chosen summary over scene to move us more quickly through the expanse of the dinner, with things like this:
_From time to time, he'd turn his blind face toward me, put his hand under his beard, ask me something. How long had I been in my present position? (Three years.) Did I like my work? (I didn't.) Was I going to stay with it? (What were my options?) Finally, when I thought he was beginning to run down, I got up and turned on the TV._
And with that final line, we move on to the next stage of the evening, watching TV and drinking. In only seven pages, Carver has moved us through several hours and we don't feel we've missed anything important. We're also probably starting to feel a little drunk. Then we get a scene in which the narrator and Robert start watching television. After a little dialogue, we skip ahead further with:
_After she'd left the room, he and I listened to the weather report and then to the sports roundup. By that time, she'd been gone so long I didn't know if she was going to come back._
We sense they've watched some prime-time TV and are now into the news. The passing of time is now being reflected in the changing of the TV programs. Soon the news ends, the wife returns but promptly falls asleep on the couch, the narrator and Robert smoke some pot (while still drinking), and they end up watching some kind of documentary on "the church and the Middle Ages." It's beginning to feel like a long night—an aimless, drunken, stoned, claustrophobic night that is probably quite similar to most nights in this house. Then we make one more jump in time:
_We didn't say anything for a time. He was leaning forward with his head turned at me, his right ear aimed in the direction of the set. Very disconcerting._
The story is now approaching its apex—the narrator's transformation—and here Carver really slows down time and gives us a long scene that takes us through practically every moment of what happens, even following the details of the TV program:
_The TV showed this one cathedral. Then there was a long, slow look at another one. Finally, the picture switched to the famous one in Paris, with its flying buttresses and its spires reaching up to the clouds._
With agonizing slowness, we watch as the narrator vainly attempts to describe a cathedral, and, failing this, he attempts to draw one while the blind man places his hands on those of the narrator, and we hear pretty much everything that is spoken between these two men in this most dramatic of moments.
In fourteen pages, Carver has expertly taken us through a very long-seeming night in which a man's life is believably altered and every single sentence is both necessary and interesting. This is what pacing is all about.
Your pacing choices will be greatly affected by the length of your fictional work. In a short story, where there is a minimum of page "real estate," you'll need to be very choosy about what you show and don't show. With a novella or novel, you have more wiggle room, but even with a longer work you should ultimately be reluctant to include anything that doesn't have a significant impact on the tale being told. Think about your work as a whole, and plan how much time (and space) you'd like to spend on each part.
FLASHBACKS
And, yes, the fiction-writing wizard has another great trick in that he can move back and forth through time at will.
Flashbacks come in handy when there is a need to relate something that took place before the chosen time frame of the story. Instead of just having the narrator briefly allude to the event, you may want to show the event with some depth of detail, which will mean actually drifting back to the event.
"The Things They Carried" by Tim O'Brien follows the journey of a lieutenant in the Vietnam War. Though the story takes place entirely in Vietnam, the protagonist is continually wandering in his mind to memories of his tentative relationship with a young woman in New Jersey. While studying a picture of the woman:
_Lieutenant Cross remembered touching that left knee. A dark theater, he remembered, and the movie was_ Bonnie and Clyde, _and Martha wore a red tweed skirt, and during the final scene, when he touched her knee, she turned and looked at him in a sad, sober way that made him pull his hand back, but he would always remember the feel of the tweed skirt and the knee beneath it and the sound of the gunfire that killed Bonnie and Clyde, how embarrassing it was, how slow and oppressive._
Though the story is about Cross's experiences while in Vietnam, the periodic flashbacks give the reader a much fuller sense of all the "things" this man is carrying with him, mentally, as he wages war.
Flashbacks are usually better off not running too long, though they may include actual scenes of dialogue. But don't rely on them too much. If you find yourself needing pages and pages of flashbacks, you may have begun your story at the wrong point in time, in which case your story may take on a disjointed quality. Flashbacks can also be quite confusing to a reader unless you clearly delineate them, remembering to anchor them to the story's present. As with setting, you always want the readers to have a sense of where they _are._ And, as with anything, there are exceptions—stories, such as Toni Morrison's _Beloved,_ that move fluidly through different time frames throughout the entire work. But such a feat requires a wizard of the highest order.
[CHAPTER 8
**VOICE: THE SOUND OF A STORY**](Facu_9781596917910_epub_c4_r1.html#aa8)
BY HARDY GRIFFIN
When I started writing, I couldn't figure out what "voice" was, so I skipped it and spent my time learning how to create characters and plots out of nothing. I figured I'd just go ahead and write and use whatever "voice" showed up naturally.
That is, until my friend showed up instead and asked me to read some fiction in a reading series he was coordinating at the corner restaurant. Suddenly my little apartment was full of _my_ nervous, frustrated, bewildered voice while I spent the better part of a week pacing from the sink to the couch and back, reading and editing a simple three-page story about a young man (half me and half fiction) at his grandparents' house for Thanksgiving. Nothing sounded right, from the Southern accents to the lemon meringue pie to the fireflies in the deep blue dusk.
I stepped into the exposed-brick basement of the restaurant where the reading was and got a gin and tonic right away (for the Southern accents, you understand). It was packed and I was freaked out. But when my turn came and I stood in front of everyone and started reading, something happened. The thing in my hands didn't feel like my story so much as my narrator's story, as if the words on the page had come together and formed a new person who was speaking and who the audience in the cafe wanted to listen to.
Three things popped out at me about voice after that night. First of all, the voice of a piece is what makes it special, what sets it apart and makes it feel lived. On the other hand, voice isn't half as ephemeral as critics and academics make it sound. But most important, it's essential that your narrative voice sounds natural. Your storyteller should be relaxed and absorbed in the fiction so your readers can be too. That's what worked in my piece for the reading—even while _I_ was in a sweat, my narrator's voice was completely involved in telling the story.
But what _is_ this mysterious thing, voice?
One of my favorite oxymorons is the often-repeated phrase "a writer's voice." Just how much sound can a bunch of black marks on a piece of paper make, anyway? I don't know about you, but the only sounds I make as I'm writing are the tap of the keys and various inarticulate groans. Obviously these aren't the voices that readers, academics, and critics are always talking about.
Simply put, voice is what readers "hear" in their heads when they're reading. Voice is the "sound" of the story.
In every strong work of fiction, one voice rises above the din to unify the piece and lead the reader through the thicket of characters' voices. This voice is the most important for the simple reason that, after finishing a good story or novel, it's this overarching voice which continues to ring in the reader's mind. And yes, you guessed it—the voice of a story is the voice of the narrator.
My fiction students often get confused between the voice of a piece and the writer's voice, and with good reason. If a number of works by the same author have a similar tone, then people often lump them together as that writer's voice or style. However, the best thing you, as a writer, can do is to concentrate on the narrator's voice of each individual piece of your fiction. Someday, a critic may see what your varied works have in common and write an admiring article that defines what your voice as a writer is like. Until then, your job is to focus on the voice in each individual story.
**TYPES OF VOICE**
The amazing truth is that your chosen voice can take an infinite variety of "sounds." So how do you pick what kind of voice to give your narrator?
More than anything else, your choice of voice is related to your point-of-view choice. If you're using a first-person narrator, then your voice will need to match the personality of that particular character. If you're using a second-or third-person narrator, then the narrator will be a storyteller, who may or may not sound like you. Also, the sound of the second-or third-person narrator will be affected by the emotional distance with which this narrator is telling the story. A first-person narrator will naturally tell a story in a way that is close to the action because he or she is inside the story. But this isn't the case with a second-or third-person narrator. Such narrators may be emotionally close to the characters, as in a first-person POV, or they may be telling the story from a more remote distance, as if they are standing outside the story's events, like those broadcasters on TV commenting on the golf match in whispered, reverant tones.
To help you get a sense of your voice options, let's break voice into several general types. Although these titles aren't as official as those for POV, placing voices into general types can help you make choices about your voice and also help you tell if a story has drifted from its originally intended route.
**CONVERSATIONAL VOICE**
Everybody knows somebody whom they don't have to dress up for, whether this person is a close friend or family member. Just like it sounds, the conversational voice feels a lot like the narrator is having a casual conversation with the reader.
Mark Twain's _The Adventures of Huckleberry Finn_ is a prime example:
_You don't know about me without you have read a book by the name of_ The Adventures of Tom Sawyer; _but that ain't no matter. That book was made by Mr. Mark Twain, and he told the truth mainly. There was things which he stretched but mainly he told the truth. That is nothing. I never seen anybody but lied one time or another, without it was Aunt Polly, or the widow, or maybe Mary._
Here Twain believably captures the voice of a hillbilly kid, namely Huckleberry Finn. Before _Huck_ appeared on the shelves in 1885, most fiction had an elevated voice, but Twain threw all that away and truly let Huck speak for himself. The result is one of the most nonpretentious and entertaining voices in all literature.
J. D. Salinger's _The Catcher in the Rye_ also belongs in this category:
_If you really want to hear about it, the first thing you'll probably want to know is where I was born, and what my lousy childhood was like, and how my parents were occupied and all before they had me, and all that David Copperfield kind of crap, but I don't feel like going into it, if you want to know the truth._
You can probably guess from these first couple of lines that the voice is that of a personable and often sarcastic teenager. As we find out at the end of the novel, Holden Caulfield has been telling his story to a psychiatrist the whole time, which is exactly how the narrative sounds, as if a real person is speaking his mind.
Another example is Dorothy Parker's "The Waltz." Here a woman agrees to dance with a man when she doesn't want to, and then she starts to think about what she could have said instead of agreeing:
_I most certainly will not dance with you, I'll see you in hell first. Why, thank you, Yd like to awfully, but I'm having labor pains. Oh, yes, do let's dance together_ — _it's so nice to meet a man who isn't a scaredy-cat about catching my beri-beri. No. There was nothing for me to do, but say I'd adore to. Well, we might as well get it over with. All right, Cannonball, let's run out on the field. You won the toss; you can lead._
Most of the story is made up of this woman's sarcastic thoughts as she struggles with her dance partner, tearing him to pieces in her mind but every now and then exchanging pleasantries with him out loud.
YOUR TURN:
Take the above passage from Dorothy Parker's "The Waltz." Rewrite and expand it using the third-person objective point of view. Remember, in the objective POV you don't enter the minds of characters. You simply show the action, as it's been recorded by a journalist. In fact, keep the writing very dry and factual, distant from the actual emotions of the woman. But strive to convey the woman's thoughts solely through her actions and, if you desire, dialogue. Then compare your version with Parker's, noting how the same event can be told with very diverse voices.
The conversational voice is almost always in the first person and it usually employs colloquial speech patterns and slang. So a conversational voice would opt for _See, this woman's following me, like she has for the past two whole weeks_ rather than _Two weeks ago, a woman started following me, and I saw her again behind me today._
The great thing about this voice is that you can let your first-person narrators go full throttle with their personalities. And they can pretty much tell the reader anything. Which can also be the downside. If you're not careful, it may sound like your narrator is blabbing out all her intimate details for no good reason.
**INFORMAL VOICE**
You can dress down, but at least tuck in that shirt. Informal voice is a fairly broad category that's not as casual as the conversational voice, but it also doesn't quite have the dressed-up feel of the more formal voices.
Take Raymond Carver's "Cathedral" for example:
_I remembered having read somewhere that the blind didn't smoke because, as speculation had it, they couldn't see the smoke they exhaled. I thought I knew that much and that much only about blind people. But this blind man smoked his cigarette down to the nubbin and then lit another one._
See how this first-person narrator isn't as chatty or colloquial as the conversational narrators, but at the same time, he's an average sort of guy who drinks, smokes pot, and thinks he knows only one thing about blind people before a blind friend of his wife's comes to visit.
Another example in the first person is John Cheever's "Goodbye, My Brother":
_I don't think about the family much, but when I remember its members and the coast where they lived and the sea salt that I think is in our blood, I am happy to recall that I am a Pommeroy_ — _that I have the nose, the coloring, and the promise of longevity_ — _and that while we are not a distinguished family, we enjoy the illusion, when we are together, that the Pommeroys are unique._
The secondary-school teacher who narrates this piece strikes a balance between the "blue-blooded" roots of his family and the openness of revealing the family's problems. The informal voice allows this line to be walked.
In the informal voice, the narrator uses casual, everyday language but isn't as personality-heavy as the conversational. But, as you can see in the difference between the Carver and Cheever examples, there's a lot of leeway in what can make up casual language for different narrators.
This informality can also come through with a third-person narrator. Check out how Amy Bloom's "Song of Solomon" has a similar voice to the Carver story:
_Sarah had stopped sucking a little sooner than usual, and Kate was so grateful she sang to her all the way through burping. Everything went smoothly; little Sarah, stoned from nursing, was completely content to lie in her crib and murmur to the world. Kate dressed like a surgeon prepping, precise and careful in every movement. She checked her watch again. Twenty-five minutes to get to the temple._
Though the narrator isn't a character, the voice sounds rather like that of a real person, someone we might know relating a story to us in their living room. And Bloom's relatively close third-person narrator conveys Kate's slightly nervous actions and thoughts, but isn't so close to Kate as not to convey a sense of the child's serenity. A first-person narrator would not have maintained quite as much balance between the emotions of Kate and the baby.
Let's return to an example that appeared in chapter 4, from "Earth to Molly" by Elizabeth Tallent. Here you'll see an informal third-person narrator who inches very close, emotionally, to the POV character:
_Molly was sorry for having needed her to climb the stairs, but of course the old woman complained her stiff-legged way up them all the time, showing lodgers to their rooms. Why, oh why, would anyone spend the night here? A prickly gray carpet ran tightly from wall to wall. It was the color of static, and seemed as hateful._
Here the voice of the third-person narrator sounds pretty similar to the voice Molly might use if she were telling this story herself in the first person. If you use a third-person informal narrator, you'll usually want to make this narrator somewhat close, or very close, to the emotions of the POV characters. Otherwise your readers will feel like someone's just told them to make themselves at home in a living room packed with priceless antiques.
The main advantage of the informal voice is that it's middle of the road. If you're working in the first person but don't want the narrator's voice to dominate the story, this is a good pick. It's also a good pick if you're working in the third or second person but don't want to sound too much like a "writer." Actually, it's hard to go wrong with the informal voice, and for this reason it's probably the most commonly used voice in contemporary fiction.
**FORMAL VOICE**
Even the word _formal_ makes me think of some boarding-school prom night with a row of girls in strapless heels and boys in navy suits, but in practice, the formal voice doesn't have to be awkward at all.
In the old days most all fiction leaned toward the formal, as in this example from Leo Tolstoy's "Master and Man":
_Suddenly a weird, startling cry sounded in his very ears, and everything beneath him seemed to heave and tremble. He clutched the horse's mane, yet found that that too was quivering, while the cry grew ever more and more piercing._
As you can see, the formal voice doesn't have the same chattiness or spoken-story qualities of the conversational or informal, often conveying, instead, a certain detachment from the characters. In this passage, even though the man is panicked and on the edge of death from freezing, the third-person narrator stays fairly observational. You'll see what I mean if you compare how close you feel to this narrator's emotions with those of Huck Finn or Holden Caulfield.
This kind of dressed-up style can work in contemporary fiction too. If you're working on an epic story that, say, covers multiple generations, a number of locations, and a large cast of characters, the formal voice is a good bet because it lends itself to the story's "big screen" sweep.
Look at the opening to Gabriel Garcia Màrquez's _One Hundred Years of Solitude:_
_Many years later, as he faced the firing squad, Colonel Aureliano Buendía was to remember that distant afternoon when his father took him to discover ice. At that time Macondo was a village of twenty adobe houses, built on the bank of a river of clear water that ran along a bed of polished stones, which were white and enormous, like prehistoric eggs. The world was so recent that many things lacked names, and in order to indicate them it was necessary to point._
As the novel moves back and forth over a century of one family's development, strange, detailed, and beautiful descriptions of Macondo village put the reader in the setting, rather than sticking us with a single character. Which is a good thing because Màrquez ends up using more than twenty major characters. And the formal voice gives the book the depth and importance of a historical (albeit fanciful) chronicle.
The formal voice is perhaps most commonly found in the third-person POV, but it's not restricted to this. It can work with the first person as long as the first-person narrator has a formal enough personality.
For example, Humbert Humbert, the narrator of Vladimir Nabokov's _Lolita,_ is the son of the owner of a luxury hotel on the French Riviera. Early in the novel, we discover that Humbert attended a good English school and later became a literature scholar. So it's certainly fitting that he speaks in a formal tone, to the point of pretentiousness:
_And less than six inches from me and my burning life, was nebulous Lolita! After a long stirless vigil, my tentacles moved towards her again, and this time the creak of the mattress did not awake her. I managed to bring my ravenous bulk so close to her that I felt the aura of her bare shoulder like a warm breath upon my cheek._
Very few people actually sound like this, but Humbert Humbert happens to be one of the people who does.
Another example is the voice of Nick Carraway in F. Scott Fitzgerald's _The Great Gatsby._ Nick is also well educated enough to pull off the formal voice. Here he is describing his first glimpse of Jay Gatsby:
_The silhouette of a moving cat wavered across the moonlight, and turning my head to watch it, I saw that I was not alone_ — _fifty feet away a figure had emerged from the shadow of my neighbor's mansion and was standing with his hands in his pockets regarding the silver pepper of the stars. Something in his leisurely movements and the secure position of his feet upon the lawn suggested that it was Mr. Gatsby himself, come out to determine what share was his of our local heavens._
It's worth noting that Nick's observations are distant enough that he sounds more like the writer than a distinctive character, and this could almost be switched into the third person. But then we would lose the sense of Nick as a witness to the story's events.
Go for the formal voice if you want a certain high style in your prose, but make sure you're not just trying to sound like a writer and, if you're using this voice for a first-person narrator, make sure it's someone more likely to write with a Montblanc pen than a chewed-up pencil.
YOUR TURN:
Two cars collide at an intersection. Write a brief passage describing this event from the POV of a teenager, then again from the POV of a socialite, then again from the POV of a cowboy type. You decide how these characters were involved in the collision. In all cases, let the character be a first-person narrator. So pick the voice type—conversational, informal, or formal—that seems most appropriate for your narrator. Conversational may work well for the teenager, but then that depends on your teenager, doesn't it? Whatever you come up with, each passage should sound different from the others because these are three very different characters.
**CEREMONIAL VOICE**
You'll have to get your tux out of storage if you want to be the master of ceremonial. A good way to get into the mind-set of this voice is to imagine you're old Abe Lincoln about to give "The Gettysburg Address": "Four score and seven years ago, our forefathers..."
You wouldn't think that ceremonial voice would come into play very much in fiction, but many writers have used it to great effect. Take this passage from Charles Dickens's _Oliver Twist:_
_Oliver Twist's ninth birthday found him a pale thin child, somewhat diminutive in stature, and decidedly small in circumference. But nature or inheritance had implanted a good sturdy spirit in Oliver's breast: it had had plenty of room to expand, thanks to the spare diet of the establishment; and perhaps to this circumstance may be attributed his having any ninth birthday at all._
Look at how detached this narrator is. Basically, Oliver's starving and abused, and the only thing that's kept him alive up to his ninth birthday is his spirit, but the narrator's far enough from the boy's suffering to be half-joking about how empty his stomach is. But perhaps this kind of mocking ceremonial is what allows the reader to absorb Oliver's painful, bleak story over the course of the novel's four-hundred-plus pages.
At the same time, the following passage from Gertrude Stein's _Melanctha,_ a novella about a black woman's life in Bridgeport, Connecticut, around the turn of the twentieth century, shows a different side of ceremonial:
_Melanctha Herbert was always losing what she had in wanting all the things she saw. Melanctha was always being left when she was not leaving others._
_Melanctha Herbert always loved too hard and much too often. She was always full with mystery and subtle movements and denials and vague distrusts and complicated disillusions. Then Melanctha would be sudden and impulsive and unbounded in some faith, and then she would suffer and be strong in her repression._
_Melanctha Herbert was always seeking rest and quiet, and always she could only find new ways to be in trouble._
Stein creates an almost biblical rhythm through the repetitious language and the odd sentence phrasing. By using a ceremonial voice, the narrator elevates Melanctha Herbert's life nearly to the level of a prophet, and her struggles suddenly don't appear to the reader as worthless or squalid in this light.
Just as conversational is hardly ever in the third person, it's also quite rare to find a first-person ceremonial narrator. The advantage of the ceremonial voice is that it slows the reader down, giving a great sense of occasion and importance to the story. The disadvantage is that it can seem stilted and suppress the story's energy.
**OTHER VOICES**
Once again, let me say that these voice types are just arbitrary terms to help you get a sense of the options and to help you stay on track. Really, the voice of a story can take on any conceivable "sound" as long as you have a reason for it. Literature is filled with unusual voices that don't fit anywhere on my clothesline of types.
Take, for instance, Helen Fielding's _Bridget Jones's Diary:_
_TUESDAY 3 JANUARY_
_130 lbs. (terrifying slide into obesity_ — _why? why?), alcohol units 6 (excellent), cigarettes 23 (v.g.), calories 2472. 9 a.m. Ugh. Cannot face thought of going to work. Only thing which makes it tolerable is thought of seeing Daniel again, but even that is inadvisable since am fat, have spot on chin, and desire only to sit on cushion eating chocolate and watching Xmas specials._
The unique voice here comes from the fact that the whole book is written as a diary, the diary of a contemporary, smart, and somewhat neurotic thirty-something woman. It's very casual and often quite embarrassing, as you would expect from a diary.
A voice can become lyrical to the point of sounding a lot like pure poetry. Listen to the narrator of Jack Kerouac's _On the Road:_
_The only people for me are the mad ones, the ones who are mad to live, mad to talk, mad to be saved, desirous of everything at the same time, the ones who never yawn or say a commonplace thing but burn, burn, burn like fabulous yellow roman candles exploding like spiders across the stars and in the middle you see the blue centerlight pop and everybody goes "Awww!"_
Here the narrator is a soul-searching, usually inebriated beatnik (with a beatnik's bent for the poetic). You can almost see this guy stumbling and rambling drunkenly down the street.
Such poetry can stretch even further into stream of consciouness, where the writer attempts to portray a character's thoughts in the random manner in which they play through the human mind. The final chapter of James Joyce's _Ulysses_ consists of a forty-five-page sentence that careens through the mind of Molly Bloom. To save paper, I'll just show the end of it:
. . _.he kissed me under the Moorish wall and I thought well as well him as another and then I asked him with my eyes to ask again yes and then he asked me would I yes to say yes my mountain flower and first I put my arms around him yes and drew him down to me so he could feel my breasts all perfume yes and his heart was going like mad and yes I said yes I will Yes._
No, those aren't typos in there. Remember, there is no editor in the deeper recesses of the psyche.
Is this getting weird enough? We can perhaps get even weirder. Take a gander at this from Anthony Burgess's _A Clockwork Orange:_
_The chelloveck sitting next to me, there being this long big plushy seat that ran round three walls, was well away with his glazzies glazed and sort of burbling slovos like "Aristotle wishy washy works outoing cyclamen get forficulate smartish."_
Huh? What? Relax, you're not going crazy. This novel is set in the future and the writer has created a whole new vocabulary (a mix of hallucinogenic and Slavic) to suit the time and the personality of the narrator.
YOUR TURN:
Return to something you have written, perhaps from a previous exercise. Rewrite a portion of it using a different voice. You may do something simple, like shifting from formal to informal. Or you might want to try something fun, like using a voice reminiscent of, say, a film noir detective story or a fairy tale. Or you may go for stream of consciousness. To accommodate the new voice, you might end up using a different POV. Look for a voice that will shed an interesting light on your story.
STYLE
You've seen a number of authors parading before you and you've checked out their checks, plaids, and stripes. Now it's time to hit the sweatshop floor and see just how these voice-suits are put together.
People often use the terms _voice_ and _style_ interchangeably, but there's an enormous difference from the writer's perspective. Style consists of various technical choices made by a writer, and the voice is the sum result of those choices. If voice is the velvet dress, style is the fabrics, threads, buttons, and such that create the garment.
The dirty truth is that a piece's voice is created by the most elemental tools in writing—namely, what words you pick, how you string them together in a sentence, and how you mix and match your sentences to form paragraphs. Hemingway used short sentences. Short sentences and repetition. Dorothy Parker liked to throw around the slang, know what I mean. Nabokov favored amplitudinous words. Though these things may seem very technical, you'll see just how closely stylistic choices relate to the personality of the narrator and the story's content. So let's take a look at how to use these very handy tools of style.
**WORDS**
To see how deeply word choice, often known as diction, affects voice, consider the following two examples that both deal with a first-person narrator musing on sleep. First up is Haruki Murakami's "Sleep":
_All I wanted was to throw myself down and sleep. But I couldn't The wakefulness was always there beside me. I could feel its chilling shadow. It was the shadow of myself. Weird, I would think as the drowsiness overtook me, I'm in my own shadow. I would walk and eat and talk to people inside my drowsiness_...
Here, Murakami has used an informal voice for this man who's remembering his casual life at university. Murakami opts for everyday one-or two-syllable words, except for _wakefulness_ and _drowsiness,_ for which there aren't any options with fewer syllables. And I'll take these two words as random examples of how nearly everything in this quote is informal: imagine if he had switched _wakefulness_ and _drowsiness_ for _insomnia_ and _lassitude_ —these new choices would have shoved the narration right out of the informal voice and into something more formal for no apparent reason.
Look at the passage again, and watch how every word is short, to the point, and fits with a certain kind of jerky insomnia. You can feel how the narrator's movements are quick and even stunted from his lack of sleep through his simple words. _Weird_ pops out as an almost conversational word, and the slight jump in voice adds to the jittery quality of the prose.
Now compare this with the opening to John Updike's "Falling Asleep Up North":
_Falling asleep has never struck me as a very natural thing to do. There is a surreal trickiness to traversing that inbetween area, when the grip of consciousness is slipping but has not quite let go and curious mutated thoughts pass as normal cogitation unless snapped into clear light by a creaking door, one's bed partner twitching, or the prematurely jubilant realization,_ I'm falling asleep. _The little fumbling larvae of nonsense that precede dreams' uninhibited butterflies are disastrously exposed to a light they cannot survive, and one must begin again, relaxing the mind into unravelling._
Throw a dart into that paragraph and chances are you'll hit a three-syllable word. This narrator uses the formal voice, marking him as a different type of person from the one in the previous example. Such words as _traversing, cogitation,_ and _larvae_ would feel out of place in the Murakami piece, but they feel quite natural here.
The words also help to show how the insomnia of Murakami's narrator is different from that of the Updike narrator; the former is much more zombie-esque in the way he stumbles about, having no energy to sound impressive, while the narrator in the Updike story is concentrating wholly on the act of falling asleep, and all the fancy words and modifiers help to show his obsessive personality.
Choosing the right words basically boils down to this: know your narrator and what sorts of words this person is inclined to use and make sure your word choices are working with the general type of voice that you have chosen. But don't worry about it too much as you're writing away. You can always go back and take out any incongruous fellas that sneak in.
**SENTENCES**
Words alone don't create the voice; how they're thrown together into a sentence is what really gives writing its flow. I'll tell you something surprising: how you place words in a sentence is the most important stylistic choice you'll make.
A sentence is just a new thought, although that can mean anything from a one-word fragment sentence to a twisting, Route 66 of a sentence. And then within any given sentence there are a thousand things that can happen. But your choices with sentences come down to two basic things: sentence length, and the structure of the sentence, which is often called syntax.
Let's check out the difference between how Hemingway and Fitzgerald handled sentences. These two contemporary writers are both credited as being voices for the Lost Generation of Americans in the years shortly after World War I. Both Hemingway's novel-in-stories, _In Our Time,_ and Fitzgerald's short story "May Day" focus on the end of the war and how it affected individuals and society as a whole.
Here's the opening of Fitzgerald's "May Day":
_There had been a war fought and won and the great city of the conquering people was crossed with triumphal arches and vivid with thrown flowers of white, red, and rose. All through the long spring days the returning soldiers marched up the chief highway behind the strump of drums and the joyous, resonant wind of the brasses, while merchants and clerks left their bickerings and figurings and, crowding to the windows, turned their white-bunched faces gravely upon the passing battalions._
These long, grandiose sentences almost give the writing a mythic quality. Fitzgerald's third-person narrator picks up right after the war's end on the night of May 1, in the midst of the victory festivities. He uses an expansive writing style as his narrator hops among the celebrations and sufferings of multiple main characters—an expansive voice precisely because "May Day" is a mini-epic, a collage encompassing half a dozen of the thousands of stories packed into ten blocks in New York City on a given night. The long sentences reflect the mythic quality of the occasion and the busyness of the festivities.
The sentences are also long because Fitzgerald loads them with plenty of adjectives. (Clearly, Fitzgerald did not agree with Henry David Thoreau's famous advice to writers: "As to adjective: when in doubt, strike it out." Either that or Fitzgerald didn't have any doubts.) For his narrator he has chosen a florid language to match all the flowers and parades, and you can feel the mood of this _great city_ through the modifiers he's chosen, such as _conquering, triumphal,_ and _joyous, resonant wind of the brasses._ And he has used a somewhat complex sentence structure to accommodate all the pomp.
Now contrast this with Hemingway's description of soldiers marching under the spell of a different kind of happiness:
_Everybody was drunk. The whole battery was drunk going along the road in the dark_... _We went along the road all night in the dark and the adjutant kept riding up alongside my kitchen and saying, "You must put it out. It is dangerous. It will be observed." We were fifty kilometers from the front but the adjutant worried about the fire in my kitchen. It was funny going along that road. That was when I was a kitchen corporal._
First, you'll notice shorter sentences. In roughly the same amount of space, Fitzgerald uses two sentences and Hemingway uses eight. Admittedly, this is partially just a stylistic difference between the two writers, but it's also shaped by the differences in these two pieces. Hemingway's first-person narrator, Nick, fights in World War I and then travels around aimlessly after the war—shell-shocked and withdrawn from the company of other human beings, finding solace only in the natural world. No wonder his sentences are short and spare.
There also isn't a single adverb, and you can count the adjectives on one hand. Hemingway's narrator keeps all of his sentences simple—simple structure and simple words, from the adjectives— _drunk, whole, funny_ —to the verbs— _was, went, kept, put._
Whether you lean toward short or long sentences as a rule, you always want to make sure that you vary your sentence lengths once in a while. If all your sentences are exactly the same length, your reader will get bored pretty quickly, just as you would if you were talking to someone who said:
_I went to the store and bought some milk. I saw a man I knew in aisle 4. We spoke about the price of figs and fish._
Even if the next sentence is a real eye-opener, like _I wondered about his two-headed child,_ the reader may skim right over it because the unchanging length has lulled her into a kind of reading trance.
Which brings us to the question of rhythm. Working together, sentence length and syntax often create a rhythm, and you can manipulate this rhythm to great effect. Look at this passage from Hemingway's _The Old Man and the Sea:_
_The shark's head was out of the water and his back was coming out and the old man could hear the noise of skin and flesh ripping on the big fish when he rammed the harpoon down onto the shark's head at a spot where the line between his eyes intersected with the line that ran straight back from his nose. There were no such lines. There was only the heavy sharp blue head and the big eyes and the clicking, thrusting all-swallowing jaws._
Here Ernest went for an especially long sentence, and a bucket-load of adjectives. Both the length of that long sentence and the structure of it, piling one thing on top of another, give a rhythm that reflects the confusion and action of this struggle between man and fish. Also notice how well Hemingway mixes up his sentence lengths, following the long sentence with a short and then a medium sentence.
Let's look at a passage from James Baldwin's "Sonny's Blues." This story is narrated by a schoolteacher whose brother, Sonny, is a jazz musician who struggles with heroin addiction and goes to prison for a short time. At the end of the story, the narrator goes to watch Sonny play his first jazz gig after he's been released from prison, and for five beautiful pages, Baldwin ushers the reader into the jazz bar and Sonny's blues:
_Sonny's fingers filled the air with life, his life. But that life contained so many others. And Sonny went all the way back, he really began with the spare, flat statement of the opening phrase of the song. Then he began to make it his. It was very beautiful because it wasn't hurried and it was no longer a lament. I seemed to hear with what burning he had made it his, with what burning we had yet to make it ours, how we could cease lamenting. Freedom lurked around us and I understood, at last, that he could help us to be free if we would listen, that he would never be free until we did._
Baldwin plays with rhythm throughout this passage, using longer and shorter sentences with a lot of strategic word repetitions. Not only that, but the narrator enhances the sentence rhythms by manipulating the sound of the words.
See how alliteration affects this sentence, letting the reader practically hear Sonny's breath coming faster as he works the music:
_Sonny's Fingers Filled the air with life, his life._
And check out this sentence that comes later:
_I seemed to hear with what burning he had made it his_...
Notice how the words sound like a pair of brushes sweeping the snare drum. At first, the words begin with soft letters— _s, t, h, w_ —until we get to the harder 6 in _burning_ and the _ds_ in _had_ and _made._ Then comes the second half of the sentence:
... _with what burning we had yet to make it ours, how we could cease lamenting._
Here it seems the drum is being tapped. The _b_ and _d_ sounds repeat in the next phrase, except that the _b_ and _d_ come sooner, and then we get the drawn-out sound of _yet_ followed by the hard _k_ sound in _make_ and _could._
Okay, all right, maybe this is a bit much, but you can see how there's a rhythmic quality to Baldwin's sentences that matches the jazz club and, more importantly, the narrator's respect for Sonny's struggle and his music.
**PARAGRAPHS**
The length of your paragraphs also has a big influence on voice. As with sentences, you want to vary the length of your paragraphs to prevent a sense of stagnation or predictability. But beyond that, you can manipulate the feel of your voice by leaning toward long, winding paragraphs or short, snappy ones or somewhere in between.
Generally a new paragraph signals a shift in thought, either major or minor, or a jump in time or space. But there is a lot of room for interpretation on when you want to make these paragraph shifts. Some writers may cram a bunch of thought shifts into a single paragraph while other writers may separate each thought in a new paragraph. Similarly, you could move freely through time and space in a single paragraph or use a new one for each shift.
You can see what I mean in the first paragraph of Joyce Carol Oates's "The Fine Mist of Winter." Right off the bat, the author has made clear decisions about the paragraph (and sentence) lengths that set up the voice for the whole piece:
_Some time ago in Eden County the sheriffs best deputy, Rafe Murray, entered what he declared to the sheriff, and to his own wife and man-grown sons, and to every person he encountered for a month, white or black, to be his second period_ — _his new period, he would say queerly, sucking at his upper lip with a series of short, damp, deliberate noises. He was thirty-eight when he had the trouble with Bethl'em Aire, he would say, thirty-eight and with three man-grown sons behind him; but he had had his eyes opened only on that day; he was born on that day; he meant to keep it fresh in his mind. When the long winter finally ended and the roads were thick and shapeless with mud, shot with sunlight, the Negro Bethl'em and his memory had both disappeared from Eden County, and_ — _to everyone's relief, especially his wife's_ — _from Murray's mind too. But up until then, in those thick, gray, mist-choked days, he did keep what had happened fresh in his mind; memories of the fine driving snow that fell on that particular day, and of his great experience, seemed to recur again and again in his thoughts._
This paragraph containing several thought shifts and one time shift could easily have been broken into two or more paragraphs. Throughout the story, the twisting, rambling voice (shown often with long paragraphs) contrasts with the simple life that the people of Eden County enjoy (shown with dialogue and humdrum actions). Oates has also made a decision regarding time throughout the piece in the first paragraph by not splitting the previous winter and the current springtime into two separate paragraphs. You can sense that time's a bit fluid in this piece because even as Murray has forgotten the day that began _his second period,_ the narrator hasn't—and the narrator makes sure to circle back to it.
At the other extreme, you shouldn't be afraid of using short or even one-line paragraphs.
For example, the main characters in Arundhati Roy's _The God of Small Things_ are a twin sister and brother who are so close that one can recall the other's dreams. But as children, they become separated. Have a look at how the short paragraphs (and sentences) work soon after brother and sister are reunited as adults:
_But what was there to say?_
_From where he sat, at the end of the bed, Estha, without turning his head, could see her. Faintly outlined. The sharp line of her jaw. Her collarbones like wings that spread from the base of her throat to the ends of her shoulders. A bird held down by skin._
_She turned her head and looked at him. He sat very straight. Waiting for the inspection. He had finished the ironing._
_She was lovely to him. Her hair. Her cheeks. Her small, clever-looking hands._
_His sister._
You can feel the push and pull of the siblings' intimacy and awkwardness simply through the spaces between the paragraphs. Then the last paragraph—containing only two words—stands out dramatically, as if it were a tear on his cheek and yet not so melodramatic.
These short paragraphs have a dramatic feel to them. Contrast the punch each one carries with the descriptive winding of the earlier Oates example. In each piece, the general paragraph lengths are largely determining the quality of the voice's energy. Oates gives a circuitous discussion, while Roy is blunt to the point of being brutal.
Another use of paragraph breaks is to separate narration from dialogue.
This shifting between narration and dialogue also offers an interesting energy to a story's voice because the narration and dialogue often have different levels of language from each other. Either the language of the narration is more formal than the dialogue or vice versa.
Take Virginia Woolf's story "Kew Gardens." In the following passage, you can see how a simple, nearly laughable conversation between a young woman and man in love can be turned into an interchange of great (and almost lewd) importance:
_"Lucky it isn't Friday," he observed._
_"Why? D'you believe in luck?"_
_"They make you pay sixpence on Friday."_
_"What's sixpence anyway? Isn't it worth sixpence?"_
_"What's 'it'_ — _what do you mean by 'it'?"_
_"O, anything_ — _I mean_ — _you know what I mean."_
_Long pauses came between each of these remarks; they were uttered in toneless and monotonous voices. The couple stood still on the edge of the flower-bed, and together pressed the end of her parasol deep down into the soft earth. The action and the fact that his hand rested on the top of hers expressed their feelings in a strange way, as these short insignificant words also expressed something, words with short wings for their heavy body of meaning, inadequate to carry them far and thus alighting awkwardly upon the very common objects that surrounded them, and were to their inexperienced touch so massive; but who knows (so they thought as they pressed the parasol into the earth) what precipices aren't concealed in them, or what slopes of ice don't shine in the sun on the other side?_
Who knew you could get so much out of pushing an umbrella into the earth and a simple conversation about a park entrance fee? And the beauty of the story is that it's all like this—spare conversations are reinvented by the narrator as deep connections are made between different pairs of people. The dynamic tension between what's actually said and the narrator's elegant interpretation continuously gives off interesting sparks.
YOUR TURN:
Find an annoyingly dry and difficult piece of writing, preferably a legal document or a manual for some kind of appliance or equipment. Then rewrite the piece, turning the writing around 180 degrees, making it ecstatically poetic or down-home friendly or anything else you like. Use the third person. But employ drastically different words and sentences and paragraphs than found in the original document. You'll begin to see the profound effect of stylistic choices. And you will certainly provide a more entertaining document than the original.
**CONSISTENCY**
In addition to being determined from words, sentences, and paragraphs, voice is a result of every type of choice made in a work of fiction, sticking its dexterous fingers in every slice of the larger craft-element pie. Is there a lot of description or are there just a few telling details? Is the language filled with imagery and poetic devices or is it straightforward? Are the characters described from their hair to their shoes, or is a lot left to the reader's imagination? How many characters are there, a multitude or a few? Does the story have a plot with tight curves, or does the plot seem to be almost meandering? What's the balance of dialogue and narrative?
It's important that all of these elements coalesce into a unified voice. The key is consistency. As with POV, you make an unwritten agreement with the reader about how the general voice of the piece is going to sound. Readers like the sense that someone is telling them a story and they want the same storyteller to be there at all times, unless you are using a multiple-vision POV. If the narrator seems to change unwittingly from Uncle Remus to Ishmael in the middle of the tale, the reader will become confused, and, worse, the reader will stop believing any of it.
Unless... you purposely take a leap away from the original voice to achieve a certain effect. The narrator won't necessarily change to another narrator, but he may change his voice to suit the moment. A great example of this is found in J. D. Salinger's "A Perfect Day for Bananafish." The story begins with the following satirical voice:
_There were ninety-seven New York advertising men in the hotel, and, the way they were monopolizing the long-distance lines, the girl in 507 had to wait from noon till almost two-thirty to get her call through. She used the time, though. She read an article in a women's pocket-size magazine, called "Sex Is Fun_ — _or Hell."_
This woman, you discover, is the fiancee of the main character, Seymour Glass. Most of the story focuses on Seymour as he jokes around with a little girl that he runs into on the beach. The voice continues in the same witty vein as the beginning until the narrator starts to walk back to the hotel. At that point, Seymour changes, moving into a much less humorous frame of mind, and the third-person narrator makes the shift with him. Check out how different the voice has become once Seymour comes into the hotel room:
_He glanced at the girl lying asleep on one of the twin beds. Then he went over to one of the pieces of luggage, opened it, and from under a pile of shorts and undershirts he took out an Ortgies calibre 7.65 automatic. He released the magazine, looked at it, then reinserted it. He cocked the piece. Then he went over and sat down on the unoccupied twin bed, looked at the girl, aimed the pistol, and fired a bullet through his right temple._
The quick turn in the voice shocks the reader, and works with the action.
**FINDING YOUR VOICE**
Salinger, ceremonial, Parker, Hemingway, Murakami, chellovecks—all these voice choices are probably about to melt your brain. But don't get all discombobulated. The biggest key to voice is not worrying about it.
Bad style often comes when a writer is trying too hard to imitate the style of other writers. You can and should admire and study the works of other writers, but if you find yourself writing in the voice of Charles Dickens or John Cheever or Toni Morrison, you're in danger of sounding like one of those phonies that Holden Caulfìeld fears so much. Those folks aren't you and their narrators aren't your narrators. If you have to go through a Hemingway phase for a while, fine, do it, but don't stay there too long.
Your own natural voice will come from regular writing practice, whether it's in writing stories or writing in your journal or doing the exercises in this book. The more you write, the more your own voice will emerge because you'll grow more confident and you won't continuously pause to edit every word.
YOUR TURN:
Write a letter to someone you know well. Not a short e-mail note but a longish letter where you really talk about something. Don't worry about voice or style or anything else. Just write the letter. Then... when you're done, analyze the voice and style of the letter. Chances are it will be a good reflection of your natural voice, which may be similar to a storytelling voice you choose to use in your fiction. If you're so inclined, go ahead and send the letter. If you get a response, you can analyze _that_ person's voice.
You can also find the voice of a story by listening to the story's narrator. If you're using a first-person narrator, look at all the choices you have made about that character and get a sense for how this person would tell his or her story, whether the character's illiterate, like Huck, or pretentious, like Humbert. If you're working with a third-person narrator, figure out what this narrator should sound like by tailoring the voice to the characters, story, POV choice, setting, and how intimate your narrator is with both the reader and what's happening in the story. Of course, if you've chosen a multiple-vision POV, you may have to juggle more than one narrative voice.
But again, don't let the details hold you back. If you have a plot or character burning in your mind but you're agonizing about the voice for the piece, stop worrying and just write the story.
When you've finished a first draft, then you can think about the voice a bit more. Go back and check to see if the voice wavers in its general level of personality and formality. Pay attention to the way your words, sentences, and paragraphs are contributing to, or detracting from, the voice. If the voice sounds terribly unnatural or ill-suited to the story, try changing it to something that's more familiar to you or something in which your story fits more comfortably. Just as you should experiment with different POVs for a story, it's not a bad idea to do the same with voice.
At some point, you may get bolder about modifying the style to fit the story—you might, say, adopt longer sentences for a story about an obsessive person, or shorter, curt sentences for a story about an unemotional parent. You might pay closer attention to when a switch in style matches the specifics of a particular moment. For example, you might change the style of a piece to accentuate a moment of tension, as in the earlier passage from _The Old Man and the Sea._
On the other hand, you can allow voice to guide you at the outset. If you're starting a piece but no plot or subject comes to mind, start writing from the perspective of someone who has a very distinctive way of speaking and thinking, and see where that voice will take you.
Last of all, you should always test your voice with a real voice. After a draft or two, you should read the whole piece from start to finish aloud, letting your actual speaking voice merge with the voice of the writing. See if the sound of the piece fits with the voice you wanted. Perhaps you could persuade someone to read the story aloud to you so you can simply listen to the voice. Either way, you should mark down places where the voice feels particularly natural and where it feels strained. Soon enough, you'll be able to spot the voice hiccups right away and wash them away with a glass of water.
[CHAPTER 9
**THEME: SO WHAT'S YOUR STORY REALLY ABOUT?**](Facu_9781596917910_epub_c4_r1.html#aa9)
BY TERRY BAIN
Some years ago, I had a lot of stories that I thought were finished and on their way to publication. I thought I was ready to become a famous writer, making appearances before jealous young writing students worldwide, answering questions from the audience, and going out for cocktails afterward. But magazines weren't accepting my stories, and when people read them, they often said things about them that I didn't understand, such as "I don't get it."
_What is there to get?_ I thought. _It's a short story._
So I set out for a writer's conference, to get some advice from a real writer, from someone who knew what he was talking about, someone who would "get it" and praise me beyond compare and introduce me to his agent, who would also "get it," and who would send my manuscript to publishers who were just dying to jump on the bandwagon, whether they "got it" or not.
Instead I met the leader of my workshop, the fiction writer Mark Richard (a last-minute replacement author who I'd never heard of). He reviewed my story, and we talked about it, and in his final analysis he ended up asking me this: "What's your story about?"
I looked at Mark Richard, my head spinning a little, wondering if he'd said what I thought he'd just said.
I didn't know what to say. I reacted as you might imagine I would react in such a situation.
"I..." I said, "I guess I don't know."
"Find out," he said. Though I know he said more than just "find out," it was these words only that I remembered. Did he really mean to say that I had to know what my story was about before I could finish writing it? He seemed to be saying this as if it were the most obvious advice in the world. This was a world with which I was not yet familiar.
Of course, I think I was confusing plot with what my story was "about." He wasn't asking, _What happens,_ but rather, _What's the big picture? Why should I care?_ He wasn't just asking what the story was about plotwise but what was the story _really about?_ He wanted me to take a closer look at this story and arrive at some conclusion about it. He wanted me to distill my story and arrive at its thematic center.
I decided to reread my story. It wasn't about anything. It was all over the place. It was about everything. It didn't hang together. It simply moved forward willy-nilly like a movie spliced out of sequence.
When I got home from the conference, I set out searching for the theme in another story I'd begun well before the conference. I had nearly abandoned this story, thinking it pretty boring. To make a long story short (though I fear it's already too late), the revised version of my story was accepted and published in _The Gettysburg Review._ And several months later the story was republished in the _O. Henry Awards._ I don't believe it would have been published in the first place had I not revised it with theme in mind, and this is why I'm here to harp on the topic of theme.
**WHAT IS THEME?**
The theme is the container for your story. Theme will attempt to hold all the elements of your story in place. It is like a cup. A vessel. A goblet. The plot and characters and dialogue and setting and voice and everything else are all shaped by the vessel. In many cases the vessel will go unnoticed by readers, but it would be very difficult to drink a glass of wine without the glass itself. The glass itself is, of course, part of the experience, but it is not one we always pay much attention to.
Okay, okay, so I used a nice metaphor. But now you want to know: what the heck is a theme? First, the word _theme_ is confusing and may do you as much harm as good. You shouldn't think of theme as the ponderous sort of explanations given by critics and academics. That doesn't have much to do with writing a story. And you'll get into an equal amount of trouble if you think of theme as synonymous with _message_ or _moral._ That kind of thing is best left to pundits and philosophers.
The novelist John Gardner wisely said: "By theme here we mean not a message—a word no good writer likes applied to his work—but the general subject, as the theme of an evening of debates may be World Wide Inflation." You see, the theme may be simply _world wide inflation_ without there being any elegant solution for inflation or even a single point of view on the subject. The great Anton Chekhov also said something smart. He said that the fiction writer does not need to solve a problem so much as state the problem correctly.
So, you see, you're off the hook. You don't have to create themes that will solve the problems of the world. You just have to shine your flashlight on some aspect of life and let the reader see what's there. Not _every_ aspect. _Some_ aspect. And that's a key point because a theme should give a story some kind of focus, in a manner similar to how plot gives a story focus.
We're probably best off by just saying that theme is some kind of unifying idea in a story. Any kind of unifying idea will do, truth be told.
Ever read the children's picture book _Goodnight Moon_ by Margaret Wise Brown? The story is simple. A bunny is going to bed, and all the things in his room are introduced to the reader. The story, or perhaps the bunny, then proceeds to say good night to all the things in his room: socks, clocks, kittens, mittens, brush, and mush. But there is a point in the book at which the page is blank, and the caption reads, "Goodnight nobody." I am always surprised and delighted when I come to this page, and it has only recently occurred to me why this is.
When I read the caption "Goodnight nobody," I see the author's hand. I see the background to the story. I begin to look for a deeper meaning there. I think to myself, _What does Margaret Wise Brown want me to be looking for when I read the line "Goodnight nobody"?_ And what I understand, eventually, is that I am moved to create meaning from this very simple book. The meaning I take from it is this—that at the precise moment we arrive at the blank page, the bunny has fallen asleep. The room is filled with the quiet breathing of sleep. The theme of _Goodnight Moon?_ Simple, silent sleep.
A well-defined theme gives a story a kind of focus, a center. A well-defined theme allows a writer to distill the ideas, to present them in a simple fashion, to tell the story that will last longer than half an hour. _Goodnight Moon_ is a classic children's book not because it has fancy pictures or a high-concept plot, but because it's a story, with a deeper, more meaningful theme than can be found on the surface.
Have you ever read a story and said to yourself, _Well, that was nice, but what does it matter?_ Don't you want someone who reads your story to instead think, _Wow, I can't stop thinking about that story!_ Of course you do. And one of the ways to achieve that effect is by cultivating a theme and making an appropriate vessel from which your reader may drink.
By working with theme, you will take what may be an okay, nice, lovely, charming story, and help it become myth—turning it into a part of the consciousness of the reader, something that lasts longer than half an hour.
You may not think it's possible to crystallize the themes of great and profound works as neatly as I did for _Goodnight Moon,_ but I maintain that you can. At the risk of sounding like a fusty old professor, let me give you a sampling of themes from some great works of fiction, and all of these are, of course, open to interpretation.
_War and Peace_ by Leo Tolstoy— _the myriad ramifications of war and peace_
_The Great Gatsby_ by F. Scott Fitzgerald— _the corruption of the American dream_
"The Lady with the Dog" by Anton Chekhov— _the contrast between romantic love and the constraints of marriage_
_1984_ by George Orwell— _a police state like this could happen_
"A Good Man Is Hard to Find" by Flannery O'Connor— _the possibility of finding grace through facing evil_
_Lolita_ by Vladimir Nabokov— _the power of desire_
"Where Are You Going, Where Have You Been?" by Joyce Carol Oates— _the forming of identity_
And sometimes a theme _is,_ more or less, a message. _A Christmas Carol_ by Charles Dickens probably falls into this category. You remember it. There's a mean old greedy geezer named Ebenezer Scrooge and he really doesn't give a damn for anything in the world but his hordes of money. Then one Christmas Eve the Ghosts of Christmas Past, Present, and Future visit Scrooge and show him that he's a lonely soul who has squandered his life and may soon lie unloved and unremembered in a cold grave. Lo and behold, Scrooge sees the error of his ways and instantly transforms into a new man. Though the story certainly deals with greed, the real theme is something like: _Learn to correct the errors of your life before it's too late._ Yes, that's a message, but Dickens gets away with it (as he gets away with so many things) because he knows how to tell an absolutely wonderful story.
YOUR TURN:
Think of one of your favorite works of fiction, perhaps one of the stories you referred to in a previous exercise. Do your best to state the dominant theme of the story in a single word, phrase, or sentence. More than anything else, what is that story _really_ about? Some hints: look for recurring images; ponder the title; examine the climax. But please don't cheat by calling up your college literature professor.
KNOW THY THEME
Most great stories have themes and your story probably needs one too. And you should know what it is. Yeah, I know you're shaking your head, hoping I'm not saying what you think I'm saying. You think that you don't have to know what your story is about any more than you have to know what the moon is made of. You think writing is too mysterious and magical to ever figure out your story's theme.
But, no, I'm telling you that you can write your story better—craft a more appealing vessel—if you actually know your story's theme. You don't have to tell anyone, even if they ask you in interviews for fancy publications, but you need to know.
For another thing, should you neglect to supply your readers with a theme, they are likely to grab one that is handy, quite possibly the incorrect one, the one that does not contain the warning _"Caution: contents may be hot."_ Provide them with a theme so that they do not mistake your story about natural consequences for a story about the cuteness of puppies.
Can a story have more than one theme? Probably. But it is best for the short story writer to have a dominant theme in mind. The novel writer will probably have a greater opportunity to allow several themes to creep into the novel, just as the novelist may use subplots. When the contents are vast, it's possible that a more complex vessel is necessary. But even the novelist might be best off working with a single, dominant theme in which everything is contained.
In her collection of essays, _Mystery and Manners,_ Flannery O'Connor gives the reader a clue as to what she has found in her own story "A Good Man Is Hard to Find." She shows us that she does, indeed, know what her stories are about, and has set out to make her readers aware of exactly what it is that's going on in the story:
_A good story is literal in the same sense that a child's drawing is literal. When a child draws, he doesn't intend to distort but to set down exactly what he sees, and as his gaze is direct, he sees the lines that create motion. Now the lines of motion that interest the writer are usually invisible. They are the lines of spiritual motion. And in this story you should be on the lookout for such things as the action of grace in the Grandmother's soul, and not for the dead bodies._
O'Connor worked with theme. You can sense that she knows what she's writing about, that she has a complete handle on what is at the center of her stories, and she's able to keep a grip on it from beginning to end. As a result, her stories expand. They are able to slip off the side of the page somehow, to work their way into your life.
Interestingly, though, in the above passage, O'Connor makes the mistake of believing that it is important for her readers to know exactly what it was she was writing about, when the truth is far more interesting, and almost magical. A reader will perhaps get something entirely different out of reading a good story than what the writer intended. Maybe the readers will be paying too much attention to the dead bodies to notice the grace. Or they may see the grandmother's actions as something entirely different from _grace._ But, if the writer was confident enough in her thematic resonance, the theme will still be absorbed, in some way, by the reader.
So while you should be aware of your theme, you should also beware forcing it down the reader's throat (which O'Connor never did). Forcing readers into understanding what we want them to understand is what will get us into trouble. We'll end up writing didactic, forgettable stories. If we overadorn our theme vessel with bright colors and too many words and signposts, the reader is likely to care less about the wine inside the vessel.
Though you are looking for theme, and you will be revising with a theme in mind, don't spend any time making sure your readers are going to "get it." Don't overclarify your theme. If you're writing about the destruction of the rain forest, it's probably enough that you've included lush scenes from just such a forest, and described how the place has changed your main character. Your protagonist does not need to stand on a soapbox and promote the welfare of the forest. In fact, your protagonist can be the enemy. He could be a forester who devotes his life to destroying the damn forests. The point will eventually come across.
After all, little children don't ponder, discuss, and write dissertations on _Goodnight Moon._ But, on some subconscious level, they most certainly "get it." Theme does not have to instruct; it merely has to connect on some kind of deep level.
YOUR TURN:
Imagine a soldier has just returned from a war and is having a strange time readjusting to his previous life. You can pick the war, even using an imaginary one. Flesh out the character and the setting. Then write a brief passage where this character is going about some everyday activity, but having difficulty with it. Whatever you do, _do not think about the theme of this piece._ Just focus on the character and what he or she is trying to do. Once you have written the passage, write down three to seven possible themes for this piece. Pick the theme that seems the most interesting. Ponder what direction the story may take using this theme in a _subtle_ way.
**SLEUTHING OUT YOUR THEME**
One way to avoid overemphasizing your theme is by not beginning there. The writer who begins to write with a theme in mind almost invariably ends up with a didactic and forgettable tract. If you begin thinking, _I'm going to write about the politics of academia,_ then you will probably end up with something. You'll have a lot of words and sentences and commas and periods, but it most likely won't be good fiction, something that lasts longer than a few moments. Begin writing elsewhere.
Just start by telling a story.
Telling a story will take you into the heart of the story, and at the heart of the story there will be a theme that you can dig out and crystallize, and I'm willing to place a pretty big bet on that.
Take A _Christmas Carol._ Did Dickens begin by thinking something like, _I wish to write a story that will instruct my numerous readers that they should correct the grave errors of their ways before it is, indeed, too late?_ Or did he simply begin by telling a story? I like to think the latter answer is correct. He began by writing about Scrooge, and arrived at the rest based on the characters in the necessary actions that take place to tell the story that is told. You have to create the world from scratch no matter what you do. So that's what you do. You start with story, and later you go back and try to retrofit the story to the theme.
YOUR TURN:
Just to see how difficult it is, write a short piece starting with a theme. Here is your theme: _faith._ Spend some time contemplating characters, situations, settings, and so forth that may illustrate this theme in an interesting, noncliché manner. Once you have some ideas in place, start writing a story with _faith_ as the theme. You can write just a passage or you can write a whole story. Who knows? Starting with a theme may just give you a focus that allows you to write a wonderful piece, in which case you're free to write a story, every now and then, that _does_ begin with theme.
Once you've written the first draft of your story, this really isn't a bad time to start thinking about theme. If something occurs to you during the first draft, don't be afraid to jot it down. Also, don't be afraid of changing your mind later. If it turns out your first impressions were wrong, no problem. The worst that could happen is that you would have to revise your story, and you have to do that anyway.
The key thing in this process is to let the theme emerge naturally from the story you are telling, not impose the theme from above. This means that you'll have to look at what you've written and sleuth out the theme.
Yes, it's possible that as you're working on a story, a theme simply comes to you. _Oh,_ you might think, _this is really about the search for truth._ If that happens, great. Go ahead and let yourself be aware of this theme, and allow it to shape your story. But it's not always quite that simple. So let me give you some tips for finding the crucial clues that will lead you to the theme of your story.
A great technique is to start asking yourself questions. Do the characters' actions imply any universal truths? Does the superhero's triumph over the green-faced man represent a broader theme of good triumphing over evil? Does the postman's role in saving your character's life imply the presence of everyday angels? Does your protagonist's hunt for her keys represent a more universal search for the keys of meaning in life?
You can also see if there is a social context to your story. Does your protagonist's relative poverty tell you that your story is about poverty in general? Does the ruthlessness with which your forester destroys tell you that your story is about destruction?
Another way to search for theme is by doing a kind of reduction or condensation. Think, as you try to discover what your story is about, _How do I reduce my story thematically?_ Try to simplify your ideas into a few words: my story is about the inevitability of love; it's about abuse of all kinds; it's about addiction; the beauty of fruit; fear; longing; loathing. Death! It's about death!
It may seem like cheating to reduce your profound work of art to a single word or phrase, but Flannery O'Connor wasn't afraid to say her story was about "the action of grace," and I assure you Ms O'Connor was no cheat.
Here's a good one. What made you start writing this story in the first place? Is the theme of your story buried in your impetus to write it? Why did you decide to write a fictional tale of your grandparents' move from North Dakota to California? Is your story about alienation? Travel? Seniors trying to break into the movie business?
Or just start looking very closely at what is already there. Remember that story I told you about? The one I revised with theme in mind and eventually published? Here's what I did in my revision process:
The story was partially set in the game room of a house—a pool table, a strobe light, a bar—and concerned the head games of teenagers. Specifically, the more dominant of two friends was trying to force his friend to kiss the neighbor girl by kissing her himself. It was a kind of game for him, to manipulate his friend. So I figured my theme must be _the games people play with one another._ Then I zeroed in a bit closer and decided that my story was about _the games of adolescence,_ which was more specific than _the games people play with one another_ and sounded less like a Top 40 hit. I titled my story "Games." I added a new ending. I removed some of the details and scenes that weren't necessary, that weren't fitting into my thematic vessel. I added a few new details and scenes that seemed appropriate, always with an eye toward _the games of adolescence._ And, as I mentioned earlier, the story got published and won some recognition. Also let me point out that the editor who accepted the story had already rejected it once, liking it but not quite enough.
Perhaps certain things keep recurring in your story, almost functioning as symbols and metaphors, as did my games. Perhaps the story occurs in a restaurant, and the symbols are mostly food items. Your theme might be related to consuming, eating, satiating, or desire, among other things.
You can also watch for repeated words or images. Or words and passages that strike you as particularly poignant. When you were writing these things, you had something in mind. Chances are good the theme is buried in there. For instance, the word _ring_ can mean more than _ring._ It can also mean marriage, boxing, entrapment, even communication. Where you go with this word depends on what theme you choose for the story.
You can also look at individual sentences and glean something from them. The kernel of your story might be in there. For instance, if you look at your first sentence and it reads, "She walked between the Dumpsters, watching for rats," there are several thematic elements to the line that you can take note of. Someone walking between the Dumpsters might be homeless. Rats and Dumpsters imply a kind of filth. Watching for rats implies a kind of fear, or perhaps hunger. Some of the thematic elements in this sentence will depend on where the rest of the story goes, whether your protagonist is homeless or not, whether this is a familiar setting or not, whether there are actually any rats or not.
Write in the margins of your story what the possible themes or clues to themes might be. When a character seems to be playing with another character in the story, you might write _games_ in the margin. When the protagonist tries to beat the sales record for the quarter, you might write _competition_ in the margin. Hopefully some kind of commonality will begin to form. Or one of your notations will strike you as particularly relevant to the story. Circle words or phrases that seem especially poignant, that seem to point to a larger theme, or that seem to be the most essential pieces to the story. Don't do anything with these notations just yet. Just circle them. You can come back to your notations later, maybe making further notations, maybe just rereading it, slapping yourself on the head and saying, _Of course, it's about the importance of everyday heroes._
Let's use _Goodnight Moon_ as an example again. If you were writing the story to _Goodnight Moon,_ you might read your text one line at a time and make the following notes:
Text: _"In the great green room"_
Notation: _Life. Depth. Solitude. Peace. Quiet._
Text: _"There was a telephone"_
Notation: _Busyness. Life._
Text: _"And a red balloon"_
Notation: _Play. Fun._
Text: _"And a picture of—the cow jumping over the moon And there were three little bears sitting on chairs"_ Notation: _Childhood. Nursery rhyme._
Later in the story you might have notations as follows:
Text: _"Goodnight comb
And goodnight brush"_
Notation: _Night / sleep for everything._
Text: _"Goodnight nobody_
_Goodnight mush"_
Notation: _Simplicity. Sleep. The bunny has fallen asleep._
Hopefully, as you accumulate these details, a theme will appear. (In this case: sleep.) This theme may not be the exact theme that you will end up with, but it will certainly be better than the vague notion you had before. _And I thought the story was about rabbits!_
However you go about finding your theme, you will find it. Then what? Well, you might write your theme in big block letters on the first page of your story or on a sign posted over your workplace. Or, if you're still zeroing in, you might keep a list of possible themes on a piece of paper or on a document in your computer. Or you might have a fabulous memory for this sort of thing.
By whatever method, you'll want to be keeping your theme in mind as you return to work on your story.
You might ask, _What if I don't get the theme right?_ And I would answer, _You will._
Why? Because it's critical that you are confident in your skills. So you will get it right. You are the single most knowledgeable person regarding the theme of your story. The theme of your story is whatever you discover. You can shout down your friends and relations by telling them emphatically that your story is about _death_ because you absolutely get to decide. They can decide too, of course, and they can disagree with you, and that's part of the fiction game.
YOUR TURN:
Take one of the pieces you wrote for an exercise in a previous chapter. Sleuth out the theme that may be lurking behind the words. Does a character's situation say anything about human nature? Is there a particular phrase or sentence that resonates for you? There's no telling where the clues lie, but they are probably there. Once you've identified a possible theme, write it down. Then revise the piece, keeping your theme in mind. If you have to alter or even throw out most of the original piece, so be it. A theme demands attention.
**THEME TOUCHES EVERYTHING**
As you may recall, I said you probably won't be working much with theme on the first draft of your story. What I'm implying, of course, is that you won't be writing just one draft of your story but many. And on the second and third and fourth (and so on) drafts, yes, sir, that's when you'll be thinking about theme a great deal.
The choices you make about theme will strongly influence how you revise your story. For one thing, the theme will help you make sense of what is there. If the protagonist's hunt for her keys does indeed represent a more universal search for truth, then perhaps her visit to the palm reader makes more sense. And maybe, while she's at her mother's house, she could be looking _for something_ in the photo album rather than just at the pictures. What is she looking for? Perhaps her flipping channels on the car radio means more than just hating the music playing on the airwaves. She's looking for something. She can't find it. Not only does this give us an excellent way to revise our story, it allows us to expand or enliven the plot in a more natural and thematic way. Continue to cultivate these thematic threads. Choose them. Allow them to shape your story.
You see, knowing your theme actually helps you make key decisions about what to keep and what not to keep. If my story was about, say, _immortality,_ then anything in it that didn't somehow relate to immortality could be cut. And I could add things to my story without fear of becoming confused or confounded. So long as what I wrote had something to do with immortality, then I was safe.
I may end up looking at a sentence for an hour, thinking, _Is this sentence about immortality?_ In the end, maybe that particular sentence doesn't matter so much. Maybe it's just the sentence you need to get a character from the door to the kitchen. Maybe whatever's in the kitchen has to do with immortality. Or maybe she just wants a cucumber sandwich and you cannot deny her a cucumber sandwich because she's hungry.
But once you find a theme and begin working with it in the revision process, you should focus the story with that theme in mind, letting most, if not all, of what's there relate, in some way, to your theme. Thus, the story will attain depth, since the repetition of thematic elements will naturally pile up, one upon another, creating a kind of resonance within the story. Yes, thematic repetition is good. If you are able to provide enough elements relating to your theme, eventually the reader will begin to have a clearer picture of a story with a center, rather than a story that proceeds as life proceeds, without much structure or resonance.
At this point, you may be wondering two things: 1) Does most everything in a story _really_ relate to its theme? 2) How do I go about revising a story so most everything relates to the theme? Good questions, both. I think the best way to answer these questions is by looking at a particular story from the viewpoint of its theme. Since we've been discussing Raymond Carver's "Cathedral" in this book, let us use that one. While we're doing this, you may be able to see how Carver made decisions joining his theme to all the major craft elements of fiction as he took this story through the revision process.
For starters, we should ponder what the theme of "Cathedral" actually is. If I had to crystallize it, and I do, I'd probably say the theme of "Cathedral" is _True vision is much deeper than the physical ability to see._
Carver gets working on the theme right from the get-go. The story begins:
_This blind man, an old friend of my wife's, he was on his way to spend the night_
And a few lines later the narrator says:
_I wasn't enthusiastic about his visit. He was no one I knew._
Already we're getting the sense that the narrator, a man who can physically see, is emotionally blind because of his lack of interest in meeting a good friend of his wife's. The wife's friend is named Robert, but the narrator is so disinterested in this fellow that he refers to him simply as "the blind man." The narrator, in fact, isn't interested in much of anything. He stumbles through life blindly oblivious to just about everything. He's blind to Robert and his life, because everything he learns about Robert is limited by his intense focus on Robert's blindness. He's shocked to see a blind man with a beard, and without dark glasses, and smoking, and having a diverse work history. The narrator is also oblivious to his wife and her interests. He doesn't get into her poetry, or poetry in general, and doesn't want to look closely at her past or, for that matter, her present. Perhaps most importantly he seems to want to be oblivious to his own life—his job, his bad habits, and so forth. We sense his ideal evening is getting drunk, getting stoned, and watching TV. Perhaps this personal obliviousness is why he never even reveals his _own_ name.
The protagonist, the narrator, fits perfectly into the story's theme. But so do the other two characters. If the man who can physically see is emotionally "blind," then it helps to have the physically blind character someone who can "see" in the emotional sense. Such is the case with Robert. He grabs life with gusto, seemingly interested in anything that comes across his path. We sense he loved his recently deceased wife deeply and has cultivated many friendships, and even his vigorous drinking and smoking seem to be social tools rather than crutches. Hey, he even has two television sets and he prefers the one with color!
The narrator's wife, the only other character in the story, seems to be an absolutely lovely person, cherishing her friendship with Robert and doing all she can to treat him graciously. If the wife were a shrew, we might understand why the narrator takes her for granted, but her loveliness reinforces the narrator's emotional "blindness."
These important character traits are revealed clearly with just about every line of dialogue in the story. For example:
_I said, "Let me get you a drink. What's your pleasure? We have a little of everything. It's one of our pastimes."_
_"Bub, I'm a Scotch man myself," he said fast enough in this big voice._
_"Right," I said. Bub! "Sure you are. I knew it."_
_He let his fingers touch his suitcase, which was sitting alongside the sofa. He was taking his bearings. I didn't blame him for that._
_"I'll move that up to your room," my wife said._
It's all there—the narrator's disinterest, Robert's gregariousness, and the wife's sweetness. What the characters say supports the theme, though, of course, they don't know it.
Carver chose the first-person point of view for this story and, in relation to the theme, it seems an inspired choice. We're not just watching the narrator's lack of "vision," we're experiencing it with him as we live inside his mind. Interestingly, the narrator's blindness makes him a somewhat unreliable narrator. We don't trust his opinions, and we're right not to. He tells us close-minded things that we, the reader, know to be silly, like:
_This blind man, feature this, he was wearing a full beard! A beard on a blind man! Too much, I say._
Also notice how the voice of the narrator reinforces his lack of "vision," the voice being tinged with cynicism and ignorance. Throughout the story, the POV and voice are working hand in hand, perfectly, to convey the narrator's limited scope.
Perhaps you're wondering about the setting. The entire story takes place in the narrator's house. But we never get a real good sense of that house. As a rule, Carver is sparing with detail, but it seems especially appropriate in this case. Even in his own home, the narrator seems unable to "see" things. This lack of giving detail, or "seeing," extends really to most of the description in this story. Nothing—the house, the food, the liquor, the wife—is described with any relish or specificity. Notice the apathy in such a description as this:
_The news program ended. I got up and changed the channel. I sat back down on the sofa._
If we didn't know better we might be tempted to think that Raymond Carver wasn't a very good writer. Oh, but he is, though, because he is merging his POV, voice, and description choices to fully reveal the narrator's "blindness," and, thus, his story's theme. And, tellingly, the descriptions become more specific and vivid when the narrator finally does begin to "see" toward the story's end, as when he is watching a documentary about cathedrals on TV:
_The camera moved to a cathedral outside of Lisbon. The differences in the Portuguese cathedral compared with the French and Italian were not that great. But they were there._
Now this incredibly oblivious narrator is even starting to notice the difference between Portuguese and French and Italian cathedrals. That's quite a leap.
The theme is certainly apparent in every progression of the plot. Plot is usually a living illustration of theme—theme in motion, you might say. If you show the theme through the actions of the characters, then you never really need to state the theme, and, in this story, Carver never does.
As we saw in chapter 3, the major dramatic question of this story is whether our narrator will ever come to truly "see," which certainly works together with the theme of _True vision is much deeper than the physical ability to see._ All the events of the story, from beginning to end, push the narrator closer and closer to the moment when he finally does come to "see." The narrator tries with all his might to remain detached, and Robert never lets up on the narrator, being charming and inquisitive and caring until finally the narrator is quite simply unable to resist Robert's life-affirming "vision." At the story's climax—where the narrator draws a cathedral—the plot and theme come together in one blindingly bright moment of glory.
At the climax, you have these two guys watching television, and they see this cathedral. Well, no. The narrator, who has physical sight, sees the cathedral. Of course Robert, the blind man, has never seen a cathedral. So, what happens? The sighted man describes the cathedral to the blind man, at the blind man's request. Only the sighted man is emotionally "blind," so he doesn't know _how_ to describe it adequately. He confesses to Robert:
_"It just isn't in me to do it."_
He's been the way he is for a very long time. He doesn't know how to be any different. Will he ever "see"?Maybe, maybe not.
But the blind man, Robert, will insist on it because, of course, he does possess true "vision." He knows that the narrator can't "see." He knows this is important, to help this man, or maybe to help his friend who has married this emotionally "blind" man. So Robert encourages the narrator to get paper and a pencil, and the two of them get down on the floor, and the narrator attempts to draw the cathedral with the blind man's hand resting on top of his own hand, the hand doing the drawing. Robert claims this will help him get a sense of the cathedral, but maybe Robert really just wants to offer the narrator the courage it will take to accomplish this task. And the narrator does accomplish the task. He ultimately manages to draw a cathedral. When he's done, he closes his eyes. And finally, miraculously, he can "see."
And perhaps by now you are seeing how every major craft element of this short story converges on its theme. But there is one more thing worth mentioning—the symbol of the cathedral. Something becomes a symbol in a story when it takes on a meaning larger than what it really is. Symbols are nothing a writer should worry about too much; rather, they should emerge naturally, as probably happened with this story. But at some point, Carver must have become aware of cathedrals as something symbolic because he chose to call the story "Cathedral."
Why a cathedral? Think about it. Cathedrals are perhaps the most magnificent and awe-inspiring man-made structures on earth, and they were built to bring humans closer to God, to elevate the human soul as high as it could possibly travel. When the narrator learns to "see" a cathedral, he has elevated himself about as far as he can go. By story's end, not only does the narrator have the physical ability to see, but he has gone deeper to achieve _true vision._ In a big way.
YOUR TURN:
Return to the favorite work of fiction for which you figured out the theme. Get your hands on a copy of it. Focus on several pages of the text. Write down everything you see there that seems to relate to your chosen theme for the work. Anything is fair game—the characters, setting, voice, title, opening line... If not much seems to relate to your theme, what does that tell you? That the theme is illustrated with great subtlety? Or not illustrated well enough? Or do you have the wrong theme?
Can you do what Carver did with this story on your own works of fiction? Can you, as they say, try this at home? Of course you can. Simply discover the theme of your story, after a draft or two, then revise and revise. And with each revision look for ways to make all your choices fluidly and delicately flow inside the vessel of your theme. If you find this difficult, do as the narrator of "Cathedral" did—simply close your eyes and "see."
[CHAPTER 10
**REVISION: REAL WRITERS REVISE**](Facu_9781596917910_epub_c4_r1.html#aa10)
BY PETER SELGIN
I write fiction for the same reason people believe in God, to give meaning and order to life, or at least to give it some shape here and there. Like many people, I'm uncomfortable with chaos and disorder. The studio where I write contains a medley of tidily arranged shelves, bookcases, and surfaces, jars neatly bristling with writing implements, and notebooks arranged by size, category, and date, all within arm's reach. I'm pathologically tidy. But I'm far from alone. For damning evidence of man's fixation with order look no farther than heaven; what are the constellations, but tidy boxes in which we've shelved the stars? The Big Dipper is cosmic fiction.
When I teach fiction, I alarm my students right up front by announcing that I'm not going to teach them how to write, that, as a matter of fact, I _can't._ I can only teach them how to rewrite, to reconsider what they've written, and then revise it. The writing, the getting something down on paper, that's really all up to them. Once they've got something written down—when they're ready to make order from chaos—then I may just have something to offer.
FIRST DRAFTS
Before we decide how to revise, it helps to have something _to_ revise, namely, a first draft. "All that matters," said Hemingway of first drafts, "is that you finish it." Elsewhere I've heard it said that a first draft should be written with the heart, whereas subsequent drafts must bring to bear that more critical organ, the brain.
Papa Hemingway, not one to mince words, also called first drafts "excrement." That's harsh, but also liberating. It's okay if a first draft sucks; it should suck; it's supposed to suck. The only thing a first draft _needs_ to do is get finished. Get something down on paper. Be reckless, be shameless, be grossly irresponsible and self-indulgent, even, but get something down.
And remember: when writing first drafts, you should not be editing. A writer friend of mine who owns a collection of hats wears one—a red baseball cap with KEROUAC stitched in gold over the visor—while writing her first drafts, and another—Chinese, tutti-frutti, shaped like a funnel—when revising them. This, I'll admit, is pushing things, and I also think it odd that the funnel should be for the editor, but it illustrates a point: that though they share the goal of creating a work of literary art, editing and writing are different disciplines requiring different temperaments, different skills. While the woman in the red baseball cap may be driven by pure instinct and emotion, half-poet, half-ape, whosoever dons the tutti-frutti Chinese cap must be an emotionless diagnostician, probing each word, sentence, and paragraph with a screwdriver in one hand and a scalpel or a hatchet in the other.
However inspired, first drafts can always stand improvement. "No sentimentality about this job," wrote Daphne du Maurier of the editor's task. And while there may be nothing sentimental about revision, though by turns terrifying, cold-blooded, and brutal, editing can be enjoyable. With experience the fiction writer learns not only how to find and solve technical problems but that solving such problems in a manuscript can be as creative as writing that first draft. In fact, editing can be edifying, so beware: you may never want to take off that conical, tutti-frutti Chinese hat.
Still, you're reluctant. You've finished the first draft. It sits on your desktop, next to your computer, a stack of pages sprinkled with words—your words. Maybe you worked in a white heat, following Jack Kerouac's famous dictum "First thought, best thought," flinging sentences like Jackson Pollock flinging paint. Or else you worked cautiously, glacially, like William Styron, who, writing his extra-long novels, felt compelled to hone each sentence to perfection before proceeding to the next, his perfected prose accruing like plaque on the teeth. Either way, you think the work done. And maybe it is.
But most likely it's not. Few of us are William Styrons, who called his own working method "hell." Be honest with yourself and you'll admit that for every sentence hurled down onto your page like a lightning bolt from
Zeus, ten others must be dug up like root vegetables from the humble soil, from which they emerge covered with dirt and manure.
Some writers may feel that once their raw genius has been spilled onto the page it's up to someone else to clean it up; that's what editors are for. In the good old days of publishing this may have been slightly true. There was a time when a guy like Thomas Wolfe could plop a manuscript as bulky and disheveled as himself onto an editor's desk and hope to have it published—and would, providing the editor was Maxwell Perkins, the legendary Scribner's editor who whipped Wolfe and notable others into print.
Those days, sadly, are gone, as are men like Perkins. Today the typical editor is a harried creature, with more urgent things to do than edit your novel or story. Darken his or her desk with a manuscript in need of editing and it will be read "very quickly." So, you should revise. In fact, if there isn't one already, there ought to be a bumper sticker: real writers revise!
**PRELUDE TO REVISION**
Before revision can begin, however, before we slice open the body of our stories or novels and muck around in search of tumors, extraneous organs, and signs of internal bleeding, our words, together with our emotions, need to have grown cold, sober, well rested. Insomnia, intoxication, frenzied passion, and/or too much caffeine are not things wanted in a surgeon.
Hence, do not revise in the throes of creative ecstasy, or when angry, upset, exhausted, depressed, or filled with self-doubt, dread, or loathing. You've printed out your first draft? Good. Let it sit. Do something else for a while, work on another project, take two weeks and tour the Greek Islands—bring your watercolors and your scuba gear. Soak your weary soul in the wine-dark sea, while letting your manuscript grow clammy-cold. Distance, we're told, makes the heart grow fonder. It also makes editing easier. A paradox: the less we recognize our own words, the better equipped we are to judge them. Just as distance makes the heart grow fonder, familiarity breeds contempt, or, worse, a false sense of inevitability, turning our sentences into ruts in an oft-traveled road.
Still, you needn't go as far as Neil Armstrong went to gain distance on your words. Some writers write in the morning, and then, in the afternoon or evening, revise what they've written. Others wait until the next day, when they can be sure they're no longer in love or hate with what they wrote the day before. But revising in too much of a hurry has its perils. Here's Virginia Woolf in her diary _The Voyage Out_ on revising:
_When I read the thing over (one very gray evening) I thought it so flat and monotonous that I did not even "feel" the atmosphere: certainly there was no character in it. Next morning I proceeded to slash and rewrite, in the hope of animating it, and (as I suspect for I have not re-read it) destroying the one virtue it had_ — _a kind of continuity; for I wrote it originally in a dream-like state, which was at any rate, unbroken_... _I have kept all the pages I cut out, so the thing can be reconstructed precisely as it was._
Woolf makes a good case for saving all drafts. And given the luxury of more time she might not have, as it were, thrown out the Wedgwood with the dishwater.
But what if, after a day or two, you still can't see clearly what you've written, and you can't afford a trip to Greece, or you don't have or want the luxury of more time? Or you're on deadline, with an editor's hot breath wilting the short hairs behind your neck? How, then, to cool a manuscript quickly, and make your all-too-familiar words less familiar?
Try reading your words aloud to yourself, sharpened pencil in hand. Read loud and clear—hurling each word like a stone at an imaginary audience. Imagine that somewhere in that audience is your ideal reader. Maybe he or she is your favorite writer, the ghost of Jane Austen or Bill Faulkner. See her sitting there with her own sharpened pencil, or, in Faulkner's case, a flask of bourbon. As you read, imagine her facial reactions; see her twitching, scowling, smiling, or wincing at certain words. And listen to yourself. Words sound different to our ears than to our eyes. You'll hear not only faulty rhythms and errors in logic but pretentious language, clichés, digressions, and a host of other sins. You'll be surprised how much editing pencil gets on your pages this way. If reading aloud to Faulkner's ghost is too intimidating, select a more benign imaginary listener, your grandmother or the freshman comp teacher who gave you an A+ on that essay about truffle hunting in Normandy with your crazy French uncle.
Some people don't even like going to the movies alone, and may balk at reading alone to themselves, in which case they should find someone to read to. Not an editor, or even a fellow writer, just someone who likes to be read to (believe it or not, such people exist). They needn't comment. In fact, better if they don't. They aren't there to critique, but to react, to help you hear in your own words what you need to hear.
Or have someone read your words _to_ you. Having your own words thrown back to you in another's voice—with their inflections, stumbles, laughter, tears, wincing, and cringing—can greatly enhance the revision process. You can also read your own words into a tape recorder and then play it back. Me, I prefer a warm body.
YOUR TURN:
Return to something you have written, perhaps from one of the previous exercises. Read it aloud. As you read, make notes on what you think can be improved. If you find yourself bored as you read, odds are the readers will be bored too. Ask yourself why the piece is less than thrilling. And any words or sentences that make you (or your imaginary reader) wince or cringe should be treated as suspect. As a bonus round, revise the piece based on your notes.
Two other solutions remain. The first is so simple it's almost embarrassing, yet it works. Print out your chapter or story in an unusual, but legible, font. Your words will seem like strangers to you, and you can begin to edit them.
The ultimate solution is to get help, that is, if you're lucky enough to know a sympathetic reader who is also a skilled editor. By _sympathetic_ I mean sympathetic to your intentions as well as to your overall style. Their ranks being swollen by struggling writers, professional editors are by no means hard to come by; for a price you can have your pick of them. But beware, even the costliest and most experienced editor may be, if not plain wrong, a poor fit for your writing. Professional or not, an editor's opinion is still an opinion. And many a good writer has had the guts yanked out of his or her prose by some "expert" editor. Remember that last botched haircut? Editors are like barbers. If you find a good one, consider yourself blessed and chain yourself to her. Better still, give her your manuscript, and go to Greece.
That said, no editor's advice should be followed slavishly. It's your work. You've got to know when to listen to suggestions, and when to say no, thanks. Sometimes an editor's efforts will tighten and clean up your prose when you want it ragged and filthy. Also bear in mind that editors tend to err on the side of caution. As Tennessee Williams said to Gore Vidal after Vidal finished editing one of his short stories, "You have corrected all my faults, and they are all I have!"
**THE REVISION PROCESS**
So, you're willing and ready to revise. But what to revise, how to revise, how much to revise?
_"Revise till your fingers bleed."_
—Donald Newlove
_"Don't f—with it too much."_
—Lawrence Durrell
Two great writers, two seemingly irreconcilable pieces of advice. How to reconcile them? Whose advice to take?
The purpose of multiple drafts is to discover what we're writing, and then to refine it into its ultimate form. Think of a painter with a canvas. She paints all day, perhaps for days, then scrapes and starts over. She may go through this process a dozen times, more, before emerging with a masterpiece. Are all those scraped efforts wasted? Of course not; they're all part of the process.
Initially, revision is often a matter of reenvisioning. From our first drafts we may take only a good character, or a scene, maybe a description, an opening sentence, possibly a theme—the rest is a shroud of disposable words. And yet, if it has served up any one of those things, all that writing wasn't in vain.
Suggestion: having finished a first draft, start over again. Put in a fresh piece of paper or open a fresh document on your computer, and start typing, this time with a sure, or surer, sense of what it is that you're writing. Refer to your draft, if and when it contains something worth referring to. Otherwise, write from scratch. D. H. Lawrence did so three times with _Lady Chatterley's Lover,_ producing three novels on the same subject, never referring to the existent versions. Old words can block fresh insights.
YOUR TURN:
Return to something you have written, perhaps from one of the previous exercises. Reenvision the piece. Read it through several times, asking yourself what is most original or powerful about this piece. It may be a character, a theme, a stray idea, even a single line. Now be bold. Toss out everything but this one promising thing. Start over, writing the piece entirely from scratch.
But two drafts may be just the beginning. It's not unheard of for a writer to go through twenty drafts, or more, on a single story. I know. I've done it. And twenty drafts later some of those stories still molder in a file drawer, unpublished. Does that make me a fool? No, because after twenty-one drafts, they may be published. On the other hand, I've had stories published that took only two drafts.
The fact is, some stories are easier to write than others. But the hard ones are no less worth writing. Be prepared to see your work through many revisions. Raymond Carver, one of this country's best short-fiction writers, has confessed to revising his stories on average no fewer than a dozen times. He understood as well as anyone that real writers revise.
**THE BIG PICTURE**
If a first draft is the place to write from the heart, free of worry, subsequent drafts are the place to worry about _everything,_ and heed all the sage advice on craft doled out in this book. You may choose to spend whole drafts focusing only on a single craft element. Perhaps you go through each scene finessing only dialogue, then spend the next draft coalescing elements of theme. Whatever your approach, before addressing little things—like whether to use a dash or parentheses—you want to make sure the Big Things are in order.
Some of the Big Things to consider:
CHARACTER
When you get down to it, people are interested in people. That's why they read fiction.
Some questions to ask about the characters in your work: First, _Do I have all the characters I need to tell my tale?_ If so, _Can I afford to lose a few? Can my protagonist do with two buddies, or one, instead of three?_ When considering the number of characters with which to tell a story, as with so many things, less is more. It's also less work.
Once you've established that you have all the characters you need and no more, then ask yourself: _Are any of my main characters too flat? Do they fulfill their roles too neatly, too glibly?_ When we assign characters narrow, predictable roles in our fiction, we are essentially condemning them to be archetypes, if not stereotypes. As F. Scott Fitzgerald wrote, "Begin with an individual, and before you know it you find that you have created a type; begin with a type, and you find that you have created—nothing."
Finally, ask yourself: _Are my main characters sufficiently motivated?_ A character with no goals to struggle toward, who exists at the mercy of outside forces, we call a cipher. Voltaire's Candide is such a character; so is Mersault in Camus's novel _The Stranger._ But unless you're writing a satirical fable or an existential novel, your characters should want things.
PLOT
If I had to choose a formula for plot, I'd go with the English poet Philip Larkin, who described a story as consisting of three parts: a beginning, a muddle, and an end.
Beginnings are crucial. If the beginning of a story is weak, chances are no one will ever get to the "muddle," let alone the end. Writers are routinely advised by editors and other meddlers to grab their readers by the throat within a paragraph or a page. Sometimes this works. There's something irresistible about:
_Hale knew they meant to murder him before he had been in Brighton three hours._
—the opening of Graham Greene's _Brighton Rock._ But not all readers want to be grabbed by their throats. Some prefer to be gently seduced, in which case a sly wink or a wiggled finger may trump a grappling hook, as with _Moby-Dick's_ come-hither opening:
_Call me Ishmael._
The point is you don't have to be sensational to be amusing, entertaining, or interesting. Think of yourself as a guest who has just arrived at a party. You wish to make a strong impression. You can strangle the hostess; that should do it. Or tinkle your wineglass and tell a story in your own beguiling voice, a story filled with charm, eccentricity, and colorful details, that takes place in a provocative and/or magical setting. In other words, you can hook your reader without breaking, or even bruising, her neck.
The other good news about beginnings—one can often be obtained simply by amputating the first paragraphs, pages, or chapters of a draft, what editors refer to smugly as "throat clearing." Ask yourself, _What's the first interesting thing that happens in my story?_ Begin there.
Having finished your first draft, you can be fairly sure the muddle's there, right where it ought to be, in the middle. The middle is the meat of the sandwich. It consists of an event or group of events, leading to the biggest event of all, the _climax._ As I've said, motivate characters sufficiently, and select a limited number of telling moments from their lives, and, providing you've chosen and shaped each of those scenes to something near perfection, the middle more or less takes care of itself.
The question to ask is, _Have I judiciously selected the necessary events with which to tell my story?_ John Gardner speaks of the "rule of elegance and efficiency," meaning if you can tell a story in four scenes, don't tell it in five. When, in _The Great Gatsby,_ Fitzgerald paints us a wonderful scene of Gatsby heaving his multicolored shirts onto his bed before Daisy's sparkling eyes, he feels no subsequent need to escort us into Gatsby's garage and have him show off his Studz Bearcat.
With endings, though we may aim straight for a point on the horizon, it's better if we don't arrive there, exactly. It's also likely that we won't, since our characters, being motivated, are apt to find their own solutions to their goals and frustrations, and these in turn will have their own dramatic repercussions. Assuming all does not go as expected, the ending of a story should be unpredictable not only for the reader but for the writer.
That said, an ending that's surprising but also unlikely, if not impossible, is by no means satisfying. The thing to aim for, in novels and stories, is the ending that's both surprising and inevitable. Ideally, the reader's first response should be _My God!_ followed shortly thereafter by _But of course!_ —since a good ending is always the direct result of everything that has come before.
POINT OF VIEW
Decisions about point of view are often made for the writer, dictated, so to speak, by the nature of the material. And most of the time our writer's instincts won't steer us wrong. But telling a story a certain way instinctively doesn't make it the right, let alone the only, or the ultimate, way.
Having finished your first draft, ask yourself: _Have I chosen the best possible point of view? Should I stick to this one character's viewpoint, or alternate between characters?_ These are big questions, indeed. Still, to not ask them would be a mistake. True, what you've written may work just fine, in which case why change it? Then again, if a story or scene isn't working, the first culprit to round up and sit under the interrogator's lamp is point of view.
Each point-of-view option has its advantages and disadvantages. The third-person POV is the least problematic. A third-person narrative is more flexible, allowing for a wide range of diction and greater perspective. On the other hand, who would want _The Catcher in the Rye_ in third person? Or _Huckleberry Finn?_ Had Melville written, "His name was Ishmael," well, what a shame. A first-person narrator is all intimacy, all voice; we're getting the goods straight from the hero's mouth.
Then again, as a first-person narrator, Madame Bovary would be insufferable, if not impossible. Nor could Jay Gatsby by any means tell his own story.
And, of course, whatever POV choice you make, keep it consistent.
DESCRIPTION
"Go in fear of abstractions," said the poet Ezra Pound. And though he went nuts, Ezra was right about some things.
When writing description, you want your reader to hear, see, smell, taste, and feel what your characters hear, see, smell, taste, and feel; you want specific sensations that grip the senses, not the intellect. Though abstract words like _beautiful_ and _mysterious_ seem to convey qualities of universality and timelessness, they leave most readers snoring. To say _Sally had beautiful strawberry-blond hair_ is to say next to nothing. But _Sally's hair streamed like turnings of steaming copper and bronze from a spinning lathe, down both sides of her face_ —now, that says something.
With description, the particular always trumps the general, and concrete always trumps abstract. Here's Shakespeare writing up a storm:
_Blow, winds, and crack your cheeks! rage! Blow!_
_You cataracts and hurricanes spout_
_Till you have drench'd our steeples, drown'd the cocks!_
_You_ sulphurous _and_ thought-executing _fires_
_Vaunt couriers of_ oak-cleaving _thunderbolts,_
_Singe my_ white _head! And thou,_ all-shaking _thunder,_
_Strike flat the_ thick rotundity _o' the world!_
_Crack nature's moulds, all germens spill at once,_
_That make_ ingrateful _man!_
_Rumble thy bellyful! Spit fire! Spout rain!_
A vast improvement over _It was a dark and stormy night,_ don't you agree? I've highlighted the modifiers; there are a few. But what modifiers! _Thought-executing_ and _oak-cleaving_ are nouns and verbs pressed into service as adjectives, and so they give us the concrete jolt of solid, moving objects. Likewise _all-shaking. White,_ like all words standing for colors, is a concrete abstraction. As adjectives go, _sulphurous_ is also concrete; you can almost smell it. The one truly abstract word here is _ingrateful._ (Even the Bard nods, occasionally.)
Still, when choosing concrete details, it pays to be selective. D. H, Lawrence, talking about details, draws a distinction between what he calls "the quick and the dead," the quick being "lifeblood," and the dead being... well, dead. The first things you notice about a person or a place are most likely the "quick" things; the rest are likely dead.
DIALOGUE
A few words on dialogue. Concise: the fewer words to make a point, the better. Subtext: it's not what characters say, but what they mean, that counts. Illogical: people are illogical, especially when they speak, especially when they argue. Adversarial: and they _should_ argue. We learn much more when characters disagree, or have different philosophies. Dialogue should never be tape-recorder real; a few hours spent in the company of a courtroom transcript will drive that point home. But it should be speakable, another reason to read your words out loud.
Try not to force dialogue into your characters' mouths. If you know your characters well, and have motivated them successfully, they should know what to say and when to say it, placing you in the humble role of stenographer.
Pay attention to the ratio of scene to summary, of dialogue to description. (This matter relates to pacing as well as to dialogue, affecting your decisions on which events to compress and which to expand.) A skilled author layers scene with summary, weaving and blending the two, aware that the best narratives are like roller-coaster rides, with slow climbs of exposition leading to swift falls of dramatic conflict. But there's no one way to build a roller coaster. And while one author favors dialogue over summary (Elmore Leonard springs to mind), another, say, Jens Christian Grondahl, author of the novel _Silence in October_ (about a man whose wife has left him and who spends the novel's 280 pages reflecting on this and other matters) eschews dialogue entirely. When we call a work of fiction fast-paced, that's a quantitative, not a qualitative, judgment. Writing fiction isn't the Indy 500. Sometimes slow and steady wins the race, else we'd all have to agree that Thomas Mann, Virginia Woolf, and Malcolm Lowry, to name but a few, are snail-paced, and hence lousy, writers.
SETTING
Context is everything, and our fates are determined as much by landscape as by geography. Set _Madame Bovary_ in Beverly Hills in the 1990s, and you have no story. Our readers should be grounded in the time and place of our stories. This can be as easy as popping in a date here and there, or as subtle as a poster promoting the Works Progress Administration. A story set in Los Angeles hangs in a buttery layer of smog, while one set in New Orleans drips wrought iron and Spanish moss; a romance pitched against a "dark and stormy night" is bound to play out differently than one set on a bright, sunny day. Setting is character, after all, and imposes its own demands on plot.
Also look for ways to use setting metaphorically. The ubiquitous fog through which we view the London of Dickens's _Bleak House_ evokes perfectly the dreary murkiness of the British court system that is the book's subject. In Marilynne Robinson's _Housekeeping,_ the novel's setting is also its primal image: a lake that literally and figuratively drowns both the past and the present.
FLASHBACKS
In his early novel _The Centaur,_ John Updike spends three tightly packed pages taking his reader on a side trip to New York City that has little, if anything, to do with the scene at hand, which takes place in a car on its way to a school on a snowy morning in Brewer, Pennsylvania. Strictly speaking, this sort of thing is what writers call a digression, except it isn't. Updike gets away with it, so we call it a flashback. Essentially, a flashback is a digression that works. How does Updike get away with a three-page flashback? First of all, by writing like John Updike, which never hurts. Second, by knowing just how far a reader's attention can be diverted from a scene before she either forgets the scene entirely or, worse, bails out.
Beginning authors often lose sight of their own scenes, letting them drift into flashbacks like Arctic explorers into snowstorms, never to be seen or heard from again. A master like Updike always knows what scene he's writing within, and how much tension it can hold. He knows he's got three pages in which to reminisce and sightsee, then the train leaves without him, Updike is also smart enough not to break into full-blown dialogue, knowing this might confuse the reader into thinking he's abandoned one scene to enter a new one, and to the same end limits his flashback to a single paragraph, however long.
Thus a general (and, to be sure, breakable) rule for flashbacks: keep them very brief. If a flashback insists on turning into a full-blown scene, consider putting it elsewhere, or giving it its own section or chapter.
VOICE
With the first few paragraphs of a story or novel, you make a contract with your reader. You agree to tell a particular kind of story in a particular voice. Whatever you contract to do, as with POV, you contract to do it _consistently._ And though it may be the hobgoblin of little minds, half of what we do as editors is done in the name of consistency.
It can even be argued that what we call _style_ is little more than a writer's tics and mannerisms rendered consistent through editing to produce a narrator's voice. Do something weird once in a while and it's a mistake; do it consistently, and it's a style. A stylist, then, is a writer who pays particular attention to what I'll call the details: to language, punctuation, the use of figurative devices, sentence rhythms, and the overall music of words. I can't teach you to be poets, I tell my students, but I can teach you to be stylists. For the fiction writer, that's close enough.
But the very thought of a style throws many good writers, and their writing, into disarray. That's because many beginning writers worship an ideal of style that has nothing to do with its practice. Think of an actor wanting to look and sound like Marlon Brando, who assuredly had no such intention himself. Just as actors are born with certain equipment, each writer has particular strengths and gifts and must learn to work with, and not against, them.
The term _journalese_ was, I think, coined by Hemingway to describe prose that, like most newspaper stories, is made to be read once, if that. If a story or book gets read twice or more, it's not for the story or even for the plot, but for the language, for the unique pleasures offered by a specific arrangement of words. Hence, if you want your stuff read only once, skip this part.
With journalese the reader gets all of the necessary information in the proper order. But the sentences just sit there; there's concision, but little if any music; there are characters, but there's no voice. As for poetry, or music, or something approximating those things, there's none to speak of. You've told your story, you've done an adequate, journeymanlike job. Which is to say your prose is dull, if not dead.
Try pitching your voice higher. Remember _His name was Ishmael?_ The third person makes it weak. But suppose Melville had stuck to his first-person guns and written, _My name is Ishmael._ Or: _They call me Ishmael._ Or even: _You can call me Ishmael._ Compared to _Call me Ishmael,_ all three versions frankly suck. This doesn't mean that by casting all of your sentences in the imperative you too can write a masterpiece. It means that writing powerfully means taking risks, daring to have a strong character like Ishmael pound out his first spoken words like a sledgehammer pounding red-hot steel. Or, at the other extreme, having the guts to let a spineless character like John Barth's protagonist in _The End of the Road_ introduce himself like so: "In a sense, I am Jacob Horner..." One way or another, the author must take a stand with his material, must assume a position of authority, even the authority of weakness, and hold it, and not let go, ever.
THEME
As I've said, we read fiction to learn about people. And though we may start with some notion of a theme, we needn't know exactly what we're writing about until we've written it. As the historian Daniel J. Boorstin famously said, "I write to discover what I think." By writing we stumble upon our themes. They are the result, not the cause.
Still, when themes emerge, as writers we're responsible for recognizing and highlighting them. For instance, in _The Great Gatsby,_ as the theme of financial greed grew out of his material, Fitzgerald chose to color the light at the end of Daisy's dock green, and to mention it not once but several times, including the most conspicuous place, at the novel's conclusion.
Writers don't plant themes, they find and nurture them, make them resonate for the reader, dress them up and display them. And, if they're as good as Fitzgerald, they do so with a subtlety bordering on the invisible.
YOUR TURN:
Return to something you have written, perhaps from one of the previous exercises. Revise the piece, making some kind of major adjustment—changing the point of view, overhauling the dialogue, altering the setting... As you revise, force yourself to focus solely on this single craft element. If so desired, you may take another round of revision, focusing on another major craft element. With so many craft elements to juggle, often it's nice to focus on just one thing at a time.
SWEATING THE SMALL STUFF
Now, with the Big Things in place, comes the time for microrevision. You've heard the saying _"Don't sweat the small stuff"_ Now's the time to sweat it. But fear not: this part can be as much fun as that fevered first draft. This is where you get to sharpen your editing pencil and line-edit yourself into the next best thing to a poet: a literary stylist.
Once again, this is a good time to read your stuff aloud. Any little thing that trips you up as you read is worth marking and reconsidering.
It's also a good time to hand your work over to a trusted colleague for some feedback, with the understanding that you are the final judge of what stays and goes.
Some Little Things to consider:
GRAMMAR AND PUNCTUATION
Grammar is a convention, something that civilized people can agree upon, and, like all conventions, creative souls are free to depart from it, with good reason. In writing this sentence, I spell the words according to Webster's dictionary, pause with a comma after the word _sentence,_ capitalize the first word, and end with a period. but what if i chooz not to dew so what if i chooz to dispense with speling an punkchewayshun an yooz ownlee lowurkaze ledderz My guess is you'll be confused, if not flabbergasted.
Grammar is one of the few things, maybe the only thing, that keeps writers civilized. Use it. Not slavishly or mindlessly, but with due respect for the powerful minds that have brought it to bear over the ages. An indented paragraph is a lovely thing; why so many choose to dispense with indents is beyond me. Punctuation marks are dramatic personae: the ebullient exclamation mark, the impulsive dash, the coy ellipse, the intellectual semicolon. A simple comma, improperly placed, can make all the difference. _Pardon, impossible to be hanged,_ wrote the king's page, when what he meant to write was, _Pardon impossible, to be hanged._ In both cases he should have used a semicolon, but let's not quibble.
But this is no place for a grammar lesson; a good book on English usage can give you that. Also, if you haven't done so already, buy a copy of _The Elements of Style,_ by William Strunk and E. B. White. This modestly slim volume takes up no more space than T. S. Eliot's poem _The Waste Land,_ and it's as good. Whatever you need to know about the uses and abuses of English is in there, and more, including such disarming advice as "Be clear."
Not that you should sell your artistic soul to Messrs. Strunk and White, or to anyone else. But before breaking conventions, know them, at least. Only once mastered can they be broken with flair. Otherwise, people may just think you're dumb.
THE LESS-THAN-PERFECT IMPERFECT TENSE
When it comes to good prose, the imperfect tense—i.e., _he was talking; she was going; they were screaming_ —could not be more aptly named. The words _is, was,_ and _were_ are all variants of the verb _to be,_ which, among dead verbs, wears the heavyweight crown. While most verbs are chosen for their evocative powers, _to be_ paints no picture in the mind, conveys no action, makes not the slightest dent in the reader's psyche. It says practically nothing. To find a deader word, one must reach for an article or a conjunction, such as _the_ or _and_ or _but._
Which begs the question: why do writers use, let alone overuse, the imperfect tense? Why write, _Sam was wearing a pink rugby shirt,_ when you could say just as easily that he _wore_ one? Why _Susan was running,_ when if she _ran_ she'd get there faster?
True, in conversation people tend to use the imperfect tense. It sounds friendlier, softer. Which explains why, in the merry, merry month of May, I didn't _walk_ down the street one day; I _was walking._ For sure, the past imperfect has its place, and not just in corny old songs. But used too frequently, out of sheer lazy habit, like a carnivorous wasp it sucks the meat out of otherwise healthy writing. That space taken up by _was_ might have held a stronger, more active verb. _My cousin Gilberto was at the dinner table._ Okay. _My cousin Gilberto slumped at the dinner table._ Better.
MIND YOUR METAPHORS
A metaphor is a poetic device whereby one thing is described in terms of another. _Lester's mouth is an open sewer_ is, we hope, a metaphor. Add the word _like,_ and you get the watered-down version, _Lester's mouth is like an open sewer_ —a simile. My rule, if there is one, being this: if you can change a simile into a metaphor without confusing people, do so. Why say what something is _like,_ when you can say what it _is?_ Your reader isn't stupid. The reader knows you're being figurative; to be told so is an insult. And you must never insult, or underestimate, your reader.
About mixing metaphors: don't. If the art deco hotel in your novel starts off looking like an ocean liner, don't turn it into a wedding cake. If a metaphor starts out watery, keep it watery. If the stage floor under the spotlights looks like _a strip of sandy beach,_ the shifting, murmuring audience may be likened to _surf,_ but not _a field of Kansas corn._ Steinbeck wrote, "Words pick up flavors and odors like butter in a refrigerator." Metaphors are onions. Be careful, or they'll stink up everything in the icebox.
MIND YOUR MODIFIERS
A modifier is a word—adjective or adverb—that modifies another word. One hopes that in modifying, the modifier adds meaning that isn't already there. The trouble with most adjectives and adverbs is that they're dead wood. _Desperately lonely_ is such a case. The desperation of loneliness is implied by the word _lonely;_ it doesn't need help; it can manage fine on its own. Choose the right nouns and verbs, and you won't need adverbs and adjectives. Go through and strike out any adjective or adverb that is either not doing much work or can be replaced by a noun or verb that will work much harder.
Of course, adjectives and adverbs needn't be shunned entirely. The reason why modifiers have earned a bad reputation is because writers use them perfunctorily, and not as they ought to be used, to boldly send a noun or adjective somewhere it's never been before. When, in his novel _Catch-22,_ Joseph Heller, who loves his modifiers, describes General Dreedle's _ruddy, monolithic face,_ he adds something to that face that wasn't there. When he modifies a silence with _austere,_ the reader hears the silence differently. And when he endows obsequious military doctors and colonels with _efficient mouths_ and _inefficient eyes,_ the reader suspects she knows precisely what he means, even if she has no idea. "Go in fear of modifiers" doesn't mean don't use them; that's the coward's way out. It means use them boldly, bravely, but sparingly, as a chef uses spices.
KILL THOSE CLICHÉS
The novelist Martin Amis calls all writing a "campaign against cliché." "Not just clichés of the pen," he writes, "but clichés of the mind and clichés of the heart."
A cliché is a figure of speech that once had its moment in the sun. Once upon a time, the phrase _It's raining cats and dogs_ was poetry worthy of Shakespeare. Now it's just a poor little tired old cliché. Were you the first to coin that expression, you'd be rightly proud. But you're not, and neither am I, and should either of us commit that particular string of words to paper, except as dialogue in the mouth of a bland character, we should be ashamed. We're supposed to be writers; we're supposed to come up with our own strings of words to describe the rain.
And that's really all there is to cliché. When, reading over your draft, your eyes come upon a familiar grouping of words, odds are you've authored a cliché. It needn't be as obvious or extravagant as _It's raining cats and dogs. A heart of stone_ is a cliché; so are _baby-blue eyes;_ so is whatever gets handed to someone _on a silver platter._ Most clichés, in fact, are fairly prosaic: _desperately lonely_ qualifies; so does _wreaked havoc;_ so do _abject poverty_ (what other kind is there?) and _sweating profusely_ and _every name in the book._ Said too often, even "the heaventree of stars hung with humid nightblue fruit," Joyce's most gorgeous line in _Ulysses,_ risks turning into a cliché.
WATCH YOUR ATTRIBUTIONS
_Said_ —that most watery of words—is the perfect host to dialogue: smooth, discreet, all but invisible, like the butler in Kazuo Ishiguro's _Remains of the Day._ Therefore stop killing yourself to come up with new, improved ways of saying _said._ No need for she _chuckled, barked, sighed, groaned._ No need to have your characters _intone, utter,_ or, worse, _opine_ things, or _spit_ or _blurt_ them out. Nor is there good cause to have them _affirm_ something _with_ or without _conviction,_ when they could just as easily say _yes._ Or, better still, nod.
I don't mean to imply that _said_ is the only allowable attribution. In its 340 pages, no character in Nelson Algren's delectably odd-ball _The Man with the Golden Arm_ ever _says_ anything. Instead they: _agreed / wanted to know / pointed out / assured / replied / demanded / told / warned / called / mourned / decided / put in cunningly._ Algren is a master at avoiding _said._ On the other hand, Robert Stone, no less an author, never uses any other attribution. Both are brave, honorable men.
EXCOMMUNICATE THOSE LATINISMS
By my definition, a Latinism is an unnecessarily bulky word, typically derived from the Latin, when a simple, plain one would do. Hence, don't have people _converse_ when they can _talk._ If Hank goes to the package store, he can _buy_ a bottle of rock and rye; he needn't _purchase_ it. As for words like _variegated, ascertain, beneficial, extrapolate, resumption, extemporaneous,_ and _preliminary_ (to give just a few _exemplary_ examples), they belong in jargony annual reports, not in good fiction.
Why is so much academic writing bad? Because it's pretentious; because it imitates clear, concise writing while being neither clear nor concise. People say lawyers write badly. But legal writing, done well, can be gorgeous (see Judge Woolsey's opinion on _Ulysses)._ Bad legal writing isn't bad because it's legal, but because it's _bad._ To paraphrase Tolstoy, all bad writing is bad in pretty much the same handful of ways, pretentiousness being the worst.
The easiest way to be pretentious is to use pretentious words, words like _ascertain_ and _perpetrate. At this point in time we have ascertained that the perpetrator was apprehended_... At this point in time I want you to forget forever the phrase _at this point in time._ Likewise forget _the fact that_ and _the question whether._ Be on the lookout for words ending in _tion._ Ditto _ism, acy, ance, ness,_ and _ment._ Such words are for politicians, not poets, and maybe for a few pretentious narrators like Nabokov's Humbert, who'd be lost without his lexicon.
When in doubt, cross out or replace the overripe words. Simplify. Your readers will ???strike???extend gratitude to???strike??? thank you.
YOUR TURN:
Return to the piece from the previous exercise, upon which you tampered with something major. Even if you're sick of it by now, stick with it. Revise the piece doing the following: 1) check the grammar; 2) weed out the _be_ verbs, modifiers, clichés, and pretentious words, reserving the right to keep any of them you find absolutely necessary; 3) unmix any mixed metaphors; 4) adjust any attributions that call attention to themselves. You may look for all of these things at once or do them one at a time. When done, congratulate yourself for graduating to the role of an editor.
**CUTTING AND TWEAKING**
Readers are rude. They'll put your story or novel down in the middle of that sublime passage you spent ten hours on and never pick it up again, without apology. The reader holds all the cards; he has no obligation to the writer, while the writer has every obligation to him. That's why writers cut and tweak, mercilessly, throughout the revision process, down to its final stages.
There comes a time when you must cast a stern, judgmental eye on each and every one of your sentences, like a hanging judge whose noose is a sharpened pencil. No mercy here. As Don Newlove, the man with the bleeding fingers, says, "It's best to cut, not just scrape." And so your lead scalpel hovers over every line, every word.
"Omit needless words," say Strunk and White. I couldn't have said it better. For sure I couldn't have said it more concisely.
So much cutting may seem masochistic, but the fact is a piece of writing that can work well in five thousand words shouldn't run to ten thousand. And you'll be surprised what you can cut. So much of what we state is implied; so much that we've spelled out can be deduced or imagined. Remember, the reader wants to participate in the story. Do all their imagining for them, and they feel left out. Furthermore, the reader's imagination is a better writer than you or I will ever be, so why not let it do some of the work? And what we cut none but ourselves will ever miss. Unlike oil paints, words cost nothing; use as many as you like, scrape them all away, use some more—no charge. There's no excuse, in other words, for saving your words.
By tweaking I mean crafting sentences and paragraphs, reorchestrating them, shifting and changing the words around until they're as clear and pungent and crisp as possible. Like nipping and tucking, cutting and tweaking go hand in hand.
Tweaking may be less painful than cutting, but it's trickier. It calls for experience. For every paragraph I've improved through tweaking, I've mutilated dozens. Literary surgeons, we practice on our own bodies, without anesthesia, and learn from our mistakes. But we learn.
Let me share with you, if I may, the evolution of a troublesome paragraph from my own novel, _Life Goes to the Movies._ The scene: a restaurant floating on the East River. In the book's earliest draft the scene is merely sketched, with no attempt to evoke mood or atmosphere. It's hardly written:
_The wedding took place in August of 1985 on a barge on the East River, Brooklyn side. As if to celebrate the occasion the stars were out. A twelve-piece jazz orchestra played._
Journalese. Now strap yourselves into a time machine and skip ahead several drafts:
_The wedding reception took place on a barge on the East River, with the Brooklyn Bridge humming its harpsong in the warm damp air high above us. Across the water, Manhattan's rhinestone tiara glittered. Carved ice statues cradled sterling caviar buckets, while a twelve-piece swing orchestra in vanilla jackets and gold derbies bounced brass noodles and spun ribbons of silver into the breezy dark night._
My God, look at all those modifiers! Here the author reaches for a Fitzgeraldian lushness, and falls on his face. The _rhinestone tiara_ is a cliché unfit for a pulp novelist. And then comes the adjectival parade— _carved, sterling, vanilla, gold, brass, silver, breezy, dark_ —that leaves this reader lurching for a private barge from which to throw up. The _brass noodles_ and _silver ribbons_ were a valiant but misguided attempt at synesthesia, to turn notes for the ear into images for the eye. But the metaphor strays too far from its subject.
Glide your time machine forward eight months, through two more drafts, and read:
_Montage. Night. A canvas-tented barge docked on the Brooklyn side of the East River. Summer drizzle softens the mucky air as the fabled bridge rasps with car traffic overhead. Ice mermaids cradle silver buckets of caviar, oysters on cracked ice squirm in ragged shells; shrimp cling for dear life above flaming seas of cocktail sauce. A swing orchestra in vanilla jackets and paper derbies weaves and thumps rhythms into the drizzling dark. Across the river, meanwhile, the Manhattan skyline wastes as much electricity as possible._
Better, but still too many modifiers. Here cutting will become part of the improvement process, as it usually does. How 'bout this:
_Montage. Night. A tented barge docked on the Brooklyn side of the East River. Summer drizzle softens the air as the bridge rasps with traffic overhead. Ice mermaids cradle buckets of caviar, oysters on cracked ice squirm in shells; shrimp cling for dear life above flaming seas of cocktail sauce. A swing orchestra weaves and thumps rhythms into the dark. Across the river, the Manhattan skyline wastes as much electricity as possible._
Now the _brass noodles_ and _silver ribbons_ have been swept off the dance floor, replaced instead by _clinging shrimp_ and _thumping rhythms._ In keeping with the novel's theme of life blending with movies, the screenplay language has been added. Here the emphasis is on verbs: _tented, softens, rasps, cradle, cracked, squirm, cling, flaming_ (used adjectivally here, but keeping its verbal punch), _weaves, thumps. Wastes as much electricity as possible_ strikes me now as passive, weak, but I could think of no active way of expressing that thought. Maybe you can.
For better or worse, that's how the passage stands in the novel. You may disagree with my choices. But I think you will agree that overall the passage has been improved.
YOUR TURN:
Return to something you have written and, yes, you may use a previous exercise. Cut it by a third. It may seem impossible, but it probably isn't. Be ruthless. Then take a break from the piece—a half hour or several days. See if the piece isn't actually better, improved simply by the act of reduction. If you're so inclined, get back in there and start tweaking, elevating what remains to a higher level of quality.
How do we know when we're done? How do we know when our fiction has been improved to the point where it can be improved no more? Some writers claim they never really finish their stories or books; they abandon them. When the law of diminishing returns sets in, that's as good a time as any to declare victory, or throw in the towel. For some writers, no story is finished until it's between the covers of a published book. And even then they can't stop tweaking.
And then there are those who say they simply know when the work is done; when all the planets seem to have aligned themselves, when form and meaning are so of a piece they seem indistinguishable, and every word feels inevitable, if not carved in stone. For me, it's like raising children; at a certain age, ready or not, out they go. They must complete themselves out in the cruel, cold world. Perhaps some stories will never please certain people; perhaps they will never be universally loved and admired. But like their author they will have done their best.
[CHAPTER 11
**THE BUSINESS OF WRITING: DRIVING YOURSELF NUTS FOR FUN AND PROFIT**](Facu_9781596917910_epub_c4_r1.html#aa11)
BY CORENE LEMAITRE
My first memory of meeting with an editor is of being soaking wet. I was sitting in the office of the director for fiction at HarperCollins UK, in damp shoes, clutching a cup of coffee and wondering what to say.
My newly acquired agent was there. She, of course, knew exactly what to say. "Eleven people for dinner tonight," she confided to the director, with the savoir faire of the literary insider. _I don't even know eleven people,_ I thought.
But there I was. I'd gotten lost en route, in the rain, and I'd lacked the foresight to bring an umbrella. Pitifully early, I'd stopped at a cafe to tidy myself up in their makeshift restroom while suspicious staff reeking of bacon hovered outside.
"This isn't really happening," I told myself. "There's no meeting. Hell, there's no building."
For the large yet discreet edifice that is HarperCollins UK had eluded me to the point where I'd ceased to believe it existed—or that, as soon as I stepped up to it, it would disappear into the mist.
But it didn't. I approached and babbled into the intercom. The gates opened. I walked by a chauffeured Jaguar that I later learned belonged to the CEO and then past the security guards—more garbled explanations—all the while feeling as though I were committing a terrible crime.
The lobby was cold. I felt very small. But I walked out of there with an offer and, yes, it changed my life.
The writing life. Why do so many crave it? Few professions hold the degree of uncertainty that writers endure. The emotional and financial ups and downs are extreme. Nevertheless, I've yet to go to a party where nine out of ten people—successful, rational people—don't want to be writers.
If you're reading this, then I assume you crave it too.
Part of me wants to say, "Go for it." The other part is shouting, "Stop! Don't do it. You'll drive yourself nuts. You'll endure crippling poverty. And however much you assert your normalcy, people will regard you as either an object of envy or a freak."
But if you're sure...
**THE HOLY GRAIL OF PUBLICATION**
"Okay," you say, "I've written some stories and I've got a novel in the works. I'm ready to launch my literary career. So, tell me how to get published."
Enthusiasm. I like it. The desire to get published can be powerfully motivating. But though the benefits are myriad, publication should not be your only goal. In an industry as competitive as writing, it's essential that you commit to the journey, the grail-quest of becoming the best writer you can be. Indeed, there are those who write solely for the love of it.
"Is it possible to make a lot of money as a writer?" is one of the questions I'm asked most often. My reply, consisting of various statistics and an explanation of royalty structures, invariably disappoints. What they want to hear is a simple _yes._
As writing is hard work, your desire to profit from your efforts is understandable. But if you want to make serious money, fiction is probably not the best way to go about it. This is a feast-or-famine industry, where famine is much more common, and even well-established authors gnaw on the carrion bone. So, if you equate writing with winning the lottery, you are bound to be disappointed.
Without the incentive of wealth, you need to be very clear about your reasons for pursuing publication—and potentially, there are many. Getting published opens doors. You'll meet lots of great people and get invited to nice parties. The prestige alone will improve your life. Opportunities to travel, give talks, and be thoroughly nosy will come your way. Writing credits will provide you with personal and professional validation. Your confidence will grow. And it will really piss off your enemies.
But you'll pay the price. Getting published takes time. Editors and agents rarely respond quickly, and when they do, chances are that they'll reject your work. Identifying markets and preparing manuscripts will consume your leisure hours and take their toll on your personal life.
"But I don't want a life," you say. "I want to write and write and then send my material out until my knuckles swell and my brain turns to petroleum jelly."
Good. I can work with that. So let's proceed.
**THE PRODUCT**
"Typescripts arrive daily in the last stages of filth and decay. Breadcrumbs, bootlaces, long hairs, tobacco ash, and all sorts of other refuse appear between the pages... The first impression, therefore, is bad."
—Frank Swinnerton, _Authors and the Book Trade_
So, what exactly does it take to get published? Think of the process as a two-part equation. First, you create the product—in other words, the novel, novella, or short story. Then, you sell it. Very simple. Write, then sell. Neither is easy, and in order to get published, you will need to be good at both.
Note the business terminology. Product. Sell. Inappropriate? Not at all. As a writer, this is what you're going to be doing—running a small business.
Let's take a look at the first part of the equation, writing. You've just read ten chapters on craft (unless you've been naughty and skipped ahead), so you should have some understanding of how to create the product. Yes, that's what a novel or short story is. _Pride and Prejudice_ is a product. "Cathedral" is a product. What makes them products? The price tag.
And as with any business proposition, you need to make that product appealing. So what does it take to make agents and publishers want what you've got?
First, you will need a completed piece of writing. Not an idea, not a few pages, but a fully finished work. This is essential, though many aspiring writers would like to believe otherwise. I was once treated to a "pitch" by an acquaintance convinced he had a great idea for a novel. He detailed every twist and turn. When finished, he turned to me.
"So, what do I do now?" he asked.
"Well, you have to write the book."
He stared at me, aghast. "The whole thing?"
Yes, the whole thing. Beginning, middle, and end. From the first agonizing word to the final draft that you cast, clammy-fingered, into the mailbox. No one is going to do it for you.
YOUR TURN:
It's time to finish something. If you have an unfinished short story lying around, finish it. If you don't have a short story under way, start and finish one. You should complete a short story even if you're mostly a novel person because you'll be able to finish it in a relatively short amount of time. If you're not sure what to work on, simply take your work from one of the exercises in this book and use that as a springboard for your project. If you want to be a fiction writer, the most important thing is to start a project and the second most important thing is to finish the first draft. Of course, if you're really serious, you'll need to take your project through numerous drafts. So do that too.
"But what about those authors who get huge advances on the basis of partial manuscripts?" you ask.
Well, those writers probably have previous best-sellers under their belts, with a sales record so good that their very name functions as a brand (there's that business terminology again). Best-selling authors achieve their status by writing books—genre or literary fiction—that for any number of reasons, including quality, sell well.
Which brings us to the next point, quality control. Editors and agents are busy people with little time to commit to development, so don't expect them to discern the hidden merit in your work. You must ensure that your product is outstandingly good.
"So, how do I do that?" you ask.
Well, it's tough, especially if you're new to the game and don't know exactly what editors and agents want. Given this difficulty, you might want to consider consulting a manuscript analyst.
Manuscript analysts, also known as freelance editors, professional readers, and book doctors, are expert in figuring out what's wrong with your writing and how to put it right. If requested, they will also evaluate the novel or short story's market potential. The advantage of working with an analyst prior to submitting to an agent or editor is that you benefit from industry-level criticism before approaching the industry. By identifying weaknesses and showing you how to fix them, an analyst may greatly increase your chances of acceptance.
There are many analysts out there—the good, the bad, and the ugly. Reputable analysts work freelance or through writing schools, and often advertise their services over the Internet or through writing magazines. Check their credentials. Published authors and individuals with a background in the industry are generally a safe bet. Don't hesitate to ask for references and sample reports.
So, let's assume that you've ensured that your work of fiction is first-rate. What next? Let's shift for a moment from content to format—in other words, how your manuscript should look.
If you want to make a good impression on an agent or editor, you must follow certain formatting guidelines. Your adherence to these rules of presentation will set you apart as a professional. Indeed, if you fail to follow standard format, you'll be dismissed as amateur and your work will not be seriously considered.
The standard format for all types of fiction is this:
* Use black type on white 8 1/2-by-11-inch paper.
* Choose a 12-point font, preferably Times New Roman, Courier, Courier New, or Arial.
* Double-space.
* Indent paragraphs.
* Place at least a one-inch margin all around.
* Number the pages in the upper right-hand corner.
* Spell-check and proofread—and I do mean old-fashioned proofreading, which is done by a human, not a computer.
* Include a title page with the title, word count, your name, address, phone number, and e-mail address. (Also include this information on the first page of the manuscript.)
* Keep your pages loose-leaf.
Some absolute no-no's:
* Fancy graphics or typesetting.
* Stapling or binding pages.
* Anything gimmicky, unless it is so apt and witty that it won't be viewed as an attention-getting device.
* Your head shot, photos of your kids or pets, pencil drawings... you'd be surprised at the things people will send to catch an agent's eye.
Having correctly formatted your manuscript, all that remains is to place it in an envelope (along with a cover letter). Although electronic submissions are becoming more popular, your initial approach will probably be by postal mail.
One more important point. Whenever you send anything to an agent or editor, _always include a stamped self-addressed envelope._ Use a large one if you'd like the manuscript back, a small one if you'd prefer a reply only, in which case specify in your accompanying letter that the manuscript need not be returned. No SASE, no response.
"But wait, how do I protect my work?" you say. "I've worked hard on this story. How do I keep some talentless schmuck from stealing my stuff?"
Not to worry. Copyright is automatic. In other words, you acquire federal copyright protection by the very act of putting words to paper. There are, however, advantages to registering your manuscript with the United States Copyright Office (www.loc.gov/copyright). While you cannot copyright an idea—for instance, the premise of a novel—it is possible to protect the execution. And for writers, this registration will allow you to file a copyright infringement action if someone plagiarizes your work.
That said, don't get too paranoid. Publishers are unlikely to steal a novel or story they can buy at minimal cost. There is no need to refer to copyright in your query letter or on the manuscript itself—indeed, as with incorrect formatting, this screams _amateur._ Send out your work with confidence. Remember, no one can steal your individual voice.
Corene Lemaitre 90,000 words
111 Little Street
Smalltown, PA 19876
Tel (987) 654-3210
writer@publishme.com
_April Rising_
by Corene Lemaitre
Chapter One
It doesn't take me long to figure out that something is wrong. To begin with, my key doesn't turn in the lock. I try all the old tricks-rattling the doorknob, lifting the door half an inch off the ground-but same verdict. No entry. Access denied.
I'm being observed. Our next-door neighbor is giving me the evil eye. She doesn't look as though she recognizes me. I wonder how long it will be, before she calls the police.
The lock has been changed. It's a good one, state of the art, the kind that could probably perform a citizen's arrest-more than could be said of the old lock, which had been more of a formality. Anyone could have broken in. No one ever had, of course. There is no crime in Philmont, or pain, or unhappiness, or anything resembling the usual range of human emotions. The streets are safe, divorce is rare and everyone has health insurance. The only thing likely to kill you is boredom.
Dragonlady is definitely on to me. She's exacting, believes in "love thy neighbor" provided it's on her own terms. Keeps her lawn trimmed down to the
_Sample manuscript page from_ April Rising _by Corene Lemaitre_
**THE QUERY LETTER**
"My manuscript is shipshape," you say, "so can I please, please put the damned thing in the mail?"
Be patient. We're almost there. You need one more item, a query letter.
A query letter is a cover letter, specifically one designed to pique an agent's or editor's interest in your work. With short fiction, this letter will be simple and will accompany the story. The query for a novel, which will probably accompany a portion of the work, is a bit more involved. Part of the query letter's function is to "hook" your target, so you must craft it in such a way that it impels the professional in question to request the entire manuscript.
Typically a query letter works something like this:
* An opening line or two introducing your short story or novel and explaining why you've contacted this particular agent or editor. Perhaps you've been referred to him by one of his authors, or maybe he handles work that resembles your own.
* A brief pitch of your story. Showcase character and situation. Pretend you're writing book-jacket copy. Intrigue, don't explain. For short stories, a line or two is sufficient, and for novels, one well-crafted paragraph.
* A bit about you. Don't sell yourself too hard; just try to convey a sense of who you are. Include anything relevant—writing credits, personal or professional expertise related to the subject matter, promotional experience... If you have none of these things, don't worry. Many don't.
* The closing. State that you've enclosed a self-addressed stamped envelope and that you look forward to the recipient's response.
The salutation must include the agent's or editor's name. _Dear Sir or Madam_ won't cut it. And while it's a good idea to craft your letter along the guidelines above, feel free to make it your own. Be professional and concise but allow your style and personality to shine through. Remember, this is as much a demonstration of your writing ability as the short story or novel itself. Ensure that the end result reads well.
Mr. Frank Miller
_Four Winds Quarterly_
321 Main Street
Yellow Plains, WI 19123
Dear Mr. Miller,
Please consider my short story "Hard Cider Seesaw" (2,300 words) for your magazine. As _Four Winds Quarterly_ publishes fiction with a strong narrative line, I thought this tale of a young man's confrontation with his childhood adversary (a most unlikely bully) might be of interest to you.
My prose has been short-listed for the Mid-States New Writer Award and the _Lexicon_ First Fiction Prize. My poetry has been published in several literary magazines, and I'd love to add _Four Winds Quarterly_ to my list of credits.
An SASE is enclosed for a reply, but I don't need the manuscript back. I wish you continued success with your magazine.
Best,
Matthew Piper
_Sample query letter for a short story_
YOUR TURN:
Pick a short story or novel that you have worked on, are working on, or would like to work on. Then write a query letter for it. Even if the project is unfinished, it will be helpful to write the query because that will force you to zero in on the essence of the story. Strive to make the query both professional and reflective of your personal voice. If your short story or novel is ready to send out, then start doing so!
THE SLUSH PILE
So, what happens when you finally send out your work? Well, your submission is probably headed for the slush pile.
"What is that, like the Dead Zone?" you ask.
Not exactly. The slush pile is the industry's pet name for the accumulation of unsolicited (meaning unasked-for and largely unwanted) manuscripts received by editors and agents—and they receive many. Fortunately
Ms. Pam Goodwin
The Goodwin Agency
123 Fourth Avenue, Suite 567
New York, N.Y. 10000
Dear Ms. Goodwin,
Please find, enclosed, the first three chapters of _April Rising_ (90,000 words). As your agency handles emerging writers, including J.J. Porter, whose work resembles my own, I wondered if this mainstream novel might be of interest to you.
Ellen Kaplan, carefree and twenty-three, returns to her affluent suburban home, only to discover that someone has taken her place. The culprit is April, former farm girl, HerbElixir saleswoman and proud owner of a porcelain Jesus collection. Rescued from destitution by Ellen's older brother, April has cast a spell of familial love on the entire Kaplan clan. Ellen sets out to topple her rival but in so doing discovers more than she bargained for, including the possibility that April may not be the enemy after all. This blackly comic tale explores friendship and family values at their most dysfunctional.
A bit about me. I have a B.F.A. from New York University, where I co-founded the Rough Draft writers' workshop, and was published in _The Minetta Review._ I've traveled extensively and currently live on a plane.
I've included a stamped self-addressed envelope for a reply only. I very much look forward to hearing from you.
Sincerely,
Corene Lemaitre
_Sample query letter for a novel_
for you, most of them will be poorly written, incorrectly formatted, or both. So although there is no guarantee of a response, a well-crafted and-presented manuscript may stand out. Initial screening will probably be done by an assistant. They know exactly what to look for, as agents and editors choose their staff with care. As your first point of contact, you could do worse than to catch the eye of a savvy editorial assistant or agent-in-training who is eager to discover new work.
Exercise rigorous quality control, and you have a better chance of surviving the slush pile than most.
**SUBMITTING SHORT STORIES TO MAGAZINES**
"I've got a few short stories ready to go," you say. "So, what do I do? Should I contact an agent?"
No. While an agent might consider a book-length collection of short stories, the effort required to market individual short stories is not worth the meager profit they command.
You're going to have to go it alone, so let's start by exploring potential markets for short stories.
At the top of the tier are the big glossies—the few large-circulation consumer magazines that (bless them) still publish fiction: _The New Yorker, Atlantic Monthly, Harper's, Esquire, Playboy, GQ, Jane, Seventeen,_ among others. There's no harm in approaching them, but these high-profile markets are difficult to crack. But fear not, because you have access to the vast and glorious world of literary magazines.
What exactly is a literary magazine, or litmag? Well, it's a smaller-scale publication, often published quarterly, with a circulation of anywhere from several hundred to a few thousand. Many are associated with universities and nonprofit organizations. As they cannot attract vast sums from advertising, they often depend on foundation grants to survive. They are run by (generally unpaid) staff who do it for the love of literature or the work experience or the high cool factor. And many have an excellent reputation.
There are hundreds of litmags, from top-tier publications like _Zoetrope_ and _The Paris Review,_ which are almost as difficult to get into as the glossies, to small-scale efforts that exude the appealing whiff of the mimeograph and are the literary equivalent of garage pop. In addition, there is a growing body of litmags on the Web. Some correspond to print versions, others are published solely on-line. What they have in common is a willingness to work with new writers.
"But if there are so many of them, how do I choose?" you ask.
Easy. You begin by consulting the market guides.
Market guides are industry directories and periodicals that provide essential information on book and magazine publishers—who they are, what they want, and how to submit work to them. (Contact names should be confirmed by telephone or e-mail, as turnover in publishing is notoriously high.)
Among the most comprehensive market guides:
* _Writer's Market._ The largest guide. Covers both fiction and non-fiction.
* _Novel and Short Story Writer's Market._ Focuses on fiction and may be your best bet.
* _The International Directory of Little Magazines and Small Presses._ Title tells all. An exhaustive source.
* _The CLMP Directory of Literary Magazines and Presses._ Produced by the Council of Literary Magazines and Presses.
* _Poets & Writers_ magazine. Plenty of market listings, plus features on writers and writing-related topics.
* _The Writer_ magazine. In existence since 1887, it contains valuable tips, news, and advice.
These guides are widely available in bookstores and public libraries, and some have an on-line presence. You might also check out _The Best American Short Stories,_ an annual anthology of short fiction gleaned largely from literary magazines. This will point you in the direction of a number of promising markets.
Your next step is to select which magazines to target. Based on the above guides, choose a few that look as though they might publish the kind of fiction you write. Send away for copies. Alternatively, check out your local bookstore or newsstand. If you're good at squatting, you may be able to keep the sum you shell out to a minimum.
Then study them to ensure that the type of fiction they publish corresponds to your own. Analyze the stories for both content and style—subject matter, tone, syntax, and so on. Do they focus on narrative? Or are they more "slice of life"?Is one POV favored over another? Does setting play a central role? How about the language and sentence structure? Simple? Or more complex?
"Once I've determined the right magazines/' you ask, "am I ready to send?"
A resounding _yes._
Note in your query letter that you've read their publication. It doesn't hurt to mention that you really liked it. If you can slip in that you've actually bought a copy... well, let's just say that these editors, overworked and underappreciated, may look upon your work that much more favorably.
With the exception of a few, litmags pay nothing or very little, but the benefits of publication are huge. Preparing a story for submission will help you to hone your work to a high standard. Seeing your work in print will provide you with a sense of accomplishment. And when you are ready to submit a novel or collection, your writing credits will increase your chances of catching a book editor's or agent's eye.
YOUR TURN:
If you're interested in writing short stories, go track down some literary magazines and read them. Do this with at least three different litmags, keeping an eye out for the publications that feel most right for your work. If none of the litmags you read look right for you, go find some others. You'll find a wonderfully diverse selection of literary magazines out there. Certainly one of them is perfect for your fiction.
BOOK PUBLISHERS
"Publishers are very simple, innocent people, so far as books are concerned. They often do not know one book from another. But they are optimists, sentimentalists, and experimentalists..."
—Frank Swinnerton, _Authors and the Book Trade_
"Speaking of books," you say, "I've got one of those too—a novel. Any chance of it finding a home?"
Certainly. To this end, let's take a closer look at the world of book publishing.
The quote above was published in 1932. As you may have guessed, things have changed a bit since then. Publishing, once widely regarded as a "gentleman's industry," has become a behemoth governed by big business. At one time, small publishing "houses" dominated the scene. But they merged, then merged again, to become the conglomerates we know today. Some of the houses survived in the form of imprints—subsidiary branches that specialize in particular types or styles of writing. Other houses survived as independents. Today, publishers can be divided into these two basic categories: conglomerates and independents.
The conglomerates are mostly located in New York City and include Random House, Simon & Schuster, HarperCollins, Penguin Putnam, Harcourt Brace, and others. They carry a lot of clout and most best-sellers display their corporate seal. The independents, among them such places as Algonquin, Soho Press, and Coffee House, range from small literary and university presses scattered throughout the country to larger New York City-based firms. In recent years, their marketing sawy has increased and distribution is often excellent.
As a writer, it's important that you acquaint yourself with this increasingly complex and profit-driven industry. Study the market guides, read books on the business, visit publishers' Web sites, peruse trade magazines. Learn as much as you can about this world you are entering. Knowing the context can help you to break in.
Some writers prefer to join forces with large publishers, finding their prestige and influence advantageous. Others swear by independents, citing the in-house attention their work receives. In the end, the size of the company has little bearing on success. You may work with a small team inside a large publishing house. You may work with virtually everyone at a small publishing house. Triumph will depend on commitment and creativity, not corporate dimensions.
If you submit a manuscript yourself to a publisher, what are your chances of acceptance? With the conglomerates, very slim. You can streak up to Simon & Schuster stark naked and hurl a manuscript over the transom, but it won't make any difference. Your manuscript will be heaved onto the slush pile or, more likely, just sent back. Independents and small presses may be more open. You will find _Literary Market Place, Writer's Market,_ and Jeff Herman's _Writer's Guide to Book Editors, Publishers and Literary Agents_ helpful in your search for a publisher willing to read unsolicited work.
But if you really want to increase the odds of acceptance, you are going to need an agent.
**AGENTS**
Agents. Those literary archangels who lift you out of the slush pile and carry you through the pearly gates of publication. They read your work, buy you hot dinners, and introduce you to influential talk-show hosts. Every waking moment (and half of their sleeping ones) they network furiously, always with you in mind. And, of course, they make you large amounts of money.
Too good to be true? I'm afraid so. Agents are, in fact, merely mortal. But they are an extraordinary breed of hardworking industry professionals who can help you get your work into print.
What exactly is an agent? An agent is someone who will sell your work to a publisher in exchange for a commission, generally 10 to 15 percent of the sale price. Why do you need one? Primarily because no one is better equipped to match and deliver a manuscript to an editor at a publishing house. Editors are often criticized for their reluctance to read unsolicited work, but the simple fact is that they can't. Their time and resources are limited, so they depend on agents to do the initial screening.
An agent's fundamental role is to sell your work. As part of this, he or she will:
* Read your work and provide suggestions for revision.
* Submit your work to the most appropriate editor.
* Negotiate the best possible terms for you.
* Exploit additional rights—foreign, serial, film, etc.
* Examine the contract, clause by clause.
* Track payments from publishers.
Once that (hopefully juicy) contract is signed and sealed, your agent may take on further roles in the areas of promotion and grievance resolution. She may buy you a celebratory drink and, if the going gets tough, give you a shoulder to cry on.
So, what does an agent look for in a writer? Given the increasingly competitive market for fiction, what would persuade a top agent to take on a first-timer? "What I look for is originality," says Suzanne Gluck, co-director of the William Morris Agency literary department. "Be it a singular literary voice, a new perspective on the human condition, or a view of a place we've not seen before. And anyone who can make me laugh is a winner in my book."
In describing the ideal author, Gluck is unequivocal. "My dream client is someone whose work I have genuine passion for, and for whom my advocacy is a natural outgrowth of my admiration. And to tell you the truth, over the years I've learned to limit myself to these folks."
So we're back to the beginning. You've got to write a good book.
You also want to hook up with an agent who is right for you. "The two most critical elements in an agent/author relationship are trust and a shared vision," notes Gluck. "Agents serve as creative sounding boards, fiduciary advocates, and, I hope, fierce professional champions. There are a lot of decisions of all sizes to be made along the way. When the client trusts that the agent has their best interests at heart, and the agent trusts that the client will always put their heart into their work, there's a greater chance the rest will fall into place."
"Sounds good," you say. "I want an agent of my own. So, how do I choose?"
Well, there are a number of reference books that provide lists. The most high-profile are _Writer's Market, Literary Market Place, The Writer's Digest Guide to Literary Agents,_ and Jeff Herman's _Writer's Guide to Book Editors, Publishers and Literary Agents._ These guides are revised regularly and some have online counterparts. If these sources prove too costly, check out your local library. Always call to confirm the contact names and addresses you find in the guides.
But the staggering amount of information these books provide about agents—who they are, what they want, and how to get in touch with them—poses a bit of a problem. How do you narrow it down?
Your best bet, in the first instance, is to identify a few writers whose work is similar to your own, and figure out who represents them. Some sleuth work may be necessary, as agency client lists are often confidential, so try these methods. Open a novel by one of these writers and check the acknowledgments—authors often thank their agents. Search the Internet—you may find the information you need in an interview or on the author's homepage. Finally, visit the publisher's Web site—you may find the agent listed under film or subrights, or mentioned in a press release. Similarly, check the publisher's catalog.
An alternative way to pick an agent is to identify a good one and then investigate the kind of writing they represent. To this end, read the "Hot Deals" column _of Publisher's Weekly,_ which details who's selling what, and for how much. Good to know.
YOUR TURN:
If you're interested in writing a novel, get the names of several agents who might be right for the kind of novel you have written, are writing, or plan to write. This will take some detective work, but you'll end up with the names of some agents who might be interested in seeing your novel when it's ready to be sent out. And maybe those names will give you the incentive you need to finish the novel.
Agencies come in three sizes—small, medium, and large. The smallest are the sole traders, whose agencies often bear their name. Medium-sized agencies will employ several people, sometimes quite a few. Others are huge, and represent screen and television writers as well, often "packaging" book-to-film deals from their own pool of clients. Throughout the entirety of this range, you will find excellent agents.
Many agencies have both established and new agents. So, which is better? Some top reps exceed their authors in notoriety, acquiring odd nicknames and granting interviews to glossy magazines. Their clients benefit from their experience and high profile. But up-and-coming agents have their own advantages. What they may lack in clout and contacts they make up for in determination and drive. Ultimately, it boils down to the individual. Look for someone who respects your work and is willing to fight on your behalf.
SUBMITTING TO AGENTS
"I've made a list of suitable agents," you say, "and now I'm ready to submit my novel. Should I send them the whole manuscript?"
No. They won't have time to read it, nor do they need to, because they can tell from just a sampling whether or not you can write well.
So, what do you send? Well, you have two options. Your first is to follow the specifications under each agent's entry in the previously listed reference books. Check these guides, and you'll find that some agents ask for a query letter only, or a query and a synopsis (a plot summary of one to several pages), while others want to see a few pages of the book itself. Very rarely do they request the whole novel.
Your second option is to ignore the agent's request and send a query letter with about fifty pages of the manuscript. The reasoning behind this approach is as follows: Many first novels have little structure, if any. They are episodic, merely a series of scenes strung together, and this is one of the primary reasons why agents reject work. Fifty pages will demonstrate that you can shape a story.
Should you include a synopsis? Perhaps, but there's a danger that they will read the synopsis and lose interest. A synopsis will not demonstrate that you can craft prose. Better to try to hook them on that first page and impel them to read on. No one can resist a good audition.
"Should I send my query to every agent on my list?" you ask. "All twenty-five?"
That's probably too many. Begin with three or four. As with every endeavor, there's a learning curve, and you don't want to use up every promising agent you've found. Select a variety and see what kind of response you get.
Some agents have a no-"multiple"-or-"simultaneous"-submission policy. This means that they would prefer an exclusive while considering your manuscript. This is understandable. Agents don't want to fall in love with your work only to discover that you've accepted another offer. But approaching agents one by one may take too long, as each will take several weeks or even months to respond. So, sending your initial query to three or four is a reasonable compromise that doesn't violate the no-multiple-submission policy.
Once an agent expresses an interest in your work, it's important to respond quickly. If she requests the entire manuscript, send it right away. Allow her eight weeks to get back to you. At this point, it's permissible to send a polite follow-up letter inquiring after the status of your manuscript, but don't pester her by e-mail or telephone.
There are several possible outcomes. You may be rejected by all of them. If this happens, try to figure out why. Your manuscript may need a bit more work, or you may have simply hit the wrong agents. Select several more, and try again. Then again, each may love your manuscript and ask to meet with you in person or at least have an extended phone conversation. Get in touch with them immediately and arrange to have this talk. Remember, a "shared vision" is desirable. Your relationship with your agent will be a crucial one, and this discussion will help you to determine whether or not he or she is right for you. After speaking with you, each may offer you representation. If this happens, try to make a decision as swiftly as possible. Remember, this is very much like a proposal of marriage (a metaphor often employed in the agent-author relationship), and you don't want to leave anyone "hanging on." Once you've selected your agent and informed him or her of your decision, send a polite thank-you note to the others. After all, your professional relationship with your agent of choice may not work out, and you don't want to burn bridges.
If only one agent expresses an interest in your work, but after meeting it doesn't feel right, then don't sign with him or her. Contact a few others. If your work is strong, you will find the right agent for you.
**BEWARE NONLEGITIMATE AGENTS**
"Hey, check out this classified ad," you say. "'Established literary agent now accepting manuscripts from new writers. All genres.' Hot damn. Should I send?"
No. The "invitation to submit" is a danger sign. Bona fide agents receive more work than they can handle. They don't need to place ads. This particular "agent" is a fraud and the request for manuscripts is a scam.
These so-called agents do not make money from sales to publishers but rather from "reading fees" solicited from the author. Contact them about the possibility of representation, and you will receive a request for money—at least several hundred dollars. Alternatively, they may refer you to an "editor" who will charge you quite a lot for not much (rest assured that the "agent" gets a kickback).
Legitimate agents don't charge you for reading your manuscript or for signing you as a client.
So, how can you be sure that your agent is legit? Well, most reputable agents are members of the Association of Authors' Representatives (www.aar-online.org). To qualify, an agent must meet professional standards specified in the AAR's bylaws and agree to subscribe to its canon of ethics. Members cannot charge reading fees. So, find out if your agent belongs, though not all legitimate agents do. You may also interview your potential agent, once she has expressed an interest in your work. An excellent list of questions can be found on the AAR's Web site. If in doubt, ask for a list of recent sales, and investigate her track record.
**RESPONSE AND REJECTION**
"Well, I've got a lot of stuff out there," you say. "I guess I'm going to get a lot of rejections, huh?"
Initially, yes. But there are different types of rejections, some of which are quite useful and constitute reason to be of good cheer. And at some point, there could be an acceptance, if you just hang in there.
The types of responses you may receive are more or less the same whether coming from agents, editors at publishing houses, or magazine editors. Let's go over some typical examples:
* _No response:_ Why not, when you'd enclosed the SASE? Who knows—too busy, too disorganized, too jaded... Whatever the reason, let it go. Their loss.
* _Form rejection:_ A standardized letter, but don't lose heart. This is the most common type of response. It does not necessarily mean that your work was poor or even inappropriate. Rather, it suggests overworked agents and editors.
* _Personalized rejection:_ This could be anything from a form rejection with a handwritten note scrawled in the margin to a neatly typed missive detailing exactly why your story doesn't work for them. This is rare, and may be a sign that they regard your writing as having promise.
* _Rejection with invitation to resubmit:_ Consider this encouraging. You'll receive suggestions for improvement and a request to see the revision. If possible, try to incorporate at least some of their recommendations into a rewrite. Then send the story back to them with a photocopy of their letter and a thank-you note.
* _Acceptance:_ This may come as a letter or a phone call. Celebrate. You've earned it.
You may have good fortune and receive an acceptance right away, but chances are that you will face a lot of rejection. If so, you will be joining an illustrious list of writers, including Herman Melville, Margaret Mitchell, Charles Dickens, Virginia Woolf, James Joyce, Beatrix Potter, Dylan Thomas, Emily Brontë, and Marcel Proust, who have faced this setback and lived to tell the tale. Over time the sting of rejection may diminish to a pang of disappointment, but it always hurts.
The best way to deal with it is by writing. Turn over the rejection slip and throw down some words. Brainstorm. Free write. Draft a short story.
Don't let the setback of rejection get under your skin. If possible, have several projects on the go. Submit manuscripts regularly and keep working on more, and rejection will cease to seem like such a big obstacle.
As you send out your writing, it's a good idea to keep track of what you're sending to whom and when. You don't want to target the same individual twice. Also, you may need to chase people, especially if others are interested in the same material. If your work is rejected, recording the reasons may help, as this may reveal a pattern. Acceptances are enlightening as well—you may discover, for instance, that you're having success, or getting warm, with a particular sort of magazine. Finally, keeping a list, even of rejections, is good for morale. Remember, every _no_ brings you closer to a _yes._
You may regard keeping track as yet one more task that robs you of time for your actual writing, but remember that you're running a business, and, as with any small business, you are responsible for day-to-day operations, including administration. So, develop good record-keeping habits. Once you've established a system, you'll find that you've actually maximized your writing time, with increased job satisfaction and peace of mind.
**MONEY AND CONTRACTS**
"So, what if an agent or editor expresses an interest in my work?" you ask. "How much money can I expect to make?"
A reasonable question. Anything from zero to upward of several hundred thousand dollars (though the latter sum is quite rare). These days a typical sum for a novel ranges from five thousand to twenty-five thousand dollars. The sum will be based on a number of factors. One is the estimated market value of your work—in other words, the amount your publishers think your book will bring in. Another is luck. But the important thing is to secure a fair deal.
Having interested a publisher, your agent will try to negotiate an advance. An advance is essentially a loan, which you will pay back through your royalties. The term _royalty_ refers not to your newfound status as a writer but to your slice of the financial pie—generally 7 to 15 percent of the cover price of the book, often stepped to reward greater sales. New writers seldom get big advances, but you will probably get something.
So, let's say that your book comes out as a paperback original, an increasingly common practice. At ten dollars a book, assuming your royalty schedule starts at 7 percent, you'll get seventy cents. Okay, it's a bit like holding out a hat, but it adds up.
Also, bear in mind that getting published has a domino effect. Having a novel or story in print can lead to paid talks, teaching engagements, and even writing commissions. If these opportunities don't come your way, create them. Think laterally and be entrepreneurial. Remember, you're running a small business.
You must also be prepared to deal with contracts. A contract is a written agreement between you and your publisher concerning the use and sale of your work. At some point, you are probably going to be given one. Not so much with short fiction, unless you are published by one of the glossies or issuing a collection, but certainly with novels.
When offered a contract, it's a good idea to have it scrutinized by an expert. A publishing contract is a complicated document, particularly with the advent of electronic rights, and needs to be examined with care, preferably by more than one person. Initially, your agent will take a look at it. This is part of her job, and she should have the necessary expertise. If you would like added reassurance, consult an entertainment or literary property lawyer. Finally, contact a professional organization such as the National Writers Union (www.nwu.org) or the Authors Guild (www.authorsguild.org). Joining fees are minimal, and members are offered free contract advice.
But though these individuals and organizations will help, you too must become an expert. Ultimately, you are responsible for protecting yourself. There are a number of guides to contract law for writers. Obtain one, and peruse the document clause by clause. Above all, don't ever feel pressured into signing a contract that you don't feel comfortable with.
**AFTER THE BOOK DEAL**
Celebrate. You've completed a long and difficult journey, and chances are that you'll be experiencing a variety of emotions, ranging from elation to bewilderment. But it's important to remind yourself that you've achieved a significant goal, and a good way to do that is to mark the occasion. Buy yourself a present, break out the champagne, do a private victory dance. Enjoy your accomplishment—you've earned it.
There are many further stages to come, and you will undertake a lot of hard work. So, once you've given yourself a chance to get used to this new state of affairs, roll your sleeves up and get ready to plunge into the prepublication process, beginning with the editorial department.
Once you've closed the deal, you will work with your editor to make your novel or short story collection the best it can be. You will receive notes, oral or written. These comments may be extensive and revision will probably require more than mere tinkering. Be prepared to dig in. But your editor will serve another important function, as your "in-house ally," ensuring that other departments, such as sales, marketing, and publicity, are aware of your upcoming release. Your publisher will have other books on their list and yours may not be prioritized. But however fierce the internal competition, your editor will champion your work and ensure that it gets as much attention as possible.
At some point, probably well before your book is published, you will be assigned a publicist who will endeavor to promote your book. Publicists will do mailings to various literary editors and critics, in an effort to secure reviews. They may recommend you to journalists as an interviewee. If you're very fortunate, they will arrange for you to participate in readings at bookstores and literary festivals. They may also contact radio and television producers with an eye toward getting you "on the air."
Should your publicist neglect you in favor of another, more profitable author, don't worry, because there is plenty you can do for yourself. There are several excellent guides to do-it-yourself book promotion and marketing, so invest in a few and make a plan. Prepare a media kit containing a press release, an author photo, your bio sheet, contact details, any prepublication endorsements, and a copy of the book or at least the jacket. Then send it to producers, journalists, and newspaper and magazine editors, with a personal note. Call booksellers and offer to do signings or conduct workshops, always with an eye toward making their job as easy as possible.
Whichever form your promotion takes, an "author Web site" is essential. In fact, you will benefit from being on the Web from the day you begin submitting your work. Acquire a domain name and establish your presence. If you can't design it yourself, pay a professional or recruit a technologically expert friend. As with the press kit, include your photo, bio, and so forth, but don't hesitate to give the contents a personal spin. Your
Web site should reflect not only your skills but your style and sense of humor.
**SELF-PUBLISHING**
A brief side note. If no publisher accepts your book _and_ you have an entrepreneurial bent, you might want to try self-publishing. With the advent of digital print-on-demand, this option has become much more affordable. You can hire a publishing service that takes care of producing your book—everything from typesetting to cover design. Bear in mind, though, that these companies are not selective—they'll publish anybody's manuscript. Their job is solely to get your work into print. If you go this route, you would do well to hire a good freelance editor, to ensure quality control. Also bear in mind that self-published books seldom turn a profit for their authors.
One word of caution. Beware "vanity publishers." Both vanity publishers and publishing services are sometimes referred to as _subsidy publishers,_ as they charge a fee for putting your work between two covers. So, what's the difference? Vanity publishers pretend to be something they're not—namely, real publishers. They will send you a letter telling you how wonderful your novel or collection is, suggesting that it has strong commercial potential. Having hooked you, they will gently introduce the delicate matter of payment. Don't fall for it. This is a cousin scam to the fee-charging agent. They will produce shoddy work and the charge will be exorbitant.
While self-publishing is a legitimate option, you will probably retain your dream of having work accepted by an established publisher.
**COMPETITIONS, GRANTS, AND AWARDS**
Getting your work published is not the only way to gain prestige and money as a writer.
Explore writing competitions. You will find details on these in the market guides and magazines. An Internet search is also a good idea. Many charge an entry fee, so make sure the competition is legitimate. Research the sponsoring organization or magazine. Awards sometimes include publication of your short story, novel, or collection. Your work may be brought to the attention of a literary agent. There is often a money prize. With high-profile competitions, even semifinalist status will enhance your query letters.
Your story may get a close reading or a cursory one. In the end, it's a lottery. Great work is often overlooked, but it's worth the risk. Someone's got to win, and it could be you.
Many writers receive some financial sustenance from literary grants, fellowships, and awards. These are administered by a variety of organizations—academic, nonprofit, and governmental. Among the best known are those administered by the National Endowment for the Arts. State arts councils often make similar provisions. An excellent guide is _Grants and Awards Available to American Writers,_ published by PEN American Center, which lists more than one thousand opportunities complete with contact details. The market guides and magazines are also good sources of information.
Though many grants, fellowships, and awards require writing credits or nomination by a publisher, they are worth looking into. After all, they're designed to assist authors in financial need, including those at the beginning of their careers.
**THE WRITING COMMUNITY**
Published or not, give some thought to joining the writing community. Launching your literary career requires extreme patience; connecting with others may help you to sustain excitement and hope, and pick up some knowledge in the process. Communities, both actual and virtual, exist all over the globe and can be accessed from as far off as Uzbekistan. So, let's take a look at ways to hook up with kindred spirits.
Writing conferences and retreats are excellent ways to access the community. But they are very different, and each serves a particular function.
Conferences offer a chance to meet publishing professionals and learn about the industry. Locations range from Maui to Prague. Unless the conference is local, you will pay travel and housing costs in addition to an attendance fee, but the benefits are significant. Seminars, workshops, and networking opportunities abound. Agents are often invited to give talks or participate in panel discussions, and may be available afterward to speak with attendees.
Retreats, on the other hand, are just that—a chance to withdraw from the world and do some writing. Also known as writers' colonies, they provide an opportunity to work in the company of like-minded people. Some are open to all, and charge a fee for room and board. Others are selective and pay all costs save travel. Two of the better known are Yaddo and the MacDowell Colony. Everything is designed to minimize distractions and facilitate writing. Lunch is often brought to your door. But though there may be a rule of silence during the day, evenings are often social.
For lists of conferences, visit www.awpwriter.org/wcc and writing.shaw-guides.com. For retreats, check out _Artists & Writers Colonies: Retreats, Residencies, and Respites for the Creative Mind. Poets & Writers_ is a great source for both.
You may also find fellow writing enthusiasts at literary readings or events in your corner of the world. These generally take place at colleges, bookstores, community centers, literary festivals, and cafes. Check your local newspaper or "what's on" guide for locations, dates, and times.
If you find your motivation flagging or yourself desperately in need of feedback on your work, consider joining a writing class or a writing group, or both. Both classes and groups are available on-line, as well as "live."
A writing class provides several advantages—specifically, expert advice, instruction, and feedback. The teacher will enlighten you on the art and the business of writing. You may be given exercises designed to strengthen your craft. You will have the opportunity to submit your work for criticism by your peers. Though classes are not a substitute for the actual writing, they can be an incredibly helpful way to keep you learning and feeling good about the process.
Writing groups, also known as "writing circles," often grow out of a class or are formed by several writers who decide to meet on a regular basis. Participants evaluate each other's work, share information on getting published, and offer all-around support. Such groups can last for years and become invaluable to their members.
Finally, consider joining a professional organization. There are many such bodies—national, regional, and local—but among the most prominent are the National Writers Union (NWU) and the Authors Guild.
The National Writers Union is the trade union for freelance writers of all genres, including fiction. Both published and unpublished writers are eligible. The benefits include networking opportunities, agent databases, market information, contract advice, grievance resolution, health insurance at competitive rates, and job banks. There are chapters all over the country. Join. Attend meetings, if time permits, and participate in forums and events. You'll be glad you did.
The Authors Guild is the oldest and largest professional society of published writers in the country. The Guild lobbies on a local and a national level on such issues as copyright, taxation, and protection of authors' rights under the First Amendment. Its considerable achievements include clearer royalty statements and improved contractual terms. While not technically a trade union, the Guild provides many of the same services as the NWU, including forums and events.
Bear in mind that there is a worldwide community of writers. If you'd like to connect with this global network, you might want to check out International PEN (www.pen.org), which campaigns on behalf of persecuted authors, much as Amnesty International crusades to free prisoners of conscience. PEN has an American branch. Learn about their activities and get involved.
For the most part, writing is a solitary journey, so consider joining forces with others who share your mania. Solidarity can help. There is strength in numbers.
YOUR TURN:
Find some ways in which you can join the writing community, either in your vicinity, or by traveling somewhere, or through the Internet. You can choose between conferences, retreats, literary readings/events, classes, writing groups, and writing organizations. Then join or participate in one of those for real.
**BEING A WRITER**
"Having read (and believed) the depressive truths I have communicated, do you still wish to become a writer?"
—Frank Swinnerton, _Authors and the Book Trade_
At some point, you will realize that you've made the transition from becoming a writer to being one. How will you know? Well, you'll find it a bit easier to sit down and work. You'll have adjusted to the daily discipline of writing. Submitting manuscripts will be accomplished swiftly. Rejection will sting a bit less. Every so often, you may receive an acceptance, and these credits will bring you personal and professional satisfaction. Most importantly, crafting fiction will have become part of the fabric of your life.
Though the rewards are abundant, adjusting to the writing life takes time. Like characters in your own stories, you will face obstacles and challenges. So, let's a take a look at how to circumnavigate some of these hazards.
You've probably heard the expression "Don't give up the day job." Your internal censor (that destructively critical voice in your head) may have told you this, or a well-meaning friend, or a colleague who just doesn't get it. Although you may have reasonable faith in your abilities as a writer, this advice rings in your ear with all the portent of a dire warning.
Relax. Though you may not be able to leave paid employment right away, it's not the end of the world. Most fiction writers, even those with impressive publication credits, supplement their income through other means, and successfully reconcile their day jobs with their writing life. Anton Chekhov, for instance, maintained a medical practice for part of his literary career, and claimed to have gathered much of his insight into human nature during this period.
You may even find that a day job gives you forced distance from your writing and helps keep feelings of isolation at bay. Also, as Chekhov found, the human comedy and tragedy of the workplace may prove a rich source of character and situation. And time away from your writing may increase your hunger and drive.
Regardless of your job, make sure to set aside a period every day, or at least several days a week, for writing. Treat that time as sacrosanct. Rise an hour earlier in the morning or go to bed a bit later. Watch less television. Try using ritual to ease the transition from the day job to your creative work. This may be as simple as preparing a cup of tea, but it will help you to drift into your fictional dream world.
"The problem is," you say, "that as soon as I enter the dream, someone I know tries to drag me out."
A common problem. The sad fact is, you will need to counteract the intrusion of family and friends. If you're lucky, your loved ones will turn cartwheels when you express your desire to write. But their initial enthusiasm may wane when they discover that you are _really going to do it_ and that this activity will make you less available to them. Anticipate sabotage.
Seriously, expect the worst. They will knock at your door. Shout your name from downstairs. Walk into your study and burst into tears. You will pick up your pen, only to be interrupted by someone who needs a favor or simply desires your company.
Talk to them. Explain that this endeavor is important to you, and that shutting yourself away in a room with a pad of paper doesn't mean that you don't care about them. Then bolt the door and get on with your work. And don't use their demands as an excuse to procrastinate. Learn to say _no._ You'll find this liberating. There may be occasions when you need to put people first. To the extent that you can, however, prioritize your writing, as those who write may become enraged and dangerous if prevented from doing so for too long.
You may encounter even darker perils...
You're at your table scribbling. Or perhaps you're at work, writing on the sly. All is well, until you stop... look up... and the bottom drops out of your soul. You feel crushed, discouraged, and utterly alone. Suddenly, you don't know _what_ to write or _how_ to write or _why_ you should write. In short, you are facing the Abyss.
The Abyss refers to the anguish that can strike you at any given moment. This feeling is as likely to come to you in the midst of a crowd as when you are by yourself, but it's a particular hazard in solitary occupations such as writing. This emotional chasm may shake your confidence and temporarily diminish your ability or desire to write. So try the following antidotes.
Stay in touch with the human race. However much you want or need to be alone, communication (with real people, not just fictional characters) is important. Make the most of any time you spend with family and friends. As discussed, connect with the writing community, where you will find others who know exactly what you're going through.
Also, develop a healthful lifestyle. Writing is intensive, all-consuming work, and there's a danger that you'll end up living on coffee and whatever convenience food is at hand. Mood is tied to health, as is productivity, so eat sensibly and try to get enough sleep. Step outside and breathe some fresh air. Take a walk. Writing is tiring, and physical activity will energize your body and mind.
YOUR TURN:
Make a list of ten things that you can do to make writing more of a priority in your life. This can include anything from waking up earlier to hiring a baby-sitter to exercising (to give yourself more energy) to signing up for a writing class. Post this list somewhere, say, above your desk or on the refrigerator. Then really _do_ those things, at least some of them. The best way to become a writer is to get serious about being a writer. And the best time to start is right now.
Finally, write. Just lose yourself in the process and let it flow. Writing is the best stimulant in the world. Not when it's going badly, perhaps, but when it's going well... oh, baby. Your mood lifts, the Abyss recedes, and your confidence returns. Creative blocks vanish and words flood the page before you. By the very act of writing, you will have accessed your subconscious and tapped into something larger than yourself. You will have become a conduit for the world.
CHEAT SHEET
**CHARACTER**
* Do your characters have desires?
* Are your characters distinctive enough not to be types?
* Do your characters have contrasting traits that make them complex?
* Are your characters consistent despite their contrasting traits?
* Do your characters have the ability to change?
* Do you know your characters well enough?
* Are the right characters "round" and the right characters "flat"?
* Are you showing your characters more than telling about them?
* Are you utilizing all four methods of showing—action, speech, appearance, thought?
* Do your characters have the right names?
**PLOT**
* Do you have a major dramatic question?
* Do you have a protagonist with a strong goal and plenty of obstacles?
* Does your protagonist have both external and internal obstacles?
* Do you have a beginning, middle, and end?
* Is your beginning not clogged with too much exposition and not too long?
* Does your conflict escalate in the middle?
* Are the events of your middle linked by cause and effect?
* Do you have crisis, climax, and consequences at the end?
* Is your ending plausible, satisfying, and not too long?
**POINT OF VIEW**
* Does your story work best in first, second, or third person?
* Does your story work best with a single-vision or multiple-vision POV?
* Is there any reason your story might work best with the omniscient or objective POV?
* If you're using a second-or third-person narrator, how close emotionally is the narrator to the story and characters?
* Are you keeping your POV consistent?
**DESCRIPTION**
* Are your descriptions utilizing all five senses?
* Are your descriptions specific enough?
* Are you overusing adjectives and adverbs?
* Are you using figurative language and lyrical techniques where appropriate?
* Are your descriptions overdone, choking your story?
* Are you using telling details?
* Are you watching out for such description traps as clichés and mixed metaphors?
* Do your descriptions reflect the consciousness of your POV character or characters?
**DIALOGUE**
* Are you using dialogue and scenes for the more important points in your story?
* Does your dialogue sound real yet also get to the point quickly?
* Do your tags call too much attention to themselves?
* Are you using stage directions to enhance your dialogue?
* Do your characters sound distinctive from one another and appropriate to who they are?
* Is there anywhere your dialogue can be improved by using subtext?
* Does your dialogue contain clunky exposition or off-putting dialect?
**SETTING/PACING**
* Have you grounded your story in a specific place, or places?
* Have you grounded your story in a specific time, or times?
* Do the place and time of your story affect the action?
* Are there opportunities to let the setting enhance the atmosphere or mood?
* Do your characters act in a way that reflects either their comfort or discomfort with their setting?
* Are you describing your settings so much that they slow down the action?
* Have you chosen the right places either to expand or to compress time?
**VOICE**
* Have you picked a voice that works in harmony with your POV choice, the personality of your narrator, and the narrator's emotional distance to the story?
* Do your word, sentence, and paragraph choices support your voice?
* Does your voice remain consistent throughout the story?
**THEME**
* Have you identified a theme for your story?
* Does your theme surround your story with a light enough touch?
* Do all the elements of your story work to support the theme?
**REVISION**
* Have you gotten enough distance from your story to begin the revision process?
* Have you considered reenvisioning your story?
* Have you looked through a magnifying glass at all the Big Things in your story?
* Have you looked through a microscope at all the Little Things in your story?
* Have you cut and tweaked as much as you possibly can?
[APPENDIX
**CATHEDRAL**](Facu_9781596917910_epub_c4_r1.html#aa13)
BY RAYMOND CARVER
This blind man, an old friend of my wife's, he was on his way to spend the night. His wife had died. So he was visiting the dead wife's relatives in Connecticut. He called my wife from his in-laws'. Arrangements were made. He would come by train, a five-hour trip, and my wife would meet him at the station. She hadn't seen him since she worked for him one summer in Seattle ten years ago. But she and the blind man had kept in touch. They made tapes and mailed them back and forth. I wasn't enthusiastic about his visit. He was no one I knew. And his being blind bothered me. My idea of blindness came from the movies. In the movies, the blind moved slowly and never laughed. Sometimes they were led by seeing-eye dogs. A blind man in my house was not something I looked forward to.
That summer in Seattle she had needed a job. She didn't have any money. The man she was going to marry at the end of the summer was in officers' training school. He didn't have any money, either. But she was in love with the guy, and he was in love with her, etc. She'd seen something in the paper: HELP WANTED— _Reading to Blind Man,_ and a telephone number. She phoned and went over, was hired on the spot. She'd worked with this blind man all summer. She read stuff to him, case studies, reports, that sort of thing. She helped him organize his little office in the county social-service department. They'd become good friends, my wife and the blind man. How do I know these things? She told me. And she told me something else. On her last day in the office, the blind man asked if he could touch her face. She agreed to this. She told me he touched his fingers to every part of her face, her nose—even her neck! She never forgot it. She even tried to write a poem about it. She was always trying to write a poem. She wrote a poem or two every year, usually after something really important had happened to her.
When we first started going out together, she showed me the poem. In the poem, she recalled his fingers and the way they had moved around over her face. In the poem, she talked about what she had felt at the time, about what went through her mind when the blind man touched her nose and lips. I can remember I didn't think much of the poem. Of course, I didn't tell her that. Maybe I just don't understand poetry. I admit it's not the first thing I reach for when I pick up something to read.
Anyway, this man who'd first enjoyed her favors, the officer-to-be, he'd been her childhood sweetheart. So okay. I'm saying that at the end of the summer she let the blind man run his hands over her face, said goodbye to him, married her childhood, etc., who was now a commissioned officer, and she moved away from Seattle. But they'd kept in touch, she and the blind man. She made the first contact after a year or so. She called him up one night from an Air Force base in Alabama. She wanted to talk. They talked. He asked her to send him a tape and tell him about her life. She did this. She sent the tape. On the tape, she told the blind man about her husband and about their life together in the military. She told the blind man she loved her husband but she didn't like it where they lived and she didn't like it that he was a part of the military-industrial thing. She told the blind man she'd written a poem and he was in it. She told him that she was writing a poem about what it was like to be an Air Force officer's wife. The poem wasn't finished yet. She was still writing it. The blind man made a tape. He sent her the tape. She made a tape. This went on for years. My wife's officer was posted to one base and then another. She sent tapes from Moody AFB, McGuire, McConnell, and finally Travis, near Sacramento, where one night she got to feeling lonely and cut off from people she kept losing in that moving-around life. She got to feeling she couldn't go it another step. She went in and swallowed all the pills and capsules in the medicine chest and washed them down with a bottle of gin. Then she got into a hot bath and passed out.
But instead of dying, she got sick. She threw up. Her officer—why should he have a name? he was the childhood sweetheart, and what more does he want?—came home from somewhere, found her, and called the ambulance. In time, she put it all on a tape and sent the tape to the blind man. Over the years, she put all kinds of stuff on tapes and sent the tapes off lickety-split. Next to writing a poem every year, I think it was her chief means of recreation. On one tape, she told the blind man she'd decided to live away from her officer for a time. On another tape, she told him about her divorce. She and I began going out, and of course she told her blind man about it. She told him everything, or so it seemed to me. Once she asked me if I'd like to hear the latest tape from the blind man. This was a year ago. I was on the tape, she said. So I said okay, I'd listen to it. I got us drinks and we settled down in the living room. We made ready to listen. First she inserted the tape into the player and adjusted a couple of dials. Then she pushed a lever. The tape squeaked and someone began to talk in this loud voice. She lowered the volume. After a few minutes of harmless chitchat, I heard my own name in the mouth of this stranger, this blind man I didn't even know! And then this: "From all you've said about him, I can only conclude—" But we were interrupted, a knock at the door, something, and we didn't ever get back to the tape. Maybe it was just as well. I'd heard all I wanted to.
Now this same blind man was coming to sleep in my house.
"Maybe I could take him bowling," I said to my wife. She was at the draining board doing scalloped potatoes. She put down the knife she was using and turned around.
"If you love me," she said, "you can do this for me. If you don't love me, okay. But if you had a friend, any friend, and the friend came to visit, I'd make him feel comfortable." She wiped her hands with the dish towel.
"I don't have any blind friends," I said.
"You don't have _any_ friends," she said. "Period. Besides," she said, "goddamn it, his wife's just died! Don't you understand that? The man's lost his wife!"
I didn't answer. She'd told me a little about the blind man's wife. Her name was Beulah. Beulah! That's a name for a colored woman.
"Was his wife a Negro?" I asked.
"Are you crazy?" my wife said. "Have you just flipped or something?" She picked up a potato. I saw it hit the floor, then roll under the stove. "What's wrong with you?" she said. "Are you drunk?"
"I'm just asking," I said.
Right then my wife filled me in with more detail than I cared to know. I made a drink and sat at the kitchen table to listen. Pieces of the story began to fall into place.
Beulah had gone to work for the blind man the summer after my wife had stopped working for him. Pretty soon Beulah and the blind man had themselves a church wedding. It was a little wedding—who'd want to go to such a wedding in the first place?—just the two of them, plus the minister and the minister's wife. But it was a church wedding just the same. It was what Beulah had wanted, he'd said. But even then Beulah must have been carrying the cancer in her glands. After they had been inseparable for eight years—my wife's word, _inseparable_ —Beulah's health went into a rapid decline. She died in a Seattle hospital room, the blind man sitting beside the bed and holding on to her hand. They'd married, lived and worked together, slept together—had sex, sure—and then the blind man had to bury her. All this without his having ever seen what the goddamned woman looked like. It was beyond my understanding. Hearing this, I felt sorry for the blind man for a little bit. And then I found myself thinking what a pitiful life this woman must have led. Imagine a woman who could never see herself as she was seen in the eyes of her loved one. A woman who could go on day after day and never receive the smallest compliment from her beloved. A woman whose husband could never read the expression on her face, be it misery or something better. Someone who could wear make-up or not—what difference to him? She could, if she wanted, wear green eye-shadow around one eye, a straight pin in her nostril, yellow slacks and purple shoes, no matter. And then to slip off into death, the blind man's hand on her hand, his blind eyes streaming tears—I'm imagining now—her last thought maybe this: that he never even knew what she looked like, and she on an express to the grave. Robert was left with a small insurance policy and half of a 20-peso Mexican coin. The other half of the coin went into the box with her. Pathetic.
So when the time rolled around, my wife went to the depot to pick him up. With nothing to do but wait—sure, I blamed him for that—I was having a drink and watching the TV when I heard the car pull into the drive. I got up from the sofa with my drink and went to the window to have a look.
I saw my wife laughing as she parked the car. I saw her get out of the car and shut the door. She was still wearing a smile. Just amazing. She went around to the other side of the car to where the blind man was already starting to get out. This blind man, feature this, he was wearing a full beard! A beard on a blind man! Too much, I say. The blind man reached into the back seat and dragged out a suitcase. My wife took his arm, shut the car door, and, talking all the way, moved him down the drive and then up the steps to the front porch. I turned off the TV.
I finished my drink, rinsed the glass, dried my hands. Then I went to the door.
My wife said, "I want you to meet Robert. Robert, this is my husband. I've told you all about him." She was beaming. She had this blind man by his coat sleeve.
The blind man let go of his suitcase and up came his hand.
I took it. He squeezed hard, held my hand, and then he let it go.
"I feel like we've already met," he boomed.
"Likewise," I said. I didn't know what else to say. Then I said, "Welcome. I've heard a lot about you." We began to move then, a little group, from the porch into the living room, my wife guiding him by the arm. The blind man was carrying his suitcase in his other hand. My wife said things like, "To your left here, Robert. That's right. Now watch it, there's a chair. That's it. Sit down right here. This is the sofa. We just bought this sofa two weeks ago."
I started to say something about the old sofa. I'd liked that old sofa. But I didn't say anything. Then I wanted to say something else, small-talk, about the scenic ride along the Hudson. How going _to_ New York, you should sit on the right-hand side of the train, and coming _from_ New York, the left-hand side.
"Did you have a good train ride?" I said. "Which side of the train did you sit on, by the way?"
"What a question, which side!" my wife said. "What's it matter which side?" she said.
"I just asked," I said.
"Right side," the blind man said. "I hadn't been on a train in nearly forty years. Not since I was a kid. With my folks. That's been a long time. I'd nearly forgotten the sensation. I have winter in my beard now," he said. "So I've been told, anyway. Do I look distinguished, my dear?" the blind man said to my wife.
"You look distinguished, Robert," she said. "Robert," she said. "Robert, it's just so good to see you."
My wife finally took her eyes off the blind man and looked at me. I had the feeling she didn't like what she saw. I shrugged.
I've never met, or personally known, anyone who was blind. This blind man was late forties, a heavy-set, balding man with stooped shoulders, as if he carried a great weight there. He wore brown slacks, brown shoes, a light-brown shirt, a tie, a sports coat. Spiffy. He also had this full beard.
But he didn't use a cane and he didn't wear dark glasses. I'd always thought dark glasses were a must for the blind. Fact was, I wished he had a pair. At first glance, his eyes looked like anyone else's eyes. But if you looked close, there was something different about them. Too much white in the iris, for one thing, and the pupils seemed to move around in the sockets without his knowing it or being able to stop it. Creepy. As I stared at his face, I saw the left pupil turn in toward his nose while the other made an effort to keep in one place. But it was only an effort, for that eye was on the roam without his knowing it or wanting it to be.
I said, "Let me get you a drink. What's your pleasure? We have a little of everything. It's one of our pastimes."
"Bub, I'm a Scotch man myself," he said fast enough in this big voice.
"Right," I said. Bub! "Sure you are. I knew it."
He let his fingers touch his suitcase, which was sitting alongside the sofa. He was taking his bearings. I didn't blame him for that.
"I'll move that up to your room," my wife said.
"No, that's fine," the blind man said loudly. "It can go up when I go up."
"A little water with the Scotch?" I said.
"Very little," he said.
"I knew it," I said.
He said, "Just a tad. The Irish actor, Barry Fitzgerald? I'm like that fellow. When I drink water, Fitzgerald said, I drink water. When I drink whiskey, I drink whiskey." My wife laughed. The blind man brought his hand up under his beard. He lifted his beard slowly and let it drop.
I did the drinks, three big glasses of Scotch with a splash of water in each. Then we made ourselves comfortable and talked about Robert's travels. First the long flight from the West Coast to Connecticut, we covered that. Then from Connecticut up here by train. We had another drink concerning that leg of the trip.
I remembered having read somewhere that the blind didn't smoke because, as speculation had it, they couldn't see the smoke they exhaled. I thought I knew that much and that much only about blind people. But this blind man smoked his cigarette down to the nubbin and then lit another one. This blind man filled his ashtray and my wife emptied it.
When we sat down at the table for dinner, we had another drink. My wife heaped Robert's plate with cube steak, scalloped potatoes, green beans.
I buttered him up two slices of bread. I said, "Here's bread and butter for you." I swallowed some of my drink. "Now let us pray," I said, and the blind man lowered his head. My wife looked at me, her mouth agape. "Pray the phone won't ring and the food doesn't get cold," I said.
We dug in. We ate everything there was to eat on the table. We ate like there was no tomorrow. We didn't talk. We ate. We scarfed. We grazed that table. We were into serious eating. The blind man had right away located his foods, he knew just where everything was on his plate. I watched with admiration as he used his knife and fork on the meat. He'd cut two pieces of meat, fork the meat into his mouth, and then go all out for the scalloped potatoes, the beans next, and then he'd tear off a hunk of buttered bread and eat that. He'd follow this up with a big drink of milk. It didn't seem to bother him to use his fingers once in a while, either.
We finished everything, including half a strawberry pie. For a few moments, we sat as if stunned. Sweat beaded on our faces. Finally, we got up from the table and left the dirty plates. We didn't look back. We took ourselves into the living room and sank into our places again. Robert and my wife sat on the sofa. I took the big chair. We had us two or three more drinks while they talked about the major things that had come to pass for them in the past ten years. For the most part, I just listened. Now and then I joined in. I didn't want him to think I'd left the room, and I didn't want her to think I was feeling left out. They talked of things that had happened to them—to them—these past ten years. I waited in vain to hear my name on my wife's sweet lips: "And then my dear husband came into my life"—something like that. But I heard nothing of the sort. More talk of Robert. Robert had done a little of everything, it seemed, a regular blind jack-of-all-trades. But most recently he and his wife had had an Amway distributorship, from which, I gathered, they'd earned their living, such as it was. The blind man was also a ham radio operator. He talked in his loud voice about conversations he'd had with fellow operators in Guam, in the Philippines, in Alaska, and even in Tahiti. He said he'd have a lot of friends there if he ever wanted to go visit those places. From time to time, he'd turn his blind face toward me, put his hand under his beard, ask me something. How long had I been in my present position? (Three years.) Did I like my work? (I didn't.) Was I going to stay with it? (What were the options?) Finally, when I thought he was beginning to run down, I got up and turned on the TV.
My wife looked at me with irritation. She was heading toward a boil. Then she looked at the blind man and said, "Robert, do you have a TV?"
The blind man said, "My dear, I have two TVs. I have a color set and a black-and-white thing, an old relic. It's funny, but if I turn the TV on, and I'm always turning it on, I turn on the color set. It's funny, don't you think?"
I didn't know what to say to that. I had absolutely nothing to say to that. No opinion. So I watched the news program and tried to listen to what the announcer was saying.
"This is a color TV," the blind man said. "Don't ask me how, but I can tell."
"We traded up a while ago," I said.
The blind man had another taste of his drink. He lifted his beard, sniffed it, and let it fall. He leaned forward on the sofa. He positioned his ashtray on the coffee table, then put the lighter to his cigarette. He leaned back on the sofa and crossed his legs at the ankles.
My wife covered her mouth, and then she yawned. She stretched. She said, "I think I'll go upstairs and put on my robe. I think I'll change into something else. Robert, you make yourself comfortable," she said.
"I'm comfortable," the blind man said.
"I want you to feel comfortable in this house," she said.
"I am comfortable," the blind man said.
After she'd left the room, he and I listened to the weather report and then to the sports roundup. By that time, she'd been gone so long I didn't know if she was going to come back. I thought she might have gone to bed. I wished she'd come back downstairs. I didn't want to be left alone with a blind man. I asked him if he wanted another drink, and he said sure. Then I asked if he wanted to smoke some dope with me. I said I'd just rolled a number. I hadn't, but I planned to do so in about two shakes.
"I'll try some with you," he said.
"Damn right," I said. "That's the stuff."
I got our drinks and sat down on the sofa with him. Then I rolled us two fat numbers. I lit one and passed it. I brought it to his fingers. He took it and inhaled.
"Hold it as long as you can," I said. I could tell he didn't know the first thing.
My wife came back downstairs wearing her pink robe and her pink slippers.
"What do I smell?" she said.
"We thought we'd have us some cannabis," I said.
My wife gave me a savage look. Then she looked at the blind man and said, "Robert, I didn't know you smoked."
He said, "I do now, my dear. There's a first time for everything. But I don't feel anything yet."
"This stuff is pretty mellow," I said. "This stuff is mild. It's dope you can reason with," I said. "It doesn't mess you up."
"Not much it doesn't, bub," he said, and laughed.
My wife sat on the sofa between the blind man and me. I passed her the number. She took it and toked and then passed it back to me. "Which way is this going?" she said. Then she said, "I shouldn't be smoking this. I can hardly keep my eyes open as it is. That dinner did me in. I shouldn't have eaten so much."
"It was the strawberry pie," the blind man said. "That's what did it," he said, and he laughed his big laugh. Then he shook his head.
"There's more strawberry pie," I said.
"Do you want some more, Robert?" my wife said.
"Maybe in a little while," he said.
We gave our attention to the TV. My wife yawned again. She said, "Your bed is made up when you feel like going to bed, Robert. I know you must have had a long day. When you're ready to go to bed, say so." She pulled his arm. "Robert?"
He came to and said, "I've had a real nice time. This beats tapes, doesn't it?"
I said, "Coming at you," and I put the number between his fingers. He inhaled, held the smoke, and then let it go. It was like he'd been doing it since he was nine years old.
"Thanks, bub," he said. "But I think this is all for me. I think I'm beginning to feel it," he said. He held the burning roach out for my wife.
"Same here," she said. "Ditto. Me, too." She took the roach and passed it to me. "I may just sit for a while between you two guys with my eyes closed. But don't let me bother you, okay? Either one of you. If it bothers you, say so. Otherwise, I may just sit here with my eyes closed until you're ready to go to bed," she said. "Your bed's made up, Robert, when you're ready. It's right next to our room at the top of the stairs. We'll show you up when you're ready. You wake me up now, you guys, if I fall asleep." She said that and then she closed her eyes and went to sleep.
The news program ended. I got up and changed the channel. I sat back down on the sofa. I wished my wife hadn't pooped out. Her head lay across the back of the sofa, her mouth open. She'd turned so that her robe had slipped away from her legs, exposing a juicy thigh. I reached to draw her robe back over her, and it was then that I glanced at the blind man. What the hell! I flipped the robe open again.
"You say when you want some strawberry pie," I said.
"I will," he said.
I said, "Are you tired? Do you want me to take you up to your bed? Are you ready to hit the hay?"
"Not yet," he said. "No, I'll stay up with you, bub. If that's all right. I'll stay up until you're ready to turn in. We haven't had a chance to talk. Know what I mean? I feel like me and her monopolized the evening." He lifted his beard and he let it fall. He picked up his cigarettes and his lighter.
"That's all right," I said. Then I said, "I'm glad for the company."
And I guess I was. Every night I smoked dope and stayed up as long as I could before I fell asleep. My wife and I hardly ever went to bed at the same time. When I did go to sleep, I had these dreams. Sometimes I'd wake up from one of them, my heart going crazy.
Something about the church and the Middle Ages was on the TV. Not your run-of-the-mill TV fare. I wanted to watch something else. I turned to the other channels. But there was nothing on them, either. So I turned back to the first channel and apologized.
"Bub, it's all right," the blind man said. "It's fine with me. Whatever you want to watch is okay. I'm always learning something. Learning never ends. It won't hurt me to learn something tonight. I got ears," he said.
We didn't say anything for a time. He was leaning forward with his head turned at me, his right ear aimed in the direction of the set. Very disconcerting. Now and then his eyelids drooped and then they snapped open again. Now and then he put his fingers into his beard and tugged, like he was thinking about something he was hearing on the television.
On the screen, a group of men wearing cowls was being set upon and tormented by men dressed in skeleton costumes and men dressed as devils.
The men dressed as devils wore devil masks, horns, and long tails. This pageant was part of a procession. The Englishman who was narrating the thing said it took place in Spain once a year. I tried to explain to the blind man what was happening.
"Skeletons," he said. "I know about skeletons," he said, and he nodded.
The TV showed this one cathedral. Then there was a long, slow look at another one. Finally, the picture switched to the famous one in Paris, with its flying buttresses and its spires reaching up to the clouds. The camera pulled away to show the whole of the cathedral rising above the skyline.
There were times when the Englishman who was telling the thing would shut up, would simply let the camera move around over the cathedrals. Or else the camera would tour the countryside, men in fields walking behind oxen. I waited as long as I could. Then I felt I had to say something. I said, "They're showing the outside of this cathedral now. Gargoyles. Little statues carved to look like monsters. Now I guess they're in Italy. There's paintings on the walls of this one church."
"Are those fresco paintings, bub?" he asked, and he sipped from his drink.
I reached for my glass. But it was empty. I tried to remember what I could remember. "You're asking me are those frescoes?" I said. "That's a good question. I don't know."
The camera moved to a cathedral outside Lisbon. The differences in the Portuguese cathedral compared with the French and Italian were not that great. But they were there. Mostly the interior stuff. Then something occurred to me, and I said, "Something has occurred to me. Do you have any idea what a cathedral is? What they look like, that is? Do you follow me? If somebody says cathedral to you, do you have any notion what they're talking about? Do you know the difference between that and a Baptist church, say?"
He let the smoke dribble from his mouth. "I know they took hundreds of workers fifty or a hundred years to build," he said. "I just heard the man say that, of course. I know generations of the same families worked on a cathedral. I heard him say that, too. The men who began their life's work on them, they never lived to see the completion of their work. In that wise, bub, they're no different from the rest of us, right?" He laughed. Then his eyelids drooped again. His head nodded. He seemed to be snoozing. Maybe he was imagining himself in Portugal. The TV was showing another cathedral now.
This one was in Germany. The Englishman's voice droned on. "Cathedrals," the blind man said. He sat up and rolled his head back and forth. "If you want the truth, bub, that's about all I know. What I just said. What I heard him say. But maybe you could describe one to me? I wish you'd do it. I'd like that. If you want to know, I really don't have a good idea."
I stared hard at the shot of the cathedral on the TV. How could I even begin to describe it? But say my life depended on it. Say my life was being threatened by an insane guy who said I had to do it or else.
I stared some more at the cathedral before the picture flipped off into the countryside. There was no use. I turned to the blind man and said, "To begin with, they're very tall." I was looking around the room for clues. "They reach way up. Up and up. Toward the sky. They're so big, some of them, they have to have these supports. To help hold them up, so to speak. These supports are called buttresses. They remind me of viaducts, for some reason. But maybe you don't know viaducts, either? Sometimes the cathedrals have devils and such carved into the front. Sometimes lords and ladies. Don't ask me why this is," I said.
He was nodding. The whole upper part of his body seemed to be moving back and forth.
"I'm not doing so good, am I?" I said.
He stopped nodding and leaned forward on the edge of the sofa. As he listened to me, he was running his fingers through his beard. I wasn't getting through to him, I could see that. But he waited for me to go on just the same. He nodded, like he was trying to encourage me. I tried to think what else to say. "They're really big," I said. "They're massive. They're built of stone. Marble, too, sometimes. In those olden days, when they built cathedrals, men wanted to be close to God. In those olden days, God was an important part of everyone's life. You could tell this from their cathedral-building. I'm sorry," I said, "but it looks like that's the best I can do for you. I'm just no good at it."
"That's all right, bub," the blind man said. "Hey, listen. I hope you don't mind my asking you. Can I ask you something? Let me ask you a simple question, yes or no. I'm just curious and there's no offense. You're my host. But let me ask if you are in any way religious? You don't mind my asking?"
I shook my head. He couldn't see that, though. A wink is the same as a nod to a blind man. "I guess I don't believe in it. In anything. Sometimes it's hard. You know what I'm saying?"
"Sure, I do," he said.
"Right," I said.
The Englishman was still holding forth. My wife sighed in her sleep. She drew a long breath and went on with her sleeping.
"You'll have to forgive me," I said. "But I can't tell you what a cathedral looks like. It just isn't in me to do it. I can't do any more than I've done."
The blind man sat very still, his head down, as he listened to me.
I said, "The truth is, cathedrals don't mean anything special to me. Nothing. Cathedrals. They're something to look at on late-night TV. That's all they are."
It was then that the blind man cleared his throat. He brought something up. He took a handkerchief from his back pocket. Then he said, "I get it, bub. It's okay. It happens. Don't worry about it," he said. "Hey, listen to me. Will you do me a favor? I got an idea. Why don't you find us some heavy paper? And a pen. We'll do something. We'll draw one together. Get us a pen and some heavy paper. Go on, bub, get the stuff," he said.
So I went upstairs. My legs felt like they didn't have any strength in them. They felt like they did after I'd done some running. In my wife's room, I looked around. I found some ballpoints in a little basket on her table. And then I tried to think where to look for the kind of paper he was talking about.
Downstairs, in the kitchen, I found a shopping bag with onion skins in the bottom of the bag. I emptied the bag and shook it. I brought it into the living room and sat down with it near his legs. I moved some things, smoothed the wrinkles from the bag, spread it out on the coffee table.
The blind man got down from the sofa and sat next to me on the carpet.
He ran his fingers over the paper. He went up and down the sides of the paper. The edges, even the edges. He fingered the corners.
"All right," he said. "All right, let's do her."
He found my hand, the hand with the pen. He closed his hand over my hand. "Go ahead, bub, draw," he said. "Draw. You'll see. I'll follow along with you. It'll be okay. Just begin now like I'm telling you. You'll see. Draw," the blind man said.
So I began. First I drew a box that looked like a house. It could have been the house I lived in. Then I put a roof on it. At either end of the roof, I drew spires. Crazy.
"Swell," he said. "Terrific. You're doing fine," he said. "Never thought anything like this could happen in your lifetime, did you, bub? Well, it's a strange life, we all know that. Go on now. Keep it up."
I put in windows with arches. I drew flying buttresses. I hung great doors. I couldn't stop. The TV station went off the air. I put down the pen and closed and opened my fingers. The blind man felt around over the paper. He moved the tips of his fingers over the paper, all over what I had drawn, and he nodded.
"Doing fine," the blind man said.
I took up the pen again, and he found my hand. I kept at it. I'm no artist. But I kept drawing just the same.
My wife opened up her eyes and gazed at us. She sat up on the sofa, her robe hanging open. She said, "What are you doing? Tell me, I want to know."
I didn't answer her.
The blind man said, "We're drawing a cathedral. Me and him are working on it. Press hard," he said to me. "That's right. That's good," he said. "Sure. You got it, bub, I can tell. You didn't think you could. But you can, can't you? You're cooking with gas now. You know what I'm saying? We're going to really have us something here in a minute. How's the old arm?" he said. "Put some people in there now. What's a cathedral without people?"
My wife said, "What's going on? Robert, what are you doing? What's going on?"
"It's all right," he said to her. "Close your eyes now," the blind man said to me.
I did it. I closed them just like he said.
"Are they closed?" he said. "Don't fudge."
"They're closed," I said.
"Keep them that way," he said. He said, "Don't stop now. Draw."
So we kept on with it. His fingers rode my fingers as my hand went over the paper, It was like nothing else in my life up to now.
Then he said, "I think that's it. I think you got it," he said. "Take a look. What do you think?"
But I had my eyes closed. I thought I'd keep them that way for a little longer. I thought it was something I ought to do.
"Well?" he said. "Are you looking?"
My eyes were still closed. I was in my house. I knew that. But I didn't feel like I was inside anything.
"It's really something," I said.
ACKNOWLEDGMENTS
_Writing Fiction_ emerged from the experience and insight of ten Gotham Writers' Workshop teachers and our dean of faculty, Alexander Steele, who edited and oversaw this project with his customary keen eye and grace under pressure.
This book also represents the concerted efforts of Andre Becker, president of Gotham Writers' Workshop, Faith Hamlin of Sanford Greenburger Associates, Colin Dickerman at Bloomsbury USA, and Nikki Moustaki, for her efforts in the initial stages of this project and for lending us her publishing and editorial expertise.
Moreover, we recognize and gratefully thank the dedicated Gotham staff: Joel Mellin, Dana Miller, Linda Novak, Betsey Odell, Stacey Panousopoulos, and Charlie Shehadi.
_Writing Fiction_ is also indebted to the talented Gotham teachers who, over the years, have helped to distill and define the craft of fiction writing into a form that is easily understood. Finally, this book is dedicated to our students, who continually challenge and inspire us all.
CONTRIBUTORS
**Allison Amend** has published short fiction in _Other Voices, Story Quarterly, One Story, Arts and Letters,_ and _Atlantic Unbound,_ and serves as fiction editor at _StoryQuarterly_ magazine. She lives in New York City.
**Terry Bain** has published short fiction in _Book Magazine, Prize Stories 1994: The O. Henry Awards,_ and _The Gettysburg Review,_ among others, and he writes humor pieces for _Sweet Fancy Moses._ He lives in Spokane, Washington.
**David Harris Ebenbach** has published short fiction in such magazines as _Denver Quarterly,_ the _Beloit Fiction Journal,_ and the _Crescent Review,_ and his poetry has appeared in _La Petite Zine_ and the _Red River Review._ He lives in New York City.
**Hardy Griffin** has published short fiction in _The Hangman's Lime_ and _Ox,_ and his nonfìction has appeared in _The Washington Post, American Letters & Commentary,_ and Fodors.com _._ He lives in Istanbul, Turkey.
**Caren Gussoff** is the author of the novel _Homecoming_ (Serpent's Tail) and the short-story collection _The Wave and Other Stories_ (Serpent's Tail). She lives in Seattle, Washington.
**Corene Lemaitre** is the author of the novel _April Rising_ (HarperCollins UK and Carroll & Graf US) and was awarded first prize in the World Wide Writers international short fiction competition. She lives in Wayne, Pennsylvania.
**Chris Lombardi** has published short fiction in _minnesota review, Anything That Moves, living room,_ and assorted anthologies, including _HEY PAESAN! Lesbians and Gays of Italian Descent._ Her nonfiction has appeared in _The Nation, Ms, Poets & Writers,_ and _Inside MS,_ among others. She lives in New York City.
**Brandi Reissenweber** has published short fiction in such magazines as _Rattapallax_ and _Aspects,_ serves as an editorial assistant at _Zoetrope: All-Story,_ and founded a therapeutic writing workshop at Safespace, a center for homeless teens. She lives in Chicago, Illinois.
**Peter Selgin** has published short fiction in _Glimmer Train_ and _Salon,_ among others. His artwork has appeared in such publications as _The New Yorker_ and _The Wall Street Journal,_ and he is the author/illustrator of the children's book S.S. _Gigantic Across the Atlantic_ (Simon & Schuster). He lives in New York City.
**Alexander Steele** serves as dean of faculty of Gotham Writers' Workshop. He has written numerous plays, screenplays, and nonfiction pieces, and has published seventeen books for children. He lives in New York City.
**Valerie Vogrin** is the author of the novel _Shebang_ (University of Mississippi Press), has published short fiction in such magazines as _New Orleans Review_ and _Black Warrior Review_ , and cofounded Smallmouth Press. She lives in St. Louis, Missouri.
**Gotham Writers' Workshop**
**Study Online with Gotham Writers' Workshop**
Since 1997, thousands of students have discovered how simple and effective it is to study online with Gotham Writers' Workshop at www.writingclasses.com. The school's online workshops feature everything you would find in a live workshop—lectures, discussions, writing exercises, teacher feedback, and in-depth critiques of your work. Every class is taught in an easy-to-understand manner by a member of the school's renowned faculty. Class size is limited so the focus remains on you and your writing. All of which may explain why _Forbes_ selected these classes as the "Best of the Web."
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Redemption details: To receive a $40 discount off the regular ten-week workshop tuition, you must mention promotion code WBK40 when you enroll. Valid for 10-week workshop registration by first-time GWW students only. Savings may not be combined with any other offer.
**Looking for exceptional stories to read or study?
_Fiction Gallery_ is the perfect companion to _Writing Fiction_**
Gotham Writers' Workshop is pleased to introduce _Fiction Gallery,_ an anthology of excellent and accessible short stories. Every story will hold the reader spellbound from first to last page, while also exemplifying the very best in literary fiction.
The twenty-five authors include such acknowledged masters of short fiction as Anton Chekhov, Dorothy Parker, John Cheever, and Raymond Carver, and such acclaimed contemporary writers as Edwidge Danticat, Pam Houston, Ethan Canin, T. C. Boyle, Jhumpa Lahiri, and ZZ Packer.
Aspiring writers will find this anthology an invaluable source of inspiration and instruction. This gallery of stories presents diverse examples of all the elements of fiction craft and demonstrates how writers seamlessly sew these elements into unforgettable tales.
As a bonus, the anthology includes original interviews with T. C Boyle, Jhumpa Lahiri, and Hannah Tinti, in which they illuminate the process of creating a short story.
For writers and readers alike, _Fiction Gallery_ is a book to be treasured.
**Preview stories and learn more atwww.FictionGaliery.com**
"Fiction Gallery _is a wonderful selection for the general reader and writing student alike. The editors have chosen stories with a great range of period, style, form, and subject, stories that brilliantly convey the possibilities of the genre."_
—Thomas C. Foster, author of _How to Read Literature Like a Professor_
_"A stellar collection of 25 stories..._ Fiction Gallery _offers both avid readers and those looking for a smorgasbord a hearty repast. Highly recommended."_ — _Library Journal_
_"If you find yourself on a long journey,_ Fiction Gallery _is the perfect companion. And even if you're not traveling anywhere, these stories will transport you."_ —Tamara Straus. Editor in Chief. _Zoetrope: All-Story_
_"Reading_ Fiction Gallery _is like taking a seminar from the masters of short story writing. Inspiring and instructive for all writers!"_ —Elfrieda Abbe, Editor, _The Writer_
Copyright © 2003 by Gotham Writers' Workshop
"Cathedral" is reprinted from _Cathedral_ by Raymond Carver, copyright © 1983 by Raymond Carver. Used by permission of
Alfred A. Knopf, a division of Random House, Inc.
All rights reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission from the Publisher except in the case of brief quotations embodied in critical articles or reviews. For information address Bloomsbury, 175 Fifth Avenue, New York, NY 10010.
Published by Bloomsbury, New York and London
Library of Congress Cataloging-in-Publication Data
Writing fiction : the practical guide from New York's acclaimed creative writing school / written by Gotham Writers' Workshop faculty ; edited by Alexander Steele.—1st U.S. ed.
p. cm.
Includes index.
ISBN 1-58234-330-6 (pbk.)
ISBN-13 978-1-58234-330-3
1. Fiction—Authorship. I. Steele, Alexander. II. Gotham Writers' Workshop.
PN3355.W75 2003
808.3—dc22
2003057739
First U.S. Edition 2003
20 19 18 17 16 15 14 13
Interior and cover design by 2x4, New York City
Typeset by Palimpsest Book Production Limited,
Polmont, Stirlingshire, Scotland
Printed in the United States of America by
R. R. Donnelley & Sons, Harrisonburg, Virginia
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superhero in New York? "
Below are the entries for Best UX Design from the Brooklyn Adobe Creative Jam. Creatives were given the theme and had 3 hours to create a UX Design for iPhone 6 and were asked to include Adobe Experience Design in their workflow. Check out their designs!
"When every second counts, only Hero Hotline gives you directions that harness your special ability to get you to your destination through free-running parkour, up shopping mall escalators, down clean chimneys, or swinging from rooftop to rooftop. Please ensure your insurance information is up to date before downloading."
Read full review from the designers.
"In NYC no one reports pedestrian related issues like construction, live wires or fire hydrants going off. We've created an app that let's you be a here of your own neighborhood by letting other know where the issues are. It would allow the police to see complaints that don't get called in as well. The below demo is of a recurring users, Jessica, reporting a live wire and seeing what other heroes are around."
"Be a hero, or call a hero with BEACON."
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"redpajama_set_name": "RedPajamaC4"
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Disorder-induced topological change of the superconducting gap structure in iron pnictides.
Mizukami Y1, Konczykowski M2, Kawamoto Y3, Kurata S1, Kasahara S3, Hashimoto K4, Mishra V5, Kreisel A6, Wang Y6, Hirschfeld PJ6, Matsuda Y3, Shibauchi T1.
1] Department of Physics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan [2] Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277-8561, Japan.
Laboratoire des Solides Irradiés, CNRS-UMR 7642 &CEA-DSM-IRAMIS, Ecole Polytechnique, F 91128 Palaiseau cedex, France.
Department of Physics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan.
1] Department of Physics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan [2] Institute for Materials Research, Tohoku University, Aoba-ku, Sendai 980-8577, Japan.
Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA.
Department of Physics, University of Florida, Gainesville, Florida 32611, USA.
In superconductors with unconventional pairing mechanisms, the energy gap in the excitation spectrum often has nodes, which allow quasiparticle excitations at low energies. In many cases, such as in d-wave cuprate superconductors, the position and topology of nodes are imposed by the symmetry, and thus the presence of gapless excitations is protected against disorder. Here we report on the observation of distinct changes in the gap structure of iron-pnictide superconductors with increasing impurity scattering. By the successive introduction of nonmagnetic point defects into BaFe2(As(1-x)P(x))(2) crystals via electron irradiation, we find from the low-temperature penetration depth measurements that the nodal state changes to a nodeless state with fully gapped excitations. Moreover, under further irradiation the gapped state evolves into another gapless state, providing bulk evidence of unconventional sign-changing s-wave superconductivity. This demonstrates that the topology of the superconducting gap can be controlled by disorder, which is a strikingly unique feature of iron pnictides.
10.1038/ncomms6657
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\section{Introduction and outlook}
As is well known, the galactic halo chiefly consists of dark matter.
A natural possibility - repeatedly considered in the past
(Silk \cite{silk}, Carr \cite{carr}) - is that
baryonic dark matter makes a substantial contribution.
A few years ago, we
recognized that the Fall \& Rees
theory for the formation of
globular clusters
(Fall \& Rees \cite{fall},
Kang et al. \cite{kang},
Vietri \& Pesce \cite{vietri})
also leads to the existence of dark clusters
- made of brown dwarfs
\footnote{
Although we concentrate our attention on brown dwarfs, it should be
mentioned that red dwarfs as well can be accommodated within the
considered scenario.}
and cold self-gravitating clouds -
at galactocentric distances $R \ut > 10$ kpc
(De Paolis et al. \cite{depaolis1}-\cite{depaolis4},
\cite{depaolisapj})
\footnote{Similar models have also been
proposed by
Ashman \& Carr (\cite{ac}), Ashman (\cite{ashman}), Fabian and Nulsen
(\cite{fn1,fn2}), Gerhard \& Silk (\cite{gs}) and
Kerins (\cite{kerins1,kerins2}).}.
Accordingly, the inner halo is populated by globular clusters whereas the
outer halo is dominated by dark clusters.
Contrary to the case of globular clusters, a large amount of residual gas
should remain clumped into the dark clusters, as brown
dwarfs fail to generate the strong stellar winds which expel the leftover
gas from globular clusters. Moreover, although the clouds under
consideration are mainly made of $H_2$,
we expect them to be surrounded by an atomic layer and a
photo-ionized ``skin'' (De Paolis et al. \cite{depaolisapj}).
We stress that the presence of cold self-gravitating clouds in the
halo is a characteristic feature of the model in question.
Remarkably enough, quite
recently it has been pointed out (Walker \& Wardle \cite{ww}) that cold
self-gravitating clouds of the considered kind naturally explain the
``extreme scattering events'' associated with compact radio quasars
(Fiedler et al. \cite{fiedler}).
Our proposal was also motivated by the discovery
of microlensing events towards the LMC (Alcock et al.
\cite{alcock1,alcock2},
Aubourg et al. \cite{aubourg}).
The first-year data were manifestly
consistent with the assumption that MACHOs are brown dwarfs
even within the standard (isothermal) galactic model. Unfortunately,
the present situation is much less clear.
An option is that the halo resembles more closely a maximal disk rather than
an isothermal sphere, in which case MACHOs can still be brown dwarfs
\footnote{
It should be kept in mind that a large fraction of MACHOs
(up to $\sim 50\%$ in mass) can be binary systems, thereby counting as
twice more massive objects (De Paolis et al. \cite{depaolismnras}).}
(see also Binney \cite{binney}).
A more intriguing possibility has recently been suggested by
Kerins \& Evans (\cite{ke})
and naturally fits within our model. As the initial mass
function evidently changes with the galactic distance $R$, it can well
happen that brown dwarfs dominate the halo mass density without
however dominating the optical depth for microlensing \footnote{Notice that
also the considered clouds can contribute to microlensing events (Draine
\cite{draine}).}.
A few months ago, Dixon et al. (\cite{dixon}) have re-analyzed the EGRET data
concerning the diffuse $\gamma$-ray flux with a wavelet-based technique.
After subtraction of the isotropic extragalactic component and of the
expected contribution from the Milky Way, they find a statistically
significant diffuse emission from the galactic halo. At high-galactic
latitude, the integrated halo flux above 1 GeV turns out to be
$\simeq 10^{-7}-10^{-6}$ $\gamma$ cm$^{-2}$ s$^{-1}$ sr$^{-1}$,
which is slightly less than the
diffuse extragalactic flux (Sreekumar et al. \cite{sreekumar}).
Our aim is to show that the diffuse $\gamma$-ray
emission from the galactic halo discovered
by Dixon et al. (\cite{dixon}) can naturally
be explained within the considered model.
Basically, the idea is that cosmic-ray (CR) protons in the
galactic halo scatter on
halo clouds, thereby producing the observed $\gamma$-ray flux.
\section{Cosmic ray confinement in the galactic halo}
Unfortunately, neither theory nor observation allow nowadays to
make sharp statements about the propagation of CRs in the galactic
halo \footnote {We stress
that - contrary to the practice used in the CR community -
by halo we mean the (almost)
spherical galactic component which extends beyond
$\sim$ 10 kpc.}.
Therefore, the only possibility to get some insight into this issue
rests upon the extrapolation from the knowledge of CR propagation in the
disk. Actually, this strategy looks sensible, since the leading effect is CR
scattering on inhomogeneities of the magnetic field over scales
from $10^2$ pc down to less than $10^{-6}$ pc
(Berezinsky et al. \cite{berezinskii})
and - according to our model - inhomogeneities
of this kind are expected to be present in the halo,
because of the existence of gas clouds - with a photo-ionized ``skin'' -
clumped into dark clusters
\footnote{
Indeed, typical values of the dark cluster radius are
$\sim 10$ pc, whereas typical values of the cloud radius
are $\sim 10^{-5}$ pc (De Paolis et al. \cite{depaolisapj}).}.
As is well known, CRs up to energies of $\sim 10^6$ GeV
are confined in the galactic disk for $\sim 10^7$ yr.
CRs escaping from the disk will further diffuse in the galactic
halo, where they can be retained for a long time,
owing to the scattering on the above-mentioned
small inhomogeneities of the halo magnetic field
\footnote{A similar idea has been proposed with a somewhat different
motivation by Wdowczyk \& Wolfendale (\cite{wolf}).}.
Indirect evidence that CRs are in fact trapped in a low-density
halo has recently been reported. For example, Simpson and Connell
(\cite{simpson}) argue that, based on measurements of isotopic abundances
of the cosmic ratio $^{26}$Al/$^{27}$Al, the CR lifetimes are perhaps a
factor of four larger than previously thought, thereby implying that CRs
traverse an average density smaller than that of the galactic disk.
A straightforward extension of the diffusion model
(Berezinsky et al. \cite{berezinskii})
implies that the CR escape time $\tau_{\rm esc}^{~H}$ from the halo
(of size $R_H$ much larger than the disk half-thickness) is given by
\begin{equation}
\tau_{\rm esc}^{~H} \simeq \frac{R_H^2}{3D_H(E)}~,
\label{tau}
\end{equation}
where $D_H(E)$ is the diffusion coefficient.
We recall that - for CR propagation in the disk - the diffusion coefficient is
$D(E) \simeq D_0~(E/7~ GeV)^{0.3}$ cm$^2$ s$^{-1}$ in the ultra-relativistic
regime, whereas it reads
$D(E) \simeq D_0 \simeq 3 \times 10^{28}$ cm$^2$ s$^{-1}$ in
the non-relativistic regime (see Berezinsky et al. \cite{berezinskii}).
As a matter of fact, radio observations in clusters of galaxies
yield for the corresponding diffusion constant $D_0$
a value similar to that found in the galactic
disk (Schlickeiser, Sievers \& Thiemann \cite{sst})
\footnote{Moreover, we note that average magnetic field
values in galactic halos
are expected to be close to those of galaxy clusters,
i.e. in the range 0.1 - 1 $\mu$G (Hillas \cite{hillas}).}.
So, it looks plausible that a similar value for $D_0$
also holds on intermediate scale lengths, namely within the galactic halo.
In the lack of any further information on the energy-dependence of $D_H(E)$,
we assume the same dependence as that established for the disk.
Hence, from eq. (\ref{tau}) - with $R_H \sim 100$ kpc -
we find that for energies $E \ut < 10^3$ GeV the escape
time of CRs from the halo is greater than the age of the Galaxy
$t_0 \simeq 10^{10}$ yr
(notice that below the ultra-relativistic regime
$\tau_{\rm esc}^{~H}$ gets even longer).
As a consequence - since the CR flux
scales like $E^{-2.7}$ (see next Section) - protons with $E \ut < 10^3$
GeV turn out to give the leading contribution to the CR flux.
We are now in position to evaluate the CR energy density in the halo, getting
\begin{equation}
\rho_{CR}^{~H} \simeq \frac{3 t_0 L_G }{4 \pi R_H^3} \simeq 0.12
~~~~~{\rm eV~cm^{-3}}~,
\label{hcrd}
\end{equation}
where
$L_G \simeq 10^{41}$ erg s$^{-1}$ is the galactic CR luminosity.
Notice, for comparison, that $\rho_{CR}^{~H}$ turns out to be about
one tenth of the disk value (Gaisser \cite{gaisser}).
We remark that we have taken specific realistic values for the various
parameters entering the above equations, in order to make a quantitative
estimate.
However, somewhat different values can be used. For instance,
$R_H$ may range up to $\sim 200$ kpc (Bahcall, Lubin \& Dorman
\cite{bld}),
whereas $D_0$ might be as large as $\simeq 10^{29}$ cm$^2$ s$^{-1}$
consistently with our assumptions. Moreover, $L_G$ can be as large
as $3 \times 10^{41}$ erg s$^{-1}$ (V\"olk, Aharonian \& Breitschwerdt
\cite{volk}). It is easy to see
that these variations do not substantially affect our previous conclusions.
\section{Gamma-ray emission from halo clouds}
We proceed to estimate the total $\gamma$-ray flux produced by halo
clouds clumped into the dark clusters through the interaction
with high-energy CR protons.
CR protons scatter on cloud protons giving rise (in particular) to pions,
which subsequently decay into photons.
We
expect negligible high-energy ($\geq$ 100 MeV) $\gamma$-ray photon
absorption outside the clouds,
since the mean free path is orders of magnitudes larger than the
halo size.
An essential ingredient is the knowledge of both $\rho^{~H}_{CR}$
and the CR spectrum $\Phi_{CR}^{~H}(E)$ in the galactic halo.
According to the discussion in the previous Section,
we take $\rho^{~H}_{CR}\simeq 0.12$ eV cm$^{-3}$.
As far as $\Phi_{CR}^{~H}(E)$ is concerned,
we adopt the following power-law
\begin{equation}
\Phi_{CR}^{~H}(E) \simeq \frac{A}{{\rm GeV}}
\left(\frac{E}{{\rm GeV}}\right)^{-\alpha}~~~
{\rm particles~cm^{-2}~s^{-1}~sr^{-1}}~,
\label{eqno:42}
\end{equation}
where the constant $A$ is fixed by the requirement that the integrated
energy flux (in the range $1~ {\rm GeV} \leq E \leq 10^3~ {\rm GeV}$)
agrees with the above value of $\rho^{~H}_{CR}$.
The choice of $\alpha$ is nontrivial. As an orientation, the observed
spectrum of primary CRs on Earth entails
$\alpha \simeq 2.7$. However, this conclusion cannot be extrapolated
to an arbitrary region in the halo (and in the disk), since
$\alpha$ crucially depends on the diffusion processes undergone by
CRs. For instance, the best fit to EGRET data in the disk towards
the galactic centre yields $\alpha \simeq 2.45$ (Mori \cite{mori}),
thereby showing that $\alpha$ increases as a consequence of diffusion.
In the lack of any direct information, we conservatively
take $\alpha \simeq 2.7$
even in the halo, but in the Table we report for comparison some results
for different
values of $\alpha$.
As can be seen, the flux does not vary substantially.
Let us next turn our attention to the evaluation of the $\gamma$-ray flux
produced in halo clouds
through the reactions $pp \rightarrow \pi^0 \rightarrow \gamma
\gamma$. The source function $q_{\gamma}(>E_{\gamma},\rho,l,b)$
- yielding the photon number density at distance
$\rho$ from Earth in the direction $(l,b)$ with energy $>E_{\gamma}$ - is
\begin{equation}
q_{\gamma}(>E_{\gamma},\rho,l,b)=
\frac{4\pi}{m_p}\rho_{H_2}(\rho,l,b)
\int_{E_p(E_{\gamma})}^{\infty}
d\bar{E}_p~ \Phi^{~H}_{CR}(\bar{E}_p)~
\sigma_{in}(p_{lab}) <n_{\gamma}(\bar{E}_p)>~~~
{\rm \gamma~cm^{-3}~s^{-1}}~,
\label{eqno:49}
\end{equation}
where the lower integration limit $E_p(E_{\gamma})$ is the minimal proton
energy necessary to produce a photon with energy $>E_{\gamma}$,
$\sigma_{in}(p_{lab})$ is the inelastic pion production cross-section,
$n_{\gamma}(\bar{E}_p)$ is the photon multiplicity
\footnote{
For the inclusive cross-section of the reaction
$pp \rightarrow \pi^{0} \rightarrow \gamma \gamma$
we employ the parameterization given by Dermer (\cite{dermer}).}
and $\rho_{H_2}(\rho,l,b)$ is the halo gas density profile
\footnote{
As it would be exceedingly difficult to keep track of the
clumpiness of the actual gas distribution, we assume that its
density goes like the dark matter density - anyhow, the very low
angular resolution of $\gamma$-ray detectors would not permit to
distinguish between the two situations.},
which reads
\footnote{
As usual, we use the coordinate transformation
$x = -\rho \cos b \cos l +R_0$,
$y = -\rho \cos b \sin l$ and
$z = \rho \sin b$,
where $R_0 = 8.5$ kpc is our galactocentric distance.}
\begin{equation}
\rho_{H_2}(x,y,z) = f~ {\rho_0 (q)} ~ \frac{a^2+R_0^2}{a^2+x^2+y^2+(z/q)^2}~,
\label{eqno:29}
\end{equation}
for $\sqrt{ x^2+y^2+z^2/q^2} > R_{min}$, otherwise it vanishes. Here
$R_{min} \simeq 10$ kpc is the minimal galactocentric distance of
the dark clusters
in the galactic halo,
$f$ denotes the fraction of halo dark matter in the form of gas,
$\rho_0(q)$ is the local dark matter density,
$a = 5.6$ kpc is the core radius and $q$ parametrizes
the halo flattening. For the standard spherical halo model
$\rho_0(q=1) \simeq 0.3$ GeV cm$^{-3}$, whereas it turns out that e.g.
$\rho_0(q=0.5) \simeq 0.6$ GeV cm$^{-3}$.
Because $dV=\rho^2 d\rho d\Omega$, it follows that the
$\gamma$-ray flux per unit solid angle produced in halo clouds and
observed on Earth from the direction $(l,b)$ is
\begin{equation}
\Phi_{\gamma}^{~ \rm DM}
(>E_{\gamma},l,b)=\frac{1}{4\pi}
\int^{\rho_2(l,b)}_{\rho_1(l,b)} d\rho~ q_{\gamma}(>E_{\gamma},\rho,l,b)
~~~\gamma~{\rm cm^{-2}~s^{-1}~sr^{-1}}~,
\label{eqno:51}
\end{equation}
where typical
values of $\rho_1$ and $\rho_2$ are 10 kpc and 100 kpc, respectively.
\section{Discussion and conclusions}
Our main result - which follows directly from eq. (\ref{eqno:51}) -
are maps for the intensity distribution of the $\gamma$-ray emission from
baryonic dark matter in the halo. In order to make the discussion definite,
we take $f \simeq 0.5$.
In Figure 1 we show the contour plots in the first quadrant of the
sky ($0^0 \le l \le 180^0$, $0^0 \le b \le 90^0$) for
the $\gamma$-ray flux at energy $E_{\gamma}>1$ GeV
$\Phi_{\gamma}^{~\rm DM} (> 1 {~\rm GeV})$.
Corresponding contour plots for $E_{\gamma}>0.1$ GeV are identical,
up to an overall constant factor equal to 8.74 (again, this follows
from eq. (\ref{eqno:51})).
Figure 1a refers to a spherical halo, whereas Figure 1b pertains to a $q=0.5$ flattened halo.
We see that - regardless of the adopted value for $q$ -
at high-galactic
latitude $\Phi_{\gamma}^{~\rm DM}(>1{\rm ~GeV})$ lies in the range
$\simeq 6-8 \times 10^{-7}$ $\gamma$ cm$^{-2}$ s$^{-1}$ sr$^{-1}$.
However, the shape of the contour lines strongly depends on
the flatness parameter.
Indeed, for $q \ut > 0.9$ there are two contour lines (for
each flux value) approximately symmetric with respect to $l=90^0$
(see Figure 1a).
On the other hand, for
$q \ut < 0.9$ there is a single contour line (for each value of the flux)
which varies much less with the longitude (see Figure 1b).
As we can see from the Table and the Figures, the predicted
value for the $\gamma$-ray flux at high-galactic latitude
is very close to that found by Dixon et al. (\cite{dixon}).
This conclusion holds almost irrespectively of the flatness parameter.
Moreover, the comparison of the overall shape of the contour lines in our
Figures 1a and 1b with the corresponding ones in Figure 3 of
Dixon et al. (\cite{dixon}) entails that models with flatness parameter
$q \ut < 0.8$ are in better agreement with data, thereby
implying that most likely
the halo dark matter is not spherically distributed.
We remark that eq. (\ref{eqno:51}) yields
$\Phi_{\gamma}^{~\rm DM} (> 0.1 {\rm~GeV}) \simeq 5.9 \times
10^{-6}$ $~\gamma$ s$^{-1}$ cm$^{-2}$ sr$^{-1}$ at high-galactic latitude
(for a spherical halo).
This value is roughly 40\% of the diffuse extragalactic $\gamma$-ray emission
of $1.45 \pm 0.05 \times 10^{-5}$
$~\gamma$ s$^{-1}$ cm$^{-2}$ sr$^{-1}$ found by the EGRET team
(Sreekumar et al. \cite{sreekumar}).
So, our result supports the conclusion of Dixon et al. (\cite{dixon})
that the halo $\gamma$-ray emission is a relevant fraction of
the isotropic diffuse flux also
for $E_{\gamma} > 0.1{\rm~GeV}.$
Before closing this Letter, we would like to briefly
address the crucial question whether the newly discovered $\gamma$-ray
halo emission really calls for a dark matter source. For, one might suspect
that a nonstandard inverse-Compton $\gamma$-ray production
mechanism could explain the data
(owing to the large uncertainties both in the electron hight scale
and in the electron injection spectral index). However, this seems
not to be the case.
Basically, the inverse-Compton contour lines decrease much more rapidly
than the observed ones. Hence, it would be impossible
to explain in this manner the $\gamma$-ray flux found
by Dixon et al. (\cite{dixon}) while still correctly accounting for the
observed disk emission (Sreekumar et al. \cite{sreekumar}).
A more detailed account of this topic will be presented elsewhere.
In conclusion, we feel that - in spite of the various uncertainties -
the remarkably good agreement
between theory and experiment makes our model for halo dark matter worth
further consideration.
In particular, the next generation of $\gamma$-ray satellites like
AGILE and GLAST can test our prediction, thanks to the higher sensitivity
and the better angular resolution. In this respect, it might be interesting
to measure whether there is an enhancement in the $\gamma$-ray
flux towards the nearby M31 galaxy, since we expect a similar mechanism for the
$\gamma$-ray production to hold in its halo as well.
\acknowledgments
The work of FDP is supported by an INFN grant.
We would like to thank G. Bignami, P. Caraveo, D. Dixon,
T. Gaisser, M. Gibilisco,
G. Kanbach, T. Stanev, A. Strong and M. Tavani for useful discussions.
|
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Konvoj je skupina vozil (kateregakoli tipa, toda po navadi motornih vozil ali plovil), ki potujejo skupaj zaradi medsebojne podpore. Pogosto je konvoj organiziran za oboroženo zaščito, lahko pa se konvoj uporablja tudi v nevojaške namene, na primer za vožnjo skozi nevarne oz. težko prehodne predele. Če se eno od vozil pokvari ali se ustavi, mu lahko druga vozila nudijo pomoč pri popravilu, vleki ali poskušajo najti vozilo, ki se je ločilo od ostalih. Če popravilo ni mogoče, lahko potnike oz. tovor iz pokvarjenega vozila premestijo na druge.
Pomorski konvoji
Obdobje jadrnic
Plovba jadrnic v konvojih je stara toliko kot zgodovina pomorskega raziskovanja. Že staroegipčanski zapisi iz časa vladavine kraljice Hačepsut (skoraj 1500 let pr. n. št) ter reliefi iz templja v Tebah v Egiptu pričajo o drznem in uspešnem potovanju egipčanskih pomorščakov v deželo Punt, na katerem je plulo več jadrnic skupaj, torej v konvoju.
Tudi sloveči portugalski pomorščaki princa Henrika Pomorščaka (1394-1460) npr. admiral Vasco da Gama, so pri svojem odkrivanju obal Zahodne in Ekvatorialne Afrike pluli po neznanih morjih v konvojih. Krištof Kolumb je na svojem potovanju, na katerem je odkril Ameriko, plul v konvoju, v katerih so bile karavele Santa Maria, Pinta in Niña.
Zunanje povezave
Oznake konvojev v obeh svetovnih vojnah
Transport
|
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{"url":"https:\/\/www.physicsforums.com\/threads\/post-measurement-states-in-generalized-probabilistic-theories.529973\/","text":"Post-measurement states in generalized probabilistic theories\n\n1. Sep 14, 2011\n\nAmeno\n\nHello everybody\n\nI am currently trying to understand attempts to create a framework of generalized probabilistic theories in which quantum theory and classical theory appear as special cases. More precisely, I try to understand the framework which is sometimes called the framework of \"convex theories\", in which the set of states is considered to be a convex subset an ordered vector space. For those not familiar with this framework, have a look at e.g. the following papers:\n\u2022 H. Barnum: Ordered linear spaces and categories as frameworks for information-processing characterizations of quantum and classical theory, Arxiv preprint arXiv:0908.2354\n\u2022 J. Barret et al.: Limits on nonlocal correlations from the structure of the local state space, New Journal of Physics 13 (2011) 063024 (24pp)\n\u2022 O. Dahlsten et al.: Unifying typical entanglement and coin tossing: on randomization in probabilistic theories, Arxiv preprint arXiv:1107.6029\n\nThere is one thing I don't understand. So far, I have not found a paper in which it is explained how performing a measurement changes the state of a system. I'm trying to establish a rule for updating the state from the pre- to the post-measurement state. Let me tell you what I have come up with so far:\n\nI think it is clear that one cannot give a rule for updating the state for every effect of a theory (see papers above to learn what it means to be an effect). If this were possible, then one could assign a post-measurement state to every element of a POVM in quantum mechanics. But in the case of a POVM which is not a projective measurement, this is not possible, because if one regards this POVM as being induced by a measurement on a larger system containing the system in question as a subsystem, then the post-measurement state might depend on the state of the ancilla system which is needed to extend the system in question to the larger system.\nBut (still talking about the quantum case) one can assign a post-measurement state to the system if the the POVM is a projective measurement. Noting that a POVM element is a projector if and only if it is an extremal point of the set of proper effects, one might hope that one can generalize the state-update-rule to the case of a generalized probabilistic theory.\n\nOne property which, I would say, is save to demand from a post-measurement state is the following one:\nIf one measures a state $\\rho$ and gets the result corresponding to effect $f$, then the post-measurement state $\\rho_{post}$ should satisfy $f(\\rho_{post}) = 1$.\nThe problem is that in general, this condition does not specify $\\rho_{post}$ uniquely. In the quantum case, the condition specifies $\\rho_{post}$ uniquely if $f$ is induced by a one-dimensional projector, but it does not in the case where it is a 2- or higher-dimensional projector.\n\nNow one might say the following (I hope this claim is correct; i haven't proved it so far): In the quantum case, $f(\\rho_{post})=1$ specifies $\\rho_{post}$ uniquely (i.e. $f$ is a one-dimensional projector) if and only if $f$ (in addition to being an extreme point of the set of proper effects) is ray extremal (i.e. lies on an extremal ray of $V^*_+$). In the classical and quantum case, restriction to the set of extremal and ray extremal points of the set of proper effects is sufficient to get unique post-measurement states by the condition $f(\\rho_{post})=1$.\nBut again, this does not suffice. In boxworld (see the n=4 polygon theory of the second paper listed above), there are extremal and ray-extremal effects for which $f(\\rho_{post})=1$ is satisfied for a whole 1-dimensional face of the set of states.\n\nAfter all, I don't know how to establish a rule for assigning post-measurement states in generalized probabilistic theories. Can anyone help me? Any ideas or even solutions to this problem?","date":"2018-12-15 23:23:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.8315339684486389, \"perplexity\": 250.19987904325524}, \"config\": {\"markdown_headings\": false, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-51\/segments\/1544376827137.61\/warc\/CC-MAIN-20181215222234-20181216004234-00621.warc.gz\"}"}
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\section{Introduction}
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/Intro_YD.pdf}
\caption{An illustration of a novel Federated Zero-Shot Learning (\texttt{FedZSL}) framework, which aims to learn a central model from multiple participants. The local data on edge devices follows the partial class-conditional distribution (\textit{p.c.c.d}), which is far from \textit{i.i.d.} that conventional federated learning models assume.}
\label{fig:intro}
\end{figure}
With the ever-growing number of novel concepts, the ability to faithfully recognize instances from the previously \textit{seen} and \textit{unseen} classes is highly desired in modern vision systems. In response to this demand, generalized zero-shot learning (GZSL) \cite{zhang2017learning,xian2018feature,liu2018generalized,schonfeld2019generalized} has been emerging as a feasible solution. GZSL approaches learn to classify \textit{seen} classes (\textit{i.e.,} appeared during training and test) and \textit{unseen} classes (\textit{i.e.,} only encountered in testing phase) based on their semantic descriptions without requiring any exemplar of novel classes. The missing piece in the visual-semantic mapping for unseen classes can be inferred by constructing an intermediate semantic space \cite{akata2015label}, wherein the knowledge from seen classes is capable of being transferred to new concepts. To obtain an effective classification model, the mainstream GZSL workflow commonly curates large-scale training data from multiple parties into a centralized server. Nevertheless, such a centralized training paradigm may be infeasible in practice. Due to increasing concerns about data privacy and security, it is highly restricted to share the training data across different organizations.
\begin{figure*}
\centering
\includegraphics[width=0.9\linewidth]{figures/distritbution_YD_croped.pdf}
\caption{Data distribution comparisons between traditional FL with \textit{i.i.d. setting}, \textit{non-i.i.d.} FL and our proposed \texttt{FedZSL} with \textit{p.c.c.d.}}
\label{fig:fedzsl}
\end{figure*}
In this paper, we resort to \textit{federated learning} (FL) \cite{mcmahan2017communication,li2021ditto,jiang2019improving}, a privacy-preserving framework for learning models from multiple decentralized data sources with no exchange of local training data. As shown in Fig. \ref{fig:fedzsl}(\textcolor{red}{a}) and Fig. \ref{fig:fedzsl}(\textcolor{red}{b}), traditional FL approaches assume all participants possess an identical set of classes with either \textit{i.i.d.} or \textit{non-i.i.d.} sampled data. Complementary to previous work, our study investigates a more practical setting namely \textit{federated zero-shot learning} (\texttt{FedZSL}), where multiple parties exclusively hold the training data from non-overlapping classes, as depicted in Fig. \ref{fig:fedzsl}(\textcolor{red}{c}). Here, we refer to this distribution that the local training data follows as a partial class-conditional distribution (\textit{p.c.c.d.}). We consider that the FedZSL is also more flexible and scalable, since it will enforce the global model to learn how to quickly adapt to the new seen classes when new participants join in the FL cluster. As shown in Fig. \ref{fig:intro}, by aggregating the local updates from various edge devices, FedZSL aims to find a global model at the server-side that can accurately recognize all the seen classes (\textit{e.g.,} black footed albatross) that are distributed locally. Meanwhile, the learned model is expected to generalize well to the unseen classes (\textit{e.g.,} American pipit) that are not presented in the training stage.
Despite the FedZSL protocol being ideal for zero-shot recognition tasks, it poses a significant challenge on system effectiveness and robustness, mainly from two perspectives: (1) the \textit{p.c.c.d.} data makes FedZSL the worst case of \textit{non-i.i.d.} FL. With no access to visual samples of any other classes, the trained models are prone to be biased to the classes that are locally available, thus failing to generalize to the unseen classes and/or seen classes appeared in other devices; (2) as each class in the training data comes from a single source, the central model is highly vulnerable to model replacement (backdoor) attacks \cite{bagdasaryan2020backdoor,ozdayi2021defending,chen2017targeted}. The erroneous data uploaded by careless or malicious participants involved in training process may lead to performance degradation of both local and global models, which becomes more severe as the communication round goes up.
To address these challenges, in this work, we investigate how to learn an effective and robust FedZSL model that is able to faithfully classify the seen and unseen classes while defending against multiple types of backdoor attacks. In particular, three local objectives are designed for visual-semantic alignment and cross-device alignment through relation distillation, which leverages the normalized class-wise covariance estimated by Graphical Lasso to regularize the consistency of the prediction across devices. In addition, to defend against the backdoor attacks, a \textit{feature magnitude defense} (FMD) method is proposed to detect and remove the potential malicious samples by examining the correlation of semantic attributes in the visual representation of each instance. As malicious samples typically have irrelevant backgrounds, styles and objects, the associated visual features are of a low magnitude. Hence, the visual features with low magnitude will be discarded, resulting in stable updates of local models.
The main contributions of this work are summarized as follows:
\begin{itemize}
\item This work is the very first attempt to engage multiple parties to collaboratively learn a unified zero-shot model with partial class-conditional distribution (\textit{p.c.c.d.}). A novel and practical setting of FedZSL is formulated, which is complementary to all existing FL frameworks.
\item To deal with the lack of global semantic information in local model training, we propose a \textit{cross-device alignment via relation distillation} (CARD) training strategy, which reconciles the visual-semantic relationship learning among local models that are trained with exclusively owned classes.
\item We improve the robustness of FedZSL by preventing backdoor attacks with the proposed \textit{feature magnitude defense} (FMD). Three different types of backdoor attacks on backgrounds, styles, and objects are studied.
\item Extensive experiments conducted on three benchmark zero-shot learning datasets evidence the effectiveness and the robustness of the proposed FedZSL
\end{itemize}
\section{Related Work}
\subsection{Zero-Shot Learning}
Zero-shot Learning \cite{romera2015embarrassingly,xian2017zero,kodirov2017semantic} addresses a challenging problem where the test set contains additional classes not presented during training. A standard solution is to leverage intermediate class-level semantic representations (\textit{e.g.,} attribute annotations \cite{lampert2013attribute} or natural language descriptions \cite{elhoseiny2013write}) to bridge the seen and unseen classes \cite{akata2015label,li2017zero}. There are roughly two streams of methods to learn the visual-semantic relationships:
embedding-based methods \cite{xu2020attribute,palatucci2009zero,norouzi2013zero,zhang2017learning} and generative methods \cite{xian2018feature,verma2018generalized,zhu2018generative,han2021contrastive}. The former group learns to project the visual and semantic information from all classes to the same space, and use the learned mapping to infer the class attributes for the unseen samples at the test time. The latter group needs to firstly train a generative model from semantic information to visual features, and then leverage the trained generative model to hallucinate the potential visual features for the unseen classes. As generative approaches generally involve large-scale data synthesis and heavy computation, it will lead to high overhead costs on edge devices if extended to a federated learning scheme. Hence, we follow the embedding-based workflow.
\subsection{Federated Learning (FL)}
Federated learning \cite{mcmahan2017communication,mohassel2017secureml} is a decentralized learning protocol that enables multiple participants to collaboratively learn a unified model without sharing the local data, which provides a promising privacy-preserving solution. Researchers in this area have been dedicated to improving efficiency and effectiveness (\textit{e.g.,} strategies for dealing with \textit{non-i.i.d.} data \cite{stich2018local,yu2018parallel,li2019convergence}), preserving the privacy of user data \cite{bogdanov2012deploying,agrawal2019quotient}, ensuring fairness and addressing sources of bias \cite{li2019fair,jiang2019improving}, and addressing system challenges \cite{sheller2020federated,bonawitz2019towards}.
Note that our proposed FedZSL is different from existing zero-shot related methods in federated learning \cite{hao2021towards,zhang2021fedzkt,gudur_interspeech21}, which are incapable of directly generalizing the federated model to unseen classes. Instead, they only consider data or model heterogeneity problems. Specifically, Fed-ZD \cite{hao2021towards} considers improving model fairness on under-representative classes that only partial clients hold. They propose to perform data augmentation for those under-representative classes. FedZKT \cite{zhang2021fedzkt} is proposed to solve the model heterogeneity by distilling the knowledge from heterogeneous local models.
Fed-NCAC \cite{gudur_interspeech21} is inspired by the data impression technique \cite{nayak2019zero} that adapts the current model to emerging new classes. However, the adaption is based on training samples of new classes.
This paper considers FedZSL with partial class-conditional distribution (\textit{p.c.c.d.}) data, which intrinsically solves the zero-shot learning problem in a federated learning scenario.
\subsection{Backdoor Attacks in FL}
Adversarial attacks can be roughly categorized as untargeted and backdoor (targeted) attacks. In untargeted attacks \cite{blanchard2017machine,damaskinos2018asynchronous}, the malicious participants aim to corrupt the model, resulting in inferior performance on the primary tasks. In contrast, backdoor (targeted) attacks \cite{chen2017targeted} are targeted explicitly on certain sub-tasks and make them work with backdoored functionalities. Meanwhile, these backdoor functionalities should not affect the performance of the primary tasks.
Bagdasaryan \textit{et al.} \cite{bagdasaryan2020backdoor} showed federated learning is generally vulnerable to backdoors as it is impossible to ensure none of the participants are malicious.
The model replacement attack is introduced to scale the backdoored model by a constant scaling factor to dominate the aggregation updates, thus preventing the backdoor functionalities from being canceled out. Sun \textit{et al.} \cite{sun2019can} leveraged differential privacy to defend against backdoor tasks by first clipping gradient updates and then adding Gaussian noise to the local model. Ditto \cite{li2021ditto} is proposed to improve both robustness and fairness in federated learning; interpolating between local and global models is proved to be effective. Ozdayi \textit{et al.} \cite{ozdayi2021defending} proposed to defend the backdoor functionalities by carefully adjusting the aggregation server's learning rate per dimension and round, based on the sign information of agents' updates. Unlike these sophisticated methods, in FedZSL, we propose a lightweight method with minimal changes to FL protocol to defend the backdoor attacks by inspecting the attribute presences in the input samples.
\section{Federated Zero-Shot Learning}
This section will first formally introduce the \texttt{FedZSL} task, and then detail the global and local training procedure. Three different alignment objectives are applied to address the partial class-conditional distribution and misalignment of visual and semantic space. Finally, the backdoor defending with a feature magnitude mechanism is introduced.
\subsection{Overview of FedZSL}
We consider a distributed system of $K$ clients $C_1, C_2,\dots, C_K$, each of which owns a local data source for training, \textit{i.e.,} $\mathcal{D}^{s} = \{\mathcal{D}^{s,1}, \mathcal{D}^{s,2}, \ldots, \mathcal{D}^{s,K}\}$. In particular, the $k$-th device has $N^k$ pairs of images with annotations, \textit{i.e.,} $\mathcal{D}^{s,k}=\{({\bm{x}}_{i}^{s,k}, y_{i}^{s,k})\}_{i=1}^{N^{k}}$, where only a part of seen classes are observable $y_{i}^{s,k}\in \mathcal{Y}^{s,k}$. Notably, the seen classes across the devices are non-overlapping, $\bigcap_{k \in [K]}\mathcal{Y}^{s,k}=\emptyset$ and $\bigcup_{k \in [K]} \mathcal{Y}^{s,k} = \mathcal{Y}^s$, where $|\mathcal{Y}^{s}|$ means the total number of the seen classes across $K$ devices. In FedZSL, we follow the standard setting \cite{mcmahan2017communication} that each client trains a local recognition model based on the local data, while a central server will collect the parameters periodically, aggregate them to update the global parameters for recognizing both seen classes $\mathcal{Y}^s$ and $\mathcal{Y}^u$. For briefty, we define $|\mathcal{Y}^{s}|+|\mathcal{Y}^{u}| = |\mathcal{Y}|$. To enable the parameter sharing between labels, the semantic information $\mathcal{A} = \{{\bm{a}}_{j} \}_{j=1}^{|\mathcal{Y}|}\in\mathbb{R}^{d_a\times |\mathcal{Y}|}$ is shared among all devices, where $d_a$ represents the attribute dimension.
Formally, we leverage the training data on $K$ devices $\mathcal{D}^{s} \triangleq \bigcup_{k \in [K]} \mathcal{D}^{s,k}$ to initiate a unified model $w$. The global learning objective is to solve:
\begin{equation}
\begin{aligned}
\underset{w}{\min} \mathcal{L}(w) = \sum_{k=1}^{K} \dfrac{|\mathcal{Y}^{s,k}|}{|\mathcal{Y}^s|} \mathcal{L}^{k}(w),
\end{aligned}
\end{equation}
where $\mathcal{L}_{k}(w) = \mathop{\mathbb{E}}_{({\bm{x}},y)\sim \mathcal{D}^{s,k}} [\ell^{k}(w;({\bm{x}},y))]$ is the empirical loss of the client $C_{k}$. Different from the weighting scheme defined in FederatedAveraging (FedAvg) \cite{mcmahan2017communication}, a seen class ratio is utilized to measure the shared information available locally. We denote the model parameters at the round $t$ by $w_t$ and the $k$-th local model update by $\bigtriangleup w_{t}^k$. Therefore, the server will update the global model by aggregating $k$-th participant's local updates by:
\begin{equation}
\begin{aligned}
w_{t+1} = w_t + \eta \frac{|\mathcal{Y}^{s,k}|\bigtriangleup w_t^k }{|\mathcal{Y}^{s}|},
\label{modelupdate}
\end{aligned}
\end{equation}
where $\eta$ is the learning rate at the server side. The overall training procedure can be found in Algorithm \ref{alg}.
\subsection{Local Training}\label{sec:local}
During the local training procedure in the communication round $t$, the client $C_k$ receives the aggregated model weight $w_t$ and applies it to the local model $w_t^k$. Given an input image ${\bm{x}}_i$, a pre-trained CNN backbone is leveraged as an image encoder $f(\cdot)$ to produce the visual features ${\bm{v}}_i = f({\bm{x}}_i) \in \mathbb{R}^{d_v}$, where $d_v$ denotes the visual features' dimension. In contrast to conventional classification models having a fully connected layer atop the backbone to produce logits, ZSL methods usually add an attribute regression layer $g(\cdot): d_v \rightarrow d_a$ to calculate the presences of the semantic attributes from the visual features, $\widehat{{\bm{a}}}_i = g(f({\bm{x}}_i))\in\mathbb{R}^{d_a}$.
\noindent\textbf{Visual-Semantic Alignment.} For simplicity, we adopt a linear layer to map the visual features to semantic attributes. To determine the correct class that the predicted semantic attributes correspond to, the dot product operation is performed between the predicted semantic attributes and all possible semantic attributes to calculate the class logits. The semantic cross-entropy (SCE) loss is the objective to encourage the input images to have the highest compatibility score with their corresponding semantic attributes, which can be formulated as:
\begin{equation}
\begin{aligned}
\ell_{sce} = - \sum_{{\bm{x}}_i \in \mathcal{D}^{s,k}} \log \frac{\exp(\widehat{{\bm{a}}}_i^{T} \cdot {\bm{a}}_j)}{\sum_{{\bm{a}}_k \in \mathcal{A}} \exp (\widehat{{\bm{a}}}_i^{T} \cdot {\bm{a}}_{k})}.
\end{aligned}
\end{equation}
The attribute regression layer $g(\cdot)$ and the backbone $f(\cdot)$ are trained jointly.
\noindent\textbf{Cross-device Alignment.} To deal with \textit{p.c.c.d.} data distribution across devices, the idea of our solution is to align the class relationships in the local models across devices, so that we can learn a consistent visual space.
However, as the training samples that involve visual information are strictly preserved in local devices, it is elusive to align the visual space with the collaborative classes. We propose a \textit{cross-device alignment via relation distillation} (CARD) training strategy to align the visual space in different local models according to the class similarities in the semantic space.
Specifically, we start with constructing a class semantic similarity matrix. Graphical Lasso \cite{friedman2008sparse} is leveraged to estimate the sparse covariance $\Sigma\in\mathbb{R}^{|\mathcal{Y}|\times|\mathcal{Y}|}$ of the semantic information $\mathcal{A}$ as the class semantic similarities. Under the assumption that the inverse covariance $\Theta = \Sigma^{-1}$ is positive semidefinite, it minimizes an $\ell_{1}$-regularized negative log-likelihood:
\begin{equation}
\begin{aligned}
\widehat{\Theta} = \underset{\Theta}{\argmin}~~\text{tr}(\mathcal{S}\Theta) - \log \text{det}(\Theta) + \delta\|\Theta\|_1,
\end{aligned}
\end{equation}
where $\mathcal{S}$ is a sample covariance matrix generated from $\mathcal{A}$, $\delta$ denotes the regularization parameter that controls the $\ell_1$ shrinkage. Thus, we have $\Sigma_{j,k}$ as the semantic similarity between classes $j$ and $k$, where $j,k \in [|\mathcal{Y}^{s,k}|]$.
We further take the semantic similarity matrix as the probability distribution that the prediction logits of a training sample should match with. A natural way of learning the probability distribution is through knowledge distillation with logits matching\cite{hinton2015distilling}.
The class similarities $\Sigma$ are provided as the source knowledge to be transferred to target local models for learning visual features. We start with obtaining the soft targets by softening the peaky distribution of source and target logits with temperature scaling:
\begin{equation}
\begin{aligned}
&p_{j}^{\Sigma}(\Sigma_j;\tau) = \mathtt{softmax}(\Sigma_j; \tau) = \frac{\exp(\Sigma_j/ \tau)}{\sum_{m}\exp(\Sigma_m/\tau )},\\
&p_{j}^{a}(\widehat{{\bm{a}}}_{j}^{T} \mathcal{A};\tau) = \mathtt{softmax}(\widehat{{\bm{a}}}_{j}^{T} \mathcal{A}; \tau) = \frac{\exp(\widehat{{\bm{a}}}_{j}^{T} \mathcal{A}/ \tau)}{\sum_{m}\exp(\widehat{{\bm{a}}}_{m}^{T} \mathcal{A}/\tau )},
\end{aligned}
\end{equation}
where $\tau$ is the temperature that can produce a softer probability distribution over classes with a high value. We use $p^{\Sigma}_{j}$ and $p^{a}_{j}$ for short. The knowledge distillation loss measured by KL-divergence is:
\begin{equation}
\begin{aligned}
\ell_{kl} &= \sum_{{\bm{x}}_i \in \mathcal{D}^{s,k}} KL(p^{a}(\cdot; \tau) \| p^{\Sigma}(\cdot; \tau))\\
& =\tau^2 \sum_{{\bm{x}}_i \in \mathcal{D}^{s,k}} \sum_{j}^{|\mathcal{Y}|} p^{\Sigma}_{j}\log \frac{p^{\Sigma}_{j}}{p^a_{j}}.
\end{aligned}
\end{equation}
\noindent\textbf{Semantic-Visual Alignment.} To ensure the learned visual-semantic mapping is invertible, we further add a semantic-consistency loss to facilitate the visual feature reconstruction. In addition to the attribute regression layer $g(\cdot)$, we add another visual feature generation layer $h(\cdot)$ to construct visual features from the attributes $\widehat{{\bm{v}}} = h({\bm{a}})\in\mathbb{R}^{d_v}$. The semantic-consistency loss is applied between the extracted visual features and the generated visual features:
\begin{equation}
\begin{aligned}
\ell_{con} = \sum_{{\bm{x}}_i \in \mathcal{D}^{s,k}} \| h({\bm{a}}_{y_i}) - f({\bm{x}}_i) \|_2,
\end{aligned}
\end{equation}
where ${\bm{a}}_{y_i}$ is the corresponding class-level attributes for data sample ${\bm{x}}_i$.
\noindent \textbf{Joint Optimization.} The overall local objective function in the client $C_k$ for FedZSL is defined as:
\begin{equation}
\begin{aligned}
\ell_{k} = \ell_{sce} + \mu\ell_{kl} + \ell_{con},
\end{aligned}
\end{equation}
where $\mu$ denotes the loss coefficient. The local models are trained with the overall local objective for a few epochs, the local update $\bigtriangleup w_t^k$ of the client $C_k$ in $t$-th communication round is submitted to the server for aggregation with Equation \ref{modelupdate}
\begin{algorithm}[t]
\hspace{-0.01em}\textbf{Input:}
clients number $K$, local datasets $\{\mathcal{D}^{s,1}, \mathcal{D}^{s,2}, \ldots, \mathcal{D}^{s,K}\}$, scaling factor $\beta$, communication round $T$, local epochs $E$, local learning rate $\lambda$, global learning rate $\eta$, loss coefficient $\mu$, batch size $B$, FMD threshold $M$ \\
\hspace{-0.01em}\textbf{Initialize:}
Server model parameters $w^{0}$
\caption{Federated Zero-Shot Learning (\texttt{FedZSL})}\label{euclid}
\begin{algorithmic}[1]
\State \textbf{Server executes:}
\For{$t = 0,1,...,T-1$}
\State The server communicates $w^{t}$ to the \textit{i}-th client
\For{$k = 1,...,K$}
\State The server communicates $w^{t}$ to the client $C_{k}$
\State $\bigtriangleup w_t^k \leftarrow$ \textbf{PartyLocalTraining($k,w^t)$}
\EndFor
\State $w_k^{t+1}$ $\leftarrow$ $w_t + \eta \frac{|\mathcal{Y}^{s,k}|\bigtriangleup w_t^k }{|\mathcal{Y}^{s}|}$
\EndFor
\State \textbf{PartyLocalTraining($k,w^t$):}
\State $w_{k}^t \leftarrow w^t$
\For{epoch $i = 0,1,\ldots,E$}
\For{batch $\textbf{b} = \{{\bm{X}}, {\bm{Y}}\} = \{({\bm{x}}_b,y_b)\}^{B}$}
\State ${\bm{V}} \leftarrow f({\bm{X}})$
\For{$b = 0,1,\ldots,B$}
\State $\gamma_{b} = \sum_{c=1}^{d_v} {\bm{V}}_{c,b}$
\If{$\gamma_{b} < M$ }
\State Discard ${\bm{V}}_{b}$;~ $B$ -= $1$
\EndIf
\EndFor
\State $\{\widehat{a}\}^{B} \leftarrow g({\bm{V}})$; $\{\widehat{v}\}^{B} \leftarrow \{h({\bm{a}}_b\}\}^{B}$
\State $\ell_k = \ell_{sce} + \mu \ell_{kd} + \ell_{con}$ by Eq. {\color{red} 3}, {\color{red} 6}, {\color{red} 7}
\State $w_{k}^t \leftarrow w_{k}^t - \lambda \bigtriangledown \ell_k$
\EndFor
\EndFor
\State $\bigtriangleup w_t^k \leftarrow \beta(w_{k}^t - w^t)$
\State Return $\bigtriangleup w_t^k$ to server
\end{algorithmic}
\label{alg}
\end{algorithm}
\subsection{Feature Magnitude Backdoor Defense}
We consider backdoor attacks with model replacement \cite{bagdasaryan2020backdoor,bhagoji2019analyzing}, which is performed by a malicious client controlling a local device. The goal is to inject a backdoor functionality into the global model so that the specific inputs will be classified as a pre-defined class with high confidence. Without loss of generality, we assume the client 1 is malicious and attempts to replace the global model with a backdoored model $\widehat{w}$ with the updates:
\begin{equation}
\begin{aligned}
\bigtriangleup w_t^1 = \beta (\widehat{w} - w_t),
\end{aligned}
\end{equation}
where $\beta$ is a scaling factor to ensure the backdoor functionality survives in server aggregation.
A conventional visual classifier produces discriminative visual features solely based on the semantic labels, so that these features are usually sparse in terms of semantics, \textit{i.e.,} various semantic information is covered. Instead, one of the properties in a zero-shot visual classifier is that the visual features are densely attended to the semantic attributes. Thus, we can regard the visual features as the entangled attribute information and the attribute regression layer is in charge of transforming or calculating the visual features' responses to each dimension in the attributes. As the dense visual features correspond to the annotated attributes, training samples with solid presences in the semantic attributes will show normal magnitude. In contrast, due to the absence of semantic information, as shown in Fig. \ref{fig:vis}, the visual features of malicious samples are usually sparse and of low magnitude.
Based on the above insights, we construct a defending strategy, termed \textit{feature magnitude defense} (FMD). This strategy requires minimal changes to the FL protocol. By leveraging the magnitude mechanism of visual features, FMD can easily detect the malicious samples and remove them from training. Given a batch of input images
${\bm{X}} = \{{\bm{x}}_b\}^B $
, the CNN backbone extracts the visual features
: ${\bm{V}} = f({\bm{X}}) \in \mathbb{R}^{d_v \times B}$
, based on which the batch feature magnitude is defined as the sum: $\{\gamma_b\}^B = \{\sum^{d_v}_{c=1} {\bm{V}}_{c,b}\}$. Within $\{\gamma_b\}^B$, if $\gamma_b$ is smaller than the magnitude threshold $M$, the corresponding training sample ${\bm{x}}_b$ is conceived as malicious and removed from training. This defending procedure is presented in step 12 to 15 in Algorithm \ref{alg}.
\section{Experiments}
\subsection{Setup}
\noindent \textbf{Datasets.} We extensively evaluate our method on several FL scenes with three zero-shot learning benchmark datasets. Caltech-UCSD Birds-200-20 (CUB) \cite{wah2011caltech} contains 11,700 images from 200 bird species with 312 manually annotated attributes. Animals with Attributes 2 (AwA2) \cite{xian2018zero} consists of 37,322 images from 50 animal classes with 85 attributes. SUN Scene Recognition (SUN) \cite{patterson2012sun} includes 14,340 images from 717 different scenes with 102 attributes. The standard splits for the seen and unseen classes, \textit{i.e.,} 150/50, 40/10, 645/72 are adopted on CUB, AwA2 and SUN respectively for the proposed split \cite{xian2018zero}.
\smallskip
\noindent \textbf{Evaluation Metrics.}
For the evaluation criteria, we use the average per-class top-1 accuracy as the primary metric in our conventional and generalized ZSL experiments (GZSL) \cite{xian2018zero}.
In conventional ZSL setting, we only evaluate the accuracy of the \textit{unseen} classes $Acc_{\mathcal{C}}$ that none of the participants hold. In GZSL setting, we calculate the accuracies on test samples from both \textit{seen} and \textit{unseen} classes, \textit{i.e.,} $Acc_{\mathcal{Y}^{s}}$ and $Acc_{\mathcal{Y}^{u}}$. The harmonic mean $Acc_{\mathcal{H}}$ between the two sets of classes are computed to evaluate the GZSL performance: $ Acc_{\mathcal{H}} = (2*Acc_{\mathcal{Y}^s}*Acc_{\mathcal{Y}^u})/(Acc_{\mathcal{Y}^s}+Acc_{\mathcal{Y}^u})$.
\smallskip
\noindent \textbf{Implementation Details.}
We use PyTorch to implement FedZSL and other baselines. Source code is made available in the supplementary material for reference. The local models are trained with an SGD optimizer with a weight decay of $1e-5$ and momentum of $0.9$. The learning rates for CUB, AwA2 and SUN are initialized as $5e-3, 3e-4$ and $1e-2$, respectively. The number of communication rounds and the default participant number are fixed to 50 and 10 unless explicitly specified. The batch size and number of local epochs in each communication round are set to 64 and 2. The coefficient $\mu$ of knowledge distillation is fixed to 3. For label smoothing, we set the temperature $\tau$ to 10. The $\ell_1$ penalty coefficient in Graphical Lasso is set to 0.01. A pre-trained lightweight CNN network ResNet18 is adopted as the backbone for all experiments.
\vspace{-5pt}
\subsection{Results and Analysis}
\vspace{-5pt}
\label{sec:exp}
\begin {table*}[t]
\caption {Performance comparisons (\%) on three datasets between various federated learning baselines and the proposed FedZSL. The best results of decentralized settings are highlighted in bold. }
\vspace{-10pt}
\begin{center}
\scalebox{0.85}{
\begin{tabular}[t]{lc ccc | cccc | cccc}
\toprule
& \multicolumn{4}{c|}{CUB} & \multicolumn{4}{c|}{AwA2} & \multicolumn{4}{c}{SUN} \\
\cmidrule(lr){2-5} \cmidrule(lr){6-9} \cmidrule(lr){10-13}
& Acc$_{\mathcal{C}}$ & Acc$_{\mathcal{Y}^{u}}$ & Acc$_{\mathcal{Y}^{s}}$ & Acc$_{\mathcal{H}}$ & Acc$_{\mathcal{C}}$ & Acc$_{\mathcal{Y}^{u}}$ & Acc$_{\mathcal{Y}^{s}}$ & Acc$_{\mathcal{H}}$ & Acc$_{\mathcal{C}}$ & Acc$_{\mathcal{Y}^{u}}$ & Acc$_{\mathcal{Y}^{s}}$ & Acc$_{\mathcal{H}}$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\texttt{Baseline($\varphi=0.1$)}
& 14.9 & 8.2 & 7.4 & 7.7
& 29.5 & 16.5 & 12.6 & 14.3
& 25.6 & 11.1 & 5.5 & 7.4
\\
\texttt{Baseline($\varphi=0.2$)}
& 27.0 & 16.1 & 13.5 & 14.7
& 32.6 & 22.6 & 17.3 & 19.6
& 34.9 & 16.9 & 8.3 & 11.1
\\
\texttt{Baseline($\varphi=0.3$)}
& 32.1 & 22.6 & 19.0 & 20.6
& 43.3 & 25.0 & 30.2 & 27.3
& 38.6 & 18.0 & 10.4 & 13.2
\\
\texttt{Baseline($\varphi=0.5$)}
& 43.9 & 30.5 & 26.5 & 28.3
& 45.7 & 26.0 & 36.0 & 30.2
& 41.3 & 25.8 & 14.4 & 18.5
\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\texttt{FedAvg}
& 44.6 & 30.3 & 30.7 & 30.5
& 39.5 & 27.5 & 34.5 & 31.0
& 42.4 & 25.4 & 12.2 & 16.5
\\
\texttt{FedAvg-decay}
& {45.8} & {31.5} & {32.1} & {31.8}
& {40.0} & {26.8} & {35.2} & {30.4}
& {43.1} & {26.8} & {13.1} & {17.6}
\\
\texttt{FedProx}
& {44.3} & {29.5} & {31.8} & {30.6}
& {39.7} & {26.2} & {34.3} & {29.7}
& {42.0} & {25.1} & {11.8} & {16.1}
\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\texttt{FedZSL}(ours)
& {55.1} &{ 43.6} & {40.3} & {41.9}
& {46.6 } & {37.8 } & {50.8} & {43.4}
& {49.2 } & {27.6} & \textbf{16.8 } & {20.9}
\\
ours+\texttt{FedAvg-decay}
& \textbf{56.3} & \textbf{43.8} & \textbf{40.4} & \textbf{42.0}
& \textbf{47.2} & {37.9} & \textbf{51.3} & \textbf{43.6}
& \textbf{49.6} \textbf& {28.9} & {16.7} & \textbf{21.2}
\\
ours+\texttt{FedProx}
& {55.6} & {42.9} & {39.7} & {41.2}
& {45.9} & \textbf{38.1} & {50.2} & {43.3}
& {48.6} & {27.3} & {16.4} & {20.5}
\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\texttt{Centralized}
& 55.4 & 42.7 & 44.9 & 43.8
& 59.0 & 35.0 & 72.6 & 47.2
& 50.1 & 26.7 & 20.2 & 23.0
\\
\noalign{\smallskip}
\hline\bottomrule
\end{tabular}}
\end{center}
\label{gzslperoformance}
\end {table*}
Table \ref{gzslperoformance} reports the performance comparisons among several federated learning models for zero-shot recognition.
\noindent \textbf{Baselines and Centralized:} Specifically, we decentralize the data onto 10 clients, with each client only having a subset of classes.
For simplicity, we define the local class ratio $\varphi = \{0.1, 0.2, 0.3, 0.5\}$ to indicate the percentage of the total seen classes available in each device. For $\varphi=1$, it degenerates to the centralized training (\texttt{Centralized}), which leverages all the seen classes to train a zero-shot classifier. From rows 1-4, it is observed that when training with more seen classes, the model can generalize better to unseen classes, which is in line with our assumption. Fig.\ref{fig:differentclasses} depicts the detailed learning curves of baselines with various portions of the seen classes available in each round. With models trained separately in local devices, it demonstrates a significant variance (the shaded area) of model performance across devices. From the solid line, we can see that more seen classes generally lead to a better performance in both ZSL and GZSL.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/local.pdf}
\vspace{-15pt}
\caption{Learning curves of decentralized baselines for the CUB dataset in the ZSL and GZSL settings. The solid line indicates the averaged performance, and the shaded area represents the performance variance across local devices.}
\label{fig:differentclasses}
\end{figure}
\smallskip
\noindent \textbf{Federated Baselines:} We compare the proposed approach with the most representative federated learning frameworks \texttt{FedAvg} \cite{mcmahan2017communication}, \texttt{FedAvg-decay} \cite{li2019convergence}, and \texttt{FedProx} \cite{li2020federated}. FedAvg learns a global model by aggregating parameters from local devices, which is nearly the same as \texttt{Baseline ($\varphi=0.1$)} but with an additional server aggregation step. Table \ref{gzslperoformance} shows that \texttt{FedAvg} with $\varphi=0.1$ can obtain comparable results with baseline ($\varphi=0.5$). As clients do not take advantage of the semantic attributes, FedAvg fails to achieve significant performance gains for the unseen classes over baselines. FedAvg-decay has an extra requirement on the global learning rate. By decaying the learning rate of global aggregation, the performance results on CUB and SUN are improved comparing with FedAvg. FedProx is a federated baseline tackling heterogeneity, which assigns higher aggregation rates to the clients that hold more training data. However, our experiments show that in our proposed \textit{p.c.c.d.} setting, simply associating aggregation rate with the amount of data fails to achieve better performance.
\smallskip
\noindent \textbf{FedZSL:} With the proposed CARD training strategy that helps reach the consensus across devices, the proposed \texttt{FedZSL} outperforms all decentralized baselines and \texttt{FedAvg} in both ZSL and GZSL scenarios. When applying the proposed CARD training strategy on \texttt{FedAvg-decay}, in ours+\texttt{FedAvg-decay}, the global learning rate decay can slightly improve the performance. In the variant ours+\texttt{FedProx}, as the clients hold the same number of training classes, assigning different global aggregation rate causes imbalanced knowledge transfer across devices, leading to inferior global performance.
\smallskip
\begin{figure}
\centering
\vspace{-8.5pt}
\includegraphics[width=1\linewidth]{figures/imb.pdf}
\vspace{-16pt}
\caption{Performance comparisons of the proposed FedZSL with (1) \textit{uniform}: uniform class distribution, (2) \textit{imb}: imbalanced class distribution on local devices. \textit{weight} plots indicate the performance with weighted aggregation instead of averaging.}
\label{fig:imbalanceperformance}
\end{figure}
\noindent \textbf{Impact of Imbalanced Class Distribution.} Considering participants are not always holding the identical number of classes in real-world cases, we conduct experiments in a more practical setting with the SUN dataset, which is to learn a global model from imbalanced class distribution. The statistics of class distribution can be found in the Appendix. In Fig. \ref{fig:imbalanceperformance}, we compare the training convergence among local and global models trained with (1) uniform class distribution and (2) imbalanced class distribution. For class imbalance, we further make a comparison of average aggregation (in orange) and weighted aggregation (in green) based on the ratio of available classes. It is shown that even with imbalanced class distribution, the global model can reach a similar performance to the one trained with the uniform class distribution. An interesting observation is that when weighting the local model updates according to the local class number, the global model can converge quickly, but the performance is inferior to average weighting.
\smallskip
\noindent \textbf{Impact of New Participants.}
To investigate the performance influence when new seen classes participate in the training, we conduct experiments by adding more seen classes when the federated model is in different stages. In Fig. \ref{fig:moreclasses}(\textcolor{red}{a}) and \ref{fig:moreclasses}(\textcolor{red}{b}), we start by training with only 5 participants, each of which contains 15 seen classes. Then, we add another 5 participants, each with 15 seen classes from round 0, 10, 20, 30, and 40. We can see that the performance can quickly surge to the performance level as training with all participants. This result indicates that \texttt{FedZSL} can quickly adapt to new seen classes, and the emerging of new participants does not cause a negative impact on training.
\smallskip
\noindent \textbf{Impact of Various Sampling Fractions.}
To investigate the impact of sampling fraction, we conduct experiments by choosing 30\%, 50\%, and 100\% of the participants in each communication round to observe the convergence rate. It can be seen from Fig. \ref{fig:moreclasses}(\textcolor{red}{c}) and \ref{fig:moreclasses}(\textcolor{red}{d}), after 100 communication rounds, the performance results on both ZSL and GZSL reach a similar level. In addition, sampling fewer participants results in a slower convergence rate and unstable performance when aggregating the local models trained on partial seen classes.
\begin {table*}[t]
\caption {Effects of different components on three datasets with various stripped-down versions of our full proposed model.}
\vspace{-2pt}
\begin{center}
\scalebox{0.85}{
\begin{tabular}[t]{lc ccc | cccc | cccc}
\toprule
& \multicolumn{4}{c|}{CUB} & \multicolumn{4}{c|}{AwA2} & \multicolumn{4}{c}{SUN} \\
\cmidrule(lr){2-5} \cmidrule(lr){6-9} \cmidrule(lr){10-13}
& Acc$_{\mathcal{C}}$ & Acc$_{\mathcal{Y}^{u}}$ & Acc$_{\mathcal{Y}^{s}}$ & Acc$_{\mathcal{H}}$
& Acc$_{\mathcal{C}}$ & Acc$_{\mathcal{Y}^{u}}$ & Acc$_{\mathcal{Y}^{s}}$ & Acc$_{\mathcal{H}}$
& Acc$_{\mathcal{C}}$ & Acc$_{\mathcal{Y}^{u}}$ & Acc$_{\mathcal{Y}^{s}}$ & Acc$_{\mathcal{H}}$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\texttt{FedAvg}
& 44.6 & 30.3 & 30.7 & 30.5
& 39.5 & 27.5 & 34.5 & 31.0
& 42.4 & 25.4 & 12.2 & 16.5
\\
{\texttt{FedZSL} w/o $\ell_{kl}$}
& {48.1} &{ 38.4} & {39.6} & {39.0}
& {43.2} & {32.5 } & {47.9} & {38.7}
& {43.8} & {27.5} & {14.9} & {19.3}
\\
{\texttt{FedZSL} w/o $\ell_{con}$ }
& {54.0} &{39.7} & {41.2} & {40.4}
& {45.8} & {35.6} & {49.1} & {41.3}
& {47.9} & {26.8} & {16.1} & {20.1}
\\
{\texttt{FedZSL}(ours)}
& \textbf{55.1} &\textbf{ 43.6} & \textbf{40.3} & \textbf{41.9}
& \textbf{46.6 } & \textbf{37.8 } & \textbf{50.8} & \textbf{43.4}
& \textbf{49.2 } & \textbf{27.6} & \textbf{16.8 } & \textbf{20.9}
\\
\noalign{\smallskip}
\hline\bottomrule
\end{tabular}}
\end{center}
\vspace{-10pt}
\label{ablation}
\end {table*}
\begin{figure*}[t]
\centering
\includegraphics[width=1\linewidth]{figures/more_class_and_more_sampling.pdf}
\caption{(a) and (b): ZSL and GZSL learning curves with more participants joining in round 10, 20, 30, 40. (c) and (d): ZSL and GZSL learning curves with various sampling fractions. }
\label{fig:moreclasses}
\end{figure*}
\smallskip
\noindent \textbf{Ablation Study.}
In this ablation study, we evaluate various stripped-down versions of our full proposed model to
validate different components of the proposed FedZSL. In Table \ref{ablation}, we report the ZSL and GZSL performance of each version on the three benchmark datasets. The best performance is achieved when the CARD training strategy with the KL loss function $\ell_{kl}$ and the semantic-visual alignment with loss $\ell_{con}$ are both applied.
\subsection{Backdoor Attacks}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/backdoorsamples.pdf}
\caption{Exemplars of three different types of backdoor attacks.}
\label{fig:backdoorsamples}
\vspace{15pt}
\end{figure}
To investigate the impact of backdoor attacks on \texttt{FedZSL}, we perform a series of experiments on how to backdoor and how to defend backdoors in \texttt{FedZSL} with the proposed feature magnitude mechanism. There are three different backdoors we consider in this paper, including background attack, style attack, similar object attack. Correspondingly, we choose sky, cartoon birds, and dragonflies as malicious samples with 50 images per class. Fig. \ref{fig:backdoorsamples} shows some samples of three different types of backdoor attacks. The experiments are mainly conducted on the CUB dataset.
\smallskip
\noindent \textbf{Feature Magnitude Visualization.}
In Fig. \ref{fig:vis}, we visualize the visual features of normal and malicious samples. It can be seen that in contrast to the normal samples, the visual features of malicious samples are of low magnitude. As we state in the methodology section, the visual features produced by a zero-shot learning classifier are essentially the presences of the semantic information. Thus, the zero-shot learning classifier could not find the visual representations corresponding to the annotated attributes for a malicious sample, resulting in visual features with low magnitude.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/vis.pdf}
\caption{Visualization of feature magnitude for normal and malicious samples.}
\label{fig:vis}
\end{figure}
\smallskip
\noindent \textbf{Impact of the Scaling Factor.}
In Fig. \ref{fig:attack}(\textcolor{red}{a})-\ref{fig:attack}(\textcolor{red}{d}), we study the impact of the scaling factor on the backdoor accuracy. Specifically, in Fig. \ref{fig:attack}(\textcolor{red}{a}), we test with scaling factor $\beta$ in $\{1,5,10\}$ in single-round attack (round 20), it turns out that the backdoor functionality could be canceled out soon after a few rounds of server aggregation with a small scaling factor. When setting $\beta$ to 10, the backdoor accuracy remains high after 30 rounds of normal aggregation. In Fig. \ref{fig:attack}(\textcolor{red}{b}), backdoor samples are injected from rounds 20 to 25. We can see that when we set $\beta$ to 10, the main task performance drops significantly and recover to the normal standard after a few rounds.
In Fig. \ref{fig:attack}(\textcolor{red}{c}), we inject attack samples in all rounds after 20, which shows when setting $\beta$ to 1, the backdoor accuracy can keep high and main task performance is not affected. In Fig. \ref{fig:attack}(\textcolor{red}{d}), our proposed FMD is used to detect and remove the malicious sample from training, and we can completely defend the backdoor attacks for single- and multi-round attacks. For the all-round attacks, we can substantially reduce the backdoor accuracies.
\smallskip
\noindent \textbf{Impact of Attack Types.}
To investigate the attack difficulty of different attack types, in Fig. \ref{fig:attack}(\textcolor{red}{e})-(\textcolor{red}{g}), we can see that the background attack is the easiest one to inject. The style attack is also considerably easy to inject, and through more communication rounds, the accuracy is increased. As for the similar object, only all-round attack can successfully inject the backdoor functionality. In Fig. \ref{fig:attack}(\textcolor{red}{h}), we show that our defending method works for all types of attacks.
\begin{figure*}
\centering
\includegraphics[width=1\linewidth]{figures/attack.pdf}
\vspace{-20pt}
\caption{Learning curves of various attacks and corresponding defending with feature magnitude estimation.}
\label{fig:attack}
\vspace{-10pt}
\end{figure*}
\smallskip
\noindent \textbf{Impact of Simultaneous Attacks.}
In Fig. \ref{fig:attack}(\textcolor{red}{i})-(\textcolor{red}{l}), we test backdoor attacks with multiple compromised participants. In Fig. \ref{fig:attack}(\textcolor{red}{i}), both background and style attacks are injected. It can be seen that even though a single-round attack, the two backdoor accuracies remain considerably high after many communication rounds. With more attack rounds in multi-round and all-round attacks, the attack accuracies increase as expected. We show that our defending method consistently prevents malicious injections in Fig. \ref{fig:attack}(\textcolor{red}{j}). A more extreme experiment is conducted to inject all three kinds of attacks at the same time. In Fig. \ref{fig:attack}(\textcolor{red}{k}) we can see that it is very hard to inject all three backdoors at the same time successfully, and by introducing FMD in Fig. \ref{fig:attack}(\textcolor{red}{l}) we can substantially reduce all attack accuracies.
\noindent \textbf{Hyper-parameter Sensitivity.}
There are mainly two hyper-parameters controlling the overall objective function for the local model training, including the weight of $\ell_{kl}$ and $\ell_{con}$. To better understand the effects of the different components in the proposed \texttt{FedZSL}, we report the sensitivity visualization of the two hyper-parameters in Fig. \ref{fig:hyper}. It can be seen that the performance of all three datasets peaks when the weight of $\ell_{kl}$ is set to 3 and $\ell_{con}$ is set to 1.
\begin{figure}
\centering
\vspace{-10pt}
\includegraphics[width=1\linewidth]{figures/hyper_re.pdf}
\caption{ Hyper-parameter sensitivity for $\ell_{kl}$ and $\ell_{con}$. }
\label{fig:hyper}
\end{figure}
\smallskip
\noindent \textbf{Generation of Imbalanced Class Distribution.}
In Section \textcolor{red}{4.2}, we discussed the impact of imbalanced class distribution on learning curves in contrast to the uniform class distribution. To help understand what the imbalanced class distribution looks like, in Fig. \ref{fig:imbalanced} we visualize the number of classes that each device possesses. It can be seen that there is a large variance of class numbers across %
devices. As for implementation, the imbalanced class partition is sampled with the Dirichlet distribution. Specifically, we sample ${\bm{p}} \sim Dir(\alpha)$ and ${\bm{p}}_k$ is the proportion of the classes to be allocated to the client $C_k$. $Dir(\alpha)$ is the Dirichlet distribution with a concentration parameter $\alpha$ (0.5 by default). The higher the $\alpha$, the more uniform the distribution. Another distribution generation constraint is that we set the minimum class number on each device to 2.
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{figures/hyper.pdf}
\vspace{-16pt}
\caption{Imbalanced class distributions on three zero-shot datasets.}
\label{fig:imbalanced}
\end{figure}
\smallskip
\noindent \textbf{Impact of \textit{i.i.d.}, non-\textit{i.i.d.} and \textit{p.c.c.d.} Distributions.}
To investigate the impact of different data distributions, we also conducted experiments with the data sampled from \textit{i.i.d.} and non-\textit{i.i.d.} distributions to compare with the model trained in our \textit{p.c.c.d.} setting. In \textit{i.i.d.}, we uniformly split the set of the seen classes onto ten devices, whereas in non-\textit{i.i.d.} we use a Dirichlet distribution $Dir(\alpha)$ with a concentration parameter $\alpha=0.5$ to sample the data partition.
The non-\textit{i.i.d.} data partition across devices is visualized in Fig. \ref{fig:noiid}, in which each device has relatively few (even 0) training samples in some classes. In Fig. \ref{fig:iid}, we illustrate the learning curves with three different types of data distributions. The dashed lines and shaded areas represent the performance and its variance across local devices before the global aggregation in each communication round;
\begin{figure*}[t]
\begin{minipage}[t]{0.35\linewidth}
\includegraphics[width=\linewidth]{figures/noniid.pdf}
\caption{The data distribution of each device using non-\textit{i.i.d.} data partition.}
\label{fig:noiid}
\end{minipage}\hfill%
\begin{minipage}[t]{0.60\linewidth}
\includegraphics[width=\linewidth]{figures/iid.pdf}
\caption{ZSL and GZSL learning curves with different data distributions (\textit{i.e.,} \textit{i.i.d.}, non-\textit{i.i.d.}, and our proposed \textit{p.c.c.d.})}
\label{fig:iid}
\end{minipage}
\end{figure*}
\noindent the solid lines indicate the global performance for different distributions.
As the devices in our \textit{p.c.c.d.} are trained on various classes, the performance variance as indicated in the shaded area is more significant. Fig. \ref{fig:iid} also confirms that the %
\noindent non-\textit{i.i.d.} setting witnesses a slightly higher variance than the \textit{i.i.d.} setting, which is resulted from that the sample numbers of different classes vary across clients in the non-\textit{i.i.d.} protocol. For the global performance, the recognition accuracy of the proposed model trained in the challenging \textit{p.c.c.d.} setting is only slightly inferior to ones trained in the \textit{i.i.d.} and non-\textit{i.i.d.} settings. This further verifies the robustness of the proposed FedZSL framework, which is agnostic to the different data distributions on local devices.
\smallskip
\noindent \textbf{Impact of Partial Data Samples.}
To study the impact of data quantity in the local classes, we conduct experiments on the CUB dataset by decentralizing the data onto ten clients. Each client only has a subset of training samples of all classes. For simplicity, we define the local data ratio $\rho = \{0.1, 0.2, 0.3, 0.5, 1.0\}$ to indicate the percentage of the total training samples in each client. In Fig. \ref{fig:lesssamples}, we depict the detailed learning curves with various $\rho$. It can be seen that even if being trained with only 10\% data (only $\sim$ 6 samples per class in CUB), the local model achieves more than 40\% on ZSL accuracy. In comparison %
to the experiments on \texttt{Baselines} with various class percentage ratios, the improvements on training with more data are relatively lower. Also, we observe that there are only slight performance variances across devices, whereas the variances in \texttt{Baselines} are significant. This phenomenon demonstrates that the variation of data samples causes less impact than the variety of classes.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/lesssamples.pdf}
\caption{Learning curves of decentralized settings with partial data samples on the CUB dataset w.r.t the performance on ZSL and GZSL.}
\label{fig:lesssamples}
\end{figure}
\section{Supplementary Backdoor Details}
In this section, we elaborate more on the backdoor attacks for testifying the robustness of the proposed \texttt{FedZSL}. We formalize the general procedure of the backdoor attacks applied as follows. We consider that there is a compromised (malicious) client $C_k$ holds a dataset $\mathcal{D}^{mal} = \{{\bm{x}}^{mal}_i\}^r_{i=1}$ of size $r$ that is expected to be classified as the target labels $\{ y_i \}_{i=1}^r$. Let $\Gamma(\cdot)$ be the complete zero-shot classifier that predicts the label of the semantic attributes in $\mathcal{A}$, which has the highest compatibility score with ${\bm{x}}_i$. Therefore, the local objective for client $C_k$ of the backdoor attacks is defined as:
\begin{equation}
\begin{aligned}
\mathcal{M}(\mathcal{D}^{s,k}\cup\mathcal{D}^{mal}, w_t^k) = \underset{w_t^k}{\max} \sum_{i=1}^{r} \mathbf{1}[\Gamma({\bm{x}}_i^{mal},\mathcal{A};w_t^k)=y_i].
\end{aligned}
\end{equation}
As for the implementation of backdoor attacks, we start to involve the malicious samples in training when the global model is close to convergence. This injection strategy is proved to be more effective as indicated in \cite{bagdasaryan2020backdoor}. The backdoors injected in the very early rounds tend to be forgotten quickly. In particular, all the backdoor experiments start in communication round 20, where we observe that the global model tends to converge.
The backdoor samples ${\bm{x}}^{mal}$ are injected by replacing the targeted training samples ${\bm{x}}$ in each training batch. The probability of the replacement is set to 0.5. If the probability is higher, the main task performance can be affected. In contrast, if the probability is too low, the backdoor effects will be easily canceled out by the global aggregation. Moreover, to ensure the local model can generalize well to the malicious samples, each malicious sample to be injected is randomly sampled from $\mathcal{D}^{mal}$. Furthermore, to help the model generalize to the backdoor functionality, following \cite{chen2017targeted}, we add Gaussian noise with $\sigma=0.05$ to the malicious samples.
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{figures/lessclasses.pdf}
\caption{ZSL and GZSL learning curves with half participants being suspended in various communication rounds.}
\label{fig:remove}
\end{figure}
\section{Discussion of Limitations}
As is the case with a novel problem investigated and a general framework proposed, our work suggests many more questions than answers. Below, we discuss mainly two limitations that are not yet touched in this work, while being worth investigating to improve FedZSL in a near future.
\smallskip
\noindent \textbf{Catastrophic Forgetting.}
First, we observe that when some participants are suspended from federated training, their contributions to the global model start to be gradually forgotten. As shown in Fig. \ref{fig:remove}, we conducted experiments with half participants (5/10) being suspended from federated training in the communication round 10, 20, 30, and 40. It can be seen that the performance significantly drops with the communication round goes up. We refer to this problem as catastrophic forgetting \cite{kirkpatrick2017overcoming} in \texttt{FedZSL}. This problem is also pervasive in continual learning \cite{hu2018overcoming}, in which a stream of incrementally arriving inputs is processed without access to the past data. Compared to continual learning, the catastrophic forgetting challenge in our learning scenario is more severe considering the decentralized training scheme. The problem statement can be described as follows.
Given a sequence of federated learning stages or rounds in \texttt{FedZSL}, in which the training classes that appeared in the preceding learning stages may be absent in the subsequent learning stages. We aim to globally aggregate the local updates without accuracy deterioration that is caused by forgetting the contributions from the absent seen classes.
\smallskip
\noindent \textbf{Performance Degeneration.}
The second limitation we identify is the performance degeneration when training with an excess of rounds, which could be triggered by overfitting the seen classes. In Fig. \ref{fig:remove}(\textcolor{red}{a}), the blue solid line represents the learning curves of ZSL in 100 communication rounds. The default communication round number in this paper is set to 50, where the global model is converged. With an excess of communication rounds up to 100, we can clearly see the minor degeneration of the performance. Therefore, we argue that preventing the global model from overfitting to the local seen classes is a critical need for further improving the FedZSL framework.
\section{Conclusion }
In this work, we investigate a novel \textit{federated zero-shot learning} (\texttt{FedZSL}) problem by applying the CARD strategy to reach the consensus of visual-semantic predictions across devices and the FMD method for preventing backdoor attacks. Experiments validate that the proposed scheme is capable of handling new participants, various sampling fractions, and imbalanced class distribution. The proposed FMD is valid to stabilize model training and identify the malicious samples from various types.
{\small
\bibliographystyle{ieee_fullname}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,640
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Q: Login Cancelled I'm using the Facebook SDK for Swift, installed via cocoapods. I've configured an example App in the following way:
*
*Added a new facebook app on the developer site;
*Added iOS application with correct bundle identifier;
*On the iOS application edited the info.plist to add:
<key>CFBundleURLTypes</key>
<array>
<dict>
<key>CFBundleURLSchemes</key>
<array>
<string>fb169432773464820</string>
</array>
</dict>
</array>
<key>CFBundleVersion</key>
<string>1</string>
<key>FacebookAppID</key>
<string>{APPID}</string>
<key>FacebookDisplayName</key>
<string>AppDisplayName</string>
<key>LSApplicationQueriesSchemes</key>
<array>
<string>fbapi</string>
<string>fbauth</string>
<string>fb-messenger-api</string>
<string>fbauth2</string>
<string>fbshareextension</string>
</array>
After this I created a custom UIButton with the following action:
@IBAction func facebookTouched(sender: UIButton) {
let loginManager = LoginManager()
loginManager.logIn([.PublicProfile], viewController: self) { loginResult in
switch loginResult {
case .Failed(let error):
print(error)
case .Cancelled:
print("User cancelled login.")
case .Success(let grantedPermissions, let declinedPermissions, let accessToken):
print("Logged in!")
}
}
}
When I click the UIButton the action is called and Safari browser opens the facebook page.
Since I already accepted this application I get the following message "ApplicatioName already accepted" when I press "Ok" the browsers moves to a blank page and nothing happens. If I press the "Done" button the .Cancelled case is called.
Any ideas? Cheers
A: The problem was on the AppDelegate were the Facebook ApplicationDelegate methods need to also be called:
func application(application: UIApplication, openURL url: NSURL, sourceApplication: String?, annotation: AnyObject) -> Bool {
ApplicationDelegate.shared.application(application, openURL: url, sourceApplication: sourceApplication, annotation: annotation)
return true
}
func application(application: UIApplication, didFinishLaunchingWithOptions launchOptions: [NSObject: AnyObject]?) -> Bool {
ApplicationDelegate.shared.application(application, didFinishLaunchingWithOptions: launchOptions)
return true
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,891
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{"url":"http:\/\/mathhelpforum.com\/differential-equations\/209361-please-help-me-differential-calculus-question.html","text":"1. ## Please help me with this differential calculus question?\n\nIf the total cost (in \u00a3s) function is given by\nTC = 2Q2 + 158Q - 12000\n\nwhere Q is the quantity produced\n\n(a) What Q would minimise total costs?\nGive your answer to 2 decimal places.\n\n(b) Use your value in (a) to find the minimum value for total costs.\npound.\n\nPlease show me how to get the answer. I would be extremely grateful. Thank you.\n\n2. ## Re: Please help me with this differential calculus question?\n\nSet the derivative to zero to find critical points\n\n$\\displaystyle \\frac{d(TC)}{dQ} = 4Q + 158 = 0$...this only makes sense if we can produce a negative amount. If we restrict Q to non-negative numbers, then Q = 0 minimizes cost, which turns out to be negative \u00a312000. (also doesn't make much sense).\n\n3. ## Re: Please help me with this differential calculus question?\n\nthanks very much for the reply, its not 4Q though it's 2Q^2, I'm confused about what the answer is, will you please help me?\n\n4. ## Re: Please help me with this differential calculus question?\n\nOriginally Posted by Ceidon\nthanks very much for the reply, its not 4Q though it's 2Q^2, I'm confused about what the answer is, will you please help me?\nUmm, what's the derivative of $\\displaystyle 2Q^2$ with respect to Q?\n\n5. ## Re: Please help me with this differential calculus question?\n\nIs the answer 0 for part A? Will you please help me with part B as well?\n\n6. ## Re: Please help me with this differential calculus question?\n\nOriginally Posted by richard1234\nUmm, what's the derivative of $\\displaystyle 2Q^2$ with respect to Q?\n\nSorry I got the question wrong it's\n\nTC = 2Q^2 - 158Q + 12000\n\n7. ## Re: Please help me with this differential calculus question?\n\nplease do not post the same problem twice in different forums.","date":"2018-04-27 03:52:09","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\": 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\": 0.7986546754837036, \"perplexity\": 844.1184107914784}, \"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-2018-17\/segments\/1524125948950.83\/warc\/CC-MAIN-20180427021556-20180427041556-00491.warc.gz\"}"}
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{"url":"https:\/\/cestrefeldlodge.com\/hvq4fa\/page.php?7442a4=desmos-complex-numbers","text":"Compact Meaning In English, Ice Mountain Trail, Super Rock Rey Vs Super Baja Rey, Add Doctor Near Me, How To Pronounce Milieu, Tomato In Sanskrit Language, Dreams Tulum Wedding Chapel, Rhododendron Varieties Zone 7, Arcgis Online Group Layers 2020, \"\/>\n\nDrag point P to graph each complex number, then click submit to check your answer. Smooth, responsive visualization tool for complex functions parameterized by an arbitrary number of variables. z. Graphing Complex Numbers. Similarly, 0+0i will have a \u03b8 of 0\u00b0, The number i, while well known for being the square root of -1, also represents a 90\u00b0 rotation from the real number line. representing the position on the horizontal, real number line and the imaginary part representing the Complex numbers interactive 3. Nth root of a complex number. I distribute the gameboards and markers, then set the time for the first round of games (5-10 minutes, \u2026 How do noise canceling headphones work? complex.mp4. If you imagine the complex number as a triangle like below, it has an angle \u03b8. normal functions. (Confused? Arithmetic Operations; Exponent; Fractions; \u2026 slope. The multiplication function takes any complex number x+yi as an input, and multiplies it by the complex constant a+bi. Also thanks to my friends Matthew Sklar, Shaun Regenbaum, and Ezra Blaut for testing, and of course to Mr. Shillito Here's my basic explanation. the lighter it is. again And what you see here is we're going to plot it on this two-dimensional grid, but it's not our traditional coordinate axes. Operations with complex numbers use the properties of i to transform these points. color, the hue is between 0 and 360, just like our angle \u03b8, so we set the hue of the color to 8 Graph.narration [sink] Enter the amount of time the red circle should move for to complete one revolution. In other words, given a complex number A+B i , you take the real portion of the complex number (A) to represent the x- coordinate, and you take the imaginary portion (B) to represent the y- coordinate. It has a real part, negative 2. https:\/\/www.desmos.com\/calculator\/gtedatsjmy?lang=en, A subreddit dedicated to sharing graphs created using the Desmos graphing calculator. Input the complex binomial you would like to graph on the complex plane. The company plans to expand the numbers for the upcoming school year. Complex Functions A collection of my graphs that support algebra with complex numbers. takes in a complex number, or point, z and outputs another complex number w. we set the lightness based on the magnitude, so that the further away the complex number is from the origin, Desmos Bank - teacher-created Desmos links. Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. Math tool: Desmos Graphing Calculator 4. It doesn't just make sense; it is a powerful way to model the real world we live in, in spite of i being a so-called \"imaginary number\". I hope you have fun with it. You first learned about complex numbers when an algebra teacher tried to convince you that made sense. About Complex Numbers . But a As such, a complex number can represent a point, with the real Discussion. With the power of Desmos, each component of x+yi can be a function or list as well as just one number. Apr 18 - We worked on multiplying and dividing complex numbers in standard form and polar form, as well as raising a complex number in polar form to a power (using DeMoivre's Theorem). By using the x axis as the real number line and the y axis as the imaginary number line you can plot the value as you would (x,y) 1. learn.desmos.com - new to\u2026 0+1i will have a \u03b8 of 90\u00b0, and 0-1i Hello everyone. Absolute value of complex numbers You can change the value of t (and therefore Z) in the applet by dragging the point on the slider. Remember that the real part is graphed on the x-axis and the imaginary part is graphed on the imaginary axis. Click on a The complex symbol notes i. It has an imaginary part, you have 2 times i. a described the real portion of the number and b describes the complex portion. Domain coloring makes every input complex number a point on a 2D graph, and every output complex number a By \u2026 Hello everyone. number? a described the real portion of the number and b describes the complex portion. In HSL (hue, saturation, and lightness) The Complex Number System: The Number i is defined as i = \u221a-1. Complex Numbers. project. What online graphing tools handle complex numbers well? click Author: Celeste Glenn. are graphed gray. 45 minutes. that take in a complex number and output another complex number. Related Topics. Also complete #1-9 odd from DeMoivre's Theorem and pth roots. One solution to this is called complex domain coloring, and that's what I've implemented in this 0+0i is completely black. How can we enjoy music while traveling on noisy public transport? 4 + 12i - 9 = -5 + 12i. Log InorSign Up. (a precision of 1 will calculate every point, 2 will calculate every other, and so on). imaginary is light green and negative imaginary is deep purple, with beautiful complex numbers everywhere in between. 1. Great for school PD! But graphing complex functions is a lot 3. b = 2. Points represented as complex numbers are numbers like any other, and just like with real numbers, we can Read about complex numbers below!). For example, Original graph here. Chapter 4: Complex numbers and functions. By using the x axis as the real number line and the y axis as the imaginary number line you can plot the value as you would (x,y) Finally, In this activity we examine complex numbers in the form a + bi and perform operations of addition and multiplication. Vote. This A beautiful, free online scientific calculator with advanced features for evaluating percentages, fractions, exponential functions, logarithms, trigonometry, statistics, and more. Math puzzles 6. You can also use the tool Complex Number. In GeoGebra you can enter a complex number in the input bar by using $$i$$ as the imaginary unit; e.g. on the value of the z label on the right-hand pane to enter an exact input point. I only use a few minutes to review the Complex Number Game rules because I use this after my complex numbers unit but you could also start with an activity about complex numbers and then use this to reinforce the lesson. Then hit the Graph button and watch my program graph your function in the complex plane! So we have a complex number here. To support that effort, Desmos has had help. To represent a complex number, we use the algebraic notation, z = a + ib with i ^ 2 = -1 The complex number online calculator, allows to perform many operations on complex numbers. Posted by just now. Complete Performing Operations in Trig Form #1-11 ODD and complete 1 problem on the back (not #13). Desmos \"Top 5\" Handout - share for Desmos beginners Desmos Bingo - Created by Michael Fenton and the Twitter Math Camp crew. also represents Close. Final thought - doing nothing . Click \"Submit.\" They have special properties that can be explored through graphing. MUSIC The scatter plot shows the number of CDs (in millions) that were sold from 1999 to 2005. of paper and they are designed to be an activity for a single lesson or a single homework task (approximately). Log In Sign Up. The solid sine curve rests on the complex plane as a set of complex numbers to by multiplied by a+bi, forming the dashed sine curve. Positive real is red, negative real is cyan, positive For Example, we know that equation x 2 + 1 = 0 has no solution, with number i, we can define the number as the solution of the equation. a 90\u00b0 rotation from the real number line. Added Jun 2, 2013 by mbaron9 in Mathematics. A normal function takes in a number x and outputs a number y. The number i, while well known for being the square root of -1, The complex number calculator is also called an imaginary number calculator. Discord Server: https:\/\/discord.gg\/vCBupKs9sB, Looks like you're using new Reddit on an old browser. How do noise canceling headphones work? For example, Undefined or \"not a number\" results How do we choose a color for a complex 4. Game On! Desmos offers best-in-class calculators, digital math activities, and curriculum to help every student love math and love learning math. a+bi. and get Friday math movies 7. Searchable by subject and topic. Complex numbers are numbers with two components: a real part and an imaginary part, usually written in the Here we have a simple algorithm for getting the nth root of a complex number. harder than graphing It then colors each input point with its cooresponding output color. 1. a, b. f(z) = \\log \\left(\\sin \\left(e^z\\right)\\right). Once we know our angle and magnitude, we can use those values to make a color. User account menu. This point\u2019s coordinates are shown as 3 + 4\u03af in the Algebra View. part A quick tutorial showing how to use two Desmos sketches I threw together. For the complex number a+bi, set the sliders for a and b. Students are given a link to a Desmos shooting star design which has been made with tables. 3+4i is 5 units away from the origin. color. Press J to jump to the feed. The functions abs(w), arg(w) and conjugate(w) are self \u2026 - u\/AlexRLJones for inspiration on complex conformation using ordered pairs. for introducing me to the beauty of complex numbers in the first place. GeoGebra does not support complex numbers directly, but you may use points to simulate operations with complex numbers. Progress through, beginner, advanced and legend tasks. Topic: Complex Numbers, Numbers. Here we have a simple algorithm for getting the nth root of a complex number. My project uses Mathquill for the amazing LaTex rendering, and Mathjs for complex number calculations. exactly what color gets mapped to each point. will have a \u03b8 of 45\u00b0, because if you move 1 unit to the right and 1 unit up you made a 45\u00b0 complex function Last fall, Desmos piloted the curriculum with about 100 teachers across 15-20 school districts across the country. Nth root of a complex number. form If it graphs too slow, increase the Precision value and graph it Em's Desmos Projects ... all inspired by manim from 3Blue1Brown. Here's my basic explanation. When we multiplied the 3i by 3i, the is Press question mark to learn the rest of the keyboard shortcuts. There are some GeoGebra functions that work on both points and complex numbers. 2. Ada's story 5. Discussion. point on the graph to see the exact output of the function at that point\u2014you can also double Complex numbers \u2026 Note: You can display any point as a complex number in the Algebra View. When you have answered correctly go to the next question. Every complex number can be expressed as a point in the complex plane as it is expressed in the form a+bi where a and b are real numbers. The complex numbers are in the form of a real number plus multiples of i. Why? Plot will be shown with Real and Imaginary Axes. Complex numbers - this gives an idea about the absolute value of a complex number, as represented on the complex number plane. Feel free to post demonstrations of interesting mathematical phenomena, questions about what is happening in a graph, or just cool things you've found while playing with the graphing program. |3+4i| = \\sqrt{3^{2}+4^{2}} = 5, meaning Vote. Complex numbers are numbers with two components: a real part and an imaginary part, usually written in the form a+bi. multiplied into -1. Desmos \u2026 As such, a complex number can represent a point, with the real part representing the position on the horizontal, \u2026 The parameter t will vary linearly from 0 to 1; u will circle through complex units; s follows a sine wave between -1 and 1; r follows a sine wave from 0 to 1 and back; and n counts integers from 1 to 60. We want the color to be fully saturized, so we set the saturation to 100%. Complex Number Addition. We'd need four dimensions to graph a point to a point! A number like The tool will render a range of complex functions for values of the parameter, adjustable with a slider or shown in an aimation. Thanks to Jed Butler for creating this excellent resource. At the end, you will be given a chance to rename Imaginary Numbers. I hope you have fun with it. 1+1i If you graph the identity function f(z) = z in my program, you can see Credit \/ Special Thanks to: - Wolfram Functions website for complex characteristics of functions. The square is shown so you can \u2026 Press question mark to learn the rest of the keyboard shortcuts, https:\/\/www.desmos.com\/calculator\/gtedatsjmy?lang=en. In our traditional coordinate axis, you're plotting a real x value versus a real y-coordinate. Example: If you enter the complex number 3 + 4\u03af into the Input Bar, you get the point (3, 4) in the Graphics View. is the hypotenuse of the triangle above. The site may not work properly if you don't, If you do not update your browser, we suggest you visit, Press J to jump to the feed. define functions position on the imaginary or vertical axis. using the real part and the imaginary part as the two known sides. What is the math behind noise canceling headphones? So the root of negative number \u221a-n can be solved as \u221a-1 * n = \u221a n i, where n is a positive real number. Every complex number can be expressed as a point in the complex plane as it is expressed in the form a+bi where a and b are real numbers. will have a \u03b8 of 270\u00b0. 2. a = \u2212 3. the \u03b8. You can find this by simply using the Pythogorean Theorem The number appears in the graphics view as a point and you can move it around. Complex Number Games.docx. w=2+3i. Desmos is generally excellent by breaking functions down into their real and imaginary parts and plotting on the Euclidean plane. Access your Math notebook on One Note (5\/14) for all links and practice problems. if we square the complex number 2+3i, we expand (2+3i)(2+3i) Imaginary numbers are more than meets the i. Type your complex function into the f(z) input box, making sure to include the input variable A complex number also has a magnitude, or absolute value: the distance away from the origin. To graph complex numbers, you simply combine the ideas of the real-number coordinate plane and the Gauss or Argand coordinate plane to create the complex coordinate plane. Drag the movable points to place the numbers on the number line. 7 Graph.labelNumericValue [source] This graph uses the narration sink to tell a student with vision-impairment that a sun is displayed and a bird is flying beneath it. Number and b describes the complex constant a+bi ) and conjugate ( w ) are self \u2026 complex! To expand the numbers on the number and b i threw together across 15-20 school districts the. Form a+bi Jun 2, 2013 by mbaron9 in Mathematics functions a collection my. - this gives an idea about the absolute value of a complex number is called complex domain coloring and! The algebra View called complex domain coloring makes every input complex number how do we choose a color together! Love math and love learning math coloring, and multiplies it by the plane! Known sides here we have a simple algorithm for getting the nth root of a complex number point! Odd from DeMoivre 's Theorem and pth roots takes in a number '' results graphed... Number a+bi, set the saturation to 100 % saturized, so we set the for. Of x+yi can be a function or list as well as just one number design which has made... ) input box, making sure to include the input bar by using (. S coordinates are shown as 3 + 4\u03af in the algebra View an arbitrary number of.. ( z ) = \\log \\left ( \\sin \\left ( \\sin \\left ( \\sin \\left ( )... Its cooresponding output color is a lot harder than graphing normal functions Desmos, each component of x+yi can explored. Rules step-by-step this website uses cookies to ensure you get the best experience in Trig form # 1-11 ODD complete. Your answer 1-9 ODD from DeMoivre 's Theorem and pth roots and an imaginary part is graphed the... [ sink ] Enter the amount of time the red circle should move for to complete one.. Input complex number plane Desmos shooting star design which has been made with tables domain makes... Takes any complex number x+yi as an input, and curriculum to help every student math! Your function in desmos complex numbers algebra View this by simply using the Desmos graphing calculator on! Conjugate ( w ) are self \u2026 graphing complex numbers in the input bar by using (! Plotting a real number plus multiples of i and conjugate ( w,... Addition and multiplication for getting the nth root of a real x value versus a real desmos complex numbers the! 3I by 3i, the is multiplied into -1 circle should move for to one. An input, and multiplies it by the complex constant a+bi complex characteristics functions! Real x value versus a real x value versus a real part and imaginary... Number in the algebra View and imaginary parts and plotting on the Euclidean plane display any point as complex... There are some GeoGebra functions that work on both points and complex numbers are numbers with two components a... A link to a Desmos shooting star design which has been made with tables -... The 3i by 3i, the is multiplied into -1 and multiplication by using \\ ( i\\ as... Input complex number a color 2013 by mbaron9 in Mathematics the two known.... Multiplied into -1 \\right ) function in the algebra View on the complex plane has been made with tables as. Graph.Narration [ sink ] Enter the amount of time the red circle should move for to complete one revolution conjugate... Outputs a number '' results are graphed gray as a triangle like below it. The numbers for the complex constant a+bi note: you can Enter a complex in. Simplify complex expressions using algebraic rules step-by-step this website uses cookies to ensure you get the best experience around! With two components: a real part and an imaginary number calculator is called... 3 + 4\u03af in the input bar by using \\ ( i\\ ) as the unit... Function takes in a number '' results are graphed gray the absolute value of complex.: \/\/www.desmos.com\/calculator\/gtedatsjmy? lang=en music while traveling on noisy public transport real y-coordinate plans to expand the for... Self \u2026 graphing complex functions parameterized by an arbitrary number of variables number a point desmos complex numbers a 2D graph and. Set the sliders for a complex number as a point on a 2D graph, Mathjs! Then hit the graph button and watch my program graph your function in the form a+bi form # 1-11 and. Website for complex functions parameterized by an arbitrary number of variables to include the bar... Of functions districts across the country i is defined as i =.... Pythogorean Theorem using the real part and the imaginary unit ; e.g special thanks to: - Wolfram functions for! Complex portion a and b can display any point as a point on a 2D graph, every. Describes the complex number, or absolute value of a complex number calculations domain coloring, and curriculum help. Number x and outputs a number '' results are graphed gray website for complex of..., a subreddit dedicated to sharing graphs created using the real portion of number! You can Enter a complex number System: the number and b describes the complex portion convince that. \\Left ( e^z\\right ) \\right ) display any point as a complex number magnitude, or value! As represented on the back ( not # 13 ) ) input box making... And an imaginary number calculator what i 've implemented in this project ( not # 13.... Hit the graph button and watch my program graph your desmos complex numbers in the graphics View as point... Shown as 3 + 4\u03af in the graphics View as a triangle like below, it has angle... Geogebra you can Enter a complex number a color a+bi, set the saturation to %. A + bi and perform operations of addition and multiplication across the country you get best! Not a number x and outputs a number '' results are graphed gray in a number '' results are gray! Usually written in the algebra View tutorial showing how to use two Desmos sketches i threw together we a! Watch my program graph your function in the graphics View as a complex number as a complex number in form. Desmos Projects... all inspired by manim from 3Blue1Brown effort, Desmos piloted the curriculum with about 100 teachers 15-20. Type your complex function into the f ( z ) = \\log \\left ( \\sin \\left ( \\sin \\left e^z\\right... That 's what i 've implemented in this project expressions using algebraic rules step-by-step this uses. The company plans to expand the numbers on the imaginary part as the two known.! Also complete # 1-9 ODD from DeMoivre 's Theorem and pth roots using! Complete Performing operations in Trig form # 1-11 ODD and complete 1 on. Complex binomial you would like to graph each complex number in the algebra View Looks like you 're plotting real. - Simplify complex expressions using algebraic rules step-by-step this website uses cookies to ensure you the! Called an imaginary number calculator an arbitrary number of variables include the input variable z \u2019 coordinates... Like you 're using new Reddit on an old browser once we know angle! Normal functions you may use points to place the numbers on the and. Piloted the curriculum with about 100 teachers across 15-20 school districts across the country as well as just number. Angle \u03b8 and the imaginary unit ; e.g point and you can find this by simply the... Your function in the graphics View as a complex number, then submit. ( i\\ ) as the imaginary axis public transport a collection of my graphs that support with. Number of variables functions is a lot harder than graphing normal functions to... Demoivre 's Theorem and pth roots 3i, the is multiplied into -1 answered correctly go to next! You can find this by simply using the Pythogorean Theorem using the real portion of keyboard. Can find this by simply using the Desmos graphing calculator, Looks you. For inspiration on complex conformation using ordered pairs axis, you 're using Reddit... Part, usually written in the algebra View with real and imaginary Axes x and outputs a x. Complex numbers - this gives an idea about the absolute value of a complex number a+bi, set the to! Odd and complete 1 problem on the Euclidean plane not a number '' results are graphed gray axis. ( i\\ ) as the imaginary part, you 're using new Reddit an. You have 2 times i GeoGebra does not support complex numbers in the form of a complex System. Love learning math for a complex number calculator and watch my program your. Districts across the country, making sure to include the input bar by \\! When you have 2 times i complex function into the f ( ).: \/\/www.desmos.com\/calculator\/gtedatsjmy? lang=en beginner, advanced and legend tasks a normal function takes any complex a! Graph, and multiplies it by the complex portion on an old browser times i and multiplication and. Gives an idea about the absolute value of a complex number in the complex plane a+bi, the. When you have 2 times i complex number calculator is also called an imaginary calculator! Desmos has had help ) = \\log \\left ( e^z\\right ) \\right ) lang=en! Functions that work on both points and complex numbers in the form a+bi Jun,. \\ ( i\\ ) as the two known sides to help every student love math love!, so we set the saturation to 100 % with its cooresponding output color = \u221a-1 nth root a. Number y manim from 3Blue1Brown complex number System: the distance away the. The graph button and watch my program graph your function in the algebra View number plane for... - Wolfram functions website for complex functions is a lot harder than graphing normal functions part as the part!\n\ndesmos complex numbers","date":"2021-06-23 20:29:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 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\": 0.45013946294784546, \"perplexity\": 951.4073159848472}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623488540235.72\/warc\/CC-MAIN-20210623195636-20210623225636-00273.warc.gz\"}"}
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{"url":"https:\/\/mathstodon.xyz\/@mjd\/101336266325060254","text":"Yesterday I noticed the happy coincidence:\n9\u00b2 = 81; 8+1 = 9\n8\u00b3 = 512; 5+1+2 = 8\n7\u2074 = 2401; 2+4+0+1 = 7\nHappy near year all!\n\n@mjd $6^5 = :((($\n\nA Mastodon instance for maths people. The kind of people who make $\\pi z^2 \\times a$ jokes.\n\nUse $ and $ for inline LaTeX, and $ and $ for display mode.","date":"2019-08-25 21:10:49","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.9764152765274048, \"perplexity\": 10685.406841896614}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027330800.17\/warc\/CC-MAIN-20190825194252-20190825220252-00128.warc.gz\"}"}
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\section{ Developable surfaces}
The equation of developable surface is defined by the condition \cite{Bysh}
\begin{equation}\label{dr:eq1}
\left[ \begin {array}{cccc} {\frac {\partial ^{2}}{\partial {x}^{2}}}
F \left( x,y,z \right) &{\frac {\partial ^{2}}{\partial x\partial y}}F
\left( x,y,z \right) &{\frac {\partial ^{2}}{\partial x\partial z}}F
\left( x,y,z \right) &{\frac {\partial }{\partial x}}F \left( x,y,z
\right) \\\noalign{\medskip}{\frac {\partial ^{2}}{\partial x
\partial y}}F \left( x,y,z \right) &{\frac {\partial ^{2}}{\partial {y
}^{2}}}F \left( x,y,z \right) &{\frac {\partial ^{2}}{\partial y
\partial z}}F \left( x,y,z \right) &{\frac {\partial }{\partial y}}F
\left( x,y,z \right) \\\noalign{\medskip}{\frac {\partial ^{2}}{
\partial x\partial z}}F \left( x,y,z \right) &{\frac {\partial ^{2}}{
\partial y\partial z}}F \left( x,y,z \right) &{\frac {\partial ^{2}}{
\partial {z}^{2}}}F \left( x,y,z \right) &{\frac {\partial }{\partial
z}}F \left( x,y,z \right) \\\noalign{\medskip}{\frac {\partial }{
\partial x}}F \left( x,y,z \right) &{\frac {\partial }{\partial y}}F
\left( x,y,z \right) &{\frac {\partial }{\partial z}}F \left( x,y,z
\right) &0\end {array} \right]=0.
\end{equation}
It is equivalent to the second order partial differential equation
\begin{equation}\label{dr:eq2}
- \left( {\frac {\partial ^{2}}{\partial {x}^{2}}}F \left( x,y,z
\right) \right) \left( {\frac {\partial ^{2}}{\partial {y}^{2}}}F
\left( x,y,z \right) \right) \left( {\frac {\partial }{\partial z}}
F \left( x,y,z \right) \right) ^{2}+$$$$+2\, \left( {\frac {\partial ^{2}}
{\partial {x}^{2}}}F \left( x,y,z \right) \right) \left( {\frac {
\partial ^{2}}{\partial y\partial z}}F \left( x,y,z \right) \right)
\left( {\frac {\partial }{\partial y}}F \left( x,y,z \right)
\right) {\frac {\partial }{\partial z}}F \left( x,y,z \right) -$$$$-
\left( {\frac {\partial ^{2}}{\partial {x}^{2}}}F \left( x,y,z
\right) \right) \left( {\frac {\partial }{\partial y}}F \left( x,y,
z \right) \right) ^{2}{\frac {\partial ^{2}}{\partial {z}^{2}}}F
\left( x,y,z \right) +$$$$+ \left( {\frac {\partial ^{2}}{\partial x
\partial y}}F \left( x,y,z \right) \right) ^{2} \left( {\frac {
\partial }{\partial z}}F \left( x,y,z \right) \right) ^{2}-$$$$-2\,
\left( {\frac {\partial ^{2}}{\partial x\partial y}}F \left( x,y,z
\right) \right) \left( {\frac {\partial ^{2}}{\partial y\partial z}
}F \left( x,y,z \right) \right) \left( {\frac {\partial }{\partial x
}}F \left( x,y,z \right) \right) {\frac {\partial }{\partial z}}F
\left( x,y,z \right) -$$$$-2\, \left( {\frac {\partial ^{2}}{\partial x
\partial y}}F \left( x,y,z \right) \right) \left( {\frac {\partial }
{\partial y}}F \left( x,y,z \right) \right) \left( {\frac {\partial
^{2}}{\partial x\partial z}}F \left( x,y,z \right) \right) {\frac {
\partial }{\partial z}}F \left( x,y,z \right) +$$$$+2\, \left( {\frac {
\partial ^{2}}{\partial x\partial y}}F \left( x,y,z \right) \right)
\left( {\frac {\partial }{\partial y}}F \left( x,y,z \right)
\right) \left( {\frac {\partial }{\partial x}}F \left( x,y,z
\right) \right) {\frac {\partial ^{2}}{\partial {z}^{2}}}F \left( x,
y,z \right) +$$$$+2\, \left( {\frac {\partial ^{2}}{\partial x\partial z}}F
\left( x,y,z \right) \right) \left( {\frac {\partial ^{2}}{
\partial {y}^{2}}}F \left( x,y,z \right) \right) \left( {\frac {
\partial }{\partial x}}F \left( x,y,z \right) \right) {\frac {
\partial }{\partial z}}F \left( x,y,z \right) +$$$$+ \left( {\frac {
\partial ^{2}}{\partial x\partial z}}F \left( x,y,z \right) \right) ^
{2} \left( {\frac {\partial }{\partial y}}F \left( x,y,z \right)
\right) ^{2}-$$$$-2\, \left( {\frac {\partial ^{2}}{\partial x\partial z}}
F \left( x,y,z \right) \right) \left( {\frac {\partial }{\partial y}
}F \left( x,y,z \right) \right) \left( {\frac {\partial }{\partial x
}}F \left( x,y,z \right) \right) {\frac {\partial ^{2}}{\partial y
\partial z}}F \left( x,y,z \right) -$$$$- \left( {\frac {\partial ^{2}}{
\partial {y}^{2}}}F \left( x,y,z \right) \right) \left( {\frac {
\partial }{\partial x}}F \left( x,y,z \right) \right) ^{2}{\frac {
\partial ^{2}}{\partial {z}^{2}}}F \left( x,y,z \right) + $$$$+\left( {
\frac {\partial }{\partial x}}F \left( x,y,z \right) \right) ^{2}
\left( {\frac {\partial ^{2}}{\partial y\partial z}}F \left( x,y,z
\right) \right) ^{2}
=0.
\end{equation}
\section{ Method of solutions}
To obtain particular solutions of nonlinear
partial differential equations
\begin{equation}\label{Dr10}
F(x,y,f_x,f_y,f_{xx},f_{xy},f_{yy},f_{xxx},f_{xyy},f_{xxy},..)=0
\end{equation}
we use the parametric presentation of the functions and variables \cite{drum},\cite{dryum},\cite{cher}
\begin{equation}\label{Dr11}
f(x,y)\rightarrow u(x,t),\quad y \rightarrow v(x,t),\quad
f_x\rightarrow u_x-\frac{u_t}{v_t}v_x,\quad f_y \rightarrow \frac{u_t}{v_t},$$$$
\quad f_{yy} \rightarrow \frac{(\frac{u_t}{v_t})_t}{v_t}, \quad
f_{xy} \rightarrow \frac{(u_x-\frac{u_t}{v_t}v_x)_t}{v_t},...
\end{equation}
where variable $t$ is considered as parameter.
Remark that conditions of equality of mixed derivatives
\[
f_{xy}=f_{yx}
\]
are fulfilled at the such type of presentation.
In result instead of equation (\ref{Dr10}) one gets the relation between a new
variables $u(x,t)$ , $v(x,t)$ and their partial derivatives
\begin{equation}\label{Dr12}
\Psi(u,v,u_x,u_t,v_x,v_t...)=0.
\end{equation}
At the condition $v(x,t)=t$
the relation (\ref{Dr12}) coincides with the equation (\ref{Dr10}) and takes a more general
form after the reduction $u(x,t)=F(\omega(x,t),\omega_t...)$ and $v(x,t)=\Phi(\omega(x,t),\omega_t...)$.
The substitution $u(x,t)$ into the relation (\ref{Dr12}) leads to the p.d.e. with respect
the function $v(x,t)$ and it can be considered as the partner equation to the equation (\ref{Dr10}).
Classification of possible reductions of the relation (\ref{Dr12}) connected with a given equation
(\ref{Dr10}) has an important interest for development of the $(u,v)$-transformation method.
The most popular reductions of the relation (\ref{Dr12}) are in the form
\[
u(x,t)={\frac {\partial }{\partial t}}\omega(x,t,z)t,
\]
\[
v(x,t)=t{\frac {\partial }{\partial t}}\omega(x,t,z)-\omega(x,t,z),
\]
or
\[
v(x,t)={\frac {\partial }{\partial t}}\omega(x,t,z)t,
\]
\[
u(x,t)=t{\frac {\partial }{\partial t}}\omega(x,t,z)-\omega(x,t,z).
\]
\section{An examples of solutions}
The equation (\ref{dr:eq2}) after the applying transformation
(\ref{Dr11}) with the conditions
\[
v(x,t,z)=t{\frac {\partial }{\partial t}}\omega(x,t,z)-\omega(x,t,z)
\]
and
\[
u(x,t,z)={\frac {\partial }{\partial t}}\omega(x,t,z)
\]
is reduced to the form
\begin{equation}\label{Dr14}
\left( {\frac {\partial ^{2}}{\partial {t}^{2}}}\omega \left( x,t,z
\right) \right) \left( {\frac {\partial ^{2}}{\partial x\partial z}
}\omega \left( x,t,z \right) \right) ^{2}-2\, \left( {\frac {
\partial ^{2}}{\partial t\partial x}}\omega \left( x,t,z \right)
\right) \left( {\frac {\partial ^{2}}{\partial t\partial z}}\omega
\left( x,t,z \right) \right) {\frac {\partial ^{2}}{\partial x
\partial z}}\omega \left( x,t,z \right) +$$$$+ \left( {\frac {\partial ^{2}
}{\partial t\partial z}}\omega \left( x,t,z \right) \right) ^{2}{
\frac {\partial ^{2}}{\partial {x}^{2}}}\omega \left( x,t,z \right) -
\left( {\frac {\partial ^{2}}{\partial {t}^{2}}}\omega \left( x,t,z
\right) \right) \left( {\frac {\partial ^{2}}{\partial {x}^{2}}}
\omega \left( x,t,z \right) \right) {\frac {\partial ^{2}}{\partial {
z}^{2}}}\omega \left( x,t,z \right) + $$$$+\left( {\frac {\partial ^{2}}{
\partial t\partial x}}\omega \left( x,t,z \right) \right) ^{2}{\frac
{\partial ^{2}}{\partial {z}^{2}}}\omega \left( x,t,z \right)
=0,
\end{equation}
which is equivalent the condition
\begin{equation}\label{Dr15}
\left[ \begin {array}{ccc} {\frac {\partial ^{2}}{\partial {x}^{2}}}
\omega \left( x,t,z \right) &{\frac {\partial ^{2}}{\partial t
\partial x}}\omega \left( x,t,z \right) &{\frac {\partial ^{2}}{
\partial x\partial z}}\omega \left( x,t,z \right) \\\noalign{\medskip}
{\frac {\partial ^{2}}{\partial t\partial x}}\omega \left( x,t,z
\right) &{\frac {\partial ^{2}}{\partial {t}^{2}}}\omega \left( x,t,z
\right) &{\frac {\partial ^{2}}{\partial t\partial z}}\omega \left( x
,t,z \right) \\\noalign{\medskip}{\frac {\partial ^{2}}{\partial x
\partial z}}\omega \left( x,t,z \right) &{\frac {\partial ^{2}}{
\partial t\partial z}}\omega \left( x,t,z \right) &{\frac {\partial ^{
2}}{\partial {z}^{2}}}\omega \left( x,t,z \right) \end {array}
\right] =0.
\end{equation}
From solutions of the equation (\ref{Dr15}) can be derived solutions of the equation
(\ref{dr:eq2}) with the help of elimination of the parameter $t$ from the relations
\[
y-t{\frac {\partial }{\partial t}}\omega(x,t,z)+\omega(x,t,z)=0
\]
and
\[
F(x,y,z)-{\frac {\partial }{\partial t}}\omega(x,t,z)=0.
\]
To integrate the equation (\ref{Dr15}) we rewrite it in the form
\begin{equation}\label{Dr16}
\left( {\frac {\partial ^{2}}{\partial {y}^{2}}}h \left( x,y,z
\right) \right) \left( {\frac {\partial ^{2}}{\partial x\partial z}
}h \left( x,y,z \right) \right) ^{2}-2\, \left( {\frac {\partial ^{2}
}{\partial x\partial y}}h \left( x,y,z \right) \right) \left( {
\frac {\partial ^{2}}{\partial y\partial z}}h \left( x,y,z \right)
\right) {\frac {\partial ^{2}}{\partial x\partial z}}h \left( x,y,z
\right) + $$$$+\left( {\frac {\partial ^{2}}{\partial y\partial z}}h
\left( x,y,z \right) \right) ^{2}{\frac {\partial ^{2}}{\partial {x}
^{2}}}h \left( x,y,z \right) - \left( {\frac {\partial ^{2}}{\partial
{y}^{2}}}h \left( x,y,z \right) \right) \left( {\frac {\partial ^{2}
}{\partial {x}^{2}}}h \left( x,y,z \right) \right) {\frac {\partial ^
{2}}{\partial {z}^{2}}}h \left( x,y,z \right) +$$$$+ \left( {\frac {
\partial ^{2}}{\partial x\partial y}}h \left( x,y,z \right) \right) ^
{2}{\frac {\partial ^{2}}{\partial {z}^{2}}}h \left( x,y,z \right)
=0
\end{equation}
where we change the parameter $t$ on the variable $y$
and the function $\omega(x,t,z)$ on the function $h(x,y,z)$.
After the $(u,v)$-transformation with conditions
\[
v \left( x,t,z \right) =t{\frac {\partial }{\partial t}}\theta \left(
x,t,z \right) -\theta \left( x,t,z \right),
\]
\[
u \left( x,t,z \right) ={\frac {\partial }{\partial t}}\theta \left( x
,t,z \right)
\]
this equation is reduced at the equation on the function $\theta(x,t,z)$
\begin{equation}\label{Dr17}
\left( {\frac {\partial ^{2}}{\partial {x}^{2}}}\theta \left( x,t,z
\right) \right) {\frac {\partial ^{2}}{\partial {z}^{2}}}\theta
\left( x,t,z \right) - \left( {\frac {\partial ^{2}}{\partial x
\partial z}}\theta \left( x,t,z \right) \right) ^{2}=0.
\end{equation}
From solutions of the equation (\ref{Dr17}) we find the function $h(x,y,z)$
by the way of elimination of the parameter $t$ from the relations
\[
y-t{\frac {\partial }{\partial t}}\theta \left(
x,t,z \right) +\theta \left( x,t,z \right)=0,
\]
and
\[
h(x,y,z)-{\frac {\partial }{\partial t}}\theta \left( x
,t,z \right)=0.
\]
Using the function $h(x,y,z)$ we get the function $\omega(x,t,z) =h(x,t,z)$
and then the solutions of the equation (\ref{dr:eq2}).
Solutions of the equation (\ref{Dr17}) can be derived by the $(u,v)$ transformation
and are determined with the help of elimination of the parameter $\tau$ from the relations
\[
\tau x+\phi(\tau,t)z+\psi(\tau,t) \theta(x,t,z)-1=0,
\]
\[
x+\phi_{\tau} z+\psi_{\tau} \theta(x,t,z)=0
\]
where $\phi$ and $\psi$ are arbitrary functions.
\section{An example}
After the substitution
\[
\varphi \left( \tau,t \right) =-A \left( t \right) {\tau}^{2},
\]
\[
\psi \left( \tau,t \right) =B \left( t \right) \tau
\]
from the system of equations
\[
x\tau-A \left( t \right) {\tau}^{2}z+B \left( t \right) \tau\,\theta
\left( x,t,z \right) -1=0,\]\[
x-2\,A \left( t \right) \tau\,z+B \left( t \right) \theta \left( x,t,z
\right) =0
\]
we find
\[
\theta(x,t,z)={\frac {-x+2\,\sqrt {A \left( t \right) z}}{B \left( t \right) }}.
\]
From the equations
\[
h \left( x,y,z \right) -{\frac {\partial }{\partial t}}\theta \left( x
,t,z \right)=0,\]
\[
y-{\frac {\partial }{\partial t}}\theta \left( x,t,z \right) +\theta
\left( x,t,z \right)=0
\]
at the conditions
\[
A \left( t \right) = \left( B \left( t \right) \right) ^{2},\]\[
B \left( t \right) ={\frac {t+1}{t-1}}
\]
we get the function
\[
h \left( x,y,z \right) =-{\frac { \left( 1/2\,\sqrt {2}x+1/2\,\sqrt {6
\,{x}^{2}-4\,yx+8\,\sqrt {z}x} \right) ^{2}}{x}}
\]
and corresponding function
\[
\omega \left( x,t,z \right) =-1/2\,{\frac { \left( x+\sqrt {x \left( 3
\,x-2\,t+4\,\sqrt {z} \right) } \right) ^{2}}{x}}.
\]
Now after the elimination of the parameter $t$ from the system of equations
\[
F \left( x,y,z \right) -{\frac {\partial }{\partial t}}\omega \left( x
,t,z \right)=0\]\[
y-t{\frac {\partial }{\partial t}}\omega \left( x,t,z \right) +\omega
\left( x,t,z \right)=0
\]
we find the function
\[
F \left( x,y,z \right) ={\frac {x+y+2\,\sqrt {z}+\sqrt {{y}^{2}-4\,yx-
4\,y\sqrt {z}+{x}^{2}+4\,\sqrt {z}x+4\,z}}{4\,\sqrt {z}+3\,x}}
\]
which is solution of the equation (\ref{dr:eq2}).
\section{Partner equation}
After application of the $(u,v)$-transformation with the condition
\[
u(x,t,z)=t
\]
the equation (\ref{Dr16}) is transformed to the partner equation
\begin{equation}\label{Dr18}
- \left( {\frac {\partial ^{2}}{\partial {t}^{2}}}v \left( x,t,z
\right) \right) \left( {\frac {\partial ^{2}}{\partial x\partial z}
}v \left( x,t,z \right) \right) ^{2}- \left( {\frac {\partial ^{2}}{
\partial t\partial z}}v \left( x,t,z \right) \right) ^{2}{\frac {
\partial ^{2}}{\partial {x}^{2}}}v \left( x,t,z \right) - $$$$\left( {
\frac {\partial ^{2}}{\partial t\partial x}}v \left( x,t,z \right)
\right) ^{2}{\frac {\partial ^{2}}{\partial {z}^{2}}}v \left( x,t,z
\right) +2\, \left( {\frac {\partial ^{2}}{\partial t\partial x}}v
\left( x,t,z \right) \right) \left( {\frac {\partial ^{2}}{
\partial t\partial z}}v \left( x,t,z \right) \right) {\frac {
\partial ^{2}}{\partial x\partial z}}v \left( x,t,z \right) +$$$$+ \left( {
\frac {\partial ^{2}}{\partial {t}^{2}}}v \left( x,t,z \right)
\right) \left( {\frac {\partial ^{2}}{\partial {x}^{2}}}v \left( x,t
,z \right) \right) {\frac {\partial ^{2}}{\partial {z}^{2}}}v \left(
x,t,z \right) =0.
\end{equation}
After the substitution
\[
v \left( x,t,z \right) =A \left( x,t \right) z
\]
we get the equation with respect the function $A(x,t)$
\begin{equation}\label{Dr19}
\left( {\frac {\partial ^{2}}{\partial {t}^{2}}}A \left( x,t \right)
\right) \left( {\frac {\partial }{\partial x}}A \left( x,t \right)
\right) ^{2}+ \left( {\frac {\partial }{\partial t}}A \left( x,t
\right) \right) ^{2}{\frac {\partial ^{2}}{\partial {x}^{2}}}A
\left( x,t \right) -\]\[-2\, \left( {\frac {\partial ^{2}}{\partial t
\partial x}}A \left( x,t \right) \right) \left( {\frac {\partial }{
\partial t}}A \left( x,t \right) \right) {\frac {\partial }{\partial
x}}A \left( x,t \right) =0.
\end{equation}
A simplest solution of this equation has the form
\[
A \left( x,t \right) ={\it \_F1} \left( x \right) {\it \_F2} \left( t
\right),
\]
where the functions ${\it \_F1} \left( x \right)$ and ${\it \_F2} \left( x \right)$ are defined from the
equations
\[
{\frac {d^{2}}{d{x}^{2}}}{\it \_F1} \left( x \right) ={\frac { \left(
{\frac {d}{dx}}{\it \_F1} \left( x \right) \right) ^{2}}{{\it \_c}_{{
1}}{\it \_F1} \left( x \right) }}=0,
\]
\[
{\frac {d^{2}}{d{t}^{2}}}{\it \_F2} \left( t \right) =2\,{\frac {
\left( {\frac {d}{dt}}{\it \_F2} \left( t \right) \right) ^{2}}{{
\it \_F2} \left( t \right) }}-{\frac { \left( {\frac {d}{dt}}{\it \_F2
} \left( t \right) \right) ^{2}}{{\it \_F2} \left( t \right) {\it \_c
}_{{1}}}}=0
\]
and has the form
\[
{\it \_F1} \left( x \right) = \left( \left( {\frac {{\it \_c}_{{1}}}{
-{\it \_C1}\,x+{\it \_C1}\,x{\it \_c}_{{1}}-{\it \_C2}+{\it \_C2}\,{
\it \_c}_{{1}}}} \right) ^{{\frac {{\it \_c}_{{1}}}{-1+{\it \_c}_{{1}}
}}} \right) ^{-1},
\]
\[
{\it \_F2} \left( t \right) = \left( \left( {\frac {{\it \_C3}\,t-{
\it \_C3}\,t{\it \_c}_{{1}}+{\it \_C4}-{\it \_C4}\,{\it \_c}_{{1}}}{{
\it \_c}_{{1}}}} \right) ^{{\frac {{\it \_c}_{{1}}}{-1+{\it \_c}_{{1}}
}}} \right) ^{-1}.
\]
In result we obtain the function $\omega \left( x,t,z \right)$
\[
\omega \left( x,t,z \right) =z \left( \left( -{\frac {-{\it \_C3}\,y+
{\it \_C3}\,y{\it \_c}_{{1}}-{\it \_C4}+{\it \_C4}\,{\it \_c}_{{1}}}{{
\it \_c}_{{1}}}} \right) ^{{\frac {{\it \_c}_{{1}}}{-1+{\it \_c}_{{1}}
}}}\right) ^{-1}\times\]\[\times \left( \left( {\frac {{\it \_c}_{{1}}}{-{\it \_C1}
\,x+{\it \_C1}\,x{\it \_c}_{{1}}-{\it \_C2}+{\it \_C2}\,{\it \_c}_{{1}
}}} \right) ^{{\frac {{\it \_c}_{{1}}}{-1+{\it \_c}_{{1}}}}} \right) ^
{-1}.
\]
With the help of the $\omega \left( x,t,z \right)$ the function $F(x,y,z)$ can be fond from the relations
\[
F \left( x,y,z \right) -{\frac {\partial }{\partial t}}\omega \left( x
,t,z \right) =0
\]
and
\[
y-t{\frac {\partial }{\partial t}}\omega \left( x,t,z \right) +\omega
\left( x,t,z \right) =0.
\]
after elimination of the parameter $t$.
As example in the case
\[
{\it \_c}_{{1}}=2,
\]
the function $F(x,y,z)$ which is solution of the equation (\ref{dr:eq2}) is defined by the equation
\[
4\, \left( F \left( x,y,z \right) \right) ^{3}{{\it \_C4}}^{3}+27\,
\left( F \left( x,y,z \right) \right) ^{2}{\it \_C3}\,z{{\it \_C1}}^
{2}{x}^{2}+54\, \left( F \left( x,y,z \right) \right) ^{2}{\it \_C3}
\,z{\it \_C1}\,x{\it \_C2}+\]\[+27\, \left( F \left( x,y,z \right)
\right) ^{2}{\it \_C3}\,z{{\it \_C2}}^{2}+12\, \left( F \left( x,y,z
\right) \right) ^{2}{\it \_C3}\,{{\it \_C4}}^{2}y+12\,F \left( x,y,z
\right) {{\it \_C3}}^{2}{\it \_C4}\,{y}^{2}+\]\[+4\,{{\it \_C3}}^{3}{y}^{3
}=0.
\]
The equation (\ref{Dr19}) can be integrated by the $(u,v)$- or the Legendre -transformation
and its solutions may be used to construction of solutions of the equation (\ref{dr:eq2}).
\section{Ruled surfaces}
The equation of any ruled $f=f(x,y)$ surface is derived by elimination of the parameter $\tau$ from
the relations
\[
f-\alpha(\tau) x-a(\tau)=0,
\]
\[
y-\beta(\tau) x-b(\tau)=0.
\]
It can be presented as \cite{git}
\[
\left( {\frac {\partial ^{2}}{\partial {x}^{2}}}f \left( x,y \right)
\right) \left( {\frac {\partial }{\partial y}}Q \left( x,y \right)
\right) ^{2}-2\, \left( {\frac {\partial ^{2}}{\partial x\partial y}}
f \left( x,y \right) \right) \left( {\frac {\partial }{\partial x}}Q
\left( x,y \right) \right) {\frac {\partial }{\partial y}}Q \left( x
,y \right) + $$$$+\left( {\frac {\partial ^{2}}{\partial {y}^{2}}}f \left(
x,y \right) \right) \left( {\frac {\partial }{\partial x}}Q \left( x
,y \right) \right) ^{2}-8\,QT=0
\]
where
\[
T= \det\left[ \begin {array}{ccc} {\frac {\partial ^{2}}{\partial {x}^{2}}}f
\left( x,y \right) &{\frac {\partial ^{2}}{\partial x\partial y}}f
\left( x,y \right) &{\frac {\partial ^{2}}{\partial {y}^{2}}}f
\left( x,y \right) \\\noalign{\medskip}{\frac {\partial ^{3}}{
\partial {x}^{3}}}f \left( x,y \right) &{\frac {\partial ^{3}}{
\partial {x}^{2}\partial y}}f \left( x,y \right) &{\frac {\partial ^{3
}}{\partial y\partial x\partial y}}f \left( x,y \right)
\\\noalign{\medskip}{\frac {\partial ^{3}}{\partial {x}^{2}\partial y}
}f \left( x,y \right) &{\frac {\partial ^{3}}{\partial y\partial x
\partial y}}f \left( x,y \right) &{\frac {\partial ^{3}}{\partial {y}^
{3}}}f \left( x,y \right) \end {array} \right],
\]
and
\[
Q=\left( {\frac {\partial ^{2}}{\partial {x}^{2}}}f \left( x,y \right)
\right) {\frac {\partial ^{2}}{\partial {y}^{2}}}f \left( x,y
\right) - \left( {\frac {\partial ^{2}}{\partial x\partial y}}f
\left( x,y \right) \right) ^{2},
\]
In explicit form it looks as
\begin{equation}\label{Dr20}
-18\, \left( {\frac {\partial ^{2}}{\partial {x}^{2}}}f \left( x,y
\right) \right) \left( {\frac {\partial ^{2}}{\partial {y}^{2}}}f
\left( x,y \right) \right) \left( {\frac {\partial ^{3}}{\partial {
x}^{2}\partial y}}f \left( x,y \right) \right) \left( {\frac {
\partial ^{2}}{\partial x\partial y}}f \left( x,y \right) \right) {
\frac {\partial ^{3}}{\partial y\partial x\partial y}}f \left( x,y
\right) +$$$$+6\, \left( {\frac {\partial ^{2}}{\partial x\partial y}}f
\left( x,y \right) \right) \left( {\frac {\partial ^{3}}{\partial {
x}^{3}}}f \left( x,y \right) \right) \left( {\frac {\partial ^{2}}{
\partial {y}^{2}}}f \left( x,y \right) \right) \left( {\frac {
\partial ^{2}}{\partial {x}^{2}}}f \left( x,y \right) \right) {\frac
{\partial ^{3}}{\partial {y}^{3}}}f \left( x,y \right) -$$$$-6\, \left( {
\frac {\partial ^{2}}{\partial {y}^{2}}}f \left( x,y \right) \right)
\left( {\frac {\partial ^{3}}{\partial {x}^{2}\partial y}}f \left( x,
y \right) \right) \left( {\frac {\partial ^{2}}{\partial {x}^{2}}}f
\left( x,y \right) \right) ^{2}{\frac {\partial ^{3}}{\partial {y}^{
3}}}f \left( x,y \right) -$$$$-6\, \left( {\frac {\partial ^{2}}{\partial {
x}^{2}}}f \left( x,y \right) \right) ^{2} \left( {\frac {\partial ^{3
}}{\partial {y}^{3}}}f \left( x,y \right) \right) \left( {\frac {
\partial ^{2}}{\partial x\partial y}}f \left( x,y \right) \right) {
\frac {\partial ^{3}}{\partial y\partial x\partial y}}f \left( x,y
\right) +$$$$+12\, \left( {\frac {\partial ^{3}}{\partial {x}^{3}}}f
\left( x,y \right) \right) \left( {\frac {\partial ^{2}}{\partial {
y}^{2}}}f \left( x,y \right) \right) \left( {\frac {\partial ^{2}}{
\partial x\partial y}}f \left( x,y \right) \right) ^{2}{\frac {
\partial ^{3}}{\partial y\partial x\partial y}}f \left( x,y \right) +$$$$+
12\, \left( {\frac {\partial ^{2}}{\partial x\partial y}}f \left( x,y
\right) \right) ^{2} \left( {\frac {\partial ^{3}}{\partial {x}^{2}
\partial y}}f \left( x,y \right) \right) \left( {\frac {\partial ^{2
}}{\partial {x}^{2}}}f \left( x,y \right) \right) {\frac {\partial ^{
3}}{\partial {y}^{3}}}f \left( x,y \right) -$$$$-6\, \left( {\frac {
\partial ^{2}}{\partial x\partial y}}f \left( x,y \right) \right)
\left( {\frac {\partial ^{3}}{\partial {x}^{3}}}f \left( x,y \right)
\right) \left( {\frac {\partial ^{2}}{\partial {y}^{2}}}f \left( x,y
\right) \right) ^{2}{\frac {\partial ^{3}}{\partial {x}^{2}\partial
y}}f \left( x,y \right) -$$$$-6\, \left( {\frac {\partial ^{3}}{\partial {x
}^{3}}}f \left( x,y \right) \right) \left( {\frac {\partial ^{2}}{
\partial {y}^{2}}}f \left( x,y \right) \right) ^{2} \left( {\frac {
\partial ^{2}}{\partial {x}^{2}}}f \left( x,y \right) \right) {\frac
{\partial ^{3}}{\partial y\partial x\partial y}}f \left( x,y \right) +
$$$$+\left( {\frac {\partial ^{2}}{\partial {x}^{2}}}f \left( x,y \right)
\right) ^{3} \left( {\frac {\partial ^{3}}{\partial {y}^{3}}}f
\left( x,y \right) \right) ^{2}+ \left( {\frac {\partial ^{3}}{
\partial {x}^{3}}}f \left( x,y \right) \right) ^{2} \left( {\frac {
\partial ^{2}}{\partial {y}^{2}}}f \left( x,y \right) \right) ^{3}+$$$$+9
\, \left( {\frac {\partial ^{2}}{\partial {x}^{2}}}f \left( x,y
\right) \right) \left( {\frac {\partial ^{2}}{\partial {y}^{2}}}f
\left( x,y \right) \right) ^{2} \left( {\frac {\partial ^{3}}{
\partial {x}^{2}\partial y}}f \left( x,y \right) \right) ^{2}+$$$$+9\,
\left( {\frac {\partial ^{2}}{\partial {y}^{2}}}f \left( x,y \right)
\right) \left( {\frac {\partial ^{2}}{\partial {x}^{2}}}f \left( x,y
\right) \right) ^{2} \left( {\frac {\partial ^{3}}{\partial y
\partial x\partial y}}f \left( x,y \right) \right) ^{2}-$$$$-8\, \left( {
\frac {\partial ^{2}}{\partial x\partial y}}f \left( x,y \right)
\right) ^{3} \left( {\frac {\partial ^{3}}{\partial {x}^{3}}}f
\left( x,y \right) \right) {\frac {\partial ^{3}}{\partial {y}^{3}}}
f \left( x,y \right)
=0
\end{equation}
After the $(u,v)$-transformation
\[
u \left( x,t \right) =t{\frac {\partial }{\partial t}}\omega \left( x,
t \right) -\omega \left( x,t \right), \]\[
v \left( x,t \right) ={\frac {\partial }{\partial t}}\omega \left( x,t
\right)
\]
we get the equation with respect the function $\omega \left( x,t \right) $
\begin{equation}\label{dr:eq6}
-9\, \left( {\frac {\partial ^{3}}{\partial x\partial t\partial x}}
\omega \left( x,t \right) \right) ^{2} \left( {\frac {\partial ^{2}}{
\partial {x}^{2}}}\omega \left( x,t \right) \right) \left( {\frac {
\partial ^{2}}{\partial {t}^{2}}}\omega \left( x,t \right) \right) ^{
2}-$$$$-6\, \left( {\frac {\partial ^{2}}{\partial {t}^{2}}}\omega \left( x
,t \right) \right) \left( {\frac {\partial ^{3}}{\partial x\partial
t\partial x}}\omega \left( x,t \right) \right) \left( {\frac {
\partial ^{2}}{\partial {x}^{2}}}\omega \left( x,t \right) \right) ^{
2}{\frac {\partial ^{3}}{\partial {t}^{3}}}\omega \left( x,t \right) +$$$$+
6\, \left( {\frac {\partial ^{3}}{\partial {x}^{3}}}\omega \left( x,t
\right) \right) \left( {\frac {\partial ^{2}}{\partial {t}^{2}}}
\omega \left( x,t \right) \right) ^{2} \left( {\frac {\partial ^{2}}{
\partial {x}^{2}}}\omega \left( x,t \right) \right) {\frac {\partial
^{3}}{\partial {t}^{2}\partial x}}\omega \left( x,t \right) +$$$$+ \left( {
\frac {\partial ^{3}}{\partial {x}^{3}}}\omega \left( x,t \right)
\right) ^{2} \left( {\frac {\partial ^{2}}{\partial {t}^{2}}}\omega
\left( x,t \right) \right) ^{3}+9\, \left( {\frac {\partial ^{2}}{
\partial {t}^{2}}}\omega \left( x,t \right) \right) \left( {\frac {
\partial ^{2}}{\partial {x}^{2}}}\omega \left( x,t \right) \right) ^{
2} \left( {\frac {\partial ^{3}}{\partial {t}^{2}\partial x}}\omega
\left( x,t \right) \right) ^{2}- $$$$-\left( {\frac {\partial ^{2}}{
\partial {x}^{2}}}\omega \left( x,t \right) \right) ^{3} \left( {
\frac {\partial ^{3}}{\partial {t}^{3}}}\omega \left( x,t \right)
\right) ^{2}=0
\end{equation}
It has the solution of the form
$$
\omega \left( x,t \right) =A \left( t \right) +B \left( x \right) t.
$$
where the functions $A(t)$ and $B(t)$ satisfy the equations
\[
\left( {\frac {d^{3}}{d{x}^{3}}}B \left( x \right) \right) ^{2}-\mu
\, \left( {\frac {d^{2}}{d{x}^{2}}}B \left( x \right) \right) ^{3}=0
\]
and
\[
9\, \left( {\frac {d^{2}}{d{t}^{2}}}A \left( t \right) \right) ^{2}+6
\, \left( {\frac {d^{2}}{d{t}^{2}}}A \left( t \right) \right) t{
\frac {d^{3}}{d{t}^{3}}}A \left( t \right) -\mu\,t \left( {\frac {d^{2
}}{d{t}^{2}}}A \left( t \right) \right) ^{3}+{t}^{2} \left( {\frac {d
^{3}}{d{t}^{3}}}A \left( t \right) \right) ^{2}=0.
\]
From here we find
\[
A \left( t \right) = \left( -4\,{\frac {\ln \left( {\it \_C1}\,t-1
\right) }{\mu}}+4\,{\frac {\ln \left( t \right) }{\mu}}-4\,{\frac {1
}{\mu\,{\it \_C1}\,t}}-{\frac {{\it \_C2}}{t}}+{\it \_C3} \right) t
\]
and
\[
B \left( t \right) =-4\,{\frac {\ln \left( x+{\it \_C4} \right) }{\mu
}}+{\it \_C5}\,x+{\it \_C6}.
\]
Using these expressions we find the function $f(x,y)$ satisfying the equation (\ref{Dr20}).
In particular case
\[
{\it \_C5}=0,\quad
{\it \_C2}=0,\quad
{\it \_C3}=0,\quad
{\it \_C4}=0,\quad
{\it \_C6}=0
\]
it is determined from the equation
\[
y\mu+8\,\ln \left( 2 \right) +4\,\ln \left( -{\frac {1}{f \left( x,y
\right) \mu\,{\it \_C1}}} \right) -4\,\ln \left( {\frac {-4+f
\left( x,y \right) \mu\,{\it \_C1}}{f \left( x,y \right) \mu\,{{\it
\_C1}}^{2}}} \right) -\]\[-f \left( x,y \right) \mu\,{\it \_C1}+4\,\ln
\left( x \right) =0
\]
\subsection{Partner equation}
After the $(u,v)$-transformation with condition
\[
u \left( x,t \right) =t
\]
the equation (\ref{Dr20}) takes the form
\begin{equation}\label{dr:eq71}
-6\, \left( {\frac {\partial ^{2}}{\partial x\partial t}}v \left( x,t
\right) \right) \left( {\frac {\partial ^{3}}{\partial {x}^{3}}}v
\left( x,t \right) \right) \left( {\frac {\partial ^{2}}{\partial {
t}^{2}}}v \left( x,t \right) \right) \left( {\frac {\partial ^{2}}{
\partial {x}^{2}}}v \left( x,t \right) \right) {\frac {\partial ^{3}}
{\partial {t}^{3}}}v \left( x,t \right) +$$$$+18\, \left( {\frac {\partial
^{2}}{\partial {x}^{2}}}v \left( x,t \right) \right) \left( {\frac {
\partial ^{2}}{\partial {t}^{2}}}v \left( x,t \right) \right)
\left( {\frac {\partial ^{3}}{\partial {x}^{2}\partial t}}v \left( x,
t \right) \right) \left( {\frac {\partial ^{2}}{\partial x\partial t
}}v \left( x,t \right) \right) {\frac {\partial ^{3}}{\partial x
\partial {t}^{2}}}v \left( x,t \right) +$$$$+6\, \left( {\frac {\partial ^{
2}}{\partial x\partial t}}v \left( x,t \right) \right) \left( {
\frac {\partial ^{3}}{\partial {x}^{3}}}v \left( x,t \right) \right)
\left( {\frac {\partial ^{2}}{\partial {t}^{2}}}v \left( x,t \right)
\right) ^{2}{\frac {\partial ^{3}}{\partial {x}^{2}\partial t}}v
\left( x,t \right) +$$$$+6\, \left( {\frac {\partial ^{2}}{\partial {t}^{2
}}}v \left( x,t \right) \right) \left( {\frac {\partial ^{3}}{
\partial {x}^{2}\partial t}}v \left( x,t \right) \right) \left( {
\frac {\partial ^{2}}{\partial {x}^{2}}}v \left( x,t \right) \right)
^{2}{\frac {\partial ^{3}}{\partial {t}^{3}}}v \left( x,t \right) -$$$$-12
\, \left( {\frac {\partial ^{2}}{\partial x\partial t}}v \left( x,t
\right) \right) ^{2} \left( {\frac {\partial ^{3}}{\partial {x}^{2}
\partial t}}v \left( x,t \right) \right) \left( {\frac {\partial ^{2
}}{\partial {x}^{2}}}v \left( x,t \right) \right) {\frac {\partial ^{
3}}{\partial {t}^{3}}}v \left( x,t \right) +$$$$+6\, \left( {\frac {
\partial ^{3}}{\partial {x}^{3}}}v \left( x,t \right) \right)
\left( {\frac {\partial ^{2}}{\partial {t}^{2}}}v \left( x,t \right)
\right) ^{2} \left( {\frac {\partial ^{2}}{\partial {x}^{2}}}v
\left( x,t \right) \right) {\frac {\partial ^{3}}{\partial x
\partial {t}^{2}}}v \left( x,t \right) +$$$$+6\, \left( {\frac {\partial ^{
2}}{\partial {x}^{2}}}v \left( x,t \right) \right) ^{2} \left( {
\frac {\partial ^{3}}{\partial {t}^{3}}}v \left( x,t \right) \right)
\left( {\frac {\partial ^{2}}{\partial x\partial t}}v \left( x,t
\right) \right) {\frac {\partial ^{3}}{\partial x\partial {t}^{2}}}v
\left( x,t \right) -$$$$-12\, \left( {\frac {\partial ^{3}}{\partial {x}^{
3}}}v \left( x,t \right) \right) \left( {\frac {\partial ^{2}}{
\partial {t}^{2}}}v \left( x,t \right) \right) \left( {\frac {
\partial ^{2}}{\partial x\partial t}}v \left( x,t \right) \right) ^{2
}{\frac {\partial ^{3}}{\partial x\partial {t}^{2}}}v \left( x,t
\right) +$$$$+8\, \left( {\frac {\partial ^{2}}{\partial x\partial t}}v
\left( x,t \right) \right) ^{3} \left( {\frac {\partial ^{3}}{
\partial {x}^{3}}}v \left( x,t \right) \right) {\frac {\partial ^{3}}
{\partial {t}^{3}}}v \left( x,t \right) -9\, \left( {\frac {\partial ^
{2}}{\partial {t}^{2}}}v \left( x,t \right) \right) \left( {\frac {
\partial ^{2}}{\partial {x}^{2}}}v \left( x,t \right) \right) ^{2}
\left( {\frac {\partial ^{3}}{\partial x\partial {t}^{2}}}v \left( x,
t \right) \right) ^{2}-$$$$-9\, \left( {\frac {\partial ^{2}}{\partial {x}
^{2}}}v \left( x,t \right) \right) \left( {\frac {\partial ^{2}}{
\partial {t}^{2}}}v \left( x,t \right) \right) ^{2} \left( {\frac {
\partial ^{3}}{\partial {x}^{2}\partial t}}v \left( x,t \right)
\right) ^{2}- \left( {\frac {\partial ^{2}}{\partial {x}^{2}}}v
\left( x,t \right) \right) ^{3} \left( {\frac {\partial ^{3}}{
\partial {t}^{3}}}v \left( x,t \right) \right) ^{2}-$$$$- \left( {\frac {
\partial ^{3}}{\partial {x}^{3}}}v \left( x,t \right) \right) ^{2}
\left( {\frac {\partial ^{2}}{\partial {t}^{2}}}v \left( x,t \right)
\right) ^{3}.
=0
\end{equation}
It has solution of the form
\[
v(x, t) = A(t)+B(x)\]
which looks as
\[
v \left( x,t \right) =-4\,{\frac {\ln \left( {\it \_C1}+x \right) }{
\mu}}+{\it \_C2}\,x+{\it \_C3}
4\,{\frac {\ln \left( {\it \_C4}+t \right) }{\mu}}+{\it \_C5}\,t+{
\it \_C6}.
\]
Using these expressions and the conditions
\[
y-v \left( x,t \right)=0,\quad t=f(x,y)
\]
we can find the function $f(x,y$ which is solution of the equation (\ref{Dr20}.
In particular case
\[
{\it \_C6}=0,\quad
{\it \_C3}=0,\quad
{\it \_C1}=0,\quad
{\it \_C4}=0,\quad
{\it \_C5}=1
\]
one gets
\[
f\left( x,y \right) =4\,{\frac {{\it LambertW} \left( 1/4\,\mu\,{
{\rm e}^{1/4\,y\mu-1/4\,{\it \_C2}\,\mu\,x}}x \right) }{\mu}},
\]
where the function ${\it LambertW} \left( x \right)$ is defined by the relation
\[
{\it LambertW} \left( x \right) {{\rm e}^{{\it LambertW} \left( x
\right) }}=x.
\]
\section{Minimal surfaces}
Minimal surfaces are defined by solutions of the equation \cite{Kur}
\begin{equation}\label{Dr21}
\left( 1+ \left( {\frac {\partial }{\partial y}}f \left( x,y \right)
\right) ^{2} \right) {\frac {\partial ^{2}}{\partial {x}^{2}}}f
\left( x,y \right) -2\, \left( {\frac {\partial }{\partial y}}f
\left( x,y \right) \right) \left( {\frac {\partial }{\partial x}}f
\left( x,y \right) \right) {\frac {\partial ^{2}}{\partial y
\partial x}}f \left( x,y \right) + \]\[+\left( 1+ \left( {\frac {\partial }
{\partial x}}f \left( x,y \right) \right) ^{2} \right) {\frac {
\partial ^{2}}{\partial {y}^{2}}}f \left( x,y \right)=0
\end{equation}
After the $(u,v)$-transformation
\[
v \left( x,t \right) =t{\frac {\partial }{\partial t}}\omega \left( x,
t \right) -\omega \left( x,t \right),\]\[
u \left( x,t \right) ={\frac {\partial }{\partial t}}\omega \left( x,t
\right).
\]
one gets the equation
\begin{equation}\label{Dr22}
- \left( {\frac {\partial ^{2}}{\partial x\partial t}}\omega \left( x,
t \right) \right) ^{2}+{t}^{2} \left( {\frac {\partial ^{2}}{
\partial {t}^{2}}}\omega \left( x,t \right) \right) {\frac {\partial
^{2}}{\partial {x}^{2}}}\omega \left( x,t \right) -{t}^{2} \left( {
\frac {\partial ^{2}}{\partial x\partial t}}\omega \left( x,t \right)
\right) ^{2}+$$$$+2\,t \left( {\frac {\partial ^{2}}{\partial x\partial t}
}\omega \left( x,t \right) \right) {\frac {\partial }{\partial x}}
\omega \left( x,t \right) -\left( {\frac {\partial }{\partial x}}
\omega \left( x,t \right) \right) ^{2}-1+ \left( {\frac {\partial ^{2
}}{\partial {t}^{2}}}\omega \left( x,t \right) \right) {\frac {
\partial ^{2}}{\partial {x}^{2}}}\omega \left( x,t \right)=0.
\end{equation}
It has the solution
\[
\omega \left( x,t \right) =1/4\,t\arctan \left( t \right) + \left( 1+{
t}^{2} \right) {x}^{2}.
\]
Corresponding solution of the equation (\ref{Dr22}) can be presented in
a parametric form
\[
f \left( x,y \right) =1/4\,{\frac {\arctan \left( t \right) +\arctan
\left( t \right) {t}^{2}+t+8\,t{x}^{2}+8\,{t}^{3}{x}^{2}}{1+{t}^{2}}},
\]
\[
t=1/4\,{\frac {\sqrt {2}\sqrt {4\,y-1+\sqrt {16\,{y}^{2}-8\,y+1+64\,{x
}^{2}y+64\,{x}^{4}}}}{x}}.
\]
\section{Partner equation}
After the $(u,v)$-transformation with the condition
\[
u \left( x,t \right) =t
\]
the equation (\ref{Dr21}) is transformed into the partner equation
\begin{equation}\label{Dr23}
- \left( {\frac {\partial }{\partial x}}v \left( x,t \right) \right)
^{2}{\frac {\partial ^{2}}{\partial {t}^{2}}}v \left( x,t \right) +2\,
\left( {\frac {\partial }{\partial x}}v \left( x,t \right) \right)
\left( {\frac {\partial ^{2}}{\partial x\partial t}}v \left( x,t
\right) \right) {\frac {\partial }{\partial t}}v \left( x,t \right)
-{\frac {\partial ^{2}}{\partial {t}^{2}}}v \left( x,t \right) -\]\[-{
\frac {\partial ^{2}}{\partial {x}^{2}}}v \left( x,t \right) - \left(
{\frac {\partial ^{2}}{\partial {x}^{2}}}v \left( x,t \right)
\right) \left( {\frac {\partial }{\partial t}}v \left( x,t \right)
\right) ^{2}=0.
\end{equation}
It has the same form with the initial equation (\ref{Dr21}).
This property can be used to construction a new solutions of the equation
(\ref{Dr21}) by the way of elimination of the parameter $t$ from the relations
\begin{equation}\label{Dr24}
y-v\left(x,t\right)=0,\quad
t-f\left(x,y\right)=0.
\end{equation}
Let us consider an examples.
The function
\[
f \left( x,y \right) =\ln \left( \sqrt {{x}^{2}+{y}^{2}}+\sqrt {{x}^{
2}+{y}^{2}-1} \right)
\]
is solution of the equation (\ref{Dr21}).
Then the function
\[
v \left( x,t \right) =\ln \left( \sqrt {{x}^{2}+{t}^{2}}+\sqrt {{x}^{
2}+{t}^{2}-1} \right)
\]
is the solution of the equation (\ref{Dr23}).
Elimination of the parameter $t$ from the conditions
\[
y-\ln \left( \sqrt {{x}^{2}+{t}^{2}}+\sqrt {{x}^{2}+{t}^{2}-1}
\right)=0,
\]
\[
t=f\left(x,y\right)
\]
leads to the relation
\[
y-\ln \left( \sqrt {{x}^{2}+ \left( f \left( x,y \right) \right) ^{2
}}+\sqrt {{x}^{2}+ \left( f \left( x,y \right) \right) ^{2}-1}
\right)=0
\]
from which we get the function
\[
f \left( x,y \right) =1/2\,\sqrt {2-4\,{x}^{2}+{{\rm e}^{-2\,y}}+{
{\rm e}^{2\,y}}}
\]
which is solution of the equation (\ref{Dr21}).
If the
\[
f \left( x,y \right) =\ln \left( {\frac {\cos \left( y \right) }{\cos
\left( x \right) }} \right)
\]
is solution of the equation (\ref{Dr21}),
then the function
\[
v \left( x,t \right) =\ln \left( {\frac {\cos \left( t \right) }{\cos
\left( x \right) }} \right)
\]
satisfies the equation (\ref{Dr23}).
Now from the conditions (\ref{Dr24}) we find the equation
\[
y-\ln \left( {\frac {\cos \left( f \left( x,y \right) \right) }{\cos
\left( x \right) }} \right)=0
\]
from which we get a new solution of the equation (\ref{Dr21})
\[
f \left( x,y \right) =\arccos \left( {{\rm e}^{y}}\cos \left( x
\right) \right).
\]
In the case of the substitution
\[
v \left( x,t \right) =H \left( {x}^{2}+{t}^{2}+1 \right) )
\]
we find the solution of the equation (\ref{Dr23})
\[
v \left( x,t \right) =\ln \left( \left( -2\,{{\it \_C1}}^{2}+{x}^{2}
+{t}^{2}+\sqrt {-{\frac { \left( {x}^{2}+{t}^{2} \right) \left( -{t}^
{2}+4\,{{\it \_C1}}^{2}-{x}^{2} \right) }{{{\it \_C1}}^{2}}}}{\it \_C1
} \right) {{\it \_C1}}^{-1} \right) {\it \_C1}.
\]
Now from the conditions (\ref{Dr24}) we obtain the equation
\[
y=\]\[=\ln \left( \left( -2\,{{\it \_C1}}^{2}\!+\!{x}^{2}\!+\!f \left( x,
y \right)^{2}\!+\!\sqrt {{\frac { \left( {x}^{2}\!+\!f
\left( x,y \right)^{2} \right) \left( f \left( x,y
\right)^{2}\!-\!4\,{{\it \_C1}}^{2}\!+\!{x}^{2} \right) }{{{\it
\_C1}}^{2}}}}{\it \_C1} \right) {{\it \_C1}}^{-1} \right) {\it \_C1}
\]
to determination of the function $f(x,y)$.
Corresponding solution has the form
\[
f \left( x,y \right) =1/2\,\sqrt {2}\sqrt {{{\rm e}^{{\frac {y}{{\it
\_C1}}}}} \left( {\it \_C1}\,{{\rm e}^{2\,{\frac {y}{{\it \_C1}}}}}+4
\,{{\it \_C1}}^{2}{{\rm e}^{{\frac {y}{{\it \_C1}}}}}-2\,{{\rm e}^{{
\frac {y}{{\it \_C1}}}}}{x}^{2}+4\,{{\it \_C1}}^{3} \right) }{{\rm e}^
{-{\frac {y}{{\it \_C1}}}}}.
\]
\section{Acknowledgements}
The research was partially supported in the framework of joint Russian-Moldavian
research project\\[1mm]
(Grant 08.820.08.07 RF of HCSTD ASM, Moldova, and RFBR grant 08-01-90104, Russia).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,035
|
{"url":"http:\/\/blogs.ethz.ch\/kowalski\/category\/language\/page\/12\/","text":"## Buffon\u2019s needle\n\nAs a result of recent moves, the (almost) complete set of Buffon\u2019s monumental Histoire naturelle belonging to my father\u2019s family has recently arrived here in Z\u00fcrich (it comes from my grand-father, who was director of the Mus\u00e9um d\u2019histoire naturelle de Nantes). I will keep these in my office for the moment, as it definitely lends it a very scholarly air\u2026\n\n(As far as I can see from the web page above, what is missing from our set is the Histoire naturelle des poissons, which was not written by Buffon anyway, but by the Comte de Lac\u00e9p\u00e8de, who also wrote the volumes about snakes, which we do have).\n\nMany probabilists know Buffon for his annoying habit of dropping needles on the parquet, and finding the value of \u03c0 after doing this sufficiently many times. This game was indeed included in his natural history, more precisely in the Essai d\u2019arithm\u00e9tique morale (or \u201cEssay of moral arithmetic\u201d) in Volume VII of the Suppl\u00e9ments \u2014 at least, it is there in my family\u2019s edition, though it is missing from the web site containing Buffon\u2019s works, where the Essai is in Supplement volume 4.\n\nHere are pictures of the first pages of the description of the problem (click for readable larger picture):\n\nand\n\nNotice the delightful typography and orthography: the \u201cs\u201d that looks like an integral sign (and is barely distinguishable from an \u201cf\u201d), the way the past tense is written demanderoit instead of the current demanderait, etc.\n\nEnlightnement came from a somewhat unexpected source (the manual for my digital camera), and a comment on the earlier post also suggested it: what may be the most natural English rendering of the French encadrement is \u201cbracket\u201d, or \u201cbracketing\u201d.\n\nIndeed, it seems the term \u201cbracket\u201d is used in photography for the operation of taking simultaneously (or as nearly so as possible) three pictures, one with a given selected exposure, and two with higher and lower settings, so that the amateur photographer can then select which is best.\n\nThe ever-helpful OED confirms that this is a good choice: we find for Bracket, n., 5(b)\n\nThe (specified) distance between a pair of shots fired, one beyond the target and one short of it, in order to find the range for artillery; chiefly in the phrase to establish a bracket.\n\nThe quotations that go with this sense are convincing (if somewhat martial); here is the first one:\n\n1899 Daily News 6 Dec. 5\/7 At first I fire at 3100 yards, and if I find that my shot is short I fire a second round, say at 3300, in order to go beyond the object. If I see that my shot does go over I am satisfied that I have established what is called \u2018a long bracket\u2019, that is to say, I have found two ranges, 200 yards apart, between which the object must lie\u2026 I..fire another shot to shorten the distance within which I can then know that the target must be. This we call, on the same principle as the other, \u2018a short bracket\u2019.\n\nThere is then a further sense 5(c) with similar meaning:\n\nA group bracketed together as of equal standing in some graded system, as income bracket: a class of persons grouped according to income.\n\nAnd then, finally, bracketing is defined as \u201cThe action of furnishing, coupling, uniting, with brackets\u201d. Altogether, it seems one can quite correctly state, in demotic English, something like:\n\n\u2026 and so we have the bracket\n\n$x-\\frac{x^2}{2}\\leq \\cos(x)\\leq x-\\frac{x^2}{2}+\\frac{x^4}{24}$\n\n(although, to my ear, the variant \u201cwe have the bracketing\u201d seems better).\n\nIs there a particular word in English for a string of two inequalities which together give both a lower and an upper bound for a certain quantity?\u00a0 French has encadrement for things like this, as in\n\nPour tout r\u00e9el x, on a l\u2019encadrement\n\n$x-\\frac{x^2}{2}\\leq \\cos(x)\\leq x-\\frac{x^2}{2}+\\frac{x^4}{24},$\n\nThis is often quite convenient \u2014 of course, one can say \u201cwe have the following inequalities\u2026\u201d, but the extra information is useful to have, and it helps avoiding too many repetitions.\n\nNote: The OED only lists encadr\u00e9 as an English word, as a technical term in crystallography:\n\nA crystal is named encadr\u00e9, when it has facets which form kinds of squares around the planes of a more simple form already existing in the same species, R. Jameson, A treatise on the external characters of minerals, 1805.\n\n## Science and mathematics\n\nQuite by chance, I\u2019ve stumbled in the archive of Nature (alas, not freely available) on a paper by J. Sylvester (dated December 30, 1869) concerning, roughly, the status of mathematics among sciences. He says his text was a reaction to earlier talks and articles by Huxley (the biologist, not the limericks writer\u2026). His esteemed opponent having stated\n\nMathematics \u201cis that study which knows nothing of observation, nothing of induction, nothing of experiment, nothing of causation\u201d, but knows only deduction,\n\nSylvester argues strongly in the opposite direction:\n\nI think no statement could have been made more opposite to the fact,\n\nand goes on to give examples from his own work, in particular, where conclusions were reached, and entire theories were constructed, based on simple apparently accidental remarks, by processes of observation, induction and imagination.\n\nBesides this discussion, reading this paper is quite fascinating. Mostly, it must be said, because it is rather incredibly hard to read. Not only physically (the font size is small, and the footnotes even smaller, and printed 2-up, it really exercises your eyesight), but also because of the language, which I believe should cause many a lover of the English language to either faint or burst out laughing; the Gothic Victorian style (cleverly ridiculed by Jane Austen in \u201cNorthanger abbey\u201d) is here put into overdrive for the purpose of scientific discussion. There are rather frightening mathematical terms which, presumably, a few living readers can still interpret,\n\ncanonisant, octodecadic skew invariant, invariantive criteria, amphigenous surface, a catena of morphological processes\n\nand there are Latin, Greek and French quotations, untranslated, and a German one (which, strangely, Sylvester feels to be in need of translation). The following passage is quite typical:\n\nNow this gigantic outcome of modern analytical thought, itself, only the precursor and progenitor of a future still more heaven-reaching theory, which will comprise a complete study of the interoperation, the actions and reactions, of algebraic forms (Analytical Morphology in its absolute sense), how did this originate? In the accidental observation by Eisenstein, some twenty or more years ago, of a single invariant (the Quadrinvariant of a Binary Quartic) which he met with in the course of certain researches just as accidentally and unexpectedly as M. Du Chaillu might meet a Gorilla in the country of the Fantees, or any one of us in London a White Polar Bear escaped from the Zoological Gardens. Fortunately, he pounced upon his prey and preserved it for the contemplation and study of future mathematicians\u2026\n\nBut there are also interesting things, like a discussion of the status of higher-dimensional geometry, and indeed a forecast of Flatland (the book of that title was only published 15 years later):\n\nfor as we can conceive beings (like infinitely attenuated book-worms in an infinitely thin sheet of paper) which possess only the notion of space of two dimensions\u2026\n\nThe follow-up paper is much in the same style (with beauties such as \u201cthe Eikosi-heptagram\u201c, and flights of fancy like \u201cmy own latest researches in a field where Geometry, Algebra and the Theory of Numbers melt in a surprising manner into one another, like sunset tints or the colours of the dying dolphin\u201d \u2013 this theory is that of \u201cthe Reducible Cyclodes\u201d), and also quite insightful sometimes. For instance, there is en passant, the following very convincing footnote:\n\nIs it not the same disregard of principles, the indifference to truth for its own sake, which prompts the question \u201cWhere\u2019s the good of it?\u201d in reference to speculative science, and \u201cWhere\u2019s the harm of it?\u201d in reference to white lies and pious frauds? In my own experience I have found that the very same people who delight to put the first question are in the habit of acting upon the denial implied in the second. Abit in mores incuria.\n\n(Sylvester writes the \u201ci\u201d in the word \u201cin\u201d in the last quotation as a dotless i; I doubt it\u2019s a typographical error, but I can\u2019t find an indication that this is proper Latin grammar; does any reader here have an insight on this?)\n\n## a long the\u2026 riverrun\n\nIf you had to write a circular novel, that can be started at any point of the narrative and that would circle back to itself, how long would it be? Probably, most mathematicians would say that either 63, 628, or 6283, pages would be the most appropriate.\n\nAnd so it\u2019s probably not surprising that the one circular novel I know, Joyce\u2019s \u201cFinnegans wake\u201c, is indeed (in all editions I\u2019ve seen) 628 pages long. Amusingly, this is not mentioned in the introduction to the one I have (which is, also, about the only thing I\u2019ve read of the book, with the exception of the first few and last paragraphs, and isolated snatches here and there).\n\nThe reason for this post is really that today is Bloomsday, the day when, fictionally, the action of the earlier \u201cUlysses\u201d evolves, and that Z\u00fcrich is well-connected with Joyce. Not only is he buried there, very close to the zoo, so I can use one appropriate quotation\n\nAs the lion in our teargarten remembers the nenuphars of his Nile\u2026\n\nfrom \u201cFinnegans wake\u201d (one of those few snatches I have actually read), but he also wrote parts of \u201cUlysses\u201d in a house on Universit\u00e4tstrasse, quite close to the main building of ETH Z\u00fcrich where I work, as memorialized by the plaque below:\n\n(I\u2019ve heard that there are a few other such places in Z\u00fcrich, the reason being that, unable to pay the rent, Joyce had to move frequently\u2026)","date":"2018-07-16 16:14:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 2, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6020039319992065, \"perplexity\": 2003.5395315277183}, \"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-2018-30\/segments\/1531676589404.50\/warc\/CC-MAIN-20180716154548-20180716174548-00549.warc.gz\"}"}
| null | null |
Q: How do I get 'cake repl' working on OSX? (I'm getting 'Exception in thread "main" java.lang.NoClassDefFoundError: clojure/main') I'm trying to get clojure/emacs/swank/cake all working together. According to assembla I need 'cake repl' to launch a repl, but I keep getting a NoClassDefFoundError. Any idea how to fix that?
Here's what I get in the command line:
$ cake ps
7876 main -Dcake.project=/Users/bmaddy/.cake -Xms16M -Xmx64M -Djava.library.path=lib/dev/native
$ echo $CLASSPATH
$ java -version
java version "1.5.0_26"
Java(TM) 2 Runtime Environment, Standard Edition (build 1.5.0_26-b03-376-9M3263)
Java HotSpot(TM) Client VM (build 1.5.0_26-156, mixed mode, sharing)
$ ruby -v
ruby 1.8.6 (2009-06-08 patchlevel 369) [universal-darwin9.0]
$ cake repl
Exception in thread "main" java.lang.NoClassDefFoundError: clojure/main
[cake] connection to bake jvm is taking a long time...
[cake] you can use ^C to abort and use 'cake kill' or 'cake kill -9' to force the jvm to restart
^C/Users/bmaddy/.gem/ruby/1.8/gems/cake-0.5.8/bin/cake:509:in `sleep': Interrupt
from /Users/bmaddy/.gem/ruby/1.8/gems/cake-0.5.8/bin/cake:509:in `with_socket'
from /Users/bmaddy/.gem/ruby/1.8/gems/cake-0.5.8/bin/cake:431:in `repl'
from /Users/bmaddy/.gem/ruby/1.8/gems/cake-0.5.8/bin/cake:430:in `loop'
from /Users/bmaddy/.gem/ruby/1.8/gems/cake-0.5.8/bin/cake:430:in `repl'
from /Users/bmaddy/.gem/ruby/1.8/gems/cake-0.5.8/bin/cake:822:in `send'
from /Users/bmaddy/.gem/ruby/1.8/gems/cake-0.5.8/bin/cake:822
from /Users/bmaddy/.gem/ruby/1.8/bin/cake:19:in `load'
from /Users/bmaddy/.gem/ruby/1.8/bin/cake:19
$ cake ps
7876 main -Dcake.project=/Users/bmaddy/.cake -Xms16M -Xmx64M -Djava.library.path=lib/dev/native
12539 main -Dcake.project=/Users/bmaddy/Documents/books/SICP -Xms16M -Xmx64M -Djava.library.path=lib/dev/native
More info:
$ cake deps force
[deps] Fetching dependencies...
[INFO] snapshot org.clojure:clojure:1.2.0-master-SNAPSHOT: checking for updates from clojure
[INFO] snapshot org.clojure:clojure:1.2.0-master-SNAPSHOT: checking for updates from clojure-snapshots
[INFO] snapshot org.clojure:clojure:1.2.0-master-SNAPSHOT: checking for updates from clojars
[INFO] snapshot org.clojure:clojure:1.2.0-master-SNAPSHOT: checking for updates from maven
Downloading: org/clojure/clojure/1.2.0-master-SNAPSHOT/clojure-1.2.0-master-SNAPSHOT.pom from repository clojure at http://build.clojure.org/releases
Unable to locate resource in repository
[INFO] Unable to find resource 'org.clojure:clojure:pom:1.2.0-master-SNAPSHOT' in repository clojure (http://build.clojure.org/releases)
Downloading: org/clojure/clojure/1.2.0-master-SNAPSHOT/clojure-1.2.0-master-SNAPSHOT.pom from repository clojure-snapshots at http://build.clojure.org/snapshots
Unable to locate resource in repository
[INFO] Unable to find resource 'org.clojure:clojure:pom:1.2.0-master-SNAPSHOT' in repository clojure-snapshots (http://build.clojure.org/snapshots)
Downloading: org/clojure/clojure/1.2.0-master-SNAPSHOT/clojure-1.2.0-master-SNAPSHOT.pom from repository clojars at http://clojars.org/repo
Unable to locate resource in repository
[INFO] Unable to find resource 'org.clojure:clojure:pom:1.2.0-master-SNAPSHOT' in repository clojars (http://clojars.org/repo)
Downloading: org/clojure/clojure/1.2.0-master-SNAPSHOT/clojure-1.2.0-master-SNAPSHOT.pom from repository maven at http://repo1.maven.org/maven2
Unable to locate resource in repository
[INFO] Unable to find resource 'org.clojure:clojure:pom:1.2.0-master-SNAPSHOT' in repository maven (http://repo1.maven.org/maven2)
[deps] Copying 3 files to /Users/bmaddy/Documents/books/SICP/build/lib
[deps] Deleting directory /Users/bmaddy/Documents/books/SICP/lib
[deps] Attempting to rename dir: /Users/bmaddy/Documents/books/SICP/build/lib to /Users/bmaddy/Documents/books/SICP/lib
[clean] Deleting /Users/bmaddy/Documents/books/SICP/.cake/run/clean
[clean] Deleting /Users/bmaddy/Documents/books/SICP/.cake/run/default
[clean] Deleting /Users/bmaddy/Documents/books/SICP/.cake/run/deps
[clean] Deleting /Users/bmaddy/Documents/books/SICP/.cake/run/help
[clean] Deleting /Users/bmaddy/Documents/books/SICP/.cake/run/pom
[deps] Restarting project jvm
A: Looks like your default cake project doesn't have the clojure jar. Try running cake deps force.
A: Turns out this was a problem with Clojure and Java 1.5. The fix on OS X was to use /Applications/Utilities/Java Preferences.app to switch to Java 1.6.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 7,429
|
1.1 Bajaj Allianz General Insurance Co. Ltd.
1.2 Tata AIG General Insurance Co. Ltd.
1.3 The Oriental Insurance Co. Ltd.
1.4 The New India Assurance Co. Ltd.
Motor Insurance Companies in India: If you are looking for an article in which you can find out the "Top 5 Motor Insurance Companies in India", Then let me tell you, you have arrived at the perfect post, cause here you will get everything which you need to know about the Top 5 Motor Insurance Companies in India. I hope that all kinds of information helpful to you. so let's check it out.
The public sector insurers have always liked the customer's choice over private sector insurers. Though, the pattern is changing in favour of private insurers over the last few years.
Bajaj Allianz General Insurance was founded in 2001 and Headquartered in Pune, India. It is a private general insurance company in India. The Bajaj Allianz company is a joint venture between Bajaj Finserv Limited (formerly part of Bajaj Auto Limited) owned by the Bajaj Group of India and Allianz SE, a German financial services company. Its main product includes Motor Insurance, Health Insurance, Travel Insurance, Home Insurance, Marine Insurance.
Tata AIG General Insurance Company Limited was founded in 2001 and headquartered in Mumbai, India, It is an Indian general insurance company and a joint venture between the Tata Group and AIG. Tata Group holds a 51% stake in the insurance venture with AIG (American International Group) holding the balance of26 %.
The Oriental Insurance Company Ltd. was founded in 1947and headquartered in New Delhi. It is a public sector general insurance company of India. It has a number of 31 regional offices and more than 1800 active branches across the country.
The Oriental Insurance Company also has branches in Kuwait, Nepal, and Dubai. The Oriental Insurance Company offers more than 170 General Insurance products. The IRDA Registration No. of The Oriental Insurance Company is "556".
The New India Assurance Co. Ltd. was founded in 1919 by Sir Dorabji Tata and It was nationalised in 1973. It is a public sector general insurance company of India. It is Headquartered in Mumbai, Maharashtra, India. It is the "largest general insurance company of India on the basis of gross premium collection inclusive of foreign operations".
HDFC ERGO General Insurance Company Limited is a joint venture between HDFC Ltd. and ERGO International AG. its headquarter in Mumbai. It is a Germany-based company that is part of the Munich Re Group. HDFC holds 51 %, and ERGO the other 49 %. The firm operates in 101 Indian cities with over 117 branches and 2,000 staff members.
Its product includes General insurance, Vehicle insurance (Private Car & Two Wheeler Insurance) Health Insurance, Personal Accident Insurance, Commercial Insurance, Rural Insurance, Travel Insurance, and Home Insurance.
I think it should be enough information about the Top 5 Motor Insurance Companies in India, I hope you like this post, but if you have any problem regarding this article, then please comment for us, I will solve your problem solution as soon and if you like it so please share with your friends and on social media, Thanks for visiting our site and stay tuned with me for more stuff like. keep Smile, Take care of your family.
|
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"redpajama_set_name": "RedPajamaC4"
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| 5,190
|
Courtesy of Telstra, who have finally succeeded in their ongoing quest to kill our phone line, it looks like I'm going to have a couple of days without Internet access. This is, of course, as inconveniently timed as possible, given the work I need to do on the GSWoC project and said work being about ten times harder without the aforementioned Internet access.
So, I'll be back in a couple of days, give or take. Guess I'll take it as an enforced break.
This entry was posted on Tuesday, July 17th, 2007 at 15:48 and is filed under General. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.
Ah, damn. I know I was very frustrated when, after leaving our house; a former housemate had the phone cut off. It cost me an arm and a leg to restore the contract with my ISP, here in Adelaide.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 4,271
|
Earthworms and arsenic: a review.
Andrew Alexander Meharg
Chemical pollution of the environment has become a major source of concern. In particular, many studies have investigated the impact of pollution on biota in the environment. Studies on metalliferous contaminated mine spoil wastes have shown that some soil organisms have the capability to become resistant to metal/metalloid toxicity. Earthworms are known to inhabit arsenic-rich metalliferous soils and, due to their intimate contact with the soil, in both the solid and aqueous phases, are likely to accumulate contaminants present in mine spoil. Earthworms that inhabit metalliferous contaminated soils must have developed mechanisms of resistance to the toxins found in these soils. The mechanisms of resistance are not fully understood; they may involve physiological adaptation (acclimation) or be genetic. This review discusses the relationships between earthworms and arsenic-rich mine spoil wastes, looking critically at resistance and possible mechanisms of resistance, in relation to soil edaphic factors and possible trophic transfer routes. (C) 2003 Elsevier Science Ltd. All rights reserved.
EISENIA-FOETIDA OLIGOCHAETA
HEAVY-METALS
LUMBRICUS-RUBELLUS
DENDRODRILUS-RUBIDUS
METALLOTHIONEIN
Meharg, A. A. (2003). Earthworms and arsenic: a review. Environmental Pollution, 124, 361-373. https://doi.org/10.1016/S0269-7491(03)00047-2
Earthworms and arsenic: a review. / Meharg, Andrew Alexander.
In: Environmental Pollution, Vol. 124, 2003, p. 361-373.
Meharg, AA 2003, 'Earthworms and arsenic: a review.', Environmental Pollution, vol. 124, pp. 361-373. https://doi.org/10.1016/S0269-7491(03)00047-2
Meharg AA. Earthworms and arsenic: a review. Environmental Pollution. 2003;124:361-373. https://doi.org/10.1016/S0269-7491(03)00047-2
Meharg, Andrew Alexander. / Earthworms and arsenic: a review. In: Environmental Pollution. 2003 ; Vol. 124. pp. 361-373.
@article{08b28a2997d34c19961158535bfdddb5,
title = "Earthworms and arsenic: a review.",
abstract = "Chemical pollution of the environment has become a major source of concern. In particular, many studies have investigated the impact of pollution on biota in the environment. Studies on metalliferous contaminated mine spoil wastes have shown that some soil organisms have the capability to become resistant to metal/metalloid toxicity. Earthworms are known to inhabit arsenic-rich metalliferous soils and, due to their intimate contact with the soil, in both the solid and aqueous phases, are likely to accumulate contaminants present in mine spoil. Earthworms that inhabit metalliferous contaminated soils must have developed mechanisms of resistance to the toxins found in these soils. The mechanisms of resistance are not fully understood; they may involve physiological adaptation (acclimation) or be genetic. This review discusses the relationships between earthworms and arsenic-rich mine spoil wastes, looking critically at resistance and possible mechanisms of resistance, in relation to soil edaphic factors and possible trophic transfer routes. (C) 2003 Elsevier Science Ltd. All rights reserved.",
keywords = "EISENIA-FOETIDA OLIGOCHAETA, HEAVY-METALS, LUMBRICUS-RUBELLUS, DENDRODRILUS-RUBIDUS, SMALL MAMMALS, TOXICITY, RESISTANCE, BIOAVAILABILITY, METALLOTHIONEIN, CONTAMINATION",
author = "Meharg, {Andrew Alexander}",
journal = "Environmental Pollution",
publisher = "ELSEVIER APPL SCI PUBL LTD",
T1 - Earthworms and arsenic: a review.
AU - Meharg, Andrew Alexander
N2 - Chemical pollution of the environment has become a major source of concern. In particular, many studies have investigated the impact of pollution on biota in the environment. Studies on metalliferous contaminated mine spoil wastes have shown that some soil organisms have the capability to become resistant to metal/metalloid toxicity. Earthworms are known to inhabit arsenic-rich metalliferous soils and, due to their intimate contact with the soil, in both the solid and aqueous phases, are likely to accumulate contaminants present in mine spoil. Earthworms that inhabit metalliferous contaminated soils must have developed mechanisms of resistance to the toxins found in these soils. The mechanisms of resistance are not fully understood; they may involve physiological adaptation (acclimation) or be genetic. This review discusses the relationships between earthworms and arsenic-rich mine spoil wastes, looking critically at resistance and possible mechanisms of resistance, in relation to soil edaphic factors and possible trophic transfer routes. (C) 2003 Elsevier Science Ltd. All rights reserved.
AB - Chemical pollution of the environment has become a major source of concern. In particular, many studies have investigated the impact of pollution on biota in the environment. Studies on metalliferous contaminated mine spoil wastes have shown that some soil organisms have the capability to become resistant to metal/metalloid toxicity. Earthworms are known to inhabit arsenic-rich metalliferous soils and, due to their intimate contact with the soil, in both the solid and aqueous phases, are likely to accumulate contaminants present in mine spoil. Earthworms that inhabit metalliferous contaminated soils must have developed mechanisms of resistance to the toxins found in these soils. The mechanisms of resistance are not fully understood; they may involve physiological adaptation (acclimation) or be genetic. This review discusses the relationships between earthworms and arsenic-rich mine spoil wastes, looking critically at resistance and possible mechanisms of resistance, in relation to soil edaphic factors and possible trophic transfer routes. (C) 2003 Elsevier Science Ltd. All rights reserved.
KW - EISENIA-FOETIDA OLIGOCHAETA
KW - HEAVY-METALS
KW - LUMBRICUS-RUBELLUS
KW - DENDRODRILUS-RUBIDUS
KW - SMALL MAMMALS
KW - TOXICITY
KW - RESISTANCE
KW - BIOAVAILABILITY
KW - METALLOTHIONEIN
KW - CONTAMINATION
JO - Environmental Pollution
JF - Environmental Pollution
|
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Q: Golang Standard Package Structure I am faily new to Go and I am trying to create a structured application using guidance from Ben Johnson's webpage. Unfortunately, his example is not a complete working application.
His webpage is https://medium.com/@benbjohnson/standard-package-layout-7cdbc8391fc1
I have tried to use his methods and I keep getting "Undefined: db" error. It doesn't tell me what line is causing the error, just the file "MSSQL.go"
Could someone help with guidance to help me fix this error?
Edited code with accepted solution.
StatementPrinter.go
package statementprinter
type Statement struct {
CustomerId string
CustomerName string
}
type StatementService interface {
Statement(id string) (*Statement, error)
}
main.go
package main
import (
"fmt"
"log"
"github.com/ybenjolin/StatementPrinter"
"github.com/ybenjolin/StatementPrinter/mssql"
"database/sql"
_ "github.com/alexbrainman/odbc"
)
const DB_INFO = "Driver={SQL Server};Server=cdc-edb2;Database=CostarReports;Trusted_Connection=yes;"
var db *sql.DB
func init() {
var err error
db, err = sql.Open("odbc", DB_INFO)
if err != nil {
log.Fatal("Error opening database connection.\n", err.Error())
}
err = db.Ping()
if err != nil {
log.Fatal("Error pinging database server.\n", err.Error())
}
fmt.Println("Database connection established.")
}
func main () {
var err error
defer db.Close()
// Create services
// Changes required here. Was ss := &statementprinter.Stat..
ss := &mssql.StatementService{DB: db}
// Use service
var s *statementprinter.Statement
s, err = ss.Statement("101583")
if err != nil {
log.Fatal("Query failed:", err.Error())
}
fmt.Printf("Statement: %+v\n", s)
}
mssql.go
package mssql
import (
_ "github.com/alexbrainman/odbc"
"database/sql"
"github.com/ybenjolin/StatementPrinter"
)
// StatementService represents a MSSQL implementation of statemenetprinter.StatementService.
type StatementService struct {
DB *sql.DB
}
// Statement returns a statement for a given customer.
func (s *StatementService) Statement(customer string) (*statementprinter.Statement, error) {
var err error
var t statementprinter.Statement
// Changes required here. Was row := db.Query......
row := s.DB.QueryRow(`Select Customer, CustomerName From AccountsReceivable.rptfARStatementHeader(?)`, customer)
if row.Scan(&t.CustomerId, &t.CustomerName); err != nil {
return nil, err
}
return &t, nil
A: This seems like it's just a typo. It seems like the problematic line is in the method
func (s *StatementService) Statement(customer string)
in mssql.go,
row := db.QueryRow(`Select Customer, CustomerName From AccountsReceivable.rptfARStatementHeader(?)`, customer)
QueryRow is supposed to be a method of db, but db is not defined. However, in the struct
type StatementService struct {
DB *sql.DB
}
there's a *sql.DB instance. The method you're using has a *StatementService parameter, s. So, my guess is the intention would be to access the sql.DB field in s like so
func (s *StatementService) Statement(customer string) (*statementprinter.Statement, error) {
var err error
var t statementprinter.Statement
//CHANGED LINE:
row := s.DB.QueryRow(`Select Customer, CustomerName From AccountsReceivable.rptfARStatementHeader(?)`, customer)
if row.Scan(&t.CustomerId, &t.CustomerName); err != nil {
return nil, err
}
return &t, nil
Then, the method is called in main.go, and is passed a StatementService instance that contains a database:
ss := &statementprinter.StatementService{DB: db}
I believe you need to change this line to
ss := &mssql.StatementService{DB: db}
becuase that's the actual interface implementation. The line you have now treats the StatementService interface like a struct which will not compile.
The global db in main.go lives for the lifetime of the application. It's just a pointer which is copied around for use.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,888
|
\section{Introduction} \label{intro}
Let $k$ be an even integer and $$\Delta_k:=y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)-iky\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}\right)$$ be the hyperbolic laplacian on the complex upper half plane $\mathbb H$. A shifted polyharmonic Maass form $f$ of weight $k$ for a Fuchsian group $\Gamma$ is a complex-valued smooth function on $\mathbb H$ satisfying the following conditions:
\begin{itemize}
\item (modular invariant) $f|_k \gamma (z)=f(z)$ for any $\gamma \in \Gamma$, where $$ (f|_k \gamma) (z) :=(cz+d)^{-k}f\left(\frac{az+b}{cz+d}\right), \quad \gamma=\begin{pmatrix} a & b \\ c & d \end{pmatrix}. $$ We call $|_k$ a slash operator.
\item (polyharmonic) $(\Delta_k-\lambda)^m f(z)=0$ for some $m \in \mathbb Z$ and $\lambda \in \mathbb C$.
\item (moderate growth at cusps) $f$ grows polynomially at cusps in $y$, i.e. for each cusp $c \in \mathbb R$ of $\Gamma$ and $\sigma \in \text{SL}_2(\mathbb R)$ that sends the cusp $c$ to $\infty$, there exists $\alpha \in \mathbb R$ such that $(f|_k \sigma^{-1}) (x+iy)=O(y^{\alpha})$ as $y \to \infty$, uniformly in $x$.
\end{itemize}
A complex number $\lambda$ is called an eigenvalue of $f$ and a non-negative integer $m$ is called the (harmonic) depth of $f$. One can define a shifted polyharmonic weak Maass form by replacing the moderate growth condition with the exponential growth.
The notion of shifted polyharmonic Maass forms includes the notion of classical Maass forms. When a depth $m$ is equal to $1$, it is a classical Maass cusp form (of weight $k$). Further if $\lambda=0$, it is called a harmonic Maass form. If we have $\lambda=0$ only, then we call it a polyharmonic Maass form. Obviously, this notion includes the notion of modular forms, which can be seen as holomorphic harmonic Maass forms.
Polyharmonicity is parallel with the literature on the polyharmonic functions for the Euclidean laplacian operator, which are classical and historic, extensive literature. See~\cite{MR0745128}, \cite{Emi99}, and \cite{MR2401623}.
Polyharmonic Maass forms for the full modular group $\text{SL}_2(\mathbb Z)$ have been studied in recent years. Lagarias and Rhoades~\cite{MR357462} studied the space of polyharmonic Maass forms (with eigenvalue zero) $V_k^m(0)$. More precisely, they explicitly exhibit a basis of $V_k^m(0)$.
One can define the notion of shifted polyharmonic Maass forms of the half-integer depth~$m+\frac{1}{2}$ by replacing the polyharmonic condition with
\begin{itemize}
\item $(\Delta_k-\lambda)^m f(z)=g(z)$ for a holomorphic modular form $g(z)$.
\end{itemize}
Lagarias and Rhoades~\cite{MR357462} also determined the space of polyharmonic Maass forms of harmonic depth, completely.
In the sequel paper~\cite{MR3856183} of that, Andersen, Lagrias and Rhoades deal with the space of shifted polyharmonic Maass forms $V_k^m(\lambda)$ for nonzero eigenvalue $\lambda$. They showed that the space $V_k^m(\lambda)$ is also finite-dimensional, found the upper bound of the dimension, decomposed the space into the direct sum of two subspaces, and exhibited a basis of one of them.
Special kinds of polyharmonic Maass forms for congruence groups are also studied by several authors. Bruinier, Funke and Imamoḡlu~\cite{MR3353542} constructed the general regularized theta lift from the weak Maass forms of weight zero to the polyharmonic Maass forms of weight $1/2$. In particular, applying the lift to the constant function $1$, we get a certain polyharmonic Maass form of weight $1/2$ and depth $3/2$ for the congruence group $\Gamma_0(4)$, whose Fourier coefficients are related to real quadratic class numbers. Ahlgren, Andersen and Samart~\cite{MR3852563} introduced the analogous form for the full modular group $\text{SL}_2(\mathbb Z)$ and studied them. After then, the studies in a similar context on the space of polyharmonic weak Maass forms, which includes the notion of polyharmonic Maass forms, have also been actively conducted. See~\cite{MR3353542}, \cite{MR4198754}, \cite{MR3893030} and \cite{MR4047158}.
On the other hand, over the Hecke triangle groups and the Fricke groups, there have been a lot of developments about the theory of modular forms and their arithmetic properties, in each field, respectively. Let us introduce them briefly. For positive integers $m_1,m_2,m_3$ which are allowed to be infinity, with $1/m_1+1/m_2+1/m_3<1$, a triangle group $\Gamma_{(m_1,m_2,m_3)}$ as an abstract group is defined by the group presentation
$$
\Gamma_{(m_1,m_2,m_3)}=\langle g_1, g_2 : g_1^{m_1}=g_2^{m_2}=(g_1 g_2)^{m_3}=1\rangle.
$$
In the presentation, we use the convention that the infinity-power is the identity. For example, $\Gamma_{(2,3,\infty)}=\{g_1, g_2 : g_1^2=g_2^3=1 \}$, which is isomorphic to $\text{PSL}_2(\mathbb Z)$. One can explicitly define $\Gamma_{(m_1,m_2,m_3)}$ as a subgroup of $\text{SL}_2(\mathbb R)$ as follows:
$$
\Gamma_{(m_1,m_2,m_3)}=\langle \gamma_1, \gamma_2, \gamma_3 \rangle,
$$
where $$\gamma_1=\begin{pmatrix} 2\cos(\pi/m_1) & 1 \\ -1 & 0 \end{pmatrix},\quad \gamma_2=\begin{pmatrix} 0 & 1 \\ -1 & 2\cos(\pi/m_2) \end{pmatrix},\quad \gamma_3=\begin{pmatrix} 1 & 2\cos(\pi/m_1)+2\cos(\pi/m_2) \\ 0 & 1 \end{pmatrix}.$$ The other generators conjugate to these matrices can be chosen.
Doran, Gannon, Movasati and Shokri \cite{MR3228299} developed the theory of modular forms for triangle groups. They displayed the basis of modular forms on this group, and moreover explored the arithmeticity of the Fourier coefficients of them. Movasati and Shokri continue their study on the Fourier coefficients for non-arithmetic triangle groups in their sequel paper \cite{MR3253293}. These modular forms and related objects are studied from various perspectives in recent years. For instance, see \cite{MR4045722}, \cite{MR4160413} and \cite{MR3962509}.
The Fricke group of level $N$ is a subgroup of $\text{SL}_2{(\mathbb R)}$ generated by the congruence group~$\Gamma_0(N)$ and the matrix $\omega_N=\begin{pmatrix} 0 & -1/\sqrt N \\ \sqrt N & 0 \end{pmatrix}.$ It is a subgroup of the well-known group $\Gamma_0^+(N)$ which is generated by $\Gamma_0(N)$ and Atkin-Lehner involutions $\omega_e$ for the Hall divisors~$e$ of $N$. $\Gamma_0^+(N)$ is closely related to the congruence group $\Gamma_0(N)$ and the Hecke operators of level $N$ by the Atkin-Lehner theory. Hecke eigenforms are classified by the eigenvalues under the involution. Its eigenvalue is $1$ (in which case it is modular under~$\Gamma_0^+(N)$) or -1. In particular, if $e=N$, we call $w_e$ the Fricke involution.
The theory of Atkin-Lehner which was first developed by Atkin and Lehner \cite{MR0268123}, in particular, the theory on modular forms for the Fricke group is also far reached.
Two different types of the Fuchsian groups we introduced coincide with each other in the low level cases. When $N=2$ or $3$, the Atkin-Lehner group $\Gamma_0^+(N)$ is exactly the Fricke group of the same level, and they are conjugate to the Hecke triangle groups of type $(2,2N,\infty)$. They are triangle groups and the corresponding modular curves have genus~$0$. Furthermore, the Hecke triangle group $\Gamma_{(2,\infty,\infty)}$ is also a well-known group and it is conjugate to $\Gamma_0(2)$.
These groups $\Gamma_0^+(2)$ and $\Gamma_0^+(3)$ have been studied independently as the first two non-trivial simple examples of the Fricke groups or the Hecke triangle groups, from various points of view. For example, there have been results not only about the space of automorphic forms itself or its members, but also concerning the zeros of automorphic forms, linear relations on the Poincare series, or the period functions, and so on. For instance, see~\cite{MR3462568}, \cite{MR2344823}, or other subsequent results.
\
In this paper, we consider a new kind of the Fuchsian groups of the first kind which coincide with the Fricke groups or the Hecke triangle groups in the low level. We introduce them in the following subsection.
\subsection{Elliptic points and genera}\label{subsec_ell}
Let $N \in \mathbb Z$ be a positive integer and let
\begin{align}\label{RN+-def}
R(N)^+=\left\{\begin{pmatrix} a & b\sqrt N \\ c\sqrt N & d \end{pmatrix} \in \text{SL}_2(\mathbb R) : a,b,c,d \in \mathbb Z \right\}, \\
R(N)^-=\left\{\begin{pmatrix} a\sqrt N & b \\ c & d\sqrt N \end{pmatrix} \in \text{SL}_2(\mathbb R) : a,b,c,d \in \mathbb Z \right\}.
\end{align}
Each of these sets is a not a group in general, but if we define
\begin{align}\label{Rndef}
R(N)=R(N)^+ \cup R(N)^-,
\end{align}
then obviously $R(N)$ is a discrete subgroup of $\text{SL}_2(\mathbb R)$. Moreover, it turns out that they are finitely generated as abstract groups and in fact, are the Fuchsian groups with cofinite areas. Also, it turns out that they coincide with the Fricke groups or the Hecke triangle groups in the low level cases (see Section \ref{sec_RN}).
Let $\text{Ell}^-(N)$ be the collection of $R(N)$-equivalence classes of elliptic points for $R(N)^-$, which means the points stabilized by some elliptic element in $R(N)^-$. Denote by $\text{Ell}^+(N)$ the set-theoretic complement of $\text{Ell}^-(N)$ in the collection of all equivalent classes of elliptic points for $R(N)$. We already know all elliptic points and cusps when $p=2,3$. For $p=2$, any elliptic point is equivalent to $i$ or $\frac{1+i}{\sqrt 2}$ and for $p=3$, is equivalent to $i$ or $\frac{\sqrt 3 +i}{2}.$ Any cusp for $R(p)$ when $p=2,3$ is equivalent to $\infty.$ Thus in most cases we assume $p \geq 5$.
We investigate where the elliptic points lie, what the cusps are, and how they are related to the class group of quadratic forms. Specifically, we prove our first main theorem:
\begin{theorem}\label{thm_ell}
Let $p \geq 5$ be a prime.
\begin{enumerate}[\normalfont(a)]
\item $\#\left(\mathrm{Ell}^+(p)\right)=\begin{cases} 2 & \text { if } p \equiv 1\pmod{12}, \\ 1 & \text { if } p \equiv 5,7 \pmod{12}, \\ 0 & \text { if } p \equiv 11 \pmod{12}. \end{cases}$
\item There is a one-to-one correspondence between $\mathrm{Ell}^-(p)$ and the set of all reduced forms of (not necessarily primitive) positive definite binary quadratic forms of discriminant $-4p$.
\end{enumerate}
\end{theorem}
In particular, there is a correspondence between the set of elliptic points for $R(p)^-$ and the ideal class group of an imaginary quadratic field. Namely, it is the ideal class group $C(-4p)$ of an imaginary quadratic field of discriminant $-4p$ when $p \equiv 1 \pmod{4}$, or a disjoint union of the two ideal class groups $C(-4p)$ and $C(-p)$ of discriminant $-4p$ and $-p$, respectively, when $p \equiv 3 \pmod 4$. Its number of elements is obtained from the class number $h(d)$ of discriminant $d=-4p$ or $d=-p$.
There is a classical result to compute the dimension of the space of holomorphic modular forms for a Fuchsian group of the first kind from the information on the modular curves. For example, see \cite{MR1021004}. In particular, if we find the elliptic points, the cusps, their orders, and the areas of curves as a $2$-dimensional Riemannian manifold whose metric is induced by the standard hyperbolic metric on the upper half plane $\mathbb H$, then we can determine the genera of the curves $X(R(p))$ which are the compactified modular curves associated with the Fuchsian group $R(p)$. One can define the area of $X(R(p))$ by the area of its fundamental domain contained in $\mathbb H$, see \cite{MR1021004}.
Together with Theorem \ref{thm_ell}, the information on the curve $X(R(p))$ implies the following Corollary.
\begin{corollary}\label{cor_genusthm}
Let $p$ be a prime number and $v_p$ be the area of $X(R(p))$. Let $$
h_p := \begin{cases} \frac{13}{6}+\frac{1}{2}h(-4p), & \text{ if }p \equiv 1 \pmod{12}, \\
\frac{3}{2}+\frac{1}{2}h(-4p), & \text{ if }p \equiv 5 \pmod{12}, \\
\frac{5}{3}+\frac{1}{2}h(-4p)+\frac{1}{2}h(-p), & \text{ if }p \equiv 7 \pmod{12}, \\
1+\frac{1}{2}h(-4p)+\frac{1}{2}h(-p), & \text{ if }p \equiv 11 \pmod{12}.
\end{cases}
$$
If $v_p <2\pi h_p$, then the genus $g_p$ of $X(R(p))$ is zero and $v_p=2\pi (h_p-2)$.
\end{corollary}
In Section \ref{sec_RN}, we find the orders of all elliptic points and cusps for $R(p)$. Corollary \ref{cor_genusthm} is obtained by applying the formula \cite[Theorem 2.4.3]{MR1021004} of the area of modular curve
\begin{align*}
\frac{1}{2\pi}v_p=2g_p-2+\sum_{z \in X(R(p))}(1-e_z^{-1}).
\end{align*}
\subsection{Non-holomorphic Eisenstein series}
Recall that the non-holomorphic Eisenstein series for the full modular group $\text{SL}_2(\mathbb Z)$ is
$$
G_k(z,s):=\frac{1}{2}\sum_{(m,n) \in \mathbb Z^2 \setminus \{(0,0)\}} \frac{y^s}{|mz+n|^{2s}(mz+n)^k}, \text{ where } \Re(s)>1-\frac{k}{2},
$$
or
$$
g(z,\overline{z},\alpha,\beta):=\sum_{(m,n) \in \mathbb Z^2 \setminus \{(0,0)\}}(mz+n)^{-\alpha}(m\overline{z}+n)^{-\beta},
$$
where $\alpha, \beta \in \mathbb C$ such that $\Re(\alpha+\beta)>2$ and $\alpha-\beta \in 2\mathbb Z$. Note that $G_k(z,s)=\frac{1}{2}y^s g(z,\overline{z},s+k,s)$. The series $G_k(z,s)$ can be normalized as
\begin{align*}
G_k(z,s)&=\frac{1}{2}\sum_{(m,n) \in \mathbb Z^2 \setminus \{(0,0)\}} \frac{y^s}{|mz+n|^{2s}(mz+n)^k} \\
&=\frac{1}{2}\sum_{t=1}^{\infty}\sum_{\substack{(m,n) \in \mathbb Z^2 \setminus \{(0,0)\} \\ \gcd(m,n)=t}} \frac{y^s}{|mz+n|^{2s}(mz+n)^k} \\
&=\frac{1}{2}\zeta(2s+k)\sum_{t=1}^{\infty}\sum_{\substack{(c,d) \in \mathbb Z^2 \setminus \{(0,0)\} \\ \gcd(c,d)=1}} \frac{y^s}{|cz+d|^{2s}(cz+d)^k} \\
&=\zeta(2s+k)E_k(z,s),
\end{align*}
and $E_k(z,s)$ in the above can be written as a Poincare series
$$
E_k(z,s)=\sum_{\gamma \in \mathrm{SL}_2(\mathbb Z)_{\infty} \backslash \mathrm{SL}_2(\mathbb Z)} \left(\Im^s |_k \gamma\right) (z), \text{ where } \Im^s(z):=y^s \text{ for } z=x+iy \text{ with } x,y\in \mathbb R.
$$
For that reason and considering the constant term of its Fourier expansion, $E_k(z,s)$ is called a normalized (non-holomorphic) Eisenstein series.
It is well known that the non-holomorphic Eisenstein series for $\operatorname{SL}_2(\mathbb Z)$ has several analytic properties. The functions $G_k(z,s)$ and $g(z,\overline{z},\alpha,\beta)$ are harmonic with respect to some elliptic operators, so in particular they are smooth. They are defined over $\Re(s)>1-\frac{k}{2}$ or $\Re(\alpha+\beta)>2$ at first, but have analytic continuation in $s$ or $q=\alpha+\beta$ on the whole plane $\mathbb C$. Also they are invariant under the slash operator for $\operatorname{SL}_2(\mathbb Z)$, and satisfy certain functional equations.
To explore the similar ones for the group $R(N)$, consider the following sets:
\begin{align*}
&(\mathbb Z \times \mathbb Z)_{N, L}:=\bigsqcup_{t \in \mathbb Z_{>0}}\{(m,n)\in t\mathbb Z \times t\mathbb Z : \mathrm{gcd}(Nm,n)=t\}, \\
&(\mathbb Z \times \mathbb Z)_{N, R}:=\bigsqcup_{t \in \mathbb Z_{>0}}\{(m,n)\in t\mathbb Z \times t\mathbb Z : \mathrm{gcd}(m,Nn)=t\},
\end{align*}
These are subsets of $(\mathbb Z \times \mathbb Z) \setminus \{(0,0)\}$. When $N=1$, they are exactly $(\mathbb Z \times \mathbb Z) \setminus \{(0,0)\}$. We define the parts of non-holomorphic Eisenstein series for $R(N)$ by
\begin{align*}
&G_{N,k,L}(z,s):=\frac{1}{2}\sum_{(m,n) \in (\mathbb Z \times \mathbb Z)_{N,L}} \frac{y^s}{|\sqrt N mz+n|^{2s}(\sqrt N mz+n)^k}, \text{ where } \Re(s)>1-\frac{k}{2}, \\
&G_{N,k,R}(z,s):=\frac{1}{2}\sum_{(m,n) \in (\mathbb Z \times \mathbb Z)_{N,R}} \frac{y^s}{| mz+\sqrt N n|^{2s}(mz+\sqrt N n)^k}, \text{ where } \Re(s)>1-\frac{k}{2},
\end{align*}
and
\begin{align*}
&g_{N,L}(z,\overline{z},\alpha,\beta):=\sum_{(m,n) \in (\mathbb Z \times \mathbb Z)_{N,L}}(\sqrt N mz+n)^{-\alpha}(\sqrt N m\overline{z}+n)^{-\beta}, \text{ where } \Re(\alpha+\beta)>2, \\
&g_{N,R}(z,\overline{z},\alpha,\beta):=\sum_{(m,n) \in (\mathbb Z \times \mathbb Z)_{N,R}}(mz+\sqrt N n)^{-\alpha}(m\overline{z}+\sqrt N n)^{-\beta}, \text{ where } \Re(\alpha+\beta)>2.
\end{align*}
We define the non-holomorphic Eisenstein series for $R(N)$ as the sum of these two types of functions, $$G_{N,k}(z,s):=G_{N,k,L}(z,s)+G_{N,k,R}(z,s),$$ correspondingly $$g_N(z,\overline{z},\alpha,\beta):=g_{N,L}(z,\overline{z},\alpha,\beta)+g_{N,R}(z,\overline{z},\alpha,\beta).$$ Note that $G_{1,k,L}(z,s)=G_{1,k,R}(z,s)=G_k(z,s)$, $g_{1,L}(z,\overline{z},\alpha,\beta)=g_{1,R}(z,\overline{z},\alpha,\beta)=g(z,\overline{z},\alpha,\beta)$ and $G_{N,k}(z,s)=\frac{1}{2}y^s g_N(z,\overline{z},s+k,s)$. If $N=1$, these functions satisfy the analytic property and the automorphy condition.
Before stating Theorem~\ref{thm_gfourier} which provides the Fourier expansion of $G_{N,k}(z,s)$, we introduce several notations. Let
\begin{align}\label{eqn_Gausssumprin}
G(b,\mathds{1}_{n}):=\sum_{a \in (\mathbb Z/n\mathbb Z)^{\times}}(\zeta_n^a)^b
\end{align}
be the Ramanujan sum, that is the sum of the $b$th powers of the $n$th primitive roots of unity. In particular, if $b=1$, then $G(1,\mathds{1}_n)=\mu(n)$, the Mobius function. We denote the Dirichlet series associated to this sum by
\begin{align*}
\mathcal{L}(b,\mathds{1}_{\bullet},s):=\sum_{n=1}^{\infty}\frac{G(b,\mathds{1}_n)}{n^s}.
\end{align*}
Similarly, we define the associated local factors,
\begin{align*}
\mathcal{L}_N(b,\mathds{1}_{\bullet},s):=\prod_{p|N}\sum_{n=0}^{\infty}\frac{G(b,\mathds{1}_{p^n})}{p^{ns}}
\end{align*}
and the associated shifted series
\begin{align*}
\mathcal{L}(b,\mathds{1}_{N\bullet},s):=\sum_{n=1}^{\infty}\frac{G(b,\mathds{1}_{Nn})}{n^s}.
\end{align*}
By definition, $\mathcal{L}(0,\mathds{1}_{\bullet},s)$ and $\mathcal{L}(1,\mathds{1}_{\bullet},s)$ are just the Dirichlet series associated to the Euler totient function and the Mobius function, respectively. It seems unnecessary at this moment to use the notation $\mathds{1}_{\bullet}$, but in Section~\ref{sec_Eisen} we generalize this notion for the other {\it dual characters} instead of $\mathds{1}_{\bullet}$, and then it would fit the current definitions. It coincides with the current definitions when the dual character is trivial.
Now we are ready to introduce our second results providing the Fourier expansions of non-holomorphic Eisenstein series for $R(N).$
\begin{theorem}\label{thm_gfourier}
For $\alpha, \beta\in \mathbb C$ such that $q:=\alpha+\beta \in \mathbb C_{\Re>2}$ and $k:=\alpha-\beta \in 2\mathbb Z$, if $N>1$, the Fourier expansion of a non-holomorphic Eisenstein series $g_{N}(z,\overline{z},\alpha,\beta)$ in $x$ is given by
\begin{align*}
g_N(z,\overline{z},\alpha,\beta)=\phi&_{N}(y,q,k) \\
&+2(-1)^k\zeta(q)\left(\frac{2\pi}{\sqrt N}\right)^q\sum_{n \neq 0}\frac{|n|^{q-1}}{\Gamma(\frac{q}{2}+\mathrm{sgn}(n)k)}W\left(2\pi n \frac{y}{\sqrt N};\alpha,\beta\right) \\
&\times \left(N^{-q/2}\mathcal{L}(n,\mathds{1}_{N\bullet},q)+\frac{\mathcal{L}(n,\mathds{1}_{\bullet},q)}{\mathcal{L}_N(n,\mathds{1}_{\bullet},q)}\right)e^{2\pi i nx/\sqrt N},
\end{align*}
where
\begin{align*}
\phi&_{N}(y,q,k)\\
&:=2+2(-1)^k N^{-q/2}\zeta(q)\left(\frac{2y}{\sqrt N}\right)^{1-q}\frac{2\pi\Gamma(q-1)}{\Gamma(\frac{q+k}{2})\Gamma(\frac{q-k}{2})}\left(N^{-q/2}\mathcal{L}(0,\mathds{1}_{N\bullet},q)+\frac{\mathcal{L}(0,\mathds{1}_{\bullet},q)}{\mathcal{L}_N(0,\mathds{1}_{\bullet},q)}\right)
\end{align*}
and $W(n;\alpha,\beta)$ is the modified Whittaker $W$-function.
\end{theorem}
For the detailed description of the modified Whittaker $W$-function, see Section~\ref{sec_Eisen}.
\
The issue of the analytic continuation of $g_N$ or $G_{N,k}$ is a bit more delicate to handle than that of the case $N=1$. If $N>1$, the $L$-series appearing in the Fourier expansion of $g_N(z,\overline{z},\alpha,\beta)$ is more complicated than one in the Fourier expansion of $g(z,\overline{z},\alpha,\beta)$. It is an unresolved question that whether $g_N$ has the analytic continuation in $q$ to the whole plane $\mathbb C$ for any positive integer~$N$. However, it is possible to continue the series analytically at least to the neighborhood of $q=0$, which admits a taylor expansion of $g_n(z,\overline{z},\alpha,\beta)$ near $q=0$, or of $G_{N,k}(z,s)$ near $s=0$. See Section \ref{sec_Eisen}.
Let $N=N_1^2 N_2$, where $N_2$ be a square-free integer. By definition, it is obvious that the group $R(N)$ is contained in $R(N_2)$, so our main interest is focused on the case when $N$ is square-free. From now on, we assume that $N$ is square-free. We define a meromorphic function in $s$,
\begin{align*}
f_{N,b}(s)&:=\frac{N^{-(s-1)-\frac{k}{2}}\prod_{p \mid N}\left(\left(\sum_{i=1}^{v_p(b)}(p-1)p^{i-1+i(2s+k-2)}\right)-p^{v_p(b)+(v_p(b)+1)(2s+k-2)}\right)+1}{\prod_{p \mid N}\zeta_p^{-1}(-2s-k+2)\zeta_p(-2s-k+1;v_p(b))},
\end{align*}
where $v_p$ is the $p$-valuation and $\zeta_p(s;v):=\sum_{n=0}^v p^{-ns}$, a truncated local zeta function. Note that when $v_p(b)=0$, i.e., $p \nmid b$, the summation part $\sum_{i=1}^{v_p(b)}(p-1)p^{i-1+v_p(b)(2s+k-2)}$ doesn't appear in $f_{N,b}(s)$. In particular, if $b=1$, $$f_N(s):=f_{N,1}(s)=\left((-1)^{\omega(N)}N^{(s-1)+\frac{k}{2}}+1\right)\prod_{p|N}(1-p^{2s+k-2})^{-1}.$$ Here $\omega(N)$ is the number of distinct prime factors of $N$.
Define the following completed and the doubly-completed Eisenstein series for $R(N)$, respectively by
\begin{align*}
\widehat{G}_{N,k}(z,s)&:=\left(\frac{\pi}{\sqrt N}\right)^{-s-\frac{k}{2}}\Gamma\left(s+\frac{k}{2}+\frac{|k|}{2}\right)f_N(s)G_{N,k}(z,s), \\
\widetilde{G}_{N,k}(z,s)&:=\left(s+\frac{k}{2}\right)\left(s+\frac{k}{2}-1\right)\widehat{G}_{N,k}(z,s).
\end{align*}
Theorem~\ref{thm_gfourier} provides the Fourier expansion of $\widehat{G}_{N,k}$ via the relation between the functions $g_N(z,\overline{z},\alpha,\beta)$ and $\widehat{G}_{N,k}(z,s)$.
\begin{corollary}\label{cor_Gfourier}
If $N>1$ is square-free, the Fourier expansion of a non-holomorphic Eisenstein series $\widehat{G}_{N,k}(z,s)$ is given by
\begin{align*}
\widehat{G}_{N,k}(z,s)=& \frac{\Gamma(s+\frac{k}{2}+\frac{|k|}{2})}{\Gamma(s+\frac{k}{2})}\hat{\zeta}(2s+k)y^s\sqrt N^{s+\frac{k}{2}}f_N(s) \\&+(-1)^{k/2}\frac{\Gamma(s+\frac{k}{2})\Gamma(s+\frac{k}{2}+\frac{|k|}{2})}{\Gamma(s+k)\Gamma(s)}\hat{\zeta}(2-2k-2s)y^{1-k-s}f_N(s) \\
&\qquad \times \left(\sqrt N^{1-s-\frac{k}{2}}\prod_{p|N}\frac{1-p^{-1}}{1-p^{-2s-k}}+\sqrt N^{-1+s+\frac{k}{2}}\prod_{p|N}\frac{1-p^{1-k-2s}}{1-p^{-2s-k}}\right) \\
&+\sum_{n \neq 0}(-1)^{k/2}y^{-k/2}\left(\frac{\Gamma(s+\frac{k}{2}+\frac{|k|}{2})}{\Gamma(s+\frac{k}{2}+\mathrm{sgn}(n)\frac{k}{2})}|n|^{-s-\frac{k}{2}}\sigma_{2s+k-1}(n)W_{\mathrm{sgn}(n)\frac{k}{2},s+\frac{k-1}{2}}\left(4\pi |n| \frac{y}{\sqrt N}\right)\right. \\
&\qquad \times \left.f_{N,n}(1-k-s)f_N(s)e^{2\pi i nx/\sqrt N}\right),
\end{align*}
where $\hat{\zeta}(s):=\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)$ is the completed Riemann zeta function.
\end{corollary}
As in the case for $\operatorname{SL}_2(\mathbb Z)$, if $N$ is a prime, then the completed Eisenstein series satisfies the following functional equation centered at line $s=\frac{1-k}{2}$.
\begin{theorem}\label{thm_funeq}
For a prime $p$, the completed and doubly-completed Eisenstein series satisfy the functional equation,
\begin{align*}
\widehat{G}_{p,k}(z,1-k-s)=\widehat{G}_{p,k}(z,s) \quad \text{ and } \quad
\widetilde{G}_{p,k}(z,1-k-s)=\widetilde{G}_{p,k}(z,s).
\end{align*}
\end{theorem}
\subsection{Roadmap}
In Section \ref{sec_RN} we prove that the group $R(N)$ is finitely generated. Most group-theoretical properties of $R(N)$, including Theorem~\ref{thm_ell} and the fact that $R(N)$ coincides with the Fricke group $\Gamma_0^+(N)$ in the low level $N=2,3$, are given in this section. We derive Corollary~\ref{cor_genusthm} using the general and classical theorem on the theory of Fuchsian groups and we exhibit a few examples with $g_p=0$.
Section \ref{sec_Eisen} is devoted to developing the theory of $L$-series for the Gauss sum of dual characters. This leads us to the analytic continuation of the non-holomorphic Eisenstein series. In this section we prove that functions $G_{k,N,L}$ and $G_{k,N,R}$ satisfy the modular invariant condition for $R(N)$, and prove Theorem~\ref{thm_gfourier} and Theorem~\ref{thm_funeq}.
In section \ref{sec_polyhar}, we investigate the structure of polyharmonic Maass forms for $R(p)$. More preciesely, we prove that any polyharmonic Maass forms can be written as the sum of the Taylor coefficients of the non-holomorphic Eisenstein series and cusp forms. Most argument used in this section is in parallel with the argument in \cite{MR357462} once if we define and construct several properties on the non-holomorphic Eisenstein series. Thus we state our results briefly, and in most cases the proofs in this section are is omitted. For a general vector-valued version of the theory, one may see \cite{MR2097357} which gives the systematic investigation on the harmonic weak Maass forms.
\
\section{Groups $R(p)$}\label{sec_RN}
Section \ref{sec_RN} is devoted to developing the fundamental properties of the groups $R(p)$ and prove Theorem \ref{thm_ell}.
\subsection{Generating sets of $R(p)$}
In this subsection we first show that the group $R(p)$ is finitely generated.
\begin{lemma}\label{lem_gen}
Let $p>2$ be a prime. The group $R(p)$ is generated by matrices $\{T_p, \omega_1\} \cup \{s_p(n) \in \operatorname{SL}_2(\mathbb R): -\frac{p+1}{2} \leq n \leq \frac{p+1}{2},\text{ } n \neq 0 \}$, where
\begin{align*}
T_p:=\begin{pmatrix}1 & \sqrt p \\ 0 & 1 \end{pmatrix}, \qquad
\omega_1 :=\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad
s_p(n):=\begin{pmatrix} \frac{n \hat{n}+1}{p}\sqrt p & n \\ \hat{n} & \sqrt p \end{pmatrix}
\end{align*}
and $\hat{n}$ is an integer satisfying $n \hat n \equiv -1 \pmod{p}$. For $p=2$, the group $R(p)$ is generated by $\{T_2,\omega_1,s_2(\pm 1)\}$.
\end{lemma}
It is possible to choice several $\hat{n}$ for each $n$, but regardless of the choice of $\hat{n}$ it generates the same group $R(p)$.
\begin{proof}
Let $p>2$ and denote by $F$ the group generated by $T_p, \omega_1$, and $s_p(n)$'s. Let
\begin{align*}
F_{col}^-:=\Big\{(a,c) \in \mathbb Z^2 : &\text{ }|c| \leq \frac{p}{2}|a|, \text{ } \gcd(pa,c)=1, \\
& \left.\text{ and there exist $b, d \in \mathbb Z$ such that }\begin{pmatrix} a\sqrt p & b \\ c & d \sqrt p\end{pmatrix} \in F \right\},
\end{align*}
\begin{align*}
F^-:=\left\{a \in \mathbb Z : (a,c) \in F_{col}^- \text{ for any } c \text{ with } |c| \leq \frac{p}{2}|a|, \text{ } \gcd(pa,c)=1\right\}.
\end{align*}
Note that if $a \in F^-$ then $-a \in F^-$, so we may assume that $a>0$. In order to prove the assertion, we prove only that
\begin{align}\label{claim_F^-}
\text{if any integer $m$ with $|m|<a$ belongs to $F^-$, then so is $a$.}
\end{align}
Indeed, this claim means that $\mathbb Z \subseteq F^-$. Note that the group $F$ is a subgroup of $\operatorname{SL}_2(\mathbb R)$. Thus if $a,b,c$ are given and $\begin{pmatrix} a\sqrt p & b \\ c & d \sqrt p\end{pmatrix} \in F$, an integer $d$ is uniquely determined. Suppose $(a_0,c_0) \in F_{col}^-$. Pick any integer $b_0$ such that $\begin{pmatrix} a_0\sqrt p & b_0 \\ c_0 & d_0 \sqrt p\end{pmatrix} \in F$, where $d_0$ is an integer determined by $a_0,c_0$ and $b_0$. Let $\begin{pmatrix} a_0\sqrt p & b \\ c_0 & d \sqrt p\end{pmatrix}$ be an arbitrary one belongs to $F$ with given the first column $(a_0\sqrt p,c_0)^t$. Then $pa_0d-bc_0=1$, so $b \equiv b_0 \pmod{pa_0}$. Write $b=pam+b_0$ for an integer $m$, then
\begin{align*}
\begin{pmatrix} a_0\sqrt p & b \\ c_0 & d \sqrt p\end{pmatrix}=\begin{pmatrix} a\sqrt p & b_0 \\ c & d_0 \sqrt p\end{pmatrix}T_p^m.
\end{align*}
Hence $\mathbb Z = F^-$ implies $R(p)^- \subseteq F$. The set $R(p)^+$ can be written as $R(p)^+=\omega_1 R(p)^-$, so it follows that $R(p) \subseteq F$.
Let's prove Claim \eqref{claim_F^-}. Suppose that any integer with absolute value less than $a$ belongs to $F^-$. To prove $a\in F^-$, we show that $(a,c) \in F_{col}^-$ for any $c$ such that $\gcd(a,c)=1$ and $|c| \leq \frac{p}{2}a$. Use the notation $\hat{c}$ to mean that it satisfies $c\hat c \equiv -1 \pmod{p}$ and $|\hat{c}| \leq \frac{p}{2}a$. We say that $(a,c) \in \mathbb Z^2$ is {\it properly reduced} if there exists an integer $m \in \mathbb Z$ such that $|a+\hat{c}m|<a$, or equivalently, a matrix
\begin{align*}
\begin{pmatrix} a\sqrt p & \hat{c} \\ c & d \sqrt p\end{pmatrix}\begin{pmatrix} \sqrt p & 1 \\ 1 & 0 \end{pmatrix}^m=\begin{pmatrix} a'\sqrt p & \hat{c} \\ c' & d \sqrt p\end{pmatrix}
\end{align*}
satisfies $|a'|<a$. Since $\begin{pmatrix} a'\sqrt p & \hat{c} \\ c' & d \sqrt p\end{pmatrix} \in F$, it is obvious that if $(a,c)$ is properly reduced, then $(a,c) \in F_{col}^-$. Therefore it is enough to show that any non-properly reduced pair $(a,c)$ is also contained in $F_{col}^-$.
Let $(a,c)$ be a non-properly reduced pair of integers with $|c| \leq \frac{p}{2}a$, $\gcd(pa,c)=1$. Since the inequality $|\hat{c}| < 2a$ implies $|a-\mathrm{sgn}(\hat{c})\hat c | <a$, we have $2a \leq |\hat{c}| \leq \frac{p}{2}a$, so there is an integer $n \in \{1, 2, \cdots, \frac{p+1}{2}\}$ with $(n-1)a < |\hat{c}| < (n+1)a.$ For such an integer $n$, we have $|an+|\hat{c}||=|\mathrm{sgn}(\hat{c})an+\hat{c}|<a$ and
\begin{align*}
\begin{pmatrix} a\sqrt p & \hat{c} \\ c & d \sqrt p\end{pmatrix}s_p(\mathrm{sgn}(\hat{c})n)=\begin{pmatrix} * & (\mathrm{sgn}(\hat{n})an+\hat{c})\sqrt p \\ * & *\end{pmatrix}=A.
\end{align*}
By inductive hypothesis we have $A\omega_1 \in F$, so $\begin{pmatrix} a\sqrt p & \hat{c} \\ c & d \sqrt p\end{pmatrix}=A s_p(\mathrm{sgn}(\hat{c})n)^{-1} \in F$ which implies $(a,c) \in F_{col}^-$.
One can obtain the lemma for $p=2$ through the same argument as above.
\end{proof}
By Lemma \ref{lem_gen}, it is clear that the Hecke triangle groups $\Gamma_{(2,4,\infty)}$ and $\Gamma_{(2,6,\infty)}$ coincide with groups $R(2)$ and $ R(3)$, respectively.
\subsection{Elliptic points and cusps}
In this subsection we prove Theorem \ref{thm_ell}. Recall that a point $z \in \mathbb H $ is called an elliptic point for a Fuchsian group $\Gamma$ if its stabilizer group $\Gamma_z$ of the action $\Gamma$ on $\mathbb H$ has a nontrivial element. A nontrivial element in $\Gamma_z$ always has trace whose absolute value is smaller than 2, and it is called an elliptic element. If two points $z_1, z_2 \in \mathbb H$ are $\Gamma$-equivalent, i.e. there exists a matrix $\gamma \in \Gamma$ such that $\gamma z_1=z_2$, then their stabilizer groups are conjugate to each other. Thus we also use the terminology `elliptic point' by meaning the equivalence class containing that point.
For the Hecke triangle groups $\Gamma_{(2,m,\infty)}$, in particular for $m=4,6$, the set of equivalence classes of the elliptic points are known to be exactly $\{i, \frac{1+i}{\sqrt 2}\}$ and $\{i,\frac{\sqrt 3 +i}{2}\}$, respectively (see, for instance, \cite{MR3228299}). Therefore, unless specifically stated, we assume $p\geq 5$.
\begin{lemma}\label{lem_Fermat}
Let $c$ be a positive integer whose prime factors are $2$ or of the form $4r+1$ for some $r \in \mathbb Z$. Let $a$ and $a_0$ be integers satisfying
\begin{align*}
\begin{cases}
a \equiv a_0 & \pmod{p}, \\
a^2 \equiv -1 & \pmod{p}, \\
\frac{a^2+1}{p} \equiv 0 & \pmod{c}.
\end{cases}
\end{align*}
There exists $(x,y) \in \mathbb Z^2$ such that
\begin{enumerate}[\normalfont(a)]
\item $c=x^2+y^2$, \text{ and }
\item $c$ divides $x-ay$ and $\frac{a-a_0}{p}x+\frac{a_0a+1}{p}y$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $a=a_0+np$, for $n \in \mathbb Z$. Then for any integers $x$ and $y$,
\begin{align*}
\frac{a-a_0}{p}x+\frac{a_0a+1}{p}y=nx-nay+\frac{a^2+1}{p}y.
\end{align*}
Since $p \mid a^2+1$ and $c \mid \frac{a^2+1}{p}$, it is enough to show that there exists $(x,y) \in \mathbb Z^2$ such that
\begin{align*}
c=x^2+y^2 \text{ and } c \mid (x-ay).
\end{align*}
Suppose $c$ is not divisible by 2. Note that we can replace $a$ with $a':=a+rpc$ for any $r \in \mathbb Z.$ Since $c$ is odd, we may assume that $a$ is even number. By considering $a$ as a Gaussian integer i.e., $a \in \mathbb Z[i]$, we have
\begin{align}\label{eq:gcd1}
\gcd_{\mathbb Z[i]}(a+i,a-i)=\gcd_{\mathbb Z[i]}(a+i,2i)=\gcd_{\mathbb Z[i]}(a+i,2)=1,
\end{align}
so $a+i$ and $a-i$ are coprime in $\mathbb Z [i]$.
Note that $c$ can be factorized into
\begin{align*}
c=\prod_{i=1}^n z_i^{e_i}\overline{z_i}^{e_i}
\end{align*}
in $\mathbb Z [i]$, where each $z_i$ is a prime in $\mathbb Z [i]$ and $\overline{z_i}$ is its complex conjugate, which is also a prime. Since $c$ divides $a^2+1=(a+i)(a-i)$, we may assume that $z_i \mid (a+i)$ for all $i=1,2,\ldots,n$, hence $z_i^{e_i} \mid (a+i)$ by \eqref{eq:gcd1}. Let $x:=\Re(\prod_{i=1}^n z_i^{e_i})$, $y:=\Im(\prod_{i=1}^n z_i^{e_i}) \in \mathbb Z$. Then we have $c=x^2+y^2$ and
\begin{align*}
c \mid (a+i)(x-iy)=(ax+y)+i(x-ay).
\end{align*}
Since $c$ is an integer, we have $c \mid (x-ay)$.
Next suppose $c$ is even. Since $a^2+1 \equiv 0 \pmod{c}$, we have $4 \nmid c$. If $a$ is even, then we can apply the same argument as in the case when $2 \mid c$. If $a$ is an odd integer, then
\begin{align*}
\gcd_{\mathbb Z[i]}(a+i,a-i)=\gcd_{\mathbb Z[i]}(a+i,2)=\gcd_{\mathbb Z[i]}(1+i,2)=1+i,
\end{align*}
so we can write $a+i=(1+i)\prod_{i=1}^n w_i^{f_i}$ for primes $w_i \in \mathbb Z[i].$ Write $c=(1+i)(1-i)z_c \overline{z_c}$ for $z_c \in \mathbb Z[i]$. By the same reason as above, we may assume $z_c \mid \prod_{i=1}^n w_i^{f_i}.$ Define $x,y \in \mathbb Z$ so that $x+iy=(1+i)z_c$, then it satisfies the assertion.
\end{proof}
\begin{lemma}\label{lem_Fermat2}
Let $c$ be a positive integer whose prime factors are $3$ or of the form $3r+1$ for some $r \in \mathbb Z$. Let $a$ be an integer satisfying
\begin{align*}
a^2-a+1 \equiv 0 \pmod{c}.
\end{align*}
There exists $(x,y) \in \mathbb Z^2$ such that
\begin{enumerate}[\normalfont(a)]
\item $c=3x^2+y^2$, \text{ and }
\item c divides $(2a-1)x+y$.
\end{enumerate}
\end{lemma}
\begin{proof}
The proof is the same as the last part of the proof of Lemma \ref{lem_Fermat} except for using the ring of integers $\mathbb Z[\zeta_3]$ of $\mathbb{Q}(\zeta_3)$ instead of the Gaussian integers $\mathbb Z[i]$, where $\zeta_3:=\frac{-1+\sqrt{-3}}{2}$. Note that $\mathbb Z[\zeta_3]$ is a UFD and $a^2-a+1=(a+\zeta_3)(a+\overline{\zeta_3})$ in $\mathbb Z[\zeta_3]$.
\end{proof}
Consider an elliptic point in $\mathrm{Ell}(p)$ that is represented by $z \in \mathbb H.$ For another point $z' \in \mathbb H$ which is $R(p)$-equivalent to $z$, the trace of its associated elliptic element is equal to the trace of $\gamma,$ where $\gamma \in R(p)_z.$ Indeed, if $\alpha \in R(p)$ sends $z$ to $z'$, then any element of $R(p)_{z'}$ can be written as $\alpha \gamma \alpha^{-1}$.
Let $\mathrm{Ell}^+(p)_t$ be the collection of elliptic points $[z] \in \mathrm{Ell}^+(p)$ such that there is an associated elliptic element $\gamma \in R(p)_z$ whose trace is $t$. Since $|t| < 2$ and $-I$ acts trivially on $\mathbb H$, for any $t \in \mathbb Z$ we have $\mathrm{Ell}(p)_t=\mathrm{Ell}(p)_{-t}$, and the set of elliptic points $\mathrm{Ell}^+(p)$ can be written as a union
\begin{align}\label{eqn_ellunion}
\mathrm{Ell}^+(p)=\mathrm{Ell}^+(p)_0 \cup \mathrm{Ell}^+(p)_{1}.
\end{align}
In fact, it turns out that there is no elliptic point whose trace is $0$ and $1$ at the same time, i.e., the union of (\ref{eqn_ellunion}) is disjoint. This is clear by the following two propositions.
\begin{proposition}\label{prop_Ell+tr0}
Let $p$ be a prime. If $p=2$ or $p=4r+1$ for some $r \in \mathbb Z$, then the set of elliptic points $\mathrm{Ell}^+(p)_0$ is a singleton. Otherwise, $\mathrm{Ell}^+(p)_0= \emptyset$.
\end{proposition}
\begin{proof}
Let $z \in \mathrm{Ell}^+(p)_0$ be an elliptic point with trace $0$ and let $\gamma = \begin{pmatrix}
a & b \sqrt p \\ c \sqrt p & -a
\end{pmatrix} \in R(p)^+$ be one of its associated elliptic elements. Then $z$ can be written as $z=\frac{a+i}{c\sqrt p}.$ Note that $\det{\gamma}=-a^2-pbc=1$ implies $p \mid (a^2+1)$. This shows that if such $z \in \mathbb H$ exists, then $p$ must be 2 or of the form $4r+1$ for an integer $r \in \mathbb Z$. Moreover, we have $c \mid (a^2+1)$, so any prime factor $q$ of $c$ is 2 or satisfies $q \equiv 1 \pmod{4}$.
For the rest of the proof, we suppose that $p=2$ or $p=4r+1$ for some $r \in \mathbb Z$. Fix an integer $a_0 \in \mathbb Z$ such that $a_0^2+1 \equiv 0 \pmod{p}.$ Let $b_0:=\frac{a_0^2+1}{p}$ and $z_0:=\frac{a_0+i}{\sqrt p}$. It is an elliptic point of the element $\begin{pmatrix}
a_0 & b_0\sqrt p \\ \sqrt p & -a_0
\end{pmatrix}$, so $[z_0] \in \mathrm{Ell}^+(p)_0.$ Note that $[-\overline{z_0}] \in \mathrm{Ell}^+(p)_0$, which is inherited from $-a_0$ instead of $a_0$. We claim that these two elliptic points are $R(p)^-$-equivalent. Indeed, it is enough to show that there exist integers $X,Y,Z,W \in \mathbb Z$ such that
\begin{align*}
\begin{pmatrix}
X\sqrt p & Y \\ Z & W\sqrt p
\end{pmatrix} \left(\frac{a_0+i}{\sqrt p}\right)=\frac{-a_0+i}{\sqrt p}.
\end{align*}
This can be reduced to the following system of equations by direct calculation,
\begin{align*}
\begin{cases}
\frac{a_0^2+1}{p}Z^2+2aZW+pW^2=1, \\
X=W, \\
Y=\frac{-a^2-1}{p}Z-2aW.
\end{cases}
\end{align*}
Notice that a quadratic form $\frac{a_0^2+1}{p}Z^2+2aZW+pW^2$ is positive definite of discriminant $-4$. Its reduced form is the principal form $Z^2+W^2,$ so it must represent~$1$. Thus the equation $\frac{a_0^2+1}{p}Z^2+2aZW+pW^2=1$ must has an integer solution $(Z,W) \in \mathbb Z^2.$
Next, we prove that any elliptic point $z \in \mathrm{Ell}^+(p)_0$ is $R(p)$-equivalent to $z_0$. Since $a^2+1 \equiv 0~\pmod{p}$, we may assume that $a \equiv a_0~\pmod{p}.$ By Lemma \ref{lem_Fermat}, there exists a pair of integers $(x_0,y_0) \in \mathbb Z^2$ such that
\begin{enumerate}[(a)]
\item $c=x_0^2+y_0^2$, \text{ and }
\item $c$ divides $x_0-ay_0$ and $\frac{a-a_0}{p}x_0+\frac{a_0a+1}{p}y_0$.
\end{enumerate}
Let \begin{align*}
Z&:=y_0, \quad W:=-x_0-a_0y_0, \\
X&:=\frac{(a_0+a)Z+W}{c}, \quad Y:=\frac{1}{c}\left(-\frac{a_0^2+1}{p}Z+\frac{a-a_0}{p}W\right).
\end{align*}
They are all integers, and it can be directly shown that $XW-pYZ=1$ and a matrix
$
\begin{pmatrix}
X & Y\sqrt p \\ Z\sqrt p & W
\end{pmatrix}
$ sends $z_0$ to $z$
\end{proof}
\begin{proposition}\label{prop_Ell+tr1}
Let $p$ be a prime. If $p=3$ or $p=3r+1$ for some $r \in \mathbb Z$, then the set of elliptic points $\mathrm{Ell}^+(p)_1$ is a singleton. Otherwise, $\mathrm{Ell}^+(p)_1= \emptyset$.
\end{proposition}
\begin{proof}
Suppose there exists an elliptic point $z\in \mathrm{Ell}^+(p)_1$ and let $\gamma=\begin{pmatrix}
a & b \sqrt p \\ c \sqrt p & -a+1
\end{pmatrix} \in R(p)^+$ be its associated elliptic element. Then we can write $z=\frac{2a-1+\sqrt{3} i}{2\sqrt p}.$ As in the proof of Proposition~\ref{prop_Ell+tr0}, we have $p\mid (a^2-a+1)$, so $p=3$ or $p=3r+1$ for some $r \in \mathbb Z$.
Suppose $p=3$ or $p=3r+1$, and fix an integer $a_0\in \mathbb Z$ with $a_0^2-a_0+1 \equiv 0 \pmod{p}$. Note that there are exactly two integers $n$ satisfying $n^2-n+1 \equiv 0 \pmod{p}$ in mod $p$, namely $n \equiv a_0 \pmod{p}$ or $n \equiv -a_0+1 \pmod{p}.$ For $b_0:=\frac{a_0^2-a_0+1}{p}=\frac{(-a_0+1)^2-(-a_0+1)+1}{p}$, we have two elliptic points $z_0=\frac{2a_0-1+\sqrt 3 i}{2\sqrt p}$ and $-\overline{z_0}=\frac{-2a_0+1+\sqrt 3 i}{2\sqrt p}$ that are associated to the elliptic elements $\begin{pmatrix}
a_0 & b_0\sqrt p \\ \sqrt p & -a_0+1
\end{pmatrix}$ and $\begin{pmatrix}
-a_0+1 & b_0\sqrt p \\ \sqrt p & a_0
\end{pmatrix}$, respectively. These two elliptic points are $R(p)^-$-equivalent. Indeed, as in the proof of Proposition \ref{prop_Ell+tr0}, it is enough to find integer solutions of the quadratic equation
\begin{align*}
\frac{a_0^2-a_0+1}{p}Z^2+(2a_0-1)ZW+pW^2=1.
\end{align*}
Since the quadratic form $\frac{a_0^2-a_0+1}{p}Z^2+(2a_0-1)ZW+pW^2$ is positive definite of discriminant~$-3$, it must have an integer solution.
It only remains to prove that any elliptic point $z \in \mathrm{Ell}^+(p)_1$ is $R(p)$-equivalent to $z_0.$ Similarly to the proof of Proposition \ref{prop_Ell+tr0}, we know that any prime factor of $c$ is 3 or of the form $3r+1$ for an integer $r \in \mathbb Z.$ Applying Lemma \ref{lem_Fermat2}, we have a pair of integers $(x_0,y_0) \in \mathbb Z^2$ such that
\begin{enumerate}[(a)]
\item $c=x_0^2+y_0^2$, \text{ and }
\item $c$ divides $x_0-ay_0$ and $\frac{a-a_0}{p}x_0+\frac{a_0a+1}{p}y_0$.
\end{enumerate}
Define
\begin{align*}
Z&:=2x_0, \quad W:=-(2a_0-1)x_0+y_0, \\
X&:=\frac{(a_0+a-1)X+W}{c}, \quad Y:=\frac{1}{c}\left(-\frac{a_0^2-a_0+1}{p}Z+\frac{a-a_0}{p}W\right).
\end{align*}
One can see that a matrix $ \begin{pmatrix}
X & Y\sqrt p \\ Z\sqrt p & W
\end{pmatrix}$ belongs to $R(p)^+$ and sends $z_0$ to $z$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm_ell}\normalfont{(a)}]
It follows from combining Propositions~\ref{prop_Ell+tr0} and \ref{prop_Ell+tr1}.
\end{proof}
\
Next we prove Theorem~\ref{thm_ell}(b). Suppose that $D$ is a negative integer such that $D \equiv 0,1 \pmod{4}$.
Consider the set of all (not~necessarily primitive) positive definite binary quadratic forms of discriminant $D.$ We say that two binary quadratic forms $f(X,Y)$ and $g(X,Y)$ are {\it equivalent} if there exist integers $a,b,c,d$ such that $f(x,y)=g(ax+by,cx+dy)$ and $ad-bc=\pm 1$. If $ad-bc=1$, we say that they are {\it properly equivalent}.
Recall that any positive definite binary quadratic form of discriminant $D$ has exactly one reduced form up to equivalence. In other words, there is a bijection between the set of all equivalence classes of positive definite binary quadratic forms of discriminant $D$ and the set of all positive definite binary quadratic reduced forms of discriminant~$D.$ If we consider the set of all proper equivalence classes of primitive forms, which is a binary quadratic form whose coefficients are relatively prime, there is a natural group structure which is given via transport of structure of the ideal class group $C(D).$ We call such a group the {\it form class group} of discriminant $D$.
We set the following notations:
\begin{align*}
(A,B,C):=&\text{ a binary quadratic form } AX^2+BXY+CY^2, \\
[(A,B,C)]:=&\text{ an equivalence class of a binary quadratic form } (A,B,C), \\
H(D):=&\text{ the set of all proper equivalence classes of positive definite binary quadratic}\\ &\text{ forms of discriminant } D.
\end{align*}
\begin{proof}[Proof of Theorem \ref{thm_ell}\normalfont{(b)}]
For a prime $p \geq 5$, any elliptic point $[z] \in \mathrm{Ell}^-(p)$ is of trace zero. We can write $z=\frac{a\sqrt p +i}{c}$, where $\begin{pmatrix}
a \sqrt p & b \\ c & -a \sqrt p
\end{pmatrix}$ is an elliptic element of $z_0$. We associate an elliptic point $z$ to a binary quadratic form $(-pb,2pa,c)=(\frac{a^2p+1}{c}p,2pa,c),$ denoted by $QF(z).$
We claim that the map defined by
\begin{align*}
\mathrm{Ell}^-(p) &\longrightarrow H(-4p) \\
[z] &\mapsto [QF(z)]
\end{align*}
is well-defined and bijective. Indeed, let $[z_1]=[z_2] \in \mathrm{Ell}^-(p)$ be $R(p)$-equivalent elliptic elements. There exists $\gamma \in R(p)$ such that $\gamma z_1=z_2$. Moreover we may assume $\gamma:=\begin{pmatrix}
r & s\sqrt p \\ t \sqrt p & u
\end{pmatrix} \in~R(p)^+$, since if $\gamma \in R(p)^-$, then for any $\alpha \in R(p)_{z_1}^-$, we have $\gamma \alpha z_1=z_2$ and $\gamma \alpha \in R(p)^+$. It is shown by direct computation that a matrix $\begin{pmatrix}
r & s \\ tp & u
\end{pmatrix} \in \operatorname{SL}_2(\mathbb Z)$ gives a proper equivalence between $QF(z_1)$ and $QF(z_2)$. In other words,
\begin{align*}
QF(z_1)(rX+sY,tpX+uY)=QF(z_2)(X,Y).
\end{align*}
The injectivity of the map follows by reversing the above argument. If $[QF(z_1)]=[QF(z_2)]$, then there exists a matrix $\begin{pmatrix}
r & s \\ t' & u
\end{pmatrix} \in \operatorname{SL}_2(\mathbb Z)$ such that
\begin{align*}
\begin{pmatrix}
r & s \\ t' & u
\end{pmatrix}
\begin{pmatrix}
b_1p & a_1p \\ a_1 p & c_1
\end{pmatrix}
\begin{pmatrix}
r & t' \\ s & u
\end{pmatrix}=
\begin{pmatrix}
b_2p & a_2p \\ a_2 p & c_2
\end{pmatrix}.
\end{align*}
We have $t'^2 c_1 \equiv r^2 b_1 p +4rt' a_1 p + t'^2c_1 = b_2 p \equiv 0 \pmod{p},$ so $t' \equiv 0 \pmod{p}.$ Letting $t'=pt,$
the matrix $\gamma=\begin{pmatrix}
r & s\sqrt p \\ t\sqrt p & u
\end{pmatrix}\in R(p)$ sends $z_1$ to $z_2.$
The remaining part is to prove surjectivity. Let $[(A,B',C)] \in H(-4p)$ be an equivalence class of the binary quadratic form $(A,B',C)$. Since $4$ divides its discriminant, $B'$ must be even, so we write $B'=2B.$ We consider two cases: either $p \mid A$ or not.
Suppose $A$ is divisible by $p.$ Again by the divisibility of its discriminant, $B$ is also divisible by $p$, so we write $pA$ and $pB$ instead of $A$ and $B$, respectively. Note that a quadratic form $(pA,2pB,C)$ is properly equivalent to a quadratic form $(C,-2pB,pA)$ via an equivalence $\omega_1=\begin{pmatrix}
0 & -1 \\ 1 & 0
\end{pmatrix}$, i.e.,
\begin{align*}
(pA,2pB,C)(-Y,X)=(C,-2pB,pA)(X,Y).
\end{align*}
Let $a=-B$ and $c=A$. By the discriminant condition, we have $-b:=C=\frac{a^2p+1}{c}$.
Next suppose $A$ is not divisible by $p$. Let $s$ be an integer such that $sA+B \equiv 0 \pmod{p}.$ Since $B^2-AC=-p$, we have $s^2 A+sB+C \equiv 0 \pmod{p}.$ In this case, we set
\begin{align*}
a:=\frac{sA+B}{p}, \quad c:=\frac{s^2A+sB+C}{p}.
\end{align*}
Then $a$ and $c$ are integers satisfying $-b:=\frac{a^2 p+1}{c}=A \in \mathbb Z.$ One can see that the matrix $\begin{pmatrix}
1 & s \\ 0 & 1
\end{pmatrix}$ gives a proper equivalence between two quadratic forms via
\begin{align*}
\begin{pmatrix}
1 & 0 \\ s & 1
\end{pmatrix}
\begin{pmatrix}
A & B \\ B & C
\end{pmatrix}
\begin{pmatrix}
1 & s \\ 0 & 1
\end{pmatrix}=
\begin{pmatrix}
A & sA+B \\ sA+B & s^2A+sB+C
\end{pmatrix},
\end{align*}
so $[(A,2B,C)]=[(-b,2ap,cp)].$
In both cases, we have shown that the quadratic form $(A,2B,C)$ is properly equivalent to the quadratic form of the form $(-b,2ap,cp).$ Finally, we point out that there is a proper equivalence $\begin{pmatrix}
a & b \\ c & -ap
\end{pmatrix}$ between two quadratic forms $(-b,2ap,cp)$ and $(-bp,2ap,c)$. Note that a class of the quadratic form $[(-bp,2ap,c)]$ is equal to $[QF(z)]$, where $z$ is an elliptic point $z=\frac{a^2p+1}{c}.$ Therefore the map $[z] \mapsto [QF(z)]$ is surjective.
\end{proof}
\
\begin{remark}
One may slightly generalize the proof of Theorem \ref{thm_ell}(b) for an arbitrary trace $t \in \{-1,0,1\}.$ If $[z] \in \mathrm{Ell}^-(p)$ is an elliptic point and $\begin{pmatrix}
a\sqrt p & b \\ c & d\sqrt p
\end{pmatrix}$ is its associated elliptic element, then we have $z=\frac{(2a-t)\sqrt p +\sqrt{4-pt^2}i}{2c}.$ If we let $QF(z):=(-pb,p(2a-t),c)$, then the map
\begin{align*}
\mathrm{Ell}^+(p) &\longrightarrow H(t^2p^2-4p) \\
[z] &\mapsto [QF(z)]
\end{align*}
is also bijective. However, the set $\mathrm{Ell}^+(p)_t$ is nonempty only if $p=2$ or $3$.
\end{remark}
\
Let $z \in \mathbb H$ be any elliptic point for $R(N).$ The order of $z$ is defined as
\begin{align*}
e_z:=|R(N)_z/\{ \pm I\}|=\frac{1}{2}|R(N)_z|.
\end{align*}
It is independent of the choice of representatives of a class $[z] \in \mathrm{Ell}(N).$
\begin{proposition}
Let $z \in \mathbb H$ be an elliptic point for $R(p).$ Then the order of $z$ is
\begin{align*}
e_z=\begin{cases}
3 & \text{ if } [z] \in \mathrm{Ell}^+(p)_1, \\
2 & \text{ otherwise.}
\end{cases}
\end{align*}
\end{proposition}
\begin{proof}
Let $[z] \in \mathrm{Ell}^-(p)$ and $\gamma=\begin{pmatrix}
a \sqrt p & b \\ c & d\sqrt p
\end{pmatrix}$ be an elliptic element associated to $z$. One can see that the map $\alpha \mapsto \gamma\alpha$ for $\alpha \in R(p)_z^+$ induces a bijection between $R(p)_z^+/\{\pm I\}$ and $R(p)_z^-/\{\pm I\}$. Thus, in order to count the number of elements of $R(p)_z/\{\pm I\}$, it suffices to count those of $R(p)_z^+.$
By definition, the number of elements of the set $R(p)_z^+$ is equal to the number of pairs of integers $(X,Y) \in \mathbb Z^2$ such that $c$ is represented by the quadratic form associated with $z$ of a pair $(X,Y)$, i.e., $QF(z)(X,Y)=c.$
Let $(A,2B,C)$ be a reduced form of discriminant $-4p.$ By Theorem \ref{thm_ell}(b), there is an elliptic point $[z] \in \mathrm{Ell}^-(p)$ such that $[QF(z)]=[(A,2B,C)]$.
Suppose $A$ and $B$ are relatively prime. Then one can choose a representative $z=\frac{\hat{B}\sqrt p+1}{A},$ where $\hat{B}$ is any integer satisfying $B\hat{B} \equiv 1 \pmod{A}.$ Indeed, if we let
\begin{align*}
r:=-\frac{B+p\hat B}{A},
\end{align*}
then it is an integer since $B^2-AC=-p.$ Two binary quadratic forms $\left(\frac{p\hat B^2+1}{A}p,2p\hat B,A\right)$ and $(A,2B,C)$ are properly equivalent via an equivalence $\begin{pmatrix}
0 & 1 \\ -1 & r
\end{pmatrix},$ and $QF\left(\frac{\hat{B}\sqrt p+1}{A}\right)=\left(\frac{p\hat B^2+1}{A}p,2p\hat B,A\right).$
Hence we need to count the number of pairs $(X,Y)$ that represent $A$ by the binary quadratic form $(A,2B,C).$ Note that a quadratic form $(A,2B,C)$ can be written as
\begin{align}\label{eqn_qfrepreA}
AX^2+2BXY+CY^2=A\left(X+\frac{B}{A}Y\right)^2+\left(C-\frac{B^2}{A^2}\right)Y^2.
\end{align}
Recall that $C \geq A$ since $(A,2B,C)$ is reduced. Here we assume $\gcd(A,B)=1$, so $C-~\frac{B^2}{A^2}>~A.$ Thus the quadratic form in (\ref{eqn_qfrepreA}) represents $A$ if only if $X=\pm 1$ and $Y=0$.
Now suppose $A$ and $B$ is not relatively prime. By the discriminant condition, we have $\gcd(A,B)=p$, so let $A=pA'$ and $B=pB'.$ Again by the discriminant condition, we have $A'=\frac{pB'^2+1}{C}$ so that $(A,2B,C)=\left(\frac{pB'^2+1}{C}p,2pB',C\right)$, which is the associated quadratic form of an elliptic point $\frac{B'^2\sqrt p +1}{C} \in \mathbb H.$ In this case, the problem reduces to count the number of pairs $(X,Y) \in \mathbb Z^2$ for which $(A,2B,C)(X,Y)=C.$ Writing the reduced form $(A,2B,C)$ in the same way as in \eqref{eqn_qfrepreA}, we have an equation
\begin{align}\label{eqn_qfrepreC}
CY^2+2pB'YX+pA'X^2=C\left(Y+\frac{pB'}{C}X\right)^2+\frac{p}{C}X^2=C.
\end{align}
From \eqref{eqn_qfrepreC}, $Y+\frac{pB'}{C}X$ must be $0$ or $\pm 1$. If $Y+\frac{pB'}{C}=0$ then $X^2=\frac{C^2}{p}$, which contradicts the fact that $X$ in an integer. Hence $Y+\frac{pB'}{C}=1$ or $-1$, and in both cases we have $X=0.$ Therefore for any $[z] \in \mathrm{Ell}^-(p),$ we conclude $e_z=2.$
In Proposition \ref{prop_Ell+tr0} and Proposition \ref{prop_Ell+tr1}, we have explicitly found out what the set $\mathrm{Ell}^+(p)$ consists of. The direct computation using the similar argument as the above shows that $e_z=2$ for $z\in \mathrm{Ell}^+(p)_0$ and $e_z=3$ for $z \in \mathrm{Ell}^+(p)_1.$
\end{proof}
\
Recall that for a Fuchsian group $\Gamma$, a real number $x \in \mathbb R$ is called the {\it cusp} for $\Gamma$ if there exists a nontrivial element $\gamma \in \Gamma$ that stabilizes $x$ as a point in the complex plane. In this case, the matrix $\gamma$ is called the {\it parabolic element}. It follows that the trace of $\gamma$ is $2$ or $-2$ by definition.
Contrary to elliptic points, there is only one cusp for any $R(p)$.
\begin{proposition}\label{prop_cusp}
Every cusp for $R(p)$ is equivalent to $\infty.$
\end{proposition}
\begin{proof}
Note that a linear fractional transformation $\begin{pmatrix}
\sqrt p & -1 \\ 1 & 0
\end{pmatrix} \in R(p)$ sends $0$ to $\infty.$ Thus it is enough to show that any cusp for $R(p)$ is equivalent to $0.$
Let $x \in \mathbb R$ be a cusp for $R(p).$ Take its parabolic element $\gamma=\begin{pmatrix}
a & b\sqrt p \\ c \sqrt p & d
\end{pmatrix}.$ Then $x \in \mathbb R$ satisfies
\begin{align*}
x^2-\frac{a-d}{c\sqrt p}x-\frac{b}{c}=0.
\end{align*}
Since $\mathrm{tr}~{\gamma}=2$, we have $a-d=2(a-1).$ Also, since $\det{\gamma}=1,$ we have
\begin{align*}
-\frac{b}{c}=\frac{-pbc}{pc^2}=\frac{-ad+1}{pc^2}=\frac{(a-1)^2}{pc^2}.
\end{align*}
Hence $x$ can be written as $x=\frac{a-1}{pc}\sqrt p.$ Moreover, one can see that $(a-1)^2 = -pbc \equiv 0~\pmod{p}$, so we write $a-1=pa'$ and then $x=\frac{a'}{c}\sqrt p$. We may assume $a'$ and $c$ are relatively prime.
Suppose that $p$ and $c$ are also relatively prime. Then $\gcd(c,-pa')=1$, so there exists a matrix of the form $\begin{pmatrix}
c & -a' \sqrt p \\ *\sqrt p & *
\end{pmatrix}$ in $R(p).$ This matrix sends $x$ to $0$.
Suppose not. Then $c$ is divisible by $p$, so we write $c=pc'.$ In this case, there exists a matrix of the form $\begin{pmatrix}
c'\sqrt p & -a' \\ * & *\sqrt p
\end{pmatrix} \in R(p)$ and this matrix sends $x$ to $0.$
\end{proof}
\
\subsection{Reducing generating sets of $R(p)$ and the genera of $X(R(p))$}
We finish this section by proving that the genus $g_p$ of the curve $X(R(p))$ is zero for each of five primes $p=5,7,11,13$ and $17$. By Corollary \ref{cor_genusthm}, it suffices to show that $v_p<2\pi h_p$ in order to prove $g_p=0.$
Recall that the fundamental domain of a Fuchsian group $\Gamma$ can be given by the Dirichlet domain centered at some point $z_0 \in \mathbb H$, which is defined as
\begin{align*}
D(z_0,\Gamma)=\{z \in \mathbb H : d_{\mathbb H}(z,z_0) \leq d_{\mathbb H}(z,\gamma z_0) \text{ for all } \gamma \in \Gamma \}.
\end{align*}
Here $z_0 \in \mathbb H$ is assumed to be a non-elliptic point for $\Gamma$ and $d_{\mathbb H}$ is the hyperbolic metric on the Poincare upper half plane $\mathbb H.$ One can define the Dirichlet domain of a subset $H$ of $\Gamma$ in the same manner and we denote it by $D(z_0,H)$. Obviously one has $D(z_0,\Gamma) \subseteq D(z_0,H)$ for any subset $H \subseteq \Gamma$.
Let $\rho : \mathbb H \to \mathbb K$ be a conformal map which is defined by the M\"{o}bius transformation sending $z_0$ to $0$, where $\mathbb K$ be the Poincare unit disc. We can define the Dirichlet domain centered at $0$ for $\Gamma^{\rho}:=\rho\Gamma \rho^{-1}$ in $\mathbb K$ similarly. Since the conformal map $\rho$ is an isometry between $\mathbb H$ and $\mathbb K$, the Dirichlet domain of $\Gamma^{\rho}$ in $\mathbb K$ is equal to the image $\rho(D(z_0,\Gamma)).$ In order to distinguish two Dirichlet domains which are contained in $\mathbb H$ and $\mathbb K$, respectively, we denote the latter one by $D(\Gamma).$
For an element $\gamma \in \Gamma^{\rho}$, the set $I(\gamma):=\{w \in \mathbb C : d_{\mathbb K}(w,0)=d_{\mathbb K}(w, \gamma(0))\}$ is called the {\it isometric circle} of $\gamma$. Equivalently, one can define the isometric circle of $\gamma=\begin{pmatrix}
a & b \\ c & d
\end{pmatrix} \in \Gamma^{\rho}$ as $I(\gamma):=\{w \in \mathbb C : |cw+d|=1\}$. Let $\mathrm{Ext}(I(\gamma)):=\{w \in \mathbb C : |cz+d| \geq 1\}$ be the exterior domain of the circle $I(\gamma)$ with boundary. By \cite{MR1561111}, the Dirichlet domain of~$H$ is given by the intersection of exterior domains of the isometric circles of elements in~$H$. Namely,
\begin{align*}
D(H)=\bigcap_{\gamma \in H} \left(\mathbb K \cap \mathrm{Ext}(I(\gamma) \right).
\end{align*}
From now on, for $\Gamma=R(p)$, we set $z_0=2i$ and $\rho(z):=\frac{z-2i}{z+2i}$.
First note that for $T_p,\omega_1$ and $s_p(n)$ given in Lemma \ref{lem_gen},
\begin{align*}
&T_p^{\rho}=\begin{pmatrix}
1+\frac{\sqrt p}{4}i & -\frac{\sqrt p}{4}i \\ \frac{\sqrt p}{4}i & 1-\frac{\sqrt p}{4}i
\end{pmatrix}, \quad
\omega^{\rho}= \begin{pmatrix}
-\frac{5}{4}i & -\frac{3}{4}i \\ \frac{3}{4}i & -\frac{5}{4}
\end{pmatrix}, \\
&s_p(n)^{\rho}=\begin{pmatrix}
\frac{n\hat{n}+p+1}{2\sqrt p}-(\hat n -\frac{n}{4})i & \frac{n\hat{n}-p+1}{2\sqrt p}-(\hat n +\frac{n}{4})i \\ \frac{n\hat{n}-p+1}{2\sqrt p}+(\hat n +\frac{n}{4})i & \frac{n\hat{n}+p+1}{2\sqrt p}+(\hat n -\frac{n}{4})i
\end{pmatrix}.
\end{align*}
If we denote $I(\gamma):=\{w \in \mathbb C : |w-c|=r\}$ by $(c,r)$ then we have
\begin{align}\label{eqn_isometric-circles1}
&I(T_p^{\rho})=\left(1+\frac{4}{\sqrt p}i, \frac{4}{\sqrt p}\right), \quad I(\omega^{\rho})=\left(\frac{5}{3}, \frac{4}{3}\right),
\end{align}
\begin{align}\label{eqn_isometric-circles2}
I(s_p(n)^{\rho})=\left(-\frac{\frac{n\hat{n}+p+1}{2\sqrt p}+(\hat n -\frac{n}{4})i}{\frac{n\hat{n}-p+1}{2\sqrt p}+(\hat n +\frac{n}{4})i}, \frac{1}{|\frac{n\hat{n}-p+1}{2\sqrt p}+(\hat n +\frac{n}{4})i|}\right) \text{ for an integer $n$ prime to $p$},
\end{align}
\
and
\begin{align}\label{eqn_isometric-circles3}
&I(\gamma^{-1})=\overline{I(\gamma)} \text{ for any } \gamma \in \Gamma^{\rho}.
\end{align}
Note that the generating set given in Lemma \ref{lem_gen} is not minimal. Before we calculate the isometric circles for small primes $p$, we introduce several technical lemmas and propositions that reduce the number of generators of the group $R(p)$.
Let $n_i$ for $1\leq i \leq m$ be integers relatively prime to $p$. We say that $n_1, \ldots, n_m$ are dependent (or $(n_1,\ldots,n_m)$ is dependent), if $s_p(n_i) \in \langle T_p, \omega_1, s_p(n_j) : 1 \leq j \leq m, j \neq i \rangle$ for any $1 \leq i \leq m.$
From now on, we often use the notation $s_p(n,\hat{n})$ for $s_p(n)$ given in Lemma~\ref{lem_gen} whenever it is necessary or efficient to indicate what $\hat{n}$ is. We note that for two different choices $n_1$ and $n_2$ with $n_2=n_1+pr$, we have $s_p(n,n_2)=s_p(n,n_1)\omega_1 T_p^{-r} \omega_1 ^{-1}$, so the notation $s_p(n)$ omitting $\hat{n}$ has been so far used.
\begin{lemma}\label{lem_genred1} Let $p$ be a prime.
\begin{enumerate}[\normalfont(a)]
\item $s_p(\pm 1)$ is contained in $\langle T_p,\omega_1 \rangle.$
\item If $n_1 \equiv n_2 \pmod{p}$, then $n_1, n_2$ are dependent.
\item If there exists an integer $\hat{n}\in \mathbb Z$ such that $n\hat{n}+1 =p$, then $n, -\hat{n}$ are dependent.
\item If there exists an integer $\hat{n}\in \mathbb Z$ such that $n\hat{n}+1 =-p$, then $n, \hat{n}$ are dependent.
\item If there exists an integer $\hat{n}\in \mathbb Z$ such that $n\hat{n}+1 =p$, then $n, -n$ are dependent.
\end{enumerate}
\end{lemma}
\begin{proof}
The first and second assertion easily follows from the calculations $s_p(1,-1)=\omega_1^{-1} T_p$, $s_p(-1,1)=\omega T_p$ and $s_p(n_2)=T_p^r s_p(n_1)$, where $n_2=n_1+pr.$ To show the remaining assertions, one can check that
\begin{align*}
\omega_1 s_p(n,\hat n)T_p \omega_1^{-1}&=s_p(-\hat n,-n-p) \quad \text{ if $n\hat n+1=p$,}\\
\omega_1 s_p(n, \hat n) T_p \omega_1&=s_p(\hat n,n-p) \quad \text{ if $n\hat n+1=-p$,}\\
s_p(-n,-\hat n)&=s_p(n,\hat n)^{-1} \quad \text{ if $n\hat n+1=p$.}
\end{align*}
The assertion follows from the definition of dependence.
\end{proof}
Define
\begin{align*}
D_p^+:=\{(n_1,n_2,n_3) \in \mathbb Z^3 : &\gcd(n_i,p)=1, n_i \neq \pm 1 \text{ for any } 1 \leq i \leq 3, \\
&\text{and there exists an integer } \hat{n_1} \in \mathbb Z \text{ such that for } \\ &a_1:=\frac{n_1 \hat{n_1}+1}{p}, \text{ }
n_2a_1+n_1=1 \text{ and } n_3=-\hat{n_1}n_2\},
\end{align*}
\begin{align*}
D_p^-:=\{(n_1,n_2,n_3) \in \mathbb Z^3 : &\gcd(n_i,p)=1, n_i \neq \pm 1 \text{ for any } 1 \leq i \leq 3, \\
&\text{and there exists an integer } \hat{n_1} \in \mathbb Z \text{ such that for } \\ &a_1:=\frac{n_1 \hat{n_1}+1}{p}, \text{ }
n_2a_1-n_1=1 \text{ and } n_3=\hat{n_1}n_2\}.
\end{align*}
\begin{lemma}\label{lem_genDp}
If $(n_1,n_2,n_3) \in D_p^+ \cup D_p^-$, then $n_1,n_2,n_3$ are dependent.
\end{lemma}
\begin{proof}
Assume $(n_1,n_2,n_3) \in D_p^+$. Let $\hat{n_2}$ be an arbitrary integer that satisfies $n_2 \hat{n_2}+1 \pmod{p}$, and set $a_2:=\frac{n_2 \hat{n_2}+1}{p}.$ Under the assumption $n_2a_1+n_1=1$, one can see that
\begin{align*}
T_p \omega_1 s_p(n_1,\hat{n_1})s_p(n_2,\hat{n_2})=s_p(-\hat{n_1}n_2,a_1 a_2p+n_1 \hat{n_2}).
\end{align*}
Similarly, if $(n_1,n_2,n_3) \in D_p^-$, then
\begin{align*}
T_p^{-1} \omega_1^{-1} s_p(n_1,\hat{n_1})s_p(n_2,\hat{n_2})=s_p(\hat{n_1}n_2,-a_1 a_2p-n_1 \hat{n_2}).
\end{align*}
Therefore $n_1,n_2,n_3$ are dependent.
\end{proof}
\begin{lemma}\label{lem_genred2}The following statements hold true:
\begin{enumerate}[\normalfont(a)]
\item If $(n_1,n_2,n_3) \in D_p^+ \cup D_p^-$, then $(-n_1,-n_2,-n_3) \in D_p^+ \cup D_p^-$.
\item If $(n_1,n_2,n_3) \in D_p^+$, then $(\hat{n_1},n_3-p,n_2') \in D_p^-$ for some integer $n_2'$ such that $n_2' \equiv n_2 \pmod{p}.$
\item If $(n_1,n_2,n_3) \in D_p^-$, then $(\hat{n_1},n_3+p,n_2') \in D_p^-$ for some integer $n_2'$ such that $n_2' \equiv n_2 \pmod{p}.$
\end{enumerate}
Consequently, if $(n_1,n_2,n_3) \in D_p^+ \cup D_p^-$, then all of triples $(n_1,n_2,n_3)$, $(-n_1,-n_2,-n_3)$ and $(\hat{n_1},n_3,n_2)$ are dependent.
\end{lemma}
\begin{proof}
(a) It follows from the definition of $D_p^{\pm}.$
(b) Let $(m_1,m_2,m_3)=(\hat{n_1},n_3-p,n_2)$. For an integer $b_1:=\frac{m_1 \hat{m_1}+1}{p}=\frac{\hat{n_1}\hat{\hat{n_1}}+1}{p}=a_1,$ we have
\begin{align*}
m_2b_1+m_1&=(-\hat{n_1} n_2-p)a_1+\hat{n_1} \\
&=(-\hat{n_1}n_2 a_1-n_1\hat{n_1}-1)+\hat{n_1} \\
&=-\hat{n_1}(n_2a_1+n_1)+\hat{n_1}-1 \\
&=-1,
\end{align*}
and $m_3=n_2 \equiv \hat{\hat{n_1}}(-\hat{n_1}n_2) \equiv \hat{m_1}m_2 \pmod{p}.$ Thus $n_2=m_3=\hat{m_1}m_2+pr$ for some $r \in \mathbb Z.$ If we let $n_2':=\hat{m_1}m_2,$ then $n_2 \equiv n_2' \pmod{p}$ and $(m_1,m_2,n_2')=(\hat{n_1},n_3-p,n_2') \in D_p^-.$
(c) The argument of the proof is in parallel with the proof of (b).
\end{proof}
Let $n$ be an integer. We say that $n$ is {\it redundant}, if there exist integers $n_j$ and $\epsilon, \epsilon_j \in \{ \pm 1\}$ for $1 \leq j \leq r$ such that $|n_j|<|n|$ for all $j$ and $\epsilon_1 n_1, \ldots, \epsilon_r n_r, \epsilon n$ are dependent.
\begin{proposition} For a positive integer $d$, the following two statements hold true:
\begin{enumerate}[\normalfont(a)]
\item If $d$ divides $p-1$ and $d > \sqrt{p-1},$ then $d$ is redundant.
\item If $d$ divides $p+1$ and $d> \sqrt{p+1}$, then $d$ is redundant.
\end{enumerate}
\end{proposition}
\begin{proof}
We only prove (a), since the proof of (b) is obtained by the same argument. Let $d'$ be an integer satisfying $dd'=p-1$. By Lemma \ref{lem_genred1}, $d$ and $d'$ are dependent and $|d'|<\sqrt{p-1}<|d|,$ so $d$ is redundant.
\end{proof}
\begin{proposition}\label{prop_p-n/d is redundant}
Let $d$ be a positive integer relatively prime to $p$ such that $d+1 <\sqrt{p},$ and let $a$ be an integer satisfying $ap \equiv 1 \pmod{d}.$
\begin{enumerate}[\normalfont(a)]
\item If there exists $n \in \mathbb Z$ such that $an+d=1,$ then $n \equiv p \pmod{d}$ and $\frac{p-n}{d}$ is redundant.
\item If there exists $n \in \mathbb Z$ such that $an+d=-1,$ then $n \equiv -p \pmod{d}$ and $\frac{p+n}{d}$ is redundant.
\end{enumerate}
\end{proposition}
\begin{proof}
We only prove(a). The proof of (b) follows similarly.
By the assumption of the proposition, we have
\begin{align*}
1=an+d \equiv p^{-1}n \pmod{d},
\end{align*}
so $n \equiv p \pmod{d}.$ Since $n \leq 0,$ one has $n < p$ and $\max \{d,|n|\}=d$ or $d+1.$ Also we have $(d+1)^2<p$, hence
\begin{align*}
(d+1)^2+n <p
\end{align*}
and so $d+1 < \frac{p-n}{d}.$ In other words, $\frac{p-n}{d}$ is the largest positive integer among the three integers $\frac{p-n}{d},d$ and $|n|.$
Let $n_1=d$, $\hat{n}_1=\frac{ap-1}{d}$ and $n_2=n.$ By Lemma \ref{lem_genDp}, $n_1,n_2,-\hat{n}_1 n_2$ are dependent and $-\hat{n}_1 n_2 \equiv -\frac{p-n}{d} \pmod{p}.$ Therefore $|-\frac{p-n}{d}|=\frac{p-n}{d}$ is redundant.
\end{proof}
We immediately get the following two corollaries by substituting $a=\pm1$ and $a=\pm 2$ in Proposition~\ref{prop_p-n/d is redundant}, respectively.
\begin{corollary}
Let $d$ be a positive integer such that $d+1 < \sqrt p.$
\begin{enumerate}[\normalfont(a)]
\item If $d\mid p-1$, then $\frac{p-1}{d} \pm 1$ are redundant.
\item If $d\mid p+1$, then $\frac{p+1}{d} \pm 1$ are redundant.
\end{enumerate}
\end{corollary}
\begin{corollary}
Let $d$ be a positive integer.
\begin{enumerate}[\normalfont(a)]
\item If $d\mid p-2$ and $2 \leq d < \sqrt{p-2},$ then $\frac{p-2}{d}$ is redundant.
\item If $d\mid p+2$ and $2 \leq d < \sqrt{p+2},$ then $\frac{p+2}{d}$ is redundant.
\end{enumerate}
\end{corollary}
We introduce a process reducing the number of generators of $R(p)$ efficiently for small primes $p$ using these lemmas and propositions.
Referring to Lemma~\ref{lem_gen}, Lemma~\ref{lem_genred1} reduces a generating set of $R(p)$ so that $R(p)=\langle T_p,\omega_1,s_p(n) : |n| \leq \frac{p-1}{2}, n\neq 0 \rangle$. Set $S=\{1,2,\ldots,\frac{p-1}{2}\}$.
\
\begin{enumerate}[\bf Step 1.]
\item Remove all divisors $d$ of $p-1$ such that $d > \sqrt{p-1}$ from $S$.
\item Remove all divisors $d$ of $p+1$ such that $d > \sqrt{p+1}$ from $S$.
\item Remove $\frac{p-1}{d} \pm 1$ for every divisor $d$ of $p-1$ with $d < \sqrt{p-1}$ from $S$.
\item Remove $\frac{p+1}{d} \pm 1$ for every divisor $d$ of $p+1$ with $d < \sqrt{p+1}$ from $S$.
\item For an integer $m=\frac{p-1}{d} \pm 1$ (or $\frac{p+1}{d}\pm 1$) which is removed in \textbf{Step 3} and \textbf{Step 4}, find an integer $m' \in \{1,2,\ldots,\frac{p-1}{2}\}$ such that $mm' \equiv \pm 1 \pmod{p}.$ If $m' > d+1,$ then remove $m'$ from~$S$.
\item Remove all divisors $d$ of $p-2$ such that $d \neq p-2$ and $d > \sqrt{p-2}$ from $S$.
\item Remove all divisors $d$ of $p+2$ such that $d \neq p+2$ and $d > \sqrt{p+2}$ from $S$.
\end{enumerate}
\
Note that \textbf{Step 5} is supported by Lemma \ref{lem_genred1}. After terminating the process, we find a set $S$ for which
\begin{align*}
R(p) = \langle T_p, \omega_1, s_p(n), s_p(-n) : n \in S\rangle.
\end{align*}
Since $R(p)$ is {\it transpose-invariant}, i.e., for any element $A \in R(p)$, its transpose matrix~$A^t$ belongs to $R(p)$ and since $T_p^t, \omega_1^t \in \langle T_p,\omega_1\rangle$, we have
\begin{align*}
R(p) = \langle T_p, \omega_1, s_p(n)^t, s_p(-n)^t : n \in S\rangle.
\end{align*}
Applying the above process, we reduce the set of generators for $R(p)$ for each $p\in \{5,7,11,13,17\}$ as follows:
\begin{align}\label{eqn_reduced-generating-set-of-R(p)}
\begin{cases}
R(5)&=\langle T_5, \omega_1, s_5(2,2)^t \rangle, \\
R(7)&=\langle T_7, \omega_1, s_7(2,3)^t \rangle, \\
R(11)&=\langle T_{11}, \omega_1, s_{11}(2,5)^t, s_{11}(3,-4)^t \rangle, \\
R(13)&=\langle T_{11}, \omega_1, s_{13}(2,6)^t, s_{13}(3,4)^t \rangle, \\
R(17)&=\langle T_{17}, \omega_1, s_{17}(2,8)^t, s_{17}(3,-6)^t, s_{17}(-3,6)^t \rangle.
\end{cases}
\end{align}
We remark that $s_p(-2)^t$ is not exhibited in \eqref{eqn_reduced-generating-set-of-R(p)}, because $2$ and $-2$ are always dependent for any $p \geq 5$ by Lemma \ref{lem_genred1}(e).
\begin{proposition}
For each $p\in \{2,3,5,7,11,13, 17\}$, $X(R(p))$ has genus zero.
\end{proposition}
\begin{proof}
We already know the cases when $p=2,3$. Suppose $p \geq 5$ and let $H_p$ be the generating set of $R(p)$ given in \eqref{eqn_reduced-generating-set-of-R(p)}. Explicit formulae for isometric circles \eqref{eqn_isometric-circles1}, \eqref{eqn_isometric-circles2} and \eqref{eqn_isometric-circles3} provide the Dirichlet domains $D(H_p)$. It is immediately verified that these Dirichlet domains are hyperbolic $m_p$-gons, where $m_p=5$ for $p=5,7$, $m_p=7$ for $p=11,13$, and $m_p=9$ for $p=17.$
On the other hand, by Corollary \ref{cor_genusthm}, one can find that $h_p$ is given by
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|}
\hline
$p$ & $5$ & $7$ & $11$ & $13$ &$17$ \\\hline
$h_p$ & $5/2$ & $8/3$ & $3$ & $19/6$& $7/2$\\
\hline
\end{tabular}\end{center}
Note that the maximum area of hyperbolic $m$-gons in $\mathbb K$ with respect to the hyperbolic measure is $(m-2)\pi.$ Moreover, if such hyperbolic $m$-gon is the Dirichlet domain $D(H_p)$, then it's area doesn't admit $(m-2)\pi$ since there exists a vertex with non-zero angle. Therefore for given $p$,
\begin{align*}
\frac{1}{2\pi}v_p=D(R(p)^{\rho}) \leq D(H_p^{\rho}) < \frac{1}{2}m_p-1 \leq h_p.
\end{align*}
Again by Corollary \ref{cor_genusthm}, $g_p=0$ and $v_p=2\pi(h_p-2)$ for $p \in \{5,7,11,13,17\}.$
\end{proof}
We remark that the argument of the proof is not applicable to $p = 19,$ since $\frac{1}{2}m_{19}-1=\frac{7}{2} > 3=h_{19}.$
\
Thompson \cite{MR0604632} proved that there are only finitely many Fuchsian groups commensurable with $\operatorname{SL}_2(\mathbb Z)$ of fixed genus, up to conjugation.
For the Atkin-Lehner groups $\Gamma_0^+(p)$, it turns out that $\Gamma_0^+(p)$ has genus zero for only finitely many $p$, more precisely, for primes dividing the order of the {\it Monster group} (see \cite{MR0417184}).
It is natural to ask a similar question for the groups $R(p)$.
\begin{question}
For which prime $p$, does the modular curve $X(R(p))$ have genus $g_p=0$?
\end{question}
\section{Non-holomorphic Eisenstein series}\label{sec_Eisen}
\subsection{For the full modular group $\operatorname{SL}_2(\mathbb Z)$}
We start this section with the proposition which states the basic properties of non-holomorphic Eisenstein series for $\operatorname{SL}_2(\mathbb Z).$
\begin{proposition}\label{prop_g}
For the non-holomorphic Eisenstein series $g(z,\overline{z},\alpha,\beta)$ as a function in $q=\alpha+\beta\in \mathbb C$ and $k=\alpha-\beta \in 2\mathbb Z$, the following holds true:
\begin{enumerate}[\normalfont(a)]
\item The function $g(z,\overline{z},\alpha,\beta)$ has analytic continuation in the variable $q$ to the whole plane.
\item The function $g(z,\overline{z},\alpha,\beta)$ is a harmonic function with respect to the elliptic operator,
$$
\Omega_{\alpha \beta}:=-y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+i(\alpha-\beta)y\frac{\partial}{\partial x}-(\alpha+\beta)y\frac{\partial}{\partial y},
$$
i.e., $\Omega_{\alpha \beta}(g)=0.$
\item For each $\gamma=\begin{pmatrix}a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb Z)$,
$$
g(z,\overline{z},\alpha,\beta)|_{\alpha,\beta}\gamma := (cz+d)^{-\alpha}(c\overline{z}+d)^{-\beta}g(\gamma z,\gamma \overline{z},\alpha,\beta)=g(z,\overline{z},\alpha,\beta).
$$
\item The Fourier expansion of $g(z,\overline{z},\alpha,\beta)$ is given by
\begin{align*}
g(z,\overline{z},\alpha,\beta)=&\phi_{k/2}(y,q) \\
&+2(-1)^{k/2}(\sqrt 2 \pi)^q \sum_{n \in \mathbb Z \setminus \{0\}} \frac{\sigma_{q-1}(n)}{\Gamma\left(\frac{q}{2}+\mathrm{sgn}(n)\frac{k}{2}\right)}W(2\pi n y ;\alpha,\beta)e^{2\pi i nx},
\end{align*}
where
\begin{align*}
&\phi_{k/2}(y,q):=2\zeta(q)+(-1)^{k/2}(4\pi) 2^{1-q}\frac{\Gamma(q-1)}{\Gamma\left(\frac{q+k}{2}\right) \Gamma\left(\frac{q-k}{2}\right)}\zeta(q-1)y^{1-q}, \\
&\sigma_{q-1}(n):=\sum_{d|n}d^{q-1}, \text{ and }\\
&W(y;\alpha,\beta):=|y|^{-q/2}W_{\mathrm{sgn}(y)\frac{k}{2},\frac{1}{2}(q-1)}(2|y|)
\text{ is the modified $W$-Whittaker function.}
\end{align*}
\item The completed non-holomorphic Eisenstein series $$g^*(z,\overline{z},\alpha,\beta):=\frac{q}{2}\left(1-\frac{q}{2}\right)\pi^{-1/2}\Gamma\left(\frac{q}{2}+\frac{|k|}{2}\right)g(z,\overline{z},\alpha,\beta)$$ satisfies the functional equations,
\begin{align*}
&g^*(z,\overline{z},\overline{\beta},\alpha)=g^*(-\overline{z},-z,\alpha,\beta), \text{ and }\\
&g^*(z,\overline{z},1-\alpha,1-\beta)=y^{q-1}g^*(z,\overline{z},\beta,\alpha).
\end{align*}
\end{enumerate}
\end{proposition}
Proposition \ref{prop_g}(b) guarantees that $g(z,\overline{z},\alpha,\beta)$ is smooth, which is deduced from the elliptic regularity, and Proposition \ref{prop_g}(c) says that $g(z,\overline{z},\alpha,\beta)$ is automorphic of weight $(\alpha,\beta)$ for $\operatorname{SL}_2(\mathbb Z)$ as we would expect. In fact, the analytic continuation of $g$ follows from the Fourier expansion of which proof is given in \cite{MR0734485}.
Obviously one can obtain the similar properties for another definition of the Eisenstein series $G_k(z,s)$ from Proposition \ref{prop_g}. In particular, let
\begin{align*}
&\widehat{G}_k(z,s):=\pi^{-\left(s+\frac{k}{2}\right)}\Gamma \left( s+\frac{k}{2}+\frac{|k|}{2}\right) G_k(z,s), \text{ and }\\
&\widetilde{G}_k(z,s):=\left( s+\frac{k}{2}\right) \left(s+\frac{k}{2}-1\right) \widehat G_k(z,s)
\end{align*}
be the completed and the doubly-completed Eisenstein series of $G_k(z,s)$, respectively. The Fourier expansion and the functional equations of these completed series follow from those of $g(z,\overline{z},\alpha,\beta)$.
\begin{proposition}\label{prop_G}
The Fourier expansion of $\widehat{G}_k(z,s)$ is given by
\begin{align*}
\widehat{G}&_k(z,s)=\frac{\Gamma\left(s+\frac{k}{2}+\frac{|k|}{2}\right)}{\Gamma\left(s+\frac{k}{2}\right)}\hat{\zeta}(2s+k)y^s+(-1)^{k/2}\frac{\Gamma\left(s+\frac{k}{2}\right)\Gamma\left(s+\frac{k}{2}+\frac{|k|}{2}\right)}{\Gamma(s+k)\Gamma(s)}\hat{\zeta}(2-2s-k)y^{1-s-k} \\
&+(-1)^{k/2}\Gamma\left(s+\frac{k}{2}+\frac{|k|}{2}\right)y^{-k/2}\sum_{n \in \mathbb Z \setminus \{0\}}\frac{|n|^{-s-\frac{k}{2}}\sigma_{2s+k-1}(n)}{\Gamma\left(s+\frac{k}{2}+\mathrm{sgn}(n)\frac{k}{2}\right)}W_{\mathrm{sgn}(n)k/2,s+\frac{k-1}{2}}(4\pi |n|y)e^{2\pi i nx},
\end{align*}
where $\hat{\zeta}(s):=\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)$ is the completed Riemann zeta function.
\end{proposition}
\begin{proposition}\label{prop_Gfunctional}
The completed and the doubly-completed Eisenstein series satisfy the functional equations, respectively,
\begin{align*}
\widehat{G}_k(z,s)=\widehat{G}_k(z,1-k-s) \quad
\text{ and } \quad
\widetilde{G}_k(z,s)=\widetilde{G}_k(z,1-k-s).
\end{align*}
\end{proposition}
The goal of this section is to establish the analogous properties for the nonholomorphic Eisenstein series for $R(p)$. In particular, it gives the Fourier expansion of the nonholomorphic Eisenstein series for the Fricke group $\Gamma_0^+(2)$ and $\Gamma_0^+(3).$
\subsection{Automorphy condition}
First of all, we prove that the Eisenstein series $g_N(z,\overline{z},\alpha,\beta)$ and $G_{N,k}(z,s)$ satisfy the automorphy condition for $R(N).$
\begin{lemma}\label{lem_corres}
Let $N$ be a positive integer, and $a_1,b_1,c_1,d_1,a_2,b_2,c_2,d_2 \in \mathbb Z$ be any integers satisfying $a_1d_1-Nb_1c_1=1$ and $Na_2d_2-b_2c_2=1$. The following maps are bijective:
\begin{align*}
f_{LL}:(\mathbb Z \times \mathbb Z)_{N,L} &\longrightarrow (\mathbb Z \times \mathbb Z)_{N,L} \\
(m,n) &\mapsto (ma_1+nc_1, Nmb_1+nd_1),
\end{align*}
\begin{align*}
f_{RR}:(\mathbb Z \times \mathbb Z)_{N,R} &\longrightarrow (\mathbb Z \times \mathbb Z)_{N,R} \\
(m,n) &\mapsto (ma_1+Nnc_1, mb_1+nd_1),
\end{align*}
\begin{align*}
f_{LR}:(\mathbb Z \times \mathbb Z)_{N,L} &\longrightarrow (\mathbb Z \times \mathbb Z)_{N,R} \\
(m,n) &\mapsto (Nma_2+nc_2, mb_2+nd_2),
\end{align*}
\begin{align*}
f_{RL}:(\mathbb Z \times \mathbb Z)_{N,R} &\longrightarrow (\mathbb Z \times \mathbb Z)_{N,L} \\
(m,n) &\mapsto (ma_2+nc_2, mb_2+Nnd_2).
\end{align*}
\end{lemma}
\begin{proof}
We only show the lemma for the first map $f_{LL}.$ For the rest of functions the proofs can be done similarly. First we show that the map is well-defined, which means $f_{LL}(m,n) \in (\mathbb Z \times \mathbb Z)_{N,L}.$ Since $(m,n) \in (\mathbb Z \times \mathbb Z)_{N,L},$ there is a positive integer $t$ such that $(m,n) \in t\mathbb Z \times t\mathbb Z$ and $\gcd(Nm,n)=t$. It is enough to show that $\gcd(N(ma_1+nc_1),Nmb_1+nc_1)=1$, so we may assume $t=1.$ Substituting $M=Nm,$ it is reduced to show $\gcd(Ma_1+Nnc_1,Mb_1+nc_1)=1$ assuming $\gcd(M,n)=1.$ It follows from the condition $a_1d_1-Nb_1c_1=1.$
The bijectivity of $f_{LL}$ follows from the existence of the inverse map $f_{LL}^{-1}:(\mathbb Z \times \mathbb Z)_{N,L} \rightarrow (\mathbb Z \times \mathbb Z)_{N,L}$ defined by $f_{LL}^{-1}(m,n) = (d_1m-c_1n,-N n b_1+n d_1)$. In fact, this map $f_{LL}^{-1}$ is also well-defined by the same reason.
\end{proof}
\begin{proposition}\label{prop_gmodular}
Let $N$ be a positive integer and $\alpha-\beta \in 2\mathbb Z.$ For any $\gamma \in R(N),$
\begin{enumerate}[\normalfont(a)]
\item if $\gamma \in R(N)^+,$ then
\begin{equation*}
\begin{cases}
g_{N,L}(z,\overline{z},\alpha,\beta)|_{\alpha,\beta} \gamma=g_{N,L}(z,\overline{z},\alpha,\beta), \\
g_{N,R}(z,\overline{z},\alpha,\beta)|_{\alpha,\beta} \gamma=g_{N,R}(z,\overline{z},\alpha,\beta).
\end{cases}
\end{equation*}
\item if $\gamma \in R(N)^-,$ then
\begin{equation*}
\begin{cases}
g_{N,L}(z,\overline{z},\alpha,\beta)|_{\alpha,\beta} \gamma=g_{N,R}(z,\overline{z},\alpha,\beta), \\
g_{N,R}(z,\overline{z},\alpha,\beta)|_{\alpha,\beta} \gamma=g_{N,L}(z,\overline{z},\alpha,\beta).
\end{cases}
\end{equation*}
\end{enumerate}
Consequently, $g_N(z,\overline{z},\alpha,\beta)|_{\alpha,\beta} \gamma =g_N(z,\overline{z},\alpha,\beta)$ for any $\gamma \in \Gamma.$
\end{proposition}
\begin{proof}
Let $\gamma=\begin{pmatrix}
a & b\sqrt N \\ c\sqrt N & d
\end{pmatrix} \in R(N)^+.$ Then by Lemma \ref{lem_corres},
\begin{align*}
g_{N,L}&(z,\overline{z},\alpha,\beta)|_{\alpha,\beta} \gamma \\
&=(c\sqrt N z+d)^{-\alpha}(c\sqrt N \overline{z}+d)^{-\beta}g_{N,L}(\gamma z,\gamma \overline{z},\alpha,\beta) \\
&=(c\sqrt N z+d)^{-\alpha}(c\sqrt N \overline{z}+d)^{-\beta}\sum_{(m,n)\in (\mathbb Z \times \mathbb Z)_{N,L}}\left(\sqrt N mz+n\right)^{-\alpha}\left(\sqrt N m\overline{z}+n\right)^{-\beta} \\
&=\sum_{(m,n)\in (\mathbb Z \times \mathbb Z)_{N,L}}\left(\sqrt N (ma+nc)z+Nmb+nd\right)^{-\alpha}\left(\sqrt N (ma+nc)\overline{z}+Nmb+nd\right)^{-\beta} \\
&=\sum_{f_{LL}^{-1}(m',n') \in (\mathbb Z \times \mathbb Z)_{N,L}}(m'z+n')^{-\alpha}(m'\overline{z}+n')^{-\beta} \\
&=g_{N,L}(z,\overline{z},\alpha,\beta).
\end{align*}
The others follow from Lemma~\ref{lem_corres} similarly.
\end{proof}
\
\subsection{Dual characters and the Dirichlet series}
We consider a special kind of a collection of functions $\chi : \mathbb Z \to \mathbb C$, whose images are contained in $U:=\{z \in \mathbb C : |z|=1\} \cup \{0\}.$
For a positive integer $n \in \mathbb Z_{>0}$, let $Q_n$ be a positive integer and $\chi_n : \mathbb Z \to U$ be a function such that
\begin{align*}
\chi_n(a+Q_n)&=\chi_n(a) \quad \text{ for all $a \in \mathbb Z$}, \\
\chi_n(a)&=0 \quad \text{ if $\gcd(n,Q_n) \neq 1$}.
\end{align*}
Let $\chi_{\bullet}$ be a sequence of pairs $(\chi_n,Q_n)$ for $n \in \mathbb Z_{>0}$. Here we call $\chi_{\bullet}$ a dual character if the following conditions hold:
\begin{enumerate}[\normalfont(i)]
\item If $m$ and $n$ are relatively prime, then $Q_m$ and $Q_n$ are also relatively prime.
\item If $m$ and $n$ are relatively prime, then $Q_{mn}=Q_m Q_n.$
\item For any integers $a,b$ such that $\gcd(a,Q_m)=1$ and $\gcd(b,Q_n)=1$, $$\chi_m(a)\chi_n(b)=\chi_{mn}(aQ_n+bQ_m).$$
\item $\chi_1=\mathds{1}_{1}$ and $Q_1=1.$
\item There exists $\sigma \in \mathbb R$ such that $\sum_{n=1}^{\infty}\phi(Q_n)n^{-\sigma} < \infty,$ where $\phi$ is the Euler totient function.
\end{enumerate}
One of the trivial example of a dual character is $\mathds{1}_{\bullet}=(\mathds{1}_n,n)_{n \in \mathbb Z_{>0}}$, where
\begin{align*}
\mathds{1}_{n}(m):=\begin{cases} 1 \quad &\text{ if $m$ is relatively prime to $n$},\\
0 \quad &\text{ otherwise},
\end{cases}
\end{align*}
\begin{lemma}\label{lem_charprime}
Any dual character $\chi_{\bullet}$ is uniquely determined by its subsequence $\{(\chi_{\ell^e},Q_{\ell^e}): \ell \text{ is a prime, } e \in~\mathbb Z_{\geq 0}\}$.
\end{lemma}
\begin{proof}
Suppose that we have $Q_n$'s for $n \in \mathbb Z_{>0}$ that satisfy the conditions (i) and (ii) of the definition of a dual character. By the Chinese remainder theorem, for two integers $m,n$ that are relatively prime, a group homomorphism
\begin{align*}
(\mathbb Z/Q_m)^{\times} \times (\mathbb Z/Q_n)^{\times} \to (\mathbb Z/Q_{mn})^{\times}
\end{align*}
which sends $(a,b)$ to $aQ_n+bQ_m$ is an isomorphism. Moreover, if $m,n$ and $r$ are pairwise relatively prime, then the following diagram commutes:
\begin{equation}\label{diag1}
\begin{tikzcd}
{(\mathbb Z/Q_m)^{\times} \times (\mathbb Z/Q_n)^{\times} \times (\mathbb Z/Q_r)^{\times}} & {(\mathbb Z/Q_{mn})^{\times} \times (\mathbb Z/Q_r)^{\times}} & {} \\
{(\mathbb Z/Q_m)^{\times} \times (\mathbb Z/Q_{nr})^{\times}} & {(\mathbb Z/Q_{mnr})^{\times}}
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=2-2]
\arrow[from=1-1, to=2-1]
\arrow[from=2-1, to=2-2]
\end{tikzcd}
\end{equation}
The map $(\mathbb Z/Q_m)^{\times} \times (\mathbb Z/Q_n)^{\times} \times (\mathbb Z/Q_r)^{\times} \to (\mathbb Z/Q_{mnr})^{\times}$ sends $(a,b,c)$ to $aQ_{nr}+bQ_{mr}+~cQ_{mn}.$ Inductively we have an isomorphism $(\mathbb Z/Q_{m_1})^{\times} \times (\mathbb Z/Q_{m_2})^{\times} \times \cdots \times (\mathbb Z/Q_{m_j})^{\times} \to (\mathbb Z/Q_{m_1m_2 \cdots m_j})^{\times}$.
For given $\chi_{\ell^e}$, we construct $\chi_n : \mathbb Z \to \mathbb C$ for arbitrary positive integers $n\in \mathbb Z_{>0}$ as follows:
\begin{align}\label{eqn_chardecomp1}
\chi_1(a)&:=\mathds{1}_1
\end{align}
and
\begin{align}\label{eqn_chardecomp2}
\chi_n(a)&:=\prod_{i=1}^r \chi_{\ell_i^{e_i}}(\pi_i(a)) \quad \text{ for } n > 1,
\end{align}
where $n=\ell_1^{e_1} \cdots \ell_r^{e_r}$ is a prime factorization of $n$ and $\pi_i: (\mathbb Z/Q_n)^{\times} \to (\mathbb Z/Q_{\ell_i^{r_i}})^{\times}$ is the projection via~\eqref{diag1}. Also we define
\begin{align*}
Q_1:=1 \quad \text{ and } \quad
Q_n:=Q_{\ell_1^{e_1}}\cdots Q_{\ell_r^{e_r}} \text{ for $n>1$}.
\end{align*}
Then $\chi_{\bullet}:=((\chi_n,Q_n))_{n \in \mathbb Z_{>0}}$ is a dual character. Conversely, if one has a dual character $\chi_{\bullet}:=((\chi_n,Q_n))_{n \in \mathbb Z_{>0}}$ with given $(\chi_{\ell^e},Q_{\ell^e})$ for primes $\ell$ and non-negative integers $e \in \mathbb Z_{\geq 0}$, then it must satisfy the equations (\ref{eqn_chardecomp1}) and (\ref{eqn_chardecomp2}), which follow from the conditions (iii) and~(iv) of the definition of a dual character. Hence $\chi_{\bullet}$ is uniquely determined.
\end{proof}
For each function $\chi_n,$ the Gauss sum of $\chi_n$ is defined as
\begin{align*}
G(b,\chi_n):=\sum_{a \in (\mathbb Z/Q_n \mathbb Z)^{\times}}\chi_n(a)(\zeta_{Q_n}^a)^b, \quad \text{ for } b \in \mathbb Z.
\end{align*}
Then we consider the associated Dirichlet series to a dual character $\chi_{\bullet}$,
\begin{align*}
\mathcal{L}(b,\chi_{\bullet},s)&:=\sum_{n=1}^{\infty}\frac{G(b,\chi_n)}{n^s}, \\
\mathcal{L}_N(b,\chi_{\bullet},s)&:=\prod_{p|N}\sum_{m=0}^{\infty}\frac{G(b,\chi_{})}{p^{ms}},\\
\mathcal{L}_p(b,\chi_{\bullet},s;v)&:=\sum_{m=0}^v \frac{G(b,\chi_{p^m})}{p^{ms}}.
\end{align*}
Note that the condition (e) of the definition of a dual character implies that there exists $\sigma \in \mathbb R$ such that the associated Dirichlet series $\mathcal{L}(b,\chi_{\bullet},s)=\sum_{n=1}^{\infty}G(b,\chi_n)n^{-s}$ converges absolutely if $\Re(s) > \sigma$. The following proposition shows that $\mathcal{L}(b,\chi_{\bullet},s)$ can be expressed as the so-called Euler product.
\begin{proposition}
For any integer $b$ and a dual character $\chi_{\bullet}$, there exists $\sigma \in \mathbb R$ such that $$\mathcal{L}(b,\chi_{\bullet},s)=\prod_{p}\mathcal{L}_N(b,\chi_{\bullet},s).$$
for all $s \in \mathbb C$ with $\Re(s)>\sigma$.
\end{proposition}
\begin{proof}
Let $m$ and $n$ be relatively prime. Then
\begin{align*}
G(b,\chi_m,s)G(b,\chi_n,s)
&=\sum_{a_1 \in (\mathbb Z/Q_m\mathbb Z)^{\times}}\chi_m(a_1)(\zeta_m^{a_1})^b\sum_{a_2 \in (\mathbb Z/Q_n\mathbb Z)^{\times}}\chi_m(a_2)(\zeta_m^{a_2})^b \\
&=\sum_{a_1, a_2}\chi_m(a_1)\chi_n(a_2)(\zeta_n^{a_2})^b \\
&=\sum_{a_1, a_2}\chi_{mn}(a_1Q_n+a_2Q_m)(\zeta_{mn}^{a_1Q_n+a_2Q_m})^b \\
&=\sum_{a \in (\mathbb Z/Q_mQ_n\mathbb Z)^{\times}}\chi_{mn}(a)(\zeta_m^{a})^b.
\end{align*}
The last two equations follows from the definition of dual characters and the isomorphism given in Lemma \ref{lem_charprime}. Hence the coefficients of $\mathcal{L}(b,\chi_{\bullet},s)$ are provided by a multiplicative function in $n$, so it equals to the Euler product by \cite[Theorem 1.9]{MR2378655}.
\end{proof}
One of the natural question is when the function $\mathcal{L}(b,\chi_{\bullet},s)$ has analytic continuation. We suggest certain strong condition for $\chi_{\bullet}$ that admits the analytic continuation of $\mathcal{L}(b,\chi_{\bullet},s)$. However, it does not characterize equivalent conditions for $\mathcal{L}(b,\chi_{\bullet},s)$ to have analytic continuation precisely.
\begin{definition}
Let $\chi_{\bullet}$ be a dual character. We call that $\chi_{\bullet}$ is finitely generated, if there exists finitely many primes $p_1, p_2, \ldots, p_r$ for which
\begin{align*}
(\chi_{p_i^{e_i}},Q_{p_i^{e_i}})=(\chi_{p_i},Q_{p_i}), \quad \text{ where } e_i \in \mathbb Z_{> 0} \text{ for } 1\leq i \leq r,
\end{align*}
and
\begin{align*}
(\chi_q,Q_q)=(\mathds{1}_1,1) \quad \text{ for any prime $q \neq p_i$},\text{ } 1\leq i \leq r.
\end{align*}
The functions $\chi_{p_1^{e_1}}, \ldots, \chi_{p_r^{e_r}}$ are called the generators of $\chi_{\bullet}$.
\end{definition}
\begin{lemma}\label{lem_fgdualcha}
Let $\chi_{\bullet}$ be the finitely generated dual character with generators $\chi_{p_1}, \chi_{p_2}, \ldots, \chi_{p_r}.$ Then $$\chi_{mp_1p_2\cdots p_{r'}}=\chi_{p_1p_2\cdots p_{r'}}$$ for any positive integers $r' \leq r$ and $m \in \mathbb Z_{>0}.$ In particular, if $n$ is an integer such that $\gcd(n,p_i)\neq 1$ for $1\leq i \leq r'$ and $\gcd(n,p_j)=1$ for $r' < j \leq r$, then
$$
\chi_n=\chi_{p_1p_2\cdots p_{r'}}.
$$
\end{lemma}
\begin{proof}
First suppose a positive integer $m$ is of the form $m=p_1^{e_1}p_2^{e_2}\cdots p_{r'}^{e_{r'}}.$ Then by Lemma~\ref{lem_charprime},
\begin{align*}
\chi_{mp_1p_2\cdots p_{r'}}(a)&=\chi_{p_1^{e_1+1}p_2^{e_2+1}\cdots p_{r'}^{e_{r'+1}}}(a)\\
&=\chi_{p_1^{e_1+1}}(\pi_1(a))\chi_{p_2^{e_2+1}}(\pi_2(a))\cdots \chi_{p_{r'}^{e_{r'}+1}}(\pi_{r'}(a)) \\
&=\chi_{p_1}(\pi_1(a))\chi_{p_2}(\pi_2(a))\cdots \chi_{p_{r'}}(\pi_{r'}(a)) \\
&=\chi_{p_1p_2\cdots p_{r'}}(a)
\end{align*}
for any $a \in \mathbb Z.$ Now let $m=m_1m_2$ where $m_1$ is the largest divisor of $m$ relatively prime to $p_i$ for $1 \leq i \leq r'.$ By the same reason,
\begin{align*}
\chi_{mp_1p_2\cdots p_{r'}}(a)&=
\chi_{m_1}(a_1)\chi_{m_2p_1p_2\cdots p_{r'}}(a_2)\\
&=\chi_{p_1p_2\cdots p_{r'}}(a_2)
\end{align*}
for integers $a_1,a_2$ such that $a \equiv a_1 Q_{mp_1p_2\cdots p_{r'}}+a_2Q_{m_1} \pmod{ Q_{mp_1p_2\cdots p_{r'}}}.$ Note that
\begin{align*}
a_1 Q_{mp_1p_2\cdots p_{r'}}+a_2Q_{m_1} = a_1 Q_{p_1}Q_{p_2} \cdots Q_{p_{r'}} +a_2 \equiv a_2 \pmod{Q_{mp_1p_2\cdots p_{r'}}},
\end{align*}
so we get the desired equation.
\end{proof}
\begin{theorem}
Let $\chi_{\bullet}$ be the dual character generated by $\chi_{p_1}, \chi_{p_2}, \ldots, \chi_{p_r}.$ Then its associated Dirichlet series $\mathcal{L}(b,\chi_{\bullet},s)$ is given by
\begin{align*}
\mathcal{L}(b,\chi_{\bullet},s)=\left(1-\frac{1}{p_1^s}\right)\left(1-\frac{1}{p_2^s}\right)\cdots \left(1-\frac{1}{p_r^s}\right)\zeta(s)\left(1+\sum_{\substack{1 \leq j \leq r \\ 1 \leq i_1 < i_2 < \cdots < i_j\leq r}}\frac{G(b,\chi_{p_{i_1}p_{i_2}\cdots p_{i_j}})}{(p_{i_1}^s-1)(p_{i_2}^s-1)\cdots(p_{i_j}^s-1)}\right).
\end{align*}
In particular, $\mathcal{L}(b,\chi_{\bullet},s)$ is the product of meromorphic functions on $\mathbb C$, so it can be continued analytically to the complex plane $\mathbb C.$
\end{theorem}
\begin{proof}
For the sake of convenience, we abuse the notation as follows: $\frac{G(b,\chi_{p_{i_1}p_{i_2}\cdots p_{i_j}})}{(p_{i_1}^s-1)(p_{i_2}^s-1)\cdots(p_{i_j}^s-1)}:=1$ if $j=0.$
Consider a function
\begin{align*}
f(s):=(p_1p_2 \ldots p_r)^s \pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\mathcal{L}(b,\chi_{\bullet},s).
\end{align*}
Then,
\begin{align*}
f(2s)&=(p_1p_2 \ldots p_r)^{2s} \pi^{-s}\sum_{n=1}^{\infty}G(b,\chi_n)n^{-2s}\int_0^{\infty}e^{-t}t^{s-1}dt \\
&=(p_1p_2 \ldots p_r)^{2s} \sum_{n=1}^{\infty}G(b,\chi_n)\int_0^{\infty}e^{-\pi n^2t}t^{s-1}dt \\
&=(p_1p_2 \ldots p_r)^{2s} \sum_{\substack{0 \leq j \leq r, \\ 1\leq i_1 < i_2 < \cdots < i_j \leq r}}G(b,\chi_{p_{i_1}p_{i_2}\cdots p_{i_j}})\sum_{\substack{\gcd(n,p_t) \neq 1 \text{ for $t=i_1,i_2,\ldots,i_j$,} \\ \gcd(n,p_t)=1 \text{ for $t \neq i_1, i_2, \ldots, i_j$}}}\int_0^{\infty}e^{-\pi n^2 t}t^{s-1}dt.
\end{align*}
The last equation follows from Lemma \ref{lem_fgdualcha}.
Since the series $\sum\limits_{p_1p_2 \cdots p_k \mid n}\int_0^{\infty}e^{-\pi n^2 t}t^{s-1}dt$ converges absolutely, one can interchange the summation as
\begin{align*}
\sum_{\substack{\gcd(n,p_t) \neq 1 \text{ for $t=i_1,i_2,\ldots,i_j$,} \\ \gcd(n,p_t)=1 \text{ for $t \neq i_1, i_2, \ldots, i_j$}}}\int_0^{\infty}e^{-\pi n^2 t}t^{s-1}dt = \sum_{\substack{j \leq k \leq r, \\ 1 \leq i_{j+1} < i_{j+2} < \cdots < i_k \leq r, \\ i_{j+1}, i_{j+2}, \ldots i_k \in \{1,2, \ldots, r\} \setminus \{i_1,i_2, \ldots, i_j\}}}\sum_{p_1p_2 \cdots p_k \mid n}\int_0^{\infty}e^{-\pi n^2 t}t^{s-1}dt.
\end{align*}
Thus,
\begin{align*}
f(2s)&=(p_1p_2 \ldots p_r)^{2s} \times \\ &\sum_{\substack{0 \leq j \leq r, \\ 1\leq i_1 < i_2 < \cdots < i_j \leq r}}G(b,\chi_{p_{i_1}p_{i_2}\cdots p_{i_j}})\sum_{\substack{j \leq k \leq r, \\ 1 \leq i_{j+1} < i_{j+2} < \cdots < i_k \leq r, \\ i_{j+1}, i_{j+2}, \ldots i_k \in \{1,2, \ldots, r\} \setminus \{i_1,i_2, \ldots, i_j\}}}(-1)^{k-j}\sum_{p_1p_2 \cdots p_k \mid n}\int_0^{\infty}e^{-\pi n^2 t}t^{s-1}dt \\
&=\sum_{\substack{0 \leq j \leq r, \\ 1\leq i_1 < i_2 < \cdots < i_j \leq r}}G(b,\chi_{p_{i_1}p_{i_2}\cdots p_{i_j}})\sum_{\substack{j \leq k \leq r ,\\ 1 \leq i_{j+1} < i_{j+2} < \cdots < i_k \leq r, \\ i_{j+1}, i_{j+2}, \ldots i_k \in \{1,2, \ldots, r\} \setminus \{i_1,i_2, \ldots, i_j\}}}(-1)^{k-j}\frac{(p_1p_2 \ldots p_r)^{2s}}{(p_{i_1}p_{i_2} \ldots p_{i_k})^{2s}} \\ & \qquad \times \sum_{p_1p_2 \cdots p_k \mid n}\int_0^{\infty}(p_{i_1}p_{i_2} \ldots p_{i_k})^{2s}e^{-\pi n^2 t}t^{s-1}dt.
\end{align*}
Note that the last summation in the above is
\begin{align*}
\sum_{p_1p_2 \cdots p_k \mid n}\int_0^{\infty}(p_{i_1}p_{i_2} \ldots p_{i_k})^{2s}e^{-\pi n^2 t}t^{s-1}dt&=\sum_{n=1}^{\infty}\int_0^{\infty}e^{-\pi n^2 t}t^{s-1}dt=\pi^{-s}\Gamma(s)\zeta(2s)=\hat{\zeta}
(2s).
\end{align*}
Also note that for arbitrary real numbers $a_1, a_2, \ldots, a_m \in \mathbb R$,
\begin{align*}
\sum_{\substack{0 \leq j' \leq m, \\ 1\leq i_1 < i_2 < \cdots < i_{j'} \leq m}}(-1)^{m-j'}(a_{i_1}a_{i_2}\cdots a_{i_{j'}})^s=(a_1^s-1)(a_2^s-1)\cdots (a_m^s-1).
\end{align*}
By substituting the integers $p_i^2$'s except for $i=i_1, i_2, \ldots, i_j$ for $a_1,a_2, \ldots,a_m$, we have
\begin{align*}
\sum_{\substack{j \leq k \leq r, \\ 1 \leq i_{j+1} < i_{j+2} < \cdots < i_k \leq r, \\ i_{j+1}, i_{j+2}, \ldots i_k \in \{1,2, \ldots, r\} \setminus \{i_1,i_2, \ldots, i_j\}}}(-1)^{k-j}\frac{(p_1p_2 \ldots p_r)^{2s}}{(p_{i_1}p_{i_2} \ldots p_{i_k})^{2s}}=\frac{(p_1^{2s}-1)(p_2^{2s}-1)\cdots (p_r^{2s}-1)}{(p_{i_1}^{2s}-1)(p_{i_2}^{2s}-1)\cdots (p_{i_j}^{2s}-1)}.
\end{align*}
Therefore,
\begin{align}\label{eqn_f2s}
f(2s)&=\hat{\zeta}(2s)\sum_{{\substack{0 \leq j \leq r, \\ 1\leq i_1 < i_2 < \cdots < i_j \leq r}}}G(b,\chi_{p_{i_1}p_{i_2}\cdots p_{i_j}})\frac{(p_1^{2s}-1)(p_2^{2s}-1)\cdots (p_r^{2s}-1)}{(p_{i_1}^{2s}-1)(p_{i_2}^{2s}-1)\cdots (p_{i_j}^{2s}-1)}.
\end{align}
The assertion follows by dividing both terms of equation (\ref{eqn_f2s}) by $(p_1p_2 \ldots p_r)^{2s} \pi^{-s}\Gamma(s).$
\end{proof}
Unfortunately, the principal character $\mathds{1}_{\bullet}$ is not finitely generated, while the Dirichlet series $\mathcal{L}(b,\mathds{1}_{\bullet},s)$ associated to the principal character also has analytic continuation.
\begin{lemma}\label{lem_trivlocalL}
The local Dirichlet series $\mathcal{L}_p(b,\mathds{1}_{\bullet},s)$ satisfies the following equation:
\begin{align*}
\mathcal{L}_p(b,\mathds{1}_{\bullet},s)=
\begin{cases}
\zeta_p(s)^{-1} & \text{ if } p \nmid b, \\
\zeta_p(s)^{-1}\zeta_p(s-1; v_p(b)) & \text{ otherwise},
\end{cases}
\end{align*}
where $v_p$ denotes the $p$-valuation and $\zeta_p(s; v_p(b)):=\sum_{m=0}^{v_p(b)}\frac{1}{p^{ms}}.$
\end{lemma}
\begin{proof}
It follows from the definition directly. If $b$ is prime to $p$, the Gauss sum of $\mathds{1}_{}$ for each integer $m \geq 1 $ is
\begin{align*}
G(b,\mathds{1}_{p^m})=
\begin{cases}
-1 & \text{ if } m=1, \\
0 & \text{ otherwise}.
\end{cases}
\end{align*}
If $b$ is divisible by $p$, then
\begin{align*}
G(b,\mathds{1}_{p^m})=
\begin{cases}
p^m-p^{m-1} & \text{ if } 1 \leq m \leq v_p(b), \\
-p^{v_p(b)} & \text { if } m=v_p(b)+1, \\
0 & \text{ otherwise}.
\end{cases}
\end{align*}
Combining them we get the result.
\end{proof}
\begin{proposition}\label{prop_Liddformula}
The Dirichlet series $\mathcal{L}(b,\mathds{1}_{\bullet},s)$ is
\begin{align*}
\mathcal{L}(b,\mathds{1}_{\bullet},s)=\frac{\sigma_{s-1}(b)}{\zeta(s)|b|^{s-1}}.
\end{align*}
Consequently, $\mathcal{L}(b,\mathds{1}_{\bullet},s)$ can be continued analytically to the complex plane $\mathbb C.$
\end{proposition}
\begin{proof}
Recall that $\mathcal{L}(b,\mathds{1}_{\bullet},s)$ can be written as a product of local series $\prod_{p}\mathcal{L}_p(b,\mathds{1}_{\bullet},s)$. The assertion follows by applying Lemma \ref{lem_trivlocalL}.
\end{proof}
Let $\chi_{\bullet}$ be any dual character. We define another sequence of functions $\chi_{N\bullet}:=(\chi_{Nn},Q_{Nn})_{n \in \mathbb Z_{>0}}$ induced from the character $\chi_{\bullet}$. Note that $\chi_{N\bullet}$ is not a dual character in general. It is just a collection of periodic functions, but we are still able to consider its Dirichlet series $\mathcal{L}(b,\chi_{N\bullet},s)$ although it is not the same as its Euler product. It is clear that the following equality holds for an arbitrary dual character~$\chi_{\bullet}$:
\begin{align}\label{eqn_localLformula}
\mathcal{L}(b,\chi_{N\bullet},s)=N^s\frac{\mathcal{L}(b,\chi_{\bullet},s)}{\mathcal{L}_N(b,\chi_{\bullet},s)}\prod_{p \mid N}\left( \mathcal{L}_p(b,\chi_{\bullet},s)-\mathcal{L}_p(b,\chi_{\bullet},s;v_p(N)-1)\right).
\end{align}
Thus if $\mathcal{L}(b,\chi_{\bullet},s)$ and its local factor $\mathcal{L}_p(b,\chi_{\bullet},s)$ can be continued analytically for any prime $p$ dividing $N$, so is $\mathcal{L}(b,\chi_{N\bullet},s)$. In the following subsection we fix $\chi_{\bullet}=\mathds{1}_{\bullet}$ and use this fact.
\subsection{Fourier expansion and analytic properties}
We return to the subject of the Eisenstein series $g_N(z,\overline{z},\alpha,\beta).$
\
Define a function
\begin{align*}
\textnormal{\textsl{g}}(z,\overline{z},\alpha,\beta):=\sum_{n=\infty}^{\infty}(z+n)^{-\alpha}(\overline{z}+n)^{-\beta}
\end{align*}
for $z \in \mathbb C$ and $\alpha, \beta \in \mathbb C$ with $\Re(\alpha+\beta)>2.$ This function is 1-periodic in the variable $x=\Re(z)$, so it is given by the Fourier expansion
\begin{align}\label{eqn_cgfourier}
\textnormal{\textsl{g}}(z,\overline{z},\alpha,\beta)=e^{\pi i (\beta-\alpha)/2}\sum_{n=\infty}^{\infty}h_n(y;\alpha,\beta)e^{2\pi i n x},
\end{align}
where
\begin{align*}
h_n(y;\alpha,\beta)=e^{\pi i (\alpha-\beta)/2}\int_0^1 \textnormal{\textsl{g}}(z,\overline{z},\alpha,\beta)e^{-2\pi i n x}.
\end{align*}
If we define another function
\begin{align*}
h(t;\alpha,\beta):=\int_{-\infty}^{\infty}(1-ix)^{-\alpha}(1+ix)^{-\beta}e^{-itx}dx,
\end{align*}
then the Fourier coefficients $h_n(y;\alpha,\beta)$ can be written as
\begin{align}\label{eqn_hn_and_h}
h_n(y;\alpha,\beta)=y^{1-\alpha-\beta}h(2\pi ny;\alpha,\beta),
\end{align}
and the function $h(t;\alpha,\beta)$ satisfies the equation given in \cite[Chapter 4]{MR0734485}
\begin{align*}
h(t;\alpha,\beta)=\frac{2\pi \cdot 2^{1-\alpha-\beta}}{\Gamma(\alpha)\Gamma(\beta)}\int_{u>|t|}e^{-u}(u+t)^{\alpha-1}(u-t)^{\beta-1}du.
\end{align*}
\
On the other hand, for $y>0$, the modified Whittaker $W$-function is defined by
\begin{align*}
W(\epsilon y;\alpha,\beta):=y^{-q/2}W_{\frac{\epsilon k}{2},\frac{q-1}{2}}(2y),
\end{align*}
with $q=\alpha+\beta$ and $k=\alpha-\beta$. Here $W_{\kappa,\mu}$ is the standard Whittaker $W$-function which is described in \cite{MR4286926}. The integral representation of the Whittaker $W$-function is
\begin{align*}
W_{\kappa,\mu}(y)=\frac{y^{\frac{1}{2}-\mu}e^{-\frac{y}{2}}}{\Gamma(\mu+\frac{1}{2}-\kappa)}\int_0^{\infty}e^{-u}u^{\mu-\kappa-\frac{1}{2}}(u+y)^{\mu+\kappa-\frac{1}{2}}du,
\end{align*}
and it relates the function $h(t;\alpha,\beta)$ with the modified Whittaker $W$-function as follows:
\begin{align}\label{eqn_h_and_W}
h(t;\alpha,\beta)=\begin{cases}
\dfrac{2\pi \cdot 2^{-q/2}|t|^{q-1}}{\Gamma(\frac{q+\mathrm{sgn}(t) k}{2})}W(t;\alpha,\beta) & \text{ if }t \neq 0, \\ \\
\dfrac{2\pi \cdot 2^{1-q}\Gamma(q-1)}{\Gamma(\alpha)\Gamma(\beta)} & \text{ if }t=0.
\end{cases}
\end{align}
\begin{proposition}\label{prop_gNLR}
Let $N>1$ be a positive integer. The Fourier expansions of the non-holomorphic Eisenstein series of $g_{N,L}(z,\overline{z},\alpha,\beta)$ and $g_{N,R}(z,\overline{z},\alpha,\beta)$ at $\infty$ are given by
\begin{align*}
g_{N,L}(z,&\overline{z}, \alpha,\beta)=\phi_{N,L}(y,q,k) \\
&+2(-1)^k\zeta(q)\left(\frac{\sqrt 2 \pi}{N}\right)^q\sum_{n\neq 0}\frac{|n|^{q-1}}{\Gamma(\frac{q}{2}+\mathrm{sgn}(n)k)}\mathcal{L}(n,\mathds{1}_{N\bullet},s)W\left(2\pi ny/\sqrt{N};\alpha,\beta\right)e^{2 \pi i nx/\sqrt N},
\end{align*}
and
\begin{align*}
g_{N,R}(z,&\overline{z}, \alpha,\beta)=\phi_{N,R}(y,q,k) \\
&+2(-1)^k\zeta(q)\left(\frac{\sqrt 2 \pi}{\sqrt N}\right)^q\sum_{n\neq 0}\frac{|n|^{q-1}}{\Gamma(\frac{q}{2}+\mathrm{sgn}(n)k)}\frac{\mathcal{L}(n,\mathds{1}_{\bullet},s)}{\mathcal{L}_N(n,\mathds{1}_{\bullet},s)}W\left(2\pi ny/\sqrt{N};\alpha,\beta\right)e^{2 \pi i nx/\sqrt N},
\end{align*}
where $q:=\alpha+\beta$, $k:=\alpha-\beta$ and
\begin{align*}
\phi_{N,L}(y,q,k)&:=2+2(-1)^k N^{-q}\zeta(q)h_0\left(\frac{y}{\sqrt N};\alpha,\beta\right)\mathcal{L}(0,\mathds{1}_{N\bullet},s), \\
\phi_{N,R}(y,q,k)&:=2(-1)^k \sqrt{N}^{-q}\zeta(q)h_0\left(\frac{y}{\sqrt N};\alpha,\beta\right)\frac{\mathcal{L}(0,\mathds{1}_{\bullet},s)}{\mathcal{L}_N(0,\mathds{1}_{\bullet},s)}.
\end{align*}
\end{proposition}
\begin{proof}
First, we note that
\begin{align*}
g_{N,L}(z,\overline{z}, \alpha,\beta)
&=\sum_{t=1}\sum_{\substack{(c,d) \in \mathbb Z \times \mathbb Z \\ \gcd(Nc,d)=1}} t^{-q}(\sqrt N cz+d)^{-\alpha}(\sqrt N c\overline{z}+d)^{-\beta} \\
&=\zeta(q)\sum_{\substack{(c,d) \in \mathbb Z \times \mathbb Z \\ \gcd(Nc,d)=1}}(\sqrt N cz+d)^{-\alpha}(\sqrt N c\overline{z}+d)^{-\beta}.
\end{align*}
We let $e_{N,L}(z,\overline{z}, \alpha,\beta)=\sum\limits_{\substack{(c,d) \in \mathbb Z \times \mathbb Z \\ \gcd(Nc,d)=1}} (\sqrt N cz+d)^{-\alpha}(\sqrt N c\overline{z}+d)^{-\beta}$. Then
\begin{align*}
e_{N,L}(z,\overline{z}, \alpha,\beta)&=2+2\sum_{\substack{c>0\\ d \in (\mathbb Z/NcZ)^{\times}}} \sum_{n \in \mathbb Z}(\sqrt N cz+d+Ncn)^{-\alpha}(\sqrt N c\overline{z}+d+Ncn)^{-\beta} \\
&=2+2\sum_{\substack{c>0\\ d \in (\mathbb Z/NcZ)^{\times}}} \sum_{n \in \mathbb Z}(Nc)^{-q}\left(\frac{z}{\sqrt N} + \frac{d}{Nc}+n\right)^{-\alpha}\left(\frac{\overline{z}}{\sqrt N} + \frac{d}{Nc}+n\right)^{-\beta} \\
&=2+2\sum_{\substack{c>0\\ d \in (\mathbb Z/NcZ)^{\times}}} \sum_{n \in \mathbb Z}(Nc)^{-q}\textnormal{\textsl{g}}\left(\frac{z}{\sqrt N} + \frac{d}{Nc},\frac{\overline{z}}{\sqrt N} + \frac{d}{Nc},\alpha,\beta\right).
\end{align*}
By (\ref{eqn_cgfourier}),
\begin{align*}
e_{N,L}(z,\overline{z}, \alpha,\beta)&=2+2\sum_{\substack{c>0\\ d \in (\mathbb Z/NcZ)^{\times}}} \sum_{n \in \mathbb Z}(Nc)^{-q}(-1)^k \sum_{n \in \mathbb Z}h_n\left(\frac{y}{\sqrt N}; \alpha, \beta\right)e^{2\pi i (\frac{x}{\sqrt N}+\frac{d}{Nc})n} \\
&=2+2(-1)^k\sum_{n \in \mathbb Z} N^{-q}h_n\left(\frac{y}{\sqrt N}; \alpha, \beta\right) \left(\sum_{c =1}^{\infty}\sum_{d \in (\mathbb Z/Nc\mathbb Z)^{\times}}\zeta_{Nc}^{dn}c^{-q} \right)e^{2\pi i nx/\sqrt N}.
\end{align*}
As we have seen in (\ref{eqn_Gausssumprin}), the sum $\sum\limits_{\substack{0 \leq d < Nc \\ \gcd(d,Nc)=1}}\zeta_{Nc}^{dn}$ is equal to $G(n,\mathds{1}_{Nc})$, so we have
\begin{align*}
e_{N,L}(z,\overline{z}, \alpha,\beta)=&2+2(-1)^k N^{-q}h_0\left(\frac{y}{\sqrt N}; \alpha, \beta\right)\mathcal{L}(0,\mathds{1}_{N\bullet},q) \\
&+2(-1)^k\sum_{n \neq 0}N^{-q}h_n\left(\frac{y}{\sqrt N}; \alpha, \beta\right)\mathcal{L}(q,\mathds{1}_{N\bullet},q)e^{2\pi i nx/\sqrt N}.
\end{align*}
Applying the equations (\ref{eqn_hn_and_h}) and (\ref{eqn_h_and_W}), and multiplying by $\zeta(q)$, we get the Fourier expansion of $ g_{N,L}(z,\overline{z}, \alpha,\beta)$ given in the proposition.
Similarly one can deduce that
\begin{align*}
g_{N,R}(z,\overline{z}, \alpha,\beta)=2\zeta(q)\sum_{\substack{\gcd(c,N)=1 \\ d \in (\mathbb Z/c\mathbb Z)^{\times}}} \sum_{n \in \mathbb Z} (\sqrt N c)^{-q} \textnormal{\textsl{g}}\left(\frac{z}{\sqrt N} + \frac{d}{c},\frac{\overline{z}}{\sqrt N} + \frac{d}{c},\alpha,\beta\right).
\end{align*}
The same argument as above completes the remaining part of the proof.
\end{proof}
\begin{remark}
When $N=1$, the non-holomorphic Eisenstein series $g(z,\overline{z},\alpha,\beta)$ is not the same as $g_1(z,\overline{z},\alpha,\beta).$ It is $g_{1,L}(z,\overline{z},\alpha,\beta)= g_{1,R}(z,\overline{z},\alpha,\beta)=\frac{1}{2}g_{1}(z,\overline{z},\alpha,\beta).$ By comparing the $n$th Fourier coefficient of $g(z,\overline{z},\alpha,\beta)$ given in Proposition \ref{prop_g}(d) and of $g_{1,L}(z,\overline{z},\alpha,\beta)$ given by Proposition \ref{prop_gNLR}, we have
\begin{align*}
\zeta(s)\mathcal{L}(n,\mathds{1}_{\bullet},s)\abs{n}^{s-1}=\sigma_{s-1}(n).
\end{align*}
In other words, if we denote by $\star$ the Dirichlet convolution, then
\begin{align*}
1 \star G(n,\mathds{1}_{\bullet})(m)=\sum_{d\mid m}G(n,\mathds{1}_m)=\begin{cases}
m & \text{ if } m\mid n, \\
0 & \text{ otherwise. }
\end{cases}
\end{align*}
This is the classical and famous property of the Ramanujan sum. Another proof of this formula is given in \cite[Theorem 4.1]{MR2378655}.
\end{remark}
\begin{proof}[Proof of Theorem \ref{thm_gfourier}]
It follows from Proposition \ref{prop_gNLR} by adding $g_{N,L}(z,\overline{z},\alpha,\beta)$ and $g_{N,R}(z,\overline{z},\alpha,\beta)$.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor_Gfourier}]
The proof is completed by applying Lemma~\ref{lem_trivlocalL} and then using the formula given in Proposition \ref{prop_Liddformula} and (\ref{eqn_localLformula}).
\end{proof}
\
For $s\in \mathbb C$, let $s^{\vee}:=1-k-s$ and
\begin{align*}
\psi_{N,b}(s):=\frac{f_{N,b}(s^{\vee})}{f_{N,b}(s)}.
\end{align*}
Then $\psi_{N,b}(s)$ is meromorphic on $\mathbb C$ and satisfies the functional equation, \begin{align*}
\psi_{N,b}(s)\psi_{N,b}(s^{\vee})=1.
\end{align*}
\begin{lemma}\label{lem_psi}
For a prime $p$, the function $\psi_{p,b}(s)$ is independent of the choice of $b \in \mathbb Z \setminus \{0\}$, i.e.,
\begin{align*}
\psi_{p,b}(s)=\psi_{p,1}(s)=:\psi_p (s)
\end{align*}
for any nonzero integer $b$.
\end{lemma}
\begin{proof}
One can verify the Lemma by direct calculations as follows: regardless of the choice of $b$,
\begin{align*}
\psi_{p,b}(s)=\frac{(-1+p^{-s-\frac{k}{2}})\zeta_p^{-1}(2-k-2s)}{(-1+p^{s+\frac{k}{2}-1})\zeta_p^{-1}(2s+k)}=\frac{1+p^{s+\frac{k}{2}-1}}{1+p^{-s-\frac{k}{2}}}=\psi_p(s).
\end{align*}
\end{proof}
Now we are ready to prove Theorem~\ref{thm_funeq}.
\begin{proof}[Proof of Theorem \ref{thm_funeq}.]
Denote by $CT_{p,k}(s)$ the constant term appearing in Corollary \ref{cor_Gfourier}. Write
$CT_{p,k}(s)=C_1(s)\sqrt p^{s+\frac{k}{2}}f_p(s)+C_2(s)C_3(s)f_p(s)$, where
\begin{align*}
C_1(s)&:=\frac{\Gamma(s+\frac{k}{2}+\frac{|k|}{2})}{\Gamma(s+\frac{k}{2})}\hat{\zeta}(2s+k)y^s, \\
C_2(s)&:=(-1)^{k/2}\frac{\Gamma(s+\frac{k}{2})\Gamma(s+\frac{k}{2}+\frac{|k|}{2})}{\Gamma(s+k)\Gamma(s)}\hat{\zeta}(2-2k-2s)y^{1-k-s}, \\
C_3(s)&:=\sqrt p^{1-s-\frac{k}{2}}\frac{1-p^{-1}}{1-p^{-2s-k}}+\sqrt p^{-1+s+\frac{k}{2}}\frac{1-p^{1-k-2s}}{1-p^{-2s-k}}.
\end{align*}
One can observe that from the Legendre relation $2^{2s-1}\Gamma(s)\Gamma(s+\frac{1}{2})=\sqrt \pi \Gamma(2s)$, the functions $C_1(s)$ and $C_2(s)$ are interchanged with each other by dualizing $s$, i.e., $$C_1(s^{\vee})=C_2(s).$$ To prove the functional equation for $CT_{p,k}(s)$, it is enough to show that
\begin{align}\label{eqn_C3}
f_p(s^{\vee})C_3(s^{\vee})=\sqrt p^{s+\frac{k}{2}}f_p(s).
\end{align}
Note that for a prime $p$,
\begin{align*}
C_3(s)&=\sqrt p^{1-s-\frac{k}{2}}\frac{1-p^{-1}}{1-p^{-2s-k}}+\sqrt p^{-1+s+\frac{k}{2}}\frac{1-p^{1-k-2s}}{1-p^{-2s-k}} \\
&=\sqrt p^{1-s-\frac{k}{2}}\frac{1+p^{s+\frac{k}{2}-1}}{1-p^{-(s+\frac{k}{2})}} \\
&=\sqrt p^{s^{\vee}+\frac{k}{2}}\psi_p(s).
\end{align*}
Multiplying by $f_p(s)$ and replacing $s$ with $s^{\vee}$, we get the equation (\ref{eqn_C3}).
For a non-zero integer $n$, the non-constant term of the Fourier series appearing in Corollary \ref{cor_Gfourier} is
\begin{align*}
c_n(s)=\kappa_n(s)f_{p,n}(s^{\vee})f_p(s),
\end{align*}
where
\begin{align*}
\kappa_n(s):=\frac{\Gamma(s+\frac{k}{2}+\frac{|k|}{2})}{\Gamma(s+\frac{k}{2}+\mathrm{sgn}(n)\frac{k}{2})}|n|^{-s-\frac{k}{2}}\sigma_{2s+k-1}(n)W_{\mathrm{sgn}(n)\frac{k}{2},s+\frac{k-1}{2}}\left(4\pi |n| \frac{y}{\sqrt p}\right)e^{2\pi i n/\sqrt p}.
\end{align*}
We claim that $c_n(s^{\vee})=c_n(s)$, which completes the proof. One can see that $h_n(s^{\vee})=h_n(s)$ by comparing the non-constant terms of the Fourier series of $\widehat{G}_k(z,s)$ in the functional equation given in Proposition~\ref{prop_Gfunctional}. It only remains to show
\begin{align*}
f_{p,n}(s)f_p(s^{\vee})=f_{p,n}(s^{\vee})f_p(s),
\end{align*}
and this follows from Lemma \ref{lem_psi}.
\end{proof}
It follows from Corollary~\ref{cor_Gfourier} that the completed Eisenstein series $\hat{G}_{p,k}(z,s)$ can be continued analytically to the neighborhood of $s=0$ if $k \neq 0,2$. In order to show those for $k=0,2$, we need to carry out additional calculations.
\section{Polyharmonic Maass forms for $R(p)$}\label{sec_polyhar}
\subsection{Properties of $T_{p,m,k}(z)$}
We start this section by calculating the constant term and the 1st Taylor coefficient of the doubly-completed Eisenstein series $\tilde{G}_{p,k}(z,s)$ for primes $p$ when $k=0,2$.
\begin{lemma}\label{lem_Tpweight0}
Let $p$ be a prime. The constant term and the 1st Taylor coefficient of the doubly-completed Eisenstein series $\tilde{G}_{p,0}(z,s)$ are given by
\begin{align*}
T_{p,0,0}(z)&=\frac{1}{2}, \\
T_{p,0,1}(z)&=-\frac{1}{2}(\gamma+1)+\frac{1}{2}\log (4\pi \sqrt p)-\frac{\pi}{12}\left(\sqrt p + \frac{1}{\sqrt p}\right)y + M_p(z),
\end{align*}
where $\gamma$ is the Euler–Mascheroni constant and $M_p(z)$ is a holomorphic function on $\mathbb H$ defined as
\begin{align*}
M_p(z):=\sum_{n=1}^{\infty}\sigma_{-1}(n)\left(2-\frac{1-p}{1-p^{v_p(n)+1}}\right)\sin \left(\frac{2\pi n z}{\sqrt p}\right).
\end{align*}
\end{lemma}
\begin{proof}
The asymptotic formula for the Whittaker function $W_{\kappa,\mu}(z) \sim e^{-\frac{1}{2}z}z^{\kappa}$ for $\abs{\text{arg}(z)} \leq \frac{3\pi}{2}-\delta$ (for some positive $\delta$) implies that $W_{0,s-\frac{1}{2}}\left(\frac{4\pi |n|y}{\sqrt N}\right) \ll_{s} 1$. By Corollary \ref{cor_Gfourier}, for general $N=p_1^{m_1}p_2^{m_2}\cdots p_{\omega}^{m_{\omega}},$
\begin{align*}
\widehat{G}&_{N,0}(z,s)=-\frac{1}{2s}+\hat{\zeta}(2)y\frac{(p_1-1)(p_2-1)\cdots(p_{\omega}-1)(p_1^{m_1}p_2^{m_2}\cdots p_{\omega}^{m_{\omega}}+(-1)^{\omega}p_1p_2\cdots p_{\omega})}{(2s)^{\omega}p_1^{1+\frac{m_1}{2}}p_2^{1+\frac{m_2}{2}}\cdots p_{\omega}^{1+\frac{m_{\omega}}{2}}\log p_1 \log p_2 \cdots \log p_{\omega}} \\
&+\sum_{n \neq 0}\sigma_{-1}(n)W_{0,s-\frac{1}{2}}\left(\frac{4\pi |n|y}{\sqrt N}\right)\frac{1+\mu(N)}{(2s)^{\omega}\frac{1-p_1^{v_{p_1}(n)+1}}{1-p_1}\frac{1-p_2^{v_{p_2}(n)+1}}{1-p_2}\cdots \frac{1-p_{\omega}^{v_{p_{\omega}}(n)+1}}{1-p_{\omega}}\log p_1\log p_2\cdots \log p_{\omega}} \\
&+(1+s^{-\omega+1})O(1)
\end{align*}
as $s$ approaches $ 0$. In particular, if $N=p$ is a prime, we have
\begin{align*}
\widehat{G}_{p,0}(z,s)=-\frac{1}{2s}+O(1).
\end{align*}
Therefore we get $T_{p,0,0}(z)=s(s-1)\widehat{G}_{p,0}(z,s)|_{s=0}=\frac{1}{2}.$
To prove the second formula, consider the function
\begin{align*}
\frac{1}{2s}+CT_{p,0}(s)=\frac{1}{2s}+\hat{\zeta}(2s)p^{\frac{s}{2}}+\hat{\zeta}(2-2s)y^{1-s}\left(p^{\frac{1-s}{2}}\frac{1-p^{-1}}{1-p^{-2s}}+p^{\frac{s-1}{2}}\frac{1-p^{1-2s}}{1-p^{2s}}\right).
\end{align*}
From the asymptotic formula for the zeta function $\zeta(s)=\frac{1}{s-1}+\gamma+O(1-s)$, we have
\begin{align*}
\hat{\zeta}(s)=\hat{\zeta}(1-s)=-\frac{1}{s}-\frac{1}{2}\log \pi +\frac{1}{2\sqrt {\pi}} \Gamma '\left(\frac{1}{2}\right)+\gamma + O(s).
\end{align*}
Here $\Gamma'(z)=\Gamma(z)\Psi(z)$, where $\Psi(z)=\frac{d}{dz}\log \Gamma(z)$ is the digamma function. Gauss' digamma theorem gives the value $\Psi(\frac{1}{2})=-\gamma-2\log 2$, which implies $\Gamma'(\frac{1}{2})=\sqrt{\pi}(-\gamma-2\log 2)$, so we have
\begin{align*}
\frac{1}{2s}+\hat{\zeta}(2s)p^{\frac{s}{2}}=\frac{1}{2}\gamma-\frac{1}{2}\log \sqrt p -\log 2 -\frac{1}{2}\log \pi +O(s).
\end{align*}
On the other hands,
\begin{align*}
\lim_{s \to 0} \hat{\zeta}(2-2s)y^{1-s}\left(\frac{1-p^{-1}}{1-p^{-2s}}+p^{\frac{s-1}{2}}\frac{1-p^{1-2s}}{1-p^{2s}}\right)=\frac{\pi}{12}\left(\sqrt p + \frac{1}{\sqrt p}\right)y,
\end{align*}
so $\lim_{s \to 0}\left(\frac{1}{2s}+CT_{p,0}(s)\right)=\frac{1}{2}\gamma-\frac{1}{2}\log 4 \pi \sqrt p+\frac{\pi}{12}\left(\sqrt p + \frac{1}{\sqrt p}\right)y.$
The non-constant term of $\widehat{G}_{p,0}(z,s)$ is given by
\begin{align}\label{eqn_nonconstofGhat}
\sum_{n \neq 0}|n|^{-s}\sigma_{2s-1}(n)W_{0,s-\frac{1}{2}}\left(4\pi |n| \frac{y}{\sqrt p}\right)f_{p,n}(1-s)f_p(s)e^{2\pi i n x/\sqrt p}.
\end{align}
For arbitrary $\delta \in \mathbb R_{>0}$, there is a constant $C>0$ such that if $y \geq \delta$, $W_{0,s-\frac{1}{2}}\left(4\pi |n| \frac{y}{\sqrt p}\right)< Ce^{-2\pi i n y/\sqrt p}$ so that the sum \eqref{eqn_nonconstofGhat} converges uniformly and absolutely as $s$ approaches $0$. Thus
\begin{align}\label{eqn_limofnonconst}
\lim_{s \to 0} \sum_{n \neq 0}|n|^{-s}&\sigma_{2s-1}(n)W_{0,s-\frac{1}{2}}\left(4\pi |n| \frac{y}{\sqrt p}\right)f_{p,n}(1-s)f_p(s)e^{2\pi i n x/\sqrt p} \\\notag
&=\sum_{n \neq 0}|n|^{-s} \lim_{s \to 0} \sigma_{2s-1}(n)W_{0,s-\frac{1}{2}}\left(4\pi |n| \frac{y}{\sqrt p}\right)f_{p,n}(1-s)f_p(s)e^{2\pi i n x/\sqrt p} \\
&= \sum_{n \neq 0}\sigma_{-1}(n)\left(1-\frac{1-p}{2(1-p^{v_p(n)+1})}\right)e^{-2\pi i n y/\sqrt p}e^{2\pi i n x/\sqrt p}=M_p(z). \notag
\end{align}
The last equality follows from that $W_{0,-\frac{1}{2}}\left(4\pi |n| \frac{y}{\sqrt p}\right)=e^{-2\pi i n y/\sqrt p}$ and $\lim_{s \to 0}f_{p,n}(1-s)f_p(s)=1-\frac{1-p}{2(1-p^{v_p(n)+1})}$.
Therefore,
\begin{align*}
T_{p,0,1}(z)&=-\lim_{s \to 0} \left(\hat{G}_{N,0}(z,s)+\frac{1}{2s}\right) -\frac{1}{2} \\
&=-\frac{1}{2}(\gamma+1)+\frac{1}{2}\log (4\pi \sqrt p)-\frac{\pi}{12}\left(\sqrt p + \frac{1}{\sqrt p}\right)y + M_p(z).
\end{align*}
\end{proof}
\begin{lemma}\label{lem_T2}
Let $p$ be a prime. The completed Eisenstein series $\widehat{G}_{p,2}(z,s)$ of weight $2$ is holomorphic in a neighborhood of $s=0$, and the 1st Taylor coefficient of the doubly-completed Eisenstein series $\widetilde{G}_{p,2}(z,s)$ is given by
\begin{align*}
T_{p,2,1}(z)&=\widetilde{G}_{p,2}(z,0) \\
&=\frac{\pi}{6}\sqrt p-\frac{p}{1+p}\frac{1}{y}-\frac{4\pi}{\sqrt p}\sum_{n=1}^{\infty}\sigma_1(n)f_{p,n}(-1)f_p(0)e^{2\pi i n z/\sqrt p}.
\end{align*}
\end{lemma}
\begin{proof}
Using the formulae $$W_{1,\frac{1}{2}}\left(4\pi |n| \frac{y}{\sqrt N}\right)=e^{-2\pi i n y/\sqrt N}4\pi |n| \frac{y}{\sqrt N} \quad \text{ and } \quad W_{-1,\frac{1}{2}}\left(4\pi |n| \frac{y}{\sqrt N}\right)=e^{-2\pi i n y/\sqrt N}\left(4\pi |n| \frac{y}{\sqrt N}\right)^{-1},$$ one can prove the lemma similarly as in the proof of Lemma \ref{lem_Tpweight0}.
\end{proof}
\begin{lemma}\label{lem_Geigenfunction}
For a prime $p$ and an even integer $k$, the Eisenstein series $$G_{p,k,L}(z,s), G_{p,k,R}(z,s), G_{p,k}(z,s), \widehat{G}_{p,k}(z,s) \text{ and } \widetilde{G}_{p,k}(z,s)$$ are eigenfunctions of the hyperbolic Laplacian $\Delta_k$ with the eigenvalue $s(s+k-1).$
\end{lemma}
\begin{proof}
Note that $\Delta_k=\xi_{2-k}\xi_k.$ The proof is obtained by following the arguments given in the proof of \cite[Proposition 7.1]{MR357462}.
\end{proof}
Using Lemma \ref{lem_Geigenfunction}, we obtain the following relation which is an analogue of \cite[Proposition 8.3]{MR357462}.
\begin{proposition}\label{prop_laplacianT}
For a prime $p$ and an even integer $k$,
\begin{align*}
\Delta_k T_{p,n,k}(z,s)=(k-1)T_{p,n-1,k}(z,s)+T_{p,n-2,k}(z,s).
\end{align*}
\end{proposition}
\
As in the case of $N=1$, we want to show that $T_{N,m,k}(z)$ is a polyharmonic Maass form of weight $k$ and depth $\leq m+1$ when $N=p$ is a prime. Note that the polyharmonicity $\Delta_k^{m+1}T_{p,m,k}(z)=0$ and the modular invariant condition $T_{p,m,k}|_k \gamma (z)=T_{p,m,k}(z)$ for $\gamma \in R(p)$ automatically follows from Proposition \ref{prop_gmodular} and Proposition \ref{prop_laplacianT}. Thus it is enough to show that $T_{p,m,k}(z,s)$ satisfies the moderate growth condition.
\begin{comment}
\begin{lemma}
The Whittaker function $W_{\kappa,\mu}(z)$ satisfies the asymptotic formula:
\begin{align*}
\frac{\partial^m}{\partial \mu^m}W_{\kappa,\mu}(z)=e^{-\frac{z}{2}}z^{\kappa}\sum_{n=0}^{\infty}\frac{\partial^m}{\partial \mu^m}\frac{Y_{\kappa,\mu}(-n)}{n!}(-z)^{-n}+O(|z|^{-M-\frac{1}{2}}) \qquad \text{as $|z| \to 0$},
\end{align*}
where $Y_{\kappa,\mu}(-n):=\left(\frac{1}{2}+\mu-\kappa\right)_n\left(\frac{1}{2}-\mu-\kappa\right)_n$ and $M$ is some positive integer bounded polynomially in $\kappa$ and $\mu$.
\end{lemma}
\begin{proof}
Following the proof of \cite[Proposition A.1]{} and \cite[Section 16.4]{}, we have that
\begin{align*}
\frac{\partial^m}{\partial \mu^m}W_{\kappa,\mu}(z)=e^{-\frac{z}{2}}z^{\kappa}\sum_{n=0}^{\infty}\frac{\partial^m}{\partial \mu^m}\frac{Y_{\kappa,\mu}(-n)}{n!}(-z)^{-n}+\int_{-N-\frac{1}{2}-i\infty}^{-N+\frac{1}{2}-i\infty}\frac{\partial^m}{\partial \mu^m}\Gamma(s)Y_{\kappa,\mu}(s)z^sds,
\end{align*}
where $Y_{\kappa,\mu}(s):=\dfrac{\Gamma(\frac{1}{2}-\mu-\kappa-s)\Gamma(\frac{1}{2}+\mu-\kappa-s)}{\Gamma(\frac{1}{2}-\mu-\kappa)\Gamma(\frac{1}{2}+\mu-\kappa)}$ for $s \in \mathbb C.$ If $|z| \leq 1$, then the integral term is
\begin{align*}
&\int_{-N-\frac{1}{2}-i\infty}^{-N+\frac{1}{2}-i\infty}\frac{\partial^m}{\partial \mu^m}\Gamma(s)Y_{\kappa,\mu}(s)z^sds \\ & = \int_{-N-\frac{1}{2}-i}^{-N+\frac{1}{2}-i}\frac{\partial^m}{\partial \mu^m}\Gamma(s)Y_{\kappa,\mu}(s)z^sds + \int_{s=-N-\frac{1}{2}+it \text{ }: \text{ } t \in (-\infty,-1) \cup (1,\infty)}\frac{\partial^m}{\partial \mu^m}\Gamma(s)Y_{\kappa,\mu}(s)z^sds \\
& \leq \int_{-N-\frac{1}{2}-i}^{-N+\frac{1}{2}-i}\left|\frac{\partial^m}{\partial \mu^m}\Gamma(s)Y_{\kappa,\mu}(s)\right|ds + \int_{\{-N-\frac{1}{2}+it \text{ }: \text{ } t \in (-\infty,-1) \cup (1,\infty)\}}C_{m,\kappa,\mu}|z|^{-N-\frac{1}{2}}e^{-\frac{3\pi}{2}|t|}|t|^{N-2\Re(\kappa)}(\log |t|)^m dt,
\end{align*}
for some constant $C_{m,\kappa,\mu} \in \mathbb R_{>0}$ depending on $m,\kappa$ and $\mu.$ The right hand-side of the inequality is $O(|z|^{-N-\frac{1}{2}})$ as $|z| \to 0.$
\end{proof}
\end{comment}
\begin{proposition}\label{prop_Tpolyhar}
The function $T_{p,m,k}(z)$ satisfies the moderate growth condition, i.e., there exists $\alpha \in \mathbb R$ such that $T_{p,m,k}(x+iy)=O(y^{\alpha})$ as $y $ approaches $ \infty$, uniformly in $x\in \mathbb R.$ Consequently, $T_{p,m,k}(z,s)$ is a polyharmonic Maass form of weight $k$ and depth $\leq m+1$ for $R(p).$
\end{proposition}
\begin{proof}
Recall the definition of $T_{p,m,k}(z).$ We only need to look at the non-constant term of $\frac{\partial^m}{\partial s^m}|_{s=0} \widetilde{G}_{p,k}(z,s),$ which is given as follows by Corollary \ref{cor_Gfourier}:
\begin{align*}
\sum_{n \neq 0}\sum_{j=0}^m \frac{\partial^j}{\partial s^j}\Big|_{s=0}F(n,s)\frac{\partial^{m-j}}{\partial s^{m-j}}\Big|_{s=0}\left(y^{-k/2}W_{\mathrm{sgn}(n)\frac{k}{2},s+\frac{k-1}{2}}(4\pi |n|y/\sqrt p) e^{2\pi i nx/\sqrt p}\right).
\end{align*}
Here, $$F(n,s):=(-1)^{k/2}\left(s+\frac{k}{2}\right)\left(s+\frac{k}{2}-1\right)\frac{\Gamma(s+\frac{k}{2}+\frac{|k|}{2})}{\Gamma(s+\frac{k}{2}+\mathrm{sgn}(n)\frac{k}{2})}|n|^{-s-\frac{k}{2}}\sigma_{2s+k-1}(n) f_{p,n}(1-k-s)f_p(s).$$ The part $\frac{\partial^{m-j}}{\partial s^{m-j}}|_{s=0}\left(y^{-k/2}W_{\mathrm{sgn}(n)\frac{k}{2},s+\frac{k-1}{2}}(4\pi |n|y/\sqrt p) e^{2\pi i nx/\sqrt p}\right)$ decays exponentially as $y$ approaches $ \infty$ by \cite[Corollary A.3]{MR3856183}. Thus it suffices to show that for any $\epsilon >0$,
\begin{align*}
\frac{\partial^j}{\partial s^j}|_{s=0}F(n,s) \ll e^{\epsilon |n|} \qquad \text{as } |n| \to \infty,
\end{align*}
for $0 \leq j \leq m.$ Let $F_0(s):=(-1)^{k/2}\left(s+\frac{k}{2}\right)\left(s+\frac{k}{2}-1\right)\frac{\Gamma(s+\frac{k}{2}+\frac{|k|}{2})}{\Gamma(s+\frac{k}{2}+\mathrm{sgn}(n)\frac{k}{2})}.$ Then
\begin{align*}
\frac{\partial^j}{\partial s^j}\Big|_{s=0}F(n,s)=\sum_{\substack{j_1+j_2+j_3=j \\ j_1,j_2,j_3 \geq 0}} \frac{\partial^{j_1}}{\partial s^{j_1}}\Big|_{s=0}F_0(s) \frac{\partial^{j_2}}{\partial s^{j_2}}\Big|_{s=0} |n|^{-s-\frac{k}{2}}\sigma_{2s+k-1}(n) \frac{\partial^{j_3}}{\partial s^{j_3}}\Big|_{s=0} f_{p,n}(1-k-s)f_p(s).
\end{align*}
The first term $\frac{\partial^{j_1}}{\partial s^{j_1}}|_{s=0}F_0(s)$ does not depend on $n$. For the second term, note that
\begin{align*}
\frac{\partial^{j}}{\partial s^{j}}\Big|_{s=0}\sigma_{2s+k-1}(n)=\sum_{d \mid |n|}2^j (\log d)^j d^{2s+k-1}.
\end{align*}
Hence we have
\begin{align*}
\frac{\partial^{j_2}}{\partial s^{j_2}}\Big|_{s=0} |n|^{-s-\frac{k}{2}}\sigma_{2s+k-1}(n) \ll |n|^{\frac{3}{2}k-1}(\log |n|)^j \qquad \text{ as } |n| \to \infty.
\end{align*}
Lastly, in the third term $f_{p,n}(1-k-s)f_p(s)=f_{p,n}(s)f_p(1-k-s)$ by Lemma \ref{lem_psi}. Note that $v_p(n) \ll \log |n|$. By the definition of $f_{p,n}(s)$, we get that
\begin{align*}
f_{p,n}(s) \leq \frac{p^{-(s-1)-\frac{k}{2}}\left(\left(\sum_{i=1}^{v_p(n)}(p-1)p^{i-1+i(2s+k-2)}\right)-p^{v_p(n)+(v_p(n)+1)(2s+k-2)}\right)+1}{\zeta_p^{-1}(-2s-k+2)},
\end{align*}
so
\begin{align*}
\frac{\partial^j f_{p,n}(s)}{\partial s^j}\Big|_{s=0} \ll_{p,k,j} (\log |n|)^{2j}.
\end{align*}
Since $f_p(s):=f_{p,1}(s)$, this proves that the third term $\frac{\partial^{j_3}}{\partial s^{j_3}}|_{s=0} f_{p,n}(1-k-s)f_p(s)$ grows at most logarithmically, so $\frac{\partial^j}{\partial s^j}|_{s=0}F(n,s) \ll e^{\epsilon |n|}$ for any $\epsilon >0$ as we have desired.
\end{proof}
\begin{remark}
By Proposition \ref{prop_Tpolyhar}, $T_{p,2,1}(z)$ is a polyharmonic Maass form of weight 2 and depth $\leq 1$ for $R(p).$ In Lemma \ref{lem_T2} we show that $T_{p,2,1}(z)$ is of the form $t_0(z)+t_1(z)y^{-1}$, where $t_0(z)$ and $t_1(z)$ are holomorphic functions. Thus $T_{p,2,1}(z)$ is in fact an almost holomorphic modular form of weight 2 for $R(p).$ The function $t_0(z)=\frac{\pi}{6}\sqrt p-\frac{4\pi}{\sqrt p}\sum_{n=1}^{\infty}\sigma_1(n)f_{p,n}(-1)f_p(0)e^{2\pi i n z/\sqrt p}$ is a holomorphic part of a harmonic Maass form, namely, a mock modular form. At the same time, it is a constant term of some almost holomorphic modular form in a variable $y^{-1},$ so it is a quasi-modular form of weight 2 and depth 1 for $R(p).$ For any cocompact Fuchsian group there exists essentially one quasi-modular form of weight 2, and that form and the ring of modular forms generates the ring of quasi-modular forms for such a group. In this case, $t_0(z)$ does play that role.
\end{remark}
By following the similar argument as in the proof of \cite[Theorem 8.4]{MR357462}, one can prove that if $k\neq 2$, then $\Delta_k^m T_{p,m,k}(z) \neq 0$. Similarly it can be also proved that $\Delta_k^{m-1} T_{p,m,2}(z) \neq 0$. Thus we obtain the following proposition.
\begin{proposition}\label{prop_T}
Let $p$ be a prime and $k \neq 2$ be an even integer.
\begin{enumerate}[\normalfont(a)]
\item The function $T_{p,k,m}(z)$ is a polyharmonic Maass form of depth $m+1$ for $R(p)$ with $\Delta_k^{m}T_{p,2,m}(z) \neq 0$.
\item The function $T_{p,2,m}(z)$ is a polyharmonic Maass form of depth $m$ for $R(p)$ with $\Delta_k^{m-1}T_{p,2,m}(z) \neq 0$.
\end{enumerate}
\end{proposition}
\
We summarize as follows.
\begin{proposition}
Let $E_k^m(R(p))$ be the $\mathbb C$-vector space spanned by the Taylor coefficients $T_{p,n,k}(z)$ for $n \leq m$ of the Eisenstein series $\widetilde{G}_{p,k}(z,s).$ Then $E_k^m(R(p))$ is a $m$-dimensional subspace of $V_k^m(R(p))$ and its basis is given by
\begin{align*}
\begin{cases}
\{T_{p,0,k}(z), T_{p,1,k}(z), \ldots, T_{p,m-1,k}(z)\} & \text{ if } k \neq 2, \\
\{T_{p,1,2}(z), T_{p,2,2}(z), \ldots, T_{p,m,2}(z)\} & \text{ if } k=2.
\end{cases}
\end{align*}
\end{proposition}
\
\subsection{Fourier expansions of polyharmonic Maass forms}
In parallel with the case of $\operatorname{SL}_2(\mathbb Z)$, we find the general form of the Fourier expansion of a polyharmonic Maass form for $R(N).$ For $s_0 \in \mathbb C$ such that $s_0 \neq \frac{1-k}{2}$ and $m \geq 0$, let
\begin{align*}
\begin{cases}
u_{k,0}^{[m],+}(y; s_0):=(\log y)^m y^{s_0}, \\
u_{k,0}^{[m],-}(y; s_0):=(-1)^m(\log y)^m y^{1-k-s_0}, \\
\end{cases}
\end{align*}
and for $\epsilon \in \{\pm 1\}$, let
\begin{align*}
u_{N,\epsilon,k,|n|}^{[m],-}(y;s_0):=y^{-\frac{k}{2}}\frac{\partial^m}{\partial s^m}\Big|_{s=s_0}W_{\frac{1}{2}\epsilon k,s+\frac{k-1}{2}}\left(4\pi |n| \frac{y}{\sqrt N}\right).
\end{align*}
\begin{proposition}
Let $N>1$ be an integer and $f(z)$ be a shifted polyharmonic Maass form of weight $k$ and depth $m$ for $R(p)$, whose eigenvalue is $c \in \mathbb C$. Let $s_0 \in \mathbb C$ satisfy $s_0(s_0+k-1)=c.$ Then the Fourier expansion of $f(z)$ is given by
\begin{align*}
f(z)=\sum_{j=0}^{m-1}\left(c_{0,j}^+ u_{k,0}^{[j],+}(y; s_0)+c_{0,j}^- u_{k,0}^{[j],-}(y; s_0)\right) + \sum_{\epsilon \in \{\pm 1\}}\sum_{n=1}^{\infty}\sum_{j=0}^{m-1} c_{\epsilon,j,n}^- u_{N,\epsilon,k,|n|}^{[j],-}(y;s_0) e^{2\pi i \epsilon n x/\sqrt N},
\end{align*}
where $c_{0,j}^+, c_{0,j}^-, c_{\epsilon,j,n}^- \in \mathbb C.$ In particular, if $f(z)$ is a harmonic Maass form of weight $k$ for $R(p)$, then the Fourier expansion of $f$ is given by
\begin{align*}
f(z)=\sum_{n=1}^{\infty}b_{-n}\Gamma\left(1-k,4\pi |n|\frac{y}{\sqrt N}\right) e^{-2\pi i nz/\sqrt N}+(b_0y^{1-k}+a_0) + \sum_{n=1}^{\infty}a_n e^{2\pi i n z/\sqrt N},
\end{align*}
for $b_{-n},a_n \in \mathbb C$.
\end{proposition}
\begin{proof}
The proof is completely in parallel with one for $\operatorname{SL}_2(\mathbb Z)$ given in \cite[Theorem 4.3]{MR357462} and \cite[Lemma 4.4]{MR357462}.
\end{proof}
\
\subsection{Structure of polyharmonic Maass forms}
Propositions in this subsection can be proved by following the same line in \cite{MR357462}, once we prove the analytic properties of the non-holomorphic Eisenstein series. There is no new idea of the proof, so we state our results briefly.
Recall that for any Fuchsian group $\Gamma$ for which $X(\Gamma)$ has genus $g$, the dimension of the space of weight $2$ modular forms for $\Gamma$ is $g+t-1$, where $t$ is the number of cusps of $\Gamma$ (the proof is given in \cite{MR1021004}). In particular, if $g_p=0$ for a prime $p$, then there is no non-zero weight $2$ modular form for $R(p).$ This observation implies the following propositions.
\begin{proposition}
Let $p$ be a prime for which the modular curve $X(R(p))$ has genus $g_p=0.$
\begin{enumerate}[\normalfont(a)]
\item The space of weight $0$ modular forms $V_0^{1/2}(R(p))$ is $1$-dimensional, that is, $V_0^{1/2}(R(p))=\mathbb C.$
\item For any integer $m \geq 0$, $V_0^{m+\frac{1}{2}}(R(p))=V_0^{m+1}(R(p)).$
\item For any integer $m \geq 0$, $\dim V_0^{m+1}(R(p)) \leq \dim V_0^{m}(R(p))+1.$ Consequently, $\dim V_0^m(R(p)) \leq m$ for $m \geq 1.$
\item The space of weight $2$ harmonic Maass forms $V_2^1(R(p))$ is $1$-dimensional, that is, $V_2^1(R(p))=\mathbb C T_{p,1,2}(z).$
\item For any integer $m \geq 0$, $V_2^{m}(R(p))=V_2^{m+\frac{1}{2}}(R(p)).$
\item For any integer $m \geq 0$, $\dim V_2^{m+1}(R(p)) \leq \dim V_2^{m}(R(p))+1.$ Consequently, $\dim V_2^m(R(p)) \leq m$ for $m \geq 1.$
\end{enumerate}
\end{proposition}
\begin{proposition}\label{prop_polyhar}
Let $k \geq 4$ be an even integer and $m \geq 0$ be an integer.
\begin{enumerate}[\normalfont(a)]
\item Any $1$-harmonic Maass form of weight $k$ for $R(p)$ is holomorphic, i.e., $V_k^1(R(p))=V_k^{1/2}(R(p))=M_k(R(p))$.
\item The space $V_{2-k}^1(R(p))$ is spanned by $T_{p,2-k,0}(z)=\widetilde{G}_{p,2-k}(z,0)$.
\item For $m \geq 1$, $V_k^m(R(p))=E_k^m(R(p))+S_k(R(p))$, where $E_k^m(R(p))$ is the $m$-dimensional so-called Eisenstein space spanned by $T_{p,k,0}(z), \cdots , T_{p,k,m-1}(z)$, and $S_k(R(p))$ is the space of cusp forms of weight $k$ for $R(p)$. Moreover, $V_k^m(R(p))=V_k^{m-\frac{1}{2}}(R(p)).$
\item For $m \geq 0$, $V_{2-k}^m(R(p))=E_{2-k}^m(R(p))$ which is the $m$-dimensional Eisenstein space spanned by $T_{p,2-k,0}(z), \cdots , T_{p,2-k,m-1}(z)$. Moreover, $V_{2-k}^m(R(p))=V_{2-k}^{m+\frac{1}{2}}(R(p)).$
\end{enumerate}
\end{proposition}
Note that for any Fuchsian group of the first kind, the cusp form of negative weight is always $0$. Hence for $k \neq 0,2$ we summarize Proposition~\ref{prop_polyhar} as
\begin{itemize}
\item $V_k^m(p)=E_k^m(p)\oplus S_k(p)=\langle T_{p,k,0}(z), \cdots , T_{p,k,m-1}(z) \rangle \oplus S_k(p)$,
\item $V_k^{m -\mathrm{sgn}(k)\frac{1}{2}}(p)=V_k^m(p)$.
\end{itemize}
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{"url":"http:\/\/gorizia-legnami.it\/svle\/ode45-matlab.html","text":"# Ode45 Matlab\n\n Compare Results of Different Solvers. Rating is available when the video has been rented. That's an old, now undocumented syntax for ode45. The \"regular\" function approach gives you the most flexibility in describing your ODEs, but MATLAB requires that functions be stored in function files. 4: Ejercicios - Jes\u00fas. ode23 is an implementation of an explicit Runge-Kutta (2,3) pair of Bogacki and Shampine. Dear Matlab community, I am hoping you can help me with what I expect to be a simple matter. Error ODE45 \"Array indices must be positive Learn more about ode45. \u2022 Matlab has several different functions (built-ins) for the numerical solution of ODEs. This is done by clicking Start-> All Programs->Math Programs-> MATLAB R2015a. Learn more about ode45. Key MATLAB commands used in this tutorial are: plot , polyval , roots , conv , deconv , inv , eig , poly , tf , zero. 2015-01-05 Matlab\u4f7f\u7528ode45\u89e3\u5fae\u5206\u65b9\u7a0b\u7ec4\u7684\u57fa\u672c\u95ee\u9898\uff0c\u4e0b\u9762\u6709\u4e00\u4e2a 7; 2009-06-19 matlab ode45 \u6c42\u89e3\u5e38\u5fae\u5206\u65b9\u7a0b\u7ec4 78; 2012-12-04 \u600e\u6837\u7528matlab\u4e2d\u7528ode45\u6c42\u89e3\u5e26\u6709\u5206\u6bb5\u53c2\u6570\u7684\u5fae\u5206\u65b9\u7a0b\u7ec4 8; 2009-09-15 \u8bf7\u95ee\u5982\u4f55\u5b9e\u73b0matlab\u89e3\u4e00\u9636\u5fae\u5206\u65b9\u7a0b\u7ec4(\u7528\u5176\u81ea\u8eab\u547d\u4ee4\u5982od 25; 2014-03-08 matlab ode45\u89e3\u5fae\u5206. Scilab Enterprises is developing the software Scilab, and offering professional services: Training Support Development. The game begins with the outbreak of the Uranus offensive in late November 1942 and ends with Manstein's counter-attack in March 1943. 5],1) and MATLAB returns two column vectors, the \ufb01rst with values of x and the second with values of y. - am304 Nov 28 '18 at 15:28 Please consider accepting the answer if it solved your problem. (flow resistance term has been linearized). It solves a system of differential equations numerically, to describe a system at different points in time. ode45 - Di erential Equation Solver This routine uses a variable step Runge-Kutta Method to solve di erential equations numerically. 001:B] Matlab misses all the pulses and does not output the correct solution. 1-based indexing is the language of Mathematics, as confirmed by Cleve Moler himself in a comment on this April Fools blog post. HomeworkQuestion. m from sourceforge), I would not say that ode45 \"is not native to Octave\". Knowing MATLAB will definitely be a plus on your resume. ode45_with_piecwise. Trying to solve motion equation using ODE45. fun is a function handle, inline function, or string containing the name of the function that defines the ODE: y' = f(t,y). How do I store the updated value of the double derivatives as the ode45 code runs? The way ode45 works, I get x and x-dot as output but not the double derivatives. k is constant and y(0)=40 and y(15)=95 solve this equation by using ode45 can someone pleaseeeeeeeeeee check the code and make it work. The ode45 function returns our packaged dependent variables and the independent variable, which we call w and t. Learn more about ode45, ode MATLAB. pdf sdof_stiff2_ode45. t is unlikely to be the exact value 1, meaning it is unlikely to be a valid index into a scalar. Polynomial curve fitting. Hi all, I am solving a set of ODE equations that in the fist equation I need the last time step (del_t) for example equations are as follows: dA\/dt=f(A,B)+G(del_t) dB\/dt=f(A,B). Polynomials as Vectors. Follow 157 views (last 30 days) Dereje on 26 Apr 2018. Simulink Basics Tutorial. It allows you to step through your code line by line. That anonymous function calls another function, myode, with FIXED INPUTS ft, f, gt, and g which are all vectors. Line 2 solves the IVP numerically using the. HomeworkQuestion. You pass it a handle to the vectorized differential equations, the time span you want to work in, and a vector of intial conditions. I need to solve two sets of coupled differential equations, dx = f(x,u) and du = g(x,u), using ode45. In this case t and y are both scalars, and that is the only interface that ode45 knows about directly. ode45 and ode15s, including event functions), curve fitting to a model, plotting options for x-y plots, and solving sets of nonlinear equations (fsolve). The existence and uniqueness theory states that a solution exists on any interval (a,b) not containing t=0. We have always had BOTH 0-based indexing and 1-based indexing. That is all that is necessary. Problem with ode15s \/ ode45. Published on Oct 20, 2015. dsolve can't solve this system. MATLAB\u4e2d\u6587\u8bba\u575bMATLAB \u6570\u5b66\u3001\u7edf\u8ba1\u4e0e\u4f18\u5316\u677f\u5757\u53d1\u8868\u7684\u5e16\u5b50\uff1aode45\u89e3\u4e8c\u9636\u5fae\u5206\u65b9\u7a0b\u3002\u5229\u7528ode45\u89e3\u4e8c\u9636\u5fae\u5206\u65b9\u7a0b\u753b\u51fa\u7684\u56fe\u50cf\u662f\u4e0d\u65ad\u4e0a\u5347\u7684\uff0c\u800c\u89e3\u6790\u89e3\u7684\u56fe\u50cf\u662f\u4e0d\u65ad\u4e0b\u964d\u7684\u56fe\u50cf\u3002\u8bf7\u95ee\u95ee\u9898\u5728\u54ea\uff1f\u7a0b\u5e8f\u4e3a\uff1afunction dydx=myode(x,y)dydx=zeros(2,1)dydx(1)=y(2)dydx. I particularly like it for graphics. Commented: Bjorn Gustavsson on 27 Apr 2018 I need to stop the ode when y(2) is 0. ODE45 is very accurate. Knowing MATLAB will definitely be a plus on your resume. Choose a web site to get translated content where available and see local events and offers. The function vdp1. This means that the solution to the differential equation may not be defined for t=0. In essence, snapshots of ft, f, gt, and g are taken at the time this line is executed and embedded within. Stiffness is a term that defies a. I do not need to check the input variables, it will be done somewhere else. I need to solve two sets of coupled differential equations, dx = f(x,u) and du = g(x,u), using ode45. Jan 29, 2017 \u00b7 The remaining task is to define f in a way that MATLAB understands. Using ODE45 to solve a system of 2nd order ODEs Learn more about ode45. p = polyfit(x,y,n) [p,S] = polyfit(x,y,n) [p,S,mu] = polyfit(x,y,n) Description. Matlab has two functions, ode23 and ode45, which are capable of numerically solving differential equations. MATLAB REVIEW The focus of this document is to review common, useful, higher-level Matlab operations that will be employed on your assignments, such as: ordinary differential equation time integrators (i. x0 can be a scalar, vector, or matrix. Active 6 years, 7 months ago. Usage Examples. In the MatLab window,. The ode45 solver takes longer for each step, but it also takes larger steps. Solving a system of differential equations using Learn more about differential equations, conditional statement MATLAB. share | cite | improve this question. We use the damped, driven simple harmonic oscillator as an example: The ode45 function returns our packaged dependent variables and the independent variable, which we call w and t. Dear Matlab community, I am hoping you can help me with what I expect to be a simple matter. The matlab function ode45 will be used. \u2022 Example syntax for variable step size of independent parameter (if we want to specify a fixed step size, use, e. [t,a]=ode45(@fun,timerange,initial); Note that MATLAB is case sensitive, so you have to call ode45 and not ODE45 (beware the forum for that, as we often capitalize function names outside of code blocks, to differentiate them from the rest of the text). This book is aimed squarely at the MATLAB beginner. A numerical ODE solver is used as the main tool to solve the ODE's. ODE45 running infinitely without solving. MATLAB\u4e2d\u6587\u8bba\u575bMATLAB \u57fa\u7840\u8ba8\u8bba\u677f\u5757\u53d1\u8868\u7684\u5e16\u5b50\uff1aMATLAB\u4e2dode45\u7684\u7528\u6cd5\u95ee\u9898\u3002\u8bf7\u95ee\u5404\u4f4d\uff0code45\u662f\u4e0d\u662f\u5fc5\u987b\u8981\u8c03\u7528function\u6587\u4ef6\uff1f\u5982\u679c\u6211\u5728function\u91cc\u52a0\u5165for\u5faa\u73af\uff0c\u4e5f\u5c31\u662f\u8bf4\u6211\u60f3\u4e00\u6b21\u6c42\u5f88\u591a\u4e2a\u76f8\u5173\u7684\u5fae\u5206\u65b9\u7a0b\u7684\u89e3\uff0c\u6709\u6ca1\u6709\u53ef\u80fd\u5b9e\u73b0\uff1f. Contribute to Mornsplendor\/Matlab-DE development by creating an account on GitHub. A brief introduction to using ode45 in MATLAB MATLAB\u2019s standard solver for ordinary di erential equations (ODEs) is the function ode45. Published on Oct 20, 2015. ode45() is a sophisticated built-in MATLAB function that gives very accurate solutions. ode45 is designed to handle the following general problem = \u20ac dy dt f (t, y. The question goes on to ask which single parmeter should be changed to obtain an asymptotically stable steady state. Changing Inputs per Time Step in ODE45 Hi, I am a new MATLAB user and I am attempting to use the ODE45 solver to solve a system of 2 1st order differential equations. Unfortunately this is not mentioned in the help or doc of ODE45, but the documentation of Matlab cannot (and should not) replace a participation in classes for numerics. The ode45 function is a matlab built in function and was designed to solve certain ode problems, it may not be suitable for a number of problems. So the first code sample needs to be saved in a file named myode. It requires six function evaluations per integration step, but may take larger steps on smooth problems than ode23 : potentially offering improved efficiency at smaller tolerances. Merin Merin. How do I store the updated value of the double derivatives as the ode45 code runs? The way ode45 works, I get x and x-dot as output but not the double derivatives. It may be more efficient than ode45 at crude tolerances and in the presence of moderate stiffness. Solution using ode45. The MATLAB documentation recommends ode45 as the first choice. Defining f is probably the most difficult part. The tspan = [t0 tf], where t0 is the starting time, and tf is the ending time. you can find a numeric solution, using the MATLAB ode45 functions (we will learn how to use it later). PS: you may want to look at the documentation for ode45 and familiarise yourself with how to solve odes in MATLAB. trange = [0 100];. The main code that utilized and presented is MATLAB\/ode45 to enable the students solving initial value DE and experience the response of the engineering systems for different applied conditions. Learn more about ode, 45, error, graph, plot. ode45 is one of the most comonly used ordinary differential equation solvers in MATLAB. y0 is close to 16. Dear Guest, Writing tutorials and maintaining a website are expensive endeavors. 005688 seconds. Follow 161 views (last 30 days) Discover what MATLAB. ode45\uff0c\u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u6570\u503c\u6c42\u89e3\u3002MATLAB\u63d0\u4f9b\u4e86\u6c42\u5e38\u5fae\u5206\u65b9\u7a0b\u6570\u503c\u89e3\u7684\u51fd\u6570\u3002\u5f53\u96be\u4ee5\u6c42\u5f97\u5fae\u5206\u65b9\u7a0b\u7684\u89e3\u6790\u89e3\u65f6\uff0c\u53ef\u4ee5\u6c42\u5176\u6570\u503c\u89e3\uff0cMatlab\u4e2d\u6c42\u5fae\u5206\u65b9\u7a0b\u6570\u503c\u89e3\u7684\u51fd\u6570\u6709\u4e03\u4e2a\uff1aode45\uff0code23\uff0code113\uff0code15s\uff0code23s\uff0code23t\uff0code23tb \u3002. Although we present ODE results for the Matlab solver ode23, we mention that similar results were easily obtained using ode45, and ode15sdue to the common design of the solvers. Solve system of ODEs MATLAB with ode45. A brief introduction to using ode45 in MATLAB. In MATLAB, functions are defined in separate files. Rating is available when the video has been rented. Nevertheless, the output of ode45 is smooth because by default the solver uses a continuous extension formula to produce output at four equally spaced time points in the span of each step taken. In the following code I do not know how to express $\\ddot{x}$, in fact MATLAB says that xp ($\\dot{x}$) is undefined. subplot(1,1,1) or clf deletes all axes objects and returns to the default subplot(1,1,1) configuration. Hello, Discover what MATLAB. To use ODE solver, MATLAB uses following Syntax [v y] = solver (@ODEfun, Vspan, y0) Where ODEfun is the function file which you have created. ode45 \u6c42\u89e3\u975e\u521a\u6027\u5fae\u5206\u65b9\u7a0b\uff0c\u4e2d\u9636\u65b9\u6cd5\u3002 [TOUT,YOUT] = ode45(ODEFUN,TSPAN,Y0), \u5176\u4e2dTSPAN = [T0 TFINAL] \u5bf9\u521d\u59cb\u6761\u4ef6\u4e3aY0\u7684y'= f(t,y)\u5fae\u5206\u65b9\u7a0b\u7cfb\u7edf\u4ece\u65f6\u95f4T0\u5230TFINAL\u8fdb\u884c\u79ef\u5206\u3002 ODEFUN\u662f\u4e00\u4e2a\u51fd\u6570\u53e5\u67c4\u3002 \u5bf9\u4e8e\u6807\u91cfT\u548c\u5411\u91cfY\uff0cODEFUN(T,Y)\u5fc5\u987b\u8fd4\u56de\u5bf9\u5e94\u4e8ef(t,y)\u7684\u5217\u5411\u91cf\u3002. ode45 usage in this case? 0. solution to non homogenous time dependent Learn more about ode45, second order equation. y0 is close to 16. Start with the State-Variable Modeling, then set the MATLAB code with the derivative function and the ODE solver. You can obtain a vector ts and a matrix ys with the coordinates of these points using [ts,ys] = ode45(f,[t0,t1],[y10;y20]). 001:5; % time scalex initial_x=0; [t,x]=ode45( @rhs, t, initial_x); plot(t,x. Learn more about ode45. Matlab ode45 Page. In your implementation, it corresponds with test_func and the additional parameters required for test_func (P, b, m, l, u). matlab ode45 event | matlab ode45 event. - am304 Nov 30 '18 at 8:33. The Matlab ODE functions are variable-step methods, meaning. Abbasi May 30, 2012 page compiled on July 1, 2015 at 11:43am Contents 1 download examples source code 1 2 description 1 3 Simulation 3 4 Using ode45 with piecewise function 5 5 Listing of source code 5 1download examples source code 1. I have my m-file labeled HW8. The \"regular\" function approach gives you the most flexibility in describing your ODEs, but MATLAB requires that functions be stored in function files. The most frequently used ODE solver in MATLAB and Simulink is ODE45. In my problem they cannot be solved separately, because phi and f are bounded together. Defining f is probably the most difficult part. HomeworkQuestion. Learn more about ode45, ode MATLAB. Solving ODEs using MatLab A command used to solve ODE's in MatLab (a \\solver\") is ode45 Enter >> help ode45 to see information about this command. Nuevo Tema << >> Vista: Ecuaciones diferenciales ODE45 Maria (22\/05\/2015 16:31:30) 2. 3 in Differential Equations with MATLAB. Like ode45, ode23 is a one-step solver. t is unlikely to be the exact value 1, meaning it is unlikely to be a valid index into a scalar. Usually, the numerical solvers in Matlab work well with the standard settings. ode45 is designed to handle the following general problem:. [T, Y] = ODE45(F, [T0 Tfinal], Y0, TOL, 1) uses tolerance TOL and displays status while the integration proceeds. The syntax for ode45 for rst order di erential equations and that for second order di erential equations are basically the same. Learn more about ode45 MATLAB. Simulating impedance using ODE45. Ordinary Di\ufb00erential Equations with MATLAB 25 1 f = inline('t-y','t','y'); 2 [t,y] = ode45(f,[0,3],1); 3 plot(t,y) (line numbers are not part of the commands!) Line 1 de\ufb01nes the function fas a function of t and y, i. The script has a for loop and solves the system for given parameters one after another(in some limited time span. ode45 with boundary conditions. Press question mark to learn the rest of the keyboard shortcuts. The main issue I had though, is that I am used to using ode45 to solve differential equations. That is all that is necessary. ODE23 is based on the integration method, Runge Kutta23, and ODE45 is based on the integration method, Runge Kutta45. ode45 - Di erential Equation Solver This routine uses a variable step Runge-Kutta Method to solve di erential equations numerically. Use the result of that first ode45 call at time t = 700 as the initial condition for a second call to ode45 that solves a system of ODEs without torque from time t = 700 to time t = 1000. Examples of ode45 with function M-file One dependent variable with a function m-file (most common) First-order reaction dy\/dt = -0. How to solve. Two example MATLAB codes using the ode45 solver are shown below. What is the problem here? Is there something wrong with my Matlab syntax or the way I am using ode45? Any explanation is greatly appreciated. I am using ode45 to numerically solve a differential equation, and I am using the event-functionality to record certain times of interest during the solving process. 5],1) and MATLAB returns two column vectors, the \ufb01rst with values of x and the second with values of y. ODE45 is very accurate. Keyword CPC PCC Volume Score; matlab ode45: 0. $\\endgroup$ - Chris Rackauckas Oct 2 '17 at 6:40. Surveys ode45 routine in Matlab. Open Script. Matlab Ode45; Affichage des r\u00e9sultats 1 \u00e0 1 sur 1 Matlab Ode45. Does anyone have suggestions on where I'm going wrong? Thanks. This means that the solution to the differential equation may not be defined for t=0. ode15s \u8ba1\u7b97\u7684\u70b9\u6570 188 [\u6559\u7a0b] \u9690\u5f0f\u5fae\u5206\u65b9\u7a0b(IDE)\u7684 Matlab \u89e3\u6cd5 \u4e0a\u5e1d\u4e0d\u4f1a\u603b\u662f\u90a3\u4e48\u4ec1\u6148\u7684\uff0c\u4e0d\u662f\u6240\u6709\u7684 ODEs \u90fd\u662f\u53ef\u4ee5\u76f4\u63a5\u663e\u5f0f\u7684\u8868\u8fbe\u6210\u4e0b\u9762\u7684\u6837\u5b50 \u6bd4\u5982\uff0c\u4e0b\u9762\u7684\u9690\u5f0f\u5fae\u5206\u65b9\u7a0b\u7ec4 \u90a3\u8be5\u5982\u4f55\u529e\u5462\uff1f. \uc194\ubc84 \ubc0f \uacc4\uc0b0 \uc9c0\uc810 \ub4f1 \ud574\uc5d0 \ub300\ud55c \uc815\ubcf4\ub97c \ud3ec\ud568\ud558\ub294 \uad6c\uc870\uccb4\ub97c \ubc18\ud658\ud558\ub294 \ub2e8\uc77c \ucd9c\ub825\uac12\uc744 \uc9c0\uc815\ud569\ub2c8\ub2e4. matlab ode45 wont accept new parameter. This function implements a Runge-Kutta method with a variable time step for e cient computation. I'm trying to write a matlab. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. share | cite | improve this question. Use the result of that first ode45 call at time t = 700 as the initial condition for a second call to ode45 that solves a system of ODEs without torque from time t = 700 to time t = 1000. ode45 or dsolve. It solves a system of differential equations numerically, to describe a system at different points in time. matlab ode45 | matlab ode45 | matlab ode45 example | matlab ode45 code | matlab ode45 plot | matlab ode45 event | matlab ode45 solver | matlab ode45 syntax | ma. m\uc740 MATLAB\u00ae\uacfc \ud568\uaed8 \uc81c\uacf5\ub418\ub294 \ud568\uc218\ub85c, \uc774 \ubc29\uc815\uc2dd\uc744 \ub2f4\uace0 \uc788\uc2b5\ub2c8\ub2e4. \u901a\u8fc7\u8fd9\u4e09\u4e2a\u4f8b\u5b50\u53ef\u4ee5\u770b\u51fa\uff0c\u7528matlab\u89e3\u5404\u7c7b\u65b9\u7a0b\u7ec4\u90fd\u662f\u53ef\u4ee5\u7684\uff0c\u65b9\u6cd5\u4e5f\u6709\u591a\u79cd\uff0c\u53ea\u662f\u7528\u5230\u89e3\u65b9\u7a0b\u7ec4\u7684\u51fd\u6570\uff0c\u6ce8\u610f\u6b63\u786e\u4e66\u5199\u53c2\u6570\u5c31\u53ef\u4ee5\u4e86\uff0c\u975e\u5e38\u65b9\u4fbf\u3002 \u7528 Matlab \u89e3\u65b9\u7a0b\u7ec4 \u7684\u65f6\u5019\uff0c\u53d1\u73b0\u5b83\u4e0d\u80fd\u81ea\u52a8\u4ee3\u5165\u7cfb\u6570\u7684\u503c\u3002\u6bd4\u5982 \u8bf4\u5982\u4e0b\u7684\u7a0b\u5e8f; a=4;. Matlab ODE45 Help?? function dy = prob_52 (t,y) dy = cos(y) - (sin (t)*y); [t,y] = ode45 (@prob_52,[0,1],0) plot (t,y) xlabel ('Time') ylabel ('Function Value') I was told my code is incorrect. I\u2019m doing an. m function (system), time-span and initial-condition (x0) only. Please detailed step by step codes with comments. Press question mark to learn the rest of the keyboard shortcuts. It is based on method published by British mathematicians JR Dormand and PJ Prince in 1980. Examples of ode45 with function M-file One dependent variable with a function m-file (most common) First-order reaction dy\/dt = -0. Therefore to solve a higher order ODE, the ODE has to be \ufb01rst converted to a set of \ufb01rst order ODE's. It is convenient to program it in a separate le. Matlab-ode45 vs Octave-lsode for a nonlinear ODE Hi, I'm getting very different results when solving the following initial value ODE problem in Matlab and Octave: dy\/dt=1\/sqrt(y^2 + 1)+y-y^2 on t \\in [0,10] with y(0) = 0 From looking at the equation, I believe that the Matlab solution is the correct one, so I'm wondering if I have not converted. Ode45 and initial conditions. program using to solve twolevel Bloch equations by matlab ode45 method - hupidong\/twolevel_ode45_matlab. The MATLAB program ode45 integrates sets of differential equations using a 4-th order Runge-Kutta method. The equation i'm trying to solve is dq(x,t)\/dt=-c*dq(x,t)\/dx with initial condition for example q(x,0)=exp(-(x-5)^2) c is a constant. Since xand yare vectors with corresponding components, we can plot the values with. I'm attempting to propagate the angular velocity, w0 over a span of 100 sec, using ode45. All of this is happening using Matlab R2015b, also no way to get around that. I put the ode45 in a loop where the rCurrent is the radius of the liquid that changes every loop, so ode45 only solves for the y(t). In MATLAB its. A brief introduction to using ode45 in MATLAB. In general, ode45 is the best function to apply as a \"first try\" for most problems. A function is a group of statements that together perform a task. I use MATLAB commands 'ode23' and 'ode45' for solving systems of differential equations and this program involves an *. Is someone able to explain to me exactly what the \"odefun\" called by the \"ode45\" ODE solver in MATLAB is supposed to do? My understanding is that you represent an n-order ODE as a system of n first-order ODEs and that, somehow, from this system, you create the \"odefun\" which \"ode45\" uses. To be valid, the call must name the function ( f ) defined in MATLAB\u00ae, and specify both the time range ( Trange ) and the initial condition vector ( IC ). 2020-04-24 matlab\u60f3\u7528ode45\u6c42\u89e3\u4e00\u4e2a\u4e8c\u9636\u5e38\u5fae\u5206\u65b9\u7a0b 1; 2015-07-18 matlab ode45\u89e3\u4e8c\u9636\u5e38\u5fae\u5206\u65b9\u7a0b\u600e\u4e48\u628aY\u7b97\u51fa\u6765 3; 2014-09-13 matlab ode45 \u6c42\u89e3\u4e8c\u9636\u5e38\u5fae\u5206\u65b9\u7a0b 15; 2015-10-02 \u5982\u4f55\u7528matlab\u6c42\u89e3\u4e00\u4e2a\u4e8c\u9636\u5e38\u7cfb\u6570\u5fae\u5206\u65b9\u7a0b\u7ec4 14; 2014-05-10 matlab\u91cc\u7684ode45\u6c42\u89e3\u4e8c\u9636\u5fae\u5206\u65b9\u7a0b\u95ee\u9898\uff01\uff01\u6c42\u5927\u795e\uff01. The matlab function ode45 will be used. Edited: Steven Taggart on 1 Nov 2017 Accepted Answer: Alan Weiss. You can obtain a vector ts and a matrix ys with the coordinates of these points using [ts,ys] = ode45(f,[t0,t1],[y10;y20]). In the first case where the damping constant 'c' is set to 0, I get somewhat expected results, as the sum of the potential (PE) and kinetic (KE) energies is pretty much constant along the entire solution time (I assume it is reasonable for the sum of energies not to be exactly constant in this case since the solver isn't perferct, but feel free to enlighten me if that's not the case). ode45 orbit glitch. update formula similar to the one applied by ODE45, and to look for a di erent adaptive strategy. In essence, snapshots of ft, f, gt, and g are taken at the time this line is executed and embedded within. In some cases involving nonlinear equations, the output is an equivalent lower order differential equation or an integral. A brief introduction to using ode45 in MATLAB. ode45 \u8ba1\u7b97\u7684\u70b9\u6570 356981 Elapsed time is 0. PS: you may want to look at the documentation for ode45 and familiarise yourself with how to solve odes in MATLAB. System that integrates an ODE using ODE45 in its discrete step function. answers\/44752-is-it-possible-to-solve-a. How to solve. Some of the typical uses of MATLAB are given below: \u2022 Math and Computation \u2022 Algorithm Development \u2022 Modeling, Simulation and Prototyping M-Files Files that contain code in MATLAB language are called M-Files. : solution = ode45 (\u2026) Solve a set of non-stiff Ordinary Differential Equations (non-stiff ODEs) with the well known explicit Dormand-Prince method of order 4. Two of these functions are the well-known ode45 and ode15s. If you complete the whole of this tutorial, you will be able to use MATLAB to integrate equations of motion. Ordinary Di\ufb00erential Equations with MATLAB 25 1 f = inline('t-y','t','y'); 2 [t,y] = ode45(f,[0,3],1); 3 plot(t,y) (line numbers are not part of the commands!) Line 1 de\ufb01nes the function fas a function of t and y, i. matlab ode45 | matlab ode45 | matlab ode45 examples | matlab ode45 code | matlab ode45 plot | matlab ode45 event | matlab ode45 solver | matlab ode45 syntax | m. ode45 and ode15s, including event functions), curve fitting to a model,. dsolve can't solve this system. And Simulink blocks set ode45 as the default solver. This function implements a Runge-Kutta method with a variable time step for e cient computation. Solve system of ODEs MATLAB with ode45. At a Matlab prompt type: [t,y]=ode45('yp',[t0,tf],y0); (your version of ode45 may not require brackets around t0,tf) 8 >< >: yp = the. you can find a numeric solution, using the MATLAB ode45 functions (we will learn how to use it later). It is possible to solve multiple-variable systems by making sure the differential function returns values for each of the variables. The ode45 function returns our packaged dependent variables and the independent variable, which we call w and t. asked Oct 25 '17 at 11:59. Learn more about ode45. My > Matlab script uses ode45 (which is not native to Octave, so I downloaded an > ode45. cellstr (Matlab function) chol (Matlab function) cla (Matlab function) clc (Matlab function) clear (Matlab function) clf (Matlab function) clock (Matlab function) close (Matlab function) closereq (Matlab function) colordef (Matlab function) complex (Matlab function) conj (Matlab function) continue (Matlab function) conv (Matlab function) conv2. ode23 is an implementation of an explicit Runge-Kutta (2,3) pair of Bogacki and Shampine. 31\/03\/2016, 14h31 #1 Frisco6969. matlab ode45 impulse | matlab ode45 impulse. ode45 with boundary conditions. t is unlikely to be the exact value 1, meaning it is unlikely to be a valid index into a scalar. If you complete the whole of this tutorial, you will be able to use MATLAB to integrate equations of motion. Hello everybody, I'm using the MATLAB ode solver ode45 to numerically solve some easy differential equations. 2016-01-07 \u5173\u4e8ematlab \u7684ode45\u7528\u6cd5; 2013-11-27 matlab\u4e2dode45 \u53caodeset\u7684\u4f7f\u7528\u3002 8; 2012-09-11 \u6c42\u6559matlab\u9ad8\u624bode45\u7684\u7528\u6cd5\uff01; 2014-01-14 matlab\u4e2dODE45\u51fd\u6570\u8be5\u5982\u4f55\u4f7f\u7528\uff1f. Solve the van der Pol equation with \u03bc = 1 using ode45. First create a MatLab function and name it fun1. The ODE solvers in MATLAB ode45 performs well with most ODE problems and should generally be your first choice of solver. The syntax for ode45 for rst order di erential equations and that for second order di erential equations are basically the same. ode45 is the anchor of the differential equation suite. matlab ode45 impulse | matlab ode45 impulse. \u901a\u8fc7\u8fd9\u4e09\u4e2a\u4f8b\u5b50\u53ef\u4ee5\u770b\u51fa\uff0c\u7528matlab\u89e3\u5404\u7c7b\u65b9\u7a0b\u7ec4\u90fd\u662f\u53ef\u4ee5\u7684\uff0c\u65b9\u6cd5\u4e5f\u6709\u591a\u79cd\uff0c\u53ea\u662f\u7528\u5230\u89e3\u65b9\u7a0b\u7ec4\u7684\u51fd\u6570\uff0c\u6ce8\u610f\u6b63\u786e\u4e66\u5199\u53c2\u6570\u5c31\u53ef\u4ee5\u4e86\uff0c\u975e\u5e38\u65b9\u4fbf\u3002 \u7528 Matlab \u89e3\u65b9\u7a0b\u7ec4 \u7684\u65f6\u5019\uff0c\u53d1\u73b0\u5b83\u4e0d\u80fd\u81ea\u52a8\u4ee3\u5165\u7cfb\u6570\u7684\u503c\u3002\u6bd4\u5982 \u8bf4\u5982\u4e0b\u7684\u7a0b\u5e8f; a=4;. \u2022 Matlab has several different functions (built-ins) for the numerical solution of ODEs. Solve the van der Pol equation with \u03bc = 1 using ode45. The ode45 solver takes longer for each step, but it also takes larger steps. In MATLAB, ode45 solves an ODE system of the form y'=f(t,y) so, in the right and side only y must appear and not its derivatives. However, ode23 and ode113 can be more efficient than ode45 for problems with looser or tighter accuracy requirements. Resuelva la ecuaci\u00f3n de van der Pol con mediante ode45. In the MatLab window, type in the following commands line by line. In the MatLab window,. Especifique una salida \u00fanica para devolver una estructura que contenga informaci\u00f3n acerca de la soluci\u00f3n, como el solver y los puntos de evaluaci\u00f3n. Let's look at step size choice on our problem with near singularity, is a quarter. See MATLAB: Debugger. Functions operate on variables within their own workspace, which is also called the local workspace, separate from the workspace you access at the MATLAB command. Elapsed time is 171. That is, we use >>[x,y]=ode45(f,[0. Choose a web site to get translated content where available and see local events and offers. Key words: Euler's methods, Euler forward, Euler modi\ufb02ed, Euler backward, MAT-LAB, Ordinary di\ufb01erential equation, ODE, ode45. 4: Ejercicios - Jes\u00fas. The basic command to call the ode45 integrator looks like this: [t,state] = ode45(@dstate,time,ICs,options); The integrator takes a vector of initial conditions (either a column or row vector) and integrates it using the dynamics given in the dstate function. It is based on method published by British mathematicians JR Dormand and PJ Prince in 1980. Please try again later. function f=fun1(t,y) f=-t*y\/sqrt(2-y^2); Now use MatLab functions ode23 and ode45 to solve the initial value problem numerically and then plot the numerical solutions y, respectively. x0 can be a scalar, vector, or matrix. ode23 is an implementation of an explicit Runge-Kutta (2,3) pair of Bogacki and Shampine. ode15s \u8ba1\u7b97\u7684\u70b9\u6570 188 [\u6559\u7a0b] \u9690\u5f0f\u5fae\u5206\u65b9\u7a0b(IDE)\u7684 Matlab \u89e3\u6cd5 \u4e0a\u5e1d\u4e0d\u4f1a\u603b\u662f\u90a3\u4e48\u4ec1\u6148\u7684\uff0c\u4e0d\u662f\u6240\u6709\u7684 ODEs \u90fd\u662f\u53ef\u4ee5\u76f4\u63a5\u663e\u5f0f\u7684\u8868\u8fbe\u6210\u4e0b\u9762\u7684\u6837\u5b50 \u6bd4\u5982\uff0c\u4e0b\u9762\u7684\u9690\u5f0f\u5fae\u5206\u65b9\u7a0b\u7ec4 \u90a3\u8be5\u5982\u4f55\u529e\u5462\uff1f. ) Then integrate it with ode45 just as you have with your. Learn more about ode45, ode, function, handles. Just FYI, the ComputeVelocity looks like this:. m \u304c\u65b9\u7a0b\u5f0f\u3092\u30a8\u30f3\u30b3\u30fc\u30c9\u3057\u307e\u3059\u3002\u30bd\u30eb\u30d0\u30fc\u3084\u8a55\u4fa1\u70b9\u306a\u3069\u306e\u89e3\u306b\u95a2\u3059\u308b\u60c5\u5831\u3092\u3082\u3064\u69cb\u9020\u4f53\u3092\u8fd4\u3059\u305f\u3081\u306b\u30011 \u3064\u306e\u51fa\u529b\u3092\u6307\u5b9a\u3057\u307e\u3059\u3002. I need to solve two sets of coupled differential equations, dx = f(x,u) and du = g(x,u), using ode45. Don't let the length of the video scare you. The matlab function ode45 will be used. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Learn more about ode45, ode MATLAB. It solves a system of differential equations numerically, to describe a system at different points in time. ode45 is designed to handle the following general problem = \u20ac dy dt f (t, y. ODE45 running infinitely without solving. fun is a function handle, inline function, or string containing the name of the function that defines the ODE: y' = f(t,y). Learn more about ode, ode45, function, matlab function, euler, plot. MATLAB has many ODE solvers available for the coder. All of this is happening using Matlab R2015b, also no way to get around that. Or, if on the right hand side a variable y appears, an additional. This function implements a Runge-Kutta method with a variable time step for e cient computation. Learn more about differential equations. Getting Help in MATLAB. I'm attempting to propagate the angular velocity, w0 over a span of 100 sec, using ode45. Hence, w e will use ode45 solver. And Simulink blocks set ode45 as the default solver. Solve Nonstiff ODEs. In MATLAB its coordinates are x(1),x(2),x(3) so I can write the right side of the system as a MATLAB function. m function (system), time-span and initial-condition (x0) only. This shows how to use Matlab to solve standard engineering problems which involves solving a standard second order ODE. Follow 12 views (last 30 days) Steven Taggart on 29 Oct 2017. 5], and MATLAB has returned a value of y at each point in this partition. How to stop ode45 when value reach certain value other-than zero. for initial condition of 0 to final condition (1. ode23 is an implementation of an explicit Runge-Kutta (2,3) pair of Bogacki and Shampine. In general, ode45 is the best function to apply as a \"first try\" for most problems. lsim(sys,u,t) lsim(sys,u,t,x0) lsim(sys,u,t,x0,'zoh') lsim(sys,u,t,x0,'foh') lsim(sys1,sys2,,sysN,u. It happens to be in the odepkg Octave-Forge package. Use the 'ode45' command in MATLAB to generate the solution v(t) of the following differential equation: 66v 130v 260 v(0 ) 1 v(0 ) 0. It may be more efficient than ode45 at crude tolerances and in the presence of moderate stiffness. 1, 10]): Input Specifications. can get MATLAB to solve them for you, using the MATLAB functions ode45() or nareul(). x0 can be a scalar, vector, or matrix. Solving ODEs using MatLab A command used to solve ODE's in MatLab (a \\solver\") is ode45 Enter >> help ode45 to see information about this command. ode45 orbit glitch. It may be beneficial to test more than one solver on a given problem. ) Since x and y are. ode23 is a single-step solver ,. 001:B] Matlab misses all the pulses and does not output the correct solution. How to solve. m les are quite di erent. The result p is a row vector of length n+1 containing the polynomial coefficients in descending powers. Trouble with ODE45 for an array of values. 397287 seconds. fun is a function handle, inline function, or string containing the name of the function that defines the ODE: y' = f(t,y). I can try with that. Defining f is probably the most difficult part. ODE(Ordinary Differential Equation)\uc774\ub780 \ub9d0\uc758 \ub2e8\uc5b4\ub4e4\uc744 \ud558\ub098\uc529 \uc54c\uc544\ubcf4\uc790. A non-linear second order ODE was solved numerically using Matlab\u2019s ode45. La funci\u00f3n vdp1. Key words: Euler's methods, Euler forward, Euler modi\ufb02ed, Euler backward, MAT-LAB, Ordinary di\ufb01erential equation, ODE, ode45. Discover Scilab Cloud. In order to distinguish between the two. Solve the ODE using the ode45 function on the time interval [0 20] with initial values [2 0]. As an example, the function ode45 is used to solve the equation of motion for a driven-damped mass\/spring system. can get MATLAB to solve them for you, using the MATLAB functions ode45() or nareul(). This function implements a Runge-Kutta method with a variable time step for ecient computation. A brief introduction to using ode45 in MATLAB MATLAB's standard solver for ordinary di erential equations (ODEs) is the function ode45. ode45 is designed to handle the following general problem = \u20ac dy dt f (t, y. Matlab ode45 numerical solution. Open Script. Toggle Main Navigation. Solving differential equation using ode45 with Learn more about ode45, second-order, differential equation. , ode45, ode23) Handle for function containing the derivatives Vector that speci\ufb01ecs the. That syntax does work for ode45 for backwards compatibility, but newer \"function functions\" (like the DDE solver ddesd) likely will not accept that syntax. System that integrates an ODE using ODE45 in its discrete step function. This means that the solution to the differential equation may not be defined for t=0. ode45 - changing parameters & function handle. m from sourceforge), I would not say that ode45 \"is not native to Octave\". Simulating impedance using ODE45. dsolve can't solve this system. It requires six function evaluations per integration step, but may take larger steps on smooth problems than ode23 : potentially offering improved efficiency at smaller tolerances. It is an easy method to use when you have a hard time solving a differential equation and are interested in approximating the behavior of the equation in a certain range. Start with the State-Variable Modeling, then set the MATLAB code with the derivative function and the ODE solver. The circles mark the values which were actually computed (the points are chosen by Matlab to optimize accuracy and efficiency). MATLAB is an interactive program for numerical computation and data visualization; it is used extensively by control. Please detailed step by step codes with comments. function first_oder_ode % SOLVE dx\/dt = -3 exp(-t). Learn more about ode45 vs dsolve, ode45. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. MATLAB ODE45() Solve the following State-variable. Use when integrating over small intervals or when accuracy is less important than speed -ode45 \u2022 High order (Runge-Kutta) solver. In the menu bar of the MatLab Command Window, select the New Script button in the File pane of the Home tab. Bucknell University Using ODE45 1 Bucknell University Using ODE45 MATLAB Help MATLAB's standard solver for ordinary differential equations (ODEs) is the function ode45. Use the result of that first ode45 call at time t = 700 as the initial condition for a second call to ode45 that solves a system of ODEs without torque from time t = 700 to time t = 1000. In approximating this solution, the algorithm ode45 has selected a certain partition of the interval [0,. loop to solve ode45 at different intial Learn more about for loop. function f=fun1(t,y) f=-t*y\/sqrt(2-y^2); Now use MatLab functions ode23 and ode45 to solve the initial value problem numerically and then plot the numerical solutions y, respectively. ode45 where odefun requires more parameters. In MATLAB its. Ode4 Matlab Ode4 Matlab. Active 6 years ago. In order to distinguish between the two. Octave ode45 vs. [\u539f\u521b]Matlab \u4e2d\u89e3\u5e38\u5fae\u5206\u65b9\u7a0b\u7684 ode45 2010-09-24 15:18 ode \u662f\u4e13\u95e8\u7528\u4e8e\u89e3\u5fae\u5206\u65b9\u7a0b\u7684\u529f\u80fd\u51fd\u6570\uff0c\u4ed6\u6709 ode23,ode45,ode23s \u7b49\u7b49\uff0c\u91c7\u7528\u7684\u662f Runge-Kutta \u7b97\u6cd5\u3002 ode45 \u8868\u793a\u91c7\u7528\u56db\u9636\uff0c\u4e94\u9636 runge-kutta \u5355\u6b65\u7b97\u6cd5,\u622a\u65ad\u8bef\u5dee\u4e3a(\u0394x)^3\u3002. 001:3; % time scale initial_x = 0; initial_dxdt = 0. Solve Nonstiff ODEs. You shouldn't be trying to index into them using the variable t. The calling sequence is [t,y] = ode45('rhs',tspan,y0) The term in quotes, 'rhs', is the name of the script which defines the problem. The Matlab ODE functions are variable-step methods, meaning. Learn more about ode45, ode, function, handles. The matlab function ode45 will be used. Hi, k is constant and MATLAB should calculate it from the 2 conditions, so can you. d y d x = f (x, y),. However, I need to vary the equation inputs, u1 and u2, at every time step. The syntax is a hybrid of C and Fortran and I often create polyglot statements which lead to the joyless task of squashing trivial bugs. 001:B] Matlab misses all the pulses and does not output the correct solution. ode45 - Di erential Equation Solver This routine uses a variable step Runge-Kutta Method to solve di erential equations numerically. fun is a function handle. m MATLAB\u2019s pre-built ode45. Follow 157 views (last 30 days) Dereje on 26 Apr 2018. Therefore to solve a higher order ODE, the ODE has to be \ufb01rst converted to a set of \ufb01rst order ODE's. We use an example of integrating an ODE that defines the van der Waal equation of an ideal gas here. Ode45 extra parameters. d y d x = f (x, y),. A cleaner (IMO) approach would be to solve your system of ODEs with torque applied from time t = 0 to time t = 700. Follow 12 views (last 30 days) Steven Taggart on 29 Oct 2017. m ships with MATLAB\u00ae and encodes the equations. Both of them use a similar numerical formula, Runge-Kutta, but to a different order of approximation. The basic method is order five. Here is the simplest possible example of what ode45 is doing, that also inputs an extra parameter. The variables you define inside your ODE function as each containing one element of the Y input with which ode45 calls your ODE function (y, v, h, x, and m) are scalars. Then use the final value of one interval as initial value of the next interval and define the new value of the parameter. Solve a 2nd Order ODE: Damped, Driven Simple Harmonic Oscillator. Description. m t0,tf = initial and terminal values of t y0 = initial value of y at t0 C. I need to solve two sets of coupled differential equations, dx = f(x,u) and du = g(x,u), using ode45. x = fminsearch(fun,x0) starts at the point x0 and finds a local minimum x of the function described in fun. Commented: Jan on 29 Feb 2016 Accepted Answer: Jan. On 17 December 2012 21:44, alexvas <[hidden email]> wrote: > I am an avid user of Matlab, but currently only have access to Octave. program using to solve twolevel Bloch equations by matlab ode45 method - hupidong\/twolevel_ode45_matlab. ODE45 - Modeling GHR Microspic Car following Learn more about ode45, car following microscopic model. We wish to solve. matlab\u4e2d\u7684ode45. Learn more about ode45 MATLAB. Functions operate on variables within their own workspace, which is also called the local workspace, separate from the workspace you access at the MATLAB command. fun is a function handle. A brief introduction to using ode45 in MATLAB MATLAB\u2019s standard solver for ordinary di erential equations (ODEs) is the function ode45. m integrator is generally the best option. program using to solve twolevel Bloch equations by matlab ode45 method - hupidong\/twolevel_ode45_matlab. In this case t and y are both scalars, and that is the only interface that ode45 knows about directly. m integrator should be faster than the smart ODE45 with stepsize control - if and only if the number of steps is smaller. matlab ode45 wont accept new parameter. (That's relatively easily done, and if you don't want to do it yourself and if you have the Symbolic Math Toolbox, you can use the odeToVectorField function and matlabFunction to do it for you. Compare Results of Different Solvers. Solve the van der Pol equation with \u03bc = 1 using ode45. The first element of the vector tv is the initial t value; the vector tv must have at least 3 elements. We wish to solve. ode45 is designed to handle the following general problem: dx dt. 4*y function dy = firstorder(t,y). Follow 140 views (last 30 days) John Greene on 22 Oct 2017. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. There are. Sometimes they do not, and it is not always obvious they have not worked! Part of using a tool like Matlab is checking how well your solution really worked. ) Since x and y are. \u2022 Example syntax for variable step size of independent parameter (if we want to specify a fixed step size, use, e. In the menu bar of the MatLab Command Window, select the New Script button in the File pane of the Home tab. It solves a system of differential equations numerically, to describe a system at different points in time. It is convenient to program it in a separate le. with y0 and y0 + sqrt(eps). The circles mark the values which were actually computed (the points are chosen by Matlab to optimize accuracy and efficiency). error ode45 (line 115). Dynamics and Vibrations MATLAB tutorial School of Engineering Brown University This tutorial is intended to provide a crash-course on using a small subset of the features of MATLAB. We wish to solve. pdf: Free Vibration of a Single-degree-of-freedom System with Nonlinear Stiffness. I am not sure about this, better go to a Matlab forum and ask there. In MATLAB, ode45 solves an ODE system of the form y'=f(t,y) so, in the right and side only y must appear and not its derivatives. Using ode45 to solve Ordinary Differential Equations Matlab's standard solver for ordinary differential equations is the function ode45. The bvp4c solver returns the final values of these unknown parameters in sol. Stiffness is a term that defies a. Now a word about this particular book. Defining f is probably the most difficult part. The system will. The result p is a row vector of length n+1 containing the polynomial coefficients in descending powers. Or, if on the right hand side a variable y appears, an additional. ode45 with an input. How to solve. The syntax for actually solving a differential equation with these functions is: [T,Y] = ode45('yprime',t0,tF,y0);. Learn more about ode45 vs dsolve, ode45. All of this is happening using Matlab R2015b, also no way to get around that. (The MATLAB output is fairly long, so I\u2019ve omitted it here. MATLAB supports the basic flow control constructs found in most high level programming languages. Commented: John Greene on 22 Oct 2017. If you use the command odeset with no inputs, then MATLAB ode45 ode23 ode113 ode15s ode23s ode23t ode23tb ode15i;. MATLAB has many ODE solvers available for the coder. Can you give me some hints? ODE function ode15s instead of. Hey Ebraheem There are many excellent methods that you can use to solve your problem, for instance, the finite difference method is a very powerful method to use. Abbasi May 30, 2012 page compiled on July 1, 2015 at 11:43am Contents 1 download examples source code 1 2 description 1 3 Simulation 3 4 Using ode45 with piecewise function 5 5 Listing of source code 5 1download examples source code 1. The name of the file and of the function should be the same. ode45 is a solver with adaptive time step, and sometimes integration can fail, so solver automatically decreases time step, and \"has to go back\". Use the 'ode45' command in MATLAB to generate the solution v(t) of the following differential equation: 66v 130v 260 v(0 ) 1 v(0 ) 0. % MCE 372 Engineering Analysis Example Code The solutions from either code are identical and the plot is shown below % First Order ODE Solution Using ode45 with User Function Method % Example From Gilat 4th Ed. I particularly like it for graphics. MATLAB includes several example files that show how to use various options. Is it possible to solve a nonlinear system with signum function using ODE45? Follow 31 views (last 30 days) field of numerics). The game is aligned with the 'Roads' series, which features the titles Liberty Roads and Victory Roads (published by Hexasim). 5],1) and MATLAB returns two column vectors, the \ufb01rst with values of x and the second with values of y. Any help will be highly appreciated. This would help you to find the mistake I made: ode45 provides the trajectory y as [numel(t) x 2] matrix, but I assumed a [2 x numel(t)] array. See \"help ode45\" for a more detailed explanation. You will see various ways of using Matlab\/Octave to solve various differential equations Octave\/Matlab - Differential Equation Home : www. d y d x = f (x, y),. ODE Solvers: Matlab \u2022Matlab contains implementations of common ODE solvers \u2022Using the correct ODE solver can save you lots of time and give more accurate results -ode23 \u2022 Low-order solver. DA: 85 PA: 59 MOZ Rank: 14. See Function Handles in the MATLAB Programming documentation for more information. (The MATLAB output is fairly long, so I\u2019ve omitted it here. Each row in y corresponds to a time returned in the corresponding row of t. dsolve can't solve this system. This system should then be used in a Simulink Matlab System block with code generation enabled (no way to get around that since it needs to run on a real time system). 1, 10]): Input Specifications. I'm not acquainted with how the ode45 works but lately I started reading about Euler's method in this book Numerical Methods for Engineers. This program solves the nonlinear equation of motion such that large angular displacements are allowed. Examples of ode45 with function M-file One dependent variable with a function m-file (most common) First-order reaction dy\/dt = -0. La funci\u00f3n vdp1. Keyword Research: People who searched matlab ode45 also searched. Follow 144 views (last 30 days) John Greene on 22 Oct 2017. In MATLAB, functions are defined in separate files. log in sign up. \ubc14\ub85c, \uc9c4\ud654\ud558\ub294 \uacf5\ub300\uc0dd\uc758 \uacc4\uc0b0\uae30\uc758 \ucd5c. Learn more about matlab, ode45 MATLAB. 2 2v 18v % First m-file to start the simulation and plot the results. This function implements a Runge-Kutta method with a variable time step for efficient computation. Learn more about ode45, input, integration. - am304 Nov 30 '18 at 8:33. 001:3; % time scale initial_x = 0; initial_dxdt = 0. ode45() is a sophisticated built-in MATLAB function that gives very accurate solutions. In this section we will demonstrate how to use the inbuilt MATLAB ODE solvers such as ode45. All of this is happening using Matlab R2015b, also no way to get around that. Learn more about differential equations. Two example MATLAB codes using the ode45 solver are shown below. The main code that utilized and presented is MATLAB\/ode45 to enable the students solving initial value DE and experience the response of the engineering systems for different applied conditions. MATLAB\u2019s standard solver for ordinary dierential equations (ODEs) is the function ode45. ODE45 is very accurate. You have to describe your second-order ODE as two first-order ODEs, just as you have with your first ODE. MATLAB supports the basic flow control constructs found in most high level programming languages. , ode45, ode23) Handle for function containing the derivatives Vector that speci\ufb01ecs the. Like ode45, ode23 is a one-step solver. : solution = ode45 (\u2026) Solve a set of non-stiff Ordinary Differential Equations (non-stiff ODEs) with the well known explicit Dormand-Prince method of order 4. A brief introduction to using ode45 in MATLAB MATLAB\u2019s standard solver for ordinary di erential equations (ODEs) is the function ode45. A non-linear second order ODE was solved numerically using Matlab\u2019s ode45. This means that the solution to the differential equation may not be defined for t=0. \u2013 horchler Sep 27 '13 at 15:28 @horchler I agree. And here we get a graph of cosine t starting at 1, and sine t starting at 0. These solvers can be used with the following syntax: [outputs] = function_handle(inputs) [t,state] = solver(@dstate,tspan,ICs,options) Matlab algorithm (e. Choose an ODE Solver Ordinary Differential Equations. My > Matlab script uses ode45 (which is not native to Octave, so I downloaded an > ode45. We have always had BOTH 0-based indexing and 1-based indexing. A function is a group of statements that together perform a task. The main issue I had though, is that I am used to using ode45 to solve differential equations. In this section we will demonstrate how to use the inbuilt MATLAB ODE solvers such as ode45. This function implements a Runge-Kutta method with a variable time step for efficient computation. \u2022 Example syntax for variable step size of independent parameter (if we want to specify a fixed step size, use, e. I'm not acquainted with how the ode45 works but lately I started reading about Euler's method in this book Numerical Methods for Engineers. This is done by clicking Start-> All Programs->Math Programs-> MATLAB R2015a. Don't let the length of the video scare you. MATLAB\u2019s standard solver for ordinary dierential equations (ODEs) is the function ode45. >> [tv1 f1]=ode23('fun1',[0 5],1);. 31\/03\/2016, 14h31 #1 Frisco6969. I'm trying to create a MATLAB script that finds the maximum point of a given 3D function with gradient descent. Use when integrating over small intervals or when accuracy is less important than speed -ode45 \u2022 High order (Runge-Kutta) solver. ode45(f,[0,20],[1;0]) This shows the two functions y 1 (t)=y(t) (blue) and y 2 (t)=y'(t) (green). matlab ode45 out of memory. ) Since x and y are. ode45 or dsolve. The numerical solution has to be in agreement with the analytic one. update formula similar to the one applied by ODE45, and to look for a di erent adaptive strategy. In this section we will demonstrate how to use the inbuilt MATLAB ODE solvers such as ode45. The solution to the Van Der Pol was found to contain a limit cycle in the phase portrait when starting from any initial conditions. Once you have all these components, ode45 is ready to use. Follow 140 views (last 30 days) John Greene on 22 Oct 2017. (This is essentially the Taylor method of order 4, though. Changing Inputs per Time Step in ODE45 Hi, I am a new MATLAB user and I am attempting to use the ODE45 solver to solve a system of 2 1st order differential equations. The circles mark the values which were actually computed (the points are chosen by Matlab to optimize accuracy and efficiency). MATLAB Examples on the use of ode23 and ode45: First create a MatLab function and name it fun1. This function implements a Runge-Kutta method with a variable time step for e cient computation. k is constant and y(0)=40 and y(15)=95 solve this equation by using ode45 can someone pleaseeeeeeeeeee check the code and make it work. It is based on method published by British mathematicians JR Dormand and PJ Prince in 1980. To begin, open the MatLab application. bvp4c produces a solution that is continuous on [a,b] and has a continuous first derivative there. nareul ode45() and nareul() are two MATLAB functions that do basically the same thing. 31\/03\/2016, 14h31 #1 Frisco6969. The basic command to call the ode45 integrator looks like this: [t,state] = ode45(@dstate,time,ICs,options); The integrator takes a vector of initial conditions (either a column or row vector) and integrates it using the dynamics given in the dstate function. % initial conditions: x(0) = 0 t=0:0. Examples of ode45 with function M-file One dependent variable with a function m-file (most common) First-order reaction dy\/dt = -0. Does anyone have suggestions on where I'm going wrong? Thanks. Or, if on the right hand side a variable y appears, an additional. ode45\uff0c\u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u6570\u503c\u6c42\u89e3\u3002MATLAB\u63d0\u4f9b\u4e86\u6c42\u5e38\u5fae\u5206\u65b9\u7a0b\u6570\u503c\u89e3\u7684\u51fd\u6570\u3002\u5f53\u96be\u4ee5\u6c42\u5f97\u5fae\u5206\u65b9\u7a0b\u7684\u89e3\u6790\u89e3\u65f6\uff0c\u53ef\u4ee5\u6c42\u5176\u6570\u503c\u89e3\uff0cMatlab\u4e2d\u6c42\u5fae\u5206\u65b9\u7a0b\u6570\u503c\u89e3\u7684\u51fd\u6570\u6709\u4e03\u4e2a\uff1aode45\uff0code23\uff0code113\uff0code15s\uff0code23s\uff0code23t\uff0code23tb \u3002. Solve the ODE using the ode45 function on the time interval [0 20] with initial values [2 0]. It requires six function evaluations per integration step, but may take larger steps on smooth problems than ode23 : potentially offering improved efficiency at smaller tolerances. ode45 is designed to handle the following general problem: dx dt. (That\u2019s relatively easily done, and if you don\u2019t want to do it yourself and if you have the Symbolic Math Toolbox, you can use the odeToVectorField function and matlabFunction to do it for you. m\uc740 MATLAB\u00ae\uacfc \ud568\uaed8 \uc81c\uacf5\ub418\ub294 \ud568\uc218\ub85c, \uc774 \ubc29\uc815\uc2dd\uc744 \ub2f4\uace0 \uc788\uc2b5\ub2c8\ub2e4. And Simulink blocks set ode45 as the default solver. The function vdp1. (constant coe cients with initial conditions and nonhomogeneous). matlab ode45 how to change a parameter inside the function continiously. This is very strange. Matlab Ode45; Affichage des r\u00e9sultats 1 \u00e0 1 sur 1 Matlab Ode45. Trouble with ODE45 for an array of values. Polynomials Using the s Variable. s9d901ffg9rs4, bfg8w33bsilovgc, i04u5o3wvwvwban, z7xvkcjo500, dwjvv9f6460bk46, qpn1lptm34rq, sz2xdqhcv4ll5, t8r0grwfi92k9el, w463x51b25xwi, ek9ywooj0dk, ct8hn1t57cu, cy1rw76jp615pjs, ckpzku033zrkz2, od3xlsnvxs3zo, kd52rqgst53, qi4s2cqwt9odvu, xlyp9oabds, q6xm4ibcg7i5, kca90rhm9o, sx86v9ftk6, 3ik7c54066r0u3r, 3gehpmukgqjp2yu, f8g342facdaai, z8i3a5hka5krhz2, y3j647gi640ew, mupq1hbalou6, hruxbz5rt5fwno, 76rp701bexzsp","date":"2020-05-28 20:56:49","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\": 0.6234309673309326, \"perplexity\": 1684.5776879396506}, \"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-2020-24\/segments\/1590347400101.39\/warc\/CC-MAIN-20200528201823-20200528231823-00517.warc.gz\"}"}
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{"url":"https:\/\/stackoverflow.com\/questions\/9735578\/building-a-notification-system\/9736277","text":"# Building a notification system [closed]\n\nI am at the start of building a Facebook style notification system for our page (social gaming type) and I'm now researching what would be the best way to design such system. I'm not interested in how to push notifications to the user or anything like that (for now even). I am researching how to build the system on the server (how to store notifications, where to store them, how to fetch them etc...).\n\nSo ... some requirements that we have:\n\n\u2022 at peak times we have about 1k concurrent logged-in users (and many more guests, but they don't matter here as they will not have notifications) that will generate many events\n\u2022 there will be different types of notifications (user A has added you as a friend, user B has commented on your profile, user C has liked your image, user D has beaten you on game X, ...)\n\u2022 most events will generate 1 notification for 1 user (user X has liked your image), but there will be cases where one event will generate many notifications (it's user Y's birthday for instance)\n\u2022 notifications should be grouped together; if for instance four different users like some image, the owner of that image should get one notification stating that four users have liked the image and not four separate notifications (just like FB does)\n\nOK so what I was thinking is that I should create some sort of queue where I would store events when they happen. Then I would have a background job (gearman?) that would look at that queue and generate notifications based on those events. This job would then store notifications in the database for each user (so if an event affects 10 users, there would be 10 separate notifications). Then when user would open a page with the list of notifications I would read all those notifications for him (we ware thinking to limiting this to 100 latest notifications) and group them together and then finally display them.\n\nThings I'm concerned about with this approach:\n\n\u2022 complex as hell :)\n\u2022 is database the best storage here (we are using MySQL) or should I use something else (redis seems like a good fit too)\n\u2022 what should I store as a notification? user ID, user ID who initiated the event, type of event (so that I can group them and display appropriate text) but then I kinda don't know how to store the actual data of the notification (for instance URL&title of the image that was liked). Should I just \"bake\" that info when I generate the notification, or should I store the ID of the record (image, profile, ...) being affected and pull the info out of the DB when displaying the notification.\n\u2022 performance should be OK here, even if I have to process 100 notifications on-the-fly when displaying the notifications page\n\u2022 possible performance problem on every request because I would have to display the number of unread notifications to the user (which could be a problem in its own since I would group notifications together). This could be avoided though if I generated the view of notifications (where they are grouped) in the background and not on-the-fly\n\nSo what do you think about my proposed solution and my concerns? Please comment if you think I should mention anything else that would be relevant here.\n\nOh, we are using PHP for our page, but that shouldn't be a big factor here I think.\n\n## closed as too broad by Stephen Kennedy, EJoshuaS, EdChum, Mark Rotteveel, greg-449Oct 31 '18 at 10:13\n\nPlease edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.\n\n## locked by Yvette ColombDec 6 '18 at 8:19\n\nThis question exists because it has historical significance, but it is not considered a good, on-topic question for this site so please do not use it as evidence that you can ask similar questions here. This question and its answers are frozen and cannot be changed. See the help center for guidance on writing a good question.\n\n\u2022 How much time it took you to build this notification system as one man efforts. I just want to have an estimation to make the timelines accordingly. \u2013\u00a0Shaharyar Nov 24 '15 at 10:55\n\u2022 @Shaharyar I think it depends on the complexity of the notification system. \u2013\u00a0tyan Apr 20 '16 at 6:43\n\u2022 I used the same system with MySQL to build a priority based notification system. The good thing is that it scales to a few thousand users, if it goes more than that, it blows up, specially with Android and GCM. I would like to know alternatives to MySQL like redis, rabbitMQ, Kafka which naturally exhibit a message queue, kind of functionality. \u2013\u00a0Ankit Marothi Jan 23 '17 at 13:57\n\nA notification is about something (object = event, friendship..) being changed (verb = added, requested..) by someone (actor) and reported to the user (subject). Here is a normalized data structure (though I've used MongoDB). You need to notify certain users about changes. So it's per-user notifications.. meaning that if there were 100 users involved, you generate 100 notifications.\n\n\u2554\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2557 \u2554\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2557 \u2554\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2557\n\u255f\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2562 \u255f\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2562 \u255f\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2562\n\u2551ID \u2551\u20141:n\u2014\u2192\u2551ID \u2551\u20141:n\u2014\u2192\u2551ID \u2551\n\u255a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255d \u2551object \u2551 \u2551verb \u2551\n\u255a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255d \u2551actor \u2551\n\u255a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255d\n\n\n(Add time fields where you see fit)\n\nThis is basically for grouping changes per object, so that you could say \"You have 3 friend requests\". And grouping per actor is useful, so that you could say \"User James Bond made changes in your bed\". This also gives ability to translate and count notifications as you like.\n\nBut, since object is just an ID, you would need to get all extra info about object you want with separate calls, unless object actually changes and you want to show that history (so for example \"user changed title of event to ...\")\n\nSince notifications are close to realtime for users on the site, I would tie them with nodejs + websockets client with php pushing update to nodejs for all listeners as change gets added.\n\n\u2022 notification_object.object identifies change type, like a string \"friendship\" The actual reference to changed object with its extra data that I talk about is in notification_change.notificationObjectID \u2013\u00a0Artjom Kurapov Apr 29 '13 at 7:35\n\u2022 This may be a dumb question but with this set up what do you do once the user has seen or acted on the notification? Do you just remove it from the database or just use dates to see if the user has logged in since the notification was created? \u2013\u00a0Jeffery Mills Feb 17 '14 at 3:16\n\u2022 I know this topic is already quite old, however i'm a bit puzzled about the first table, what excactly is the purpose of this table? what is the advantage of having this as a separate table versus putting the userID in the notification_object table? In other words when will you create a new entry in notification and when will you just add an object and change to an existing notification with this structure? \u2013\u00a0Bas Goossen Feb 27 '14 at 18:48\n\u2022 @JefferyMills You could have a status field like is_notification_read in the notification table and mark it appropriately if it is unread, read or deleted. \u2013\u00a0Kevin Jun 15 '14 at 14:42\n\u2022 I also have struggled to understand some aspects of this solution, and made a separate question about it: dba.stackexchange.com\/questions\/99401\/\u2026 \u2013\u00a0user45623 May 4 '15 at 6:37\n\nThis is really an abstract question, so I guess we are just going to have to discuss it instead of pointing out what you should or shouldn't do.\n\n\u2022 Yes, a notification system is complex, but not as hell though. You can have many different approaches on modeling and implementing such systems, and they can have from a medium to a high-level of complexity;\n\n\u2022 Pesonally, I always try to make stuff database-driven. Why? Because I can guarantee having full control of everything that's going on - but that's just me, you can have control without a database-driven approach; trust me, you are gonna want control on that case;\n\n\u2022 Let me exemplify a real case for you, so you can start from somewhere. In the past year I've modeled and implemented a notification system in some kind of a social network (not like facebook, of course). The way I used to store notifications there? I had a notifications table, where I kept the generator_user_id (the ID of the user that is generating the notification), the target_user_id (kind of obvious, isn't it?), the notification_type_id (that referenced to a different table with notification types), and all that necessary stuff we need to fill our tables with (timestamps, flags, etc). My notification_types table used to have a relation with a notification_templates table, that stored specific templates for each type of notification. For instance, I had a POST_REPLY type, that had a template kind of like {USER} HAS REPLIED ONE OF YOUR #POSTS. From there, I just treated the {} as a variable and the # as a reference link;\n\n\u2022 Yes, performance should and must be ok. When you think of notifications you think of server pushing from head to toe. Either if you are going to do it with ajax requests or whatever, you are gonna have to worry about performance. But I think that's a second time concern;\n\nThat model that I've designed is, of course, not the only one that you can follow, neither the best as well. I hope my answer, at least, follows you into the right direction.\n\n\u2022 Why wouldn't I have control with some other data store? \u2013\u00a0Jan Han\u010di\u010d Mar 16 '12 at 11:40\n\u2022 Well, I didn't say that. What I said is that I can only guarantee data control with a database-driven approach; but that's just me. I'm gonna rephrase that. \u2013\u00a0Daniel Ribeiro Mar 16 '12 at 11:44\n\u2022 @DanielRibeiro the placeholders ({...}) in the notification template need to replace data of placeholders from the different set of tables in the database for the different type of notifications. E.g. one template is \"{user} has liked your photo.\", another template is \"Your {Pagename} has new like.\" Etc. {PageName} and {user} and other placeholders will map from the different database table, so what should be the schema to get the placeholders value dynamically. \u2013\u00a0Ashish Shukla Mar 27 '17 at 14:43\n\u2022 DanielRibeiro how you replaced placeholders as asked by @Ashish Shukla, \u2013\u00a0Shantaram Tupe Nov 16 '17 at 6:36\n\u2022 @AshishShukla have you used or replaced placeholders, and how ? \u2013\u00a0Shantaram Tupe Nov 16 '17 at 6:36\n\u2554\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2557\n\u255f\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2562\n\u2551Object \u2551\n\u2551verb \u2551\n\u2551actor \u2551\n\u255a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255d\n\n\nThis looks a good answer rather than having 2 collections. You can query by username, object and isRead to get new events(like 3 pending friend requests, 4 questions asked etc...)\n\nLet me know if there is problem with this schema.\n\n\u2022 The top answer used a normalized data structure, which means no redundancies in the tables. Does your answer do that? \u2013\u00a0Aaron Hall Oct 8 '14 at 13:52\n\nI personally don't understand very well the diagram for the accepted answer, So I'm going to attach a database diagram base on what I could learn from the accepted answer and other pages.","date":"2019-10-16 16:03:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.23335377871990204, \"perplexity\": 1588.4283264522062}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986668994.39\/warc\/CC-MAIN-20191016135759-20191016163259-00427.warc.gz\"}"}
| null | null |
Q: Close tab after redirecting to a new tab in Firefox I am trying to close a tab using JavaScript on Firefox. I went through a few older questions posted here on Stackoverflow and I understand that Firefox doesn't allow a tab to be closed using JavaScript unless the tab has been opened from JavaScript itself.
However, I could open a blank tab in place of the existing tab using
window.open('about:blank','_self','');.
My question is since the blank tab has been opened through JavaScript, why can't I close it using window.close();?
A:
My question is since the blank tab has been opened through JavaScript
The tab wasn't opened with JS. A window.open call just used the name of an existing tab so the document was loaded there instead of opening a new window.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 9,299
|
The first miracle was a wedding miracle. That's right; you read it correctly. The first miracle performed by Jesus was at a wedding. Jesus made His debut of performing miracles at a wedding. He turned water into wine.
Weddings instill love, excitement and joy. And some stress too. It's always been a big deal no matter how big or small, private or public, simple or over the top, weddings are a big deal. And that's a good thing. They are a covenant and a vow. These things should not be taken lightly.
Remember when you are planning your big day that there'll be unforeseen problems. There always have been and always will be problems that come up. How will you handle these hiccups? I recommend saying a little prayer before your day starts. Then lean into your faith. As I told you, it was at wedding that Jesus performed his first miracle.
Did you know this? Ever wonder why he chose a wedding?
Imagine your wedding being the talk of the town – possibly the talk of the region. You've invited all the important folks. You even invited a "highly favored & important" mother, her amazing son and 12 of his friends. These guys were very popular in those days. The ceremony had taken place and the reception was underway. Wine was being shared. Toasts were being made with lots of celebrating and lots of toasting. I am certain it was a festive celebration.
Suddenly no more wine! Wait, what? No more wine! How could this happen? Panic sets in. Lot's of folks looking at their feet. No one wants to take responsibility for such an oversight. The servants began to search the room for someone who can make this right. Mary shares this with Jesus. By now she knows His capabilities. Everyone knows how important weddings are, even Mary.
Jesus then asks her why would she involve Him. He tells her His time is not yet. But I suspect even He knew how important weddings are. Mary walks over to the servants and tells them to do exactly what her son tells them.
Jesus tells them to fill their best stone pitchers with water – to the brim. Then he tells them to serve the most important guest, the governor nonetheless, first. The governor drinks and is astonished. He calls over the groom. I wonder if the groom knew the dilemma. I like to think he knew and was wondering what was about to happen. The governor thanked the groom for serving the best wine last. The governor realized most weddings served the best first and served cheaper wine for the toasts of the lessor guests.
So what's all this have to do with your wedding? Glad you asked. If you are lucky (blessed) your wedding will be perfect. But don't count on it. Hiccups happen at every wedding. But when they do happen, how will you handle them? I suggest you don't make a fuss. Lean into your faith. Let a miracle happen. After all, the first miracle was a wedding miracle.
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<li class="navelem"><b>sf</b></li><li class="navelem"><a class="el" href="classsf_1_1IpAddress.html">IpAddress</a></li> </ul>
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<div class="title">sf::IpAddress Member List</div> </div>
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<p>This is the complete list of members for <a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a>, including all inherited members.</p>
<table class="directory">
<tr class="even"><td class="entry"><a class="el" href="classsf_1_1IpAddress.html#a3dbc10b0dc6804cc69e29342f7406907">Any</a></td><td class="entry"><a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a></td><td class="entry"><span class="mlabel">static</span></td></tr>
<tr><td class="entry"><a class="el" href="classsf_1_1IpAddress.html#aa93d1d57b65d243f2baf804b6035465c">Broadcast</a></td><td class="entry"><a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a></td><td class="entry"><span class="mlabel">static</span></td></tr>
<tr class="even"><td class="entry"><a class="el" href="classsf_1_1IpAddress.html#a4c31622ad87edca48adbb8e8ed00ee4a">getLocalAddress</a>()</td><td class="entry"><a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a></td><td class="entry"><span class="mlabel">static</span></td></tr>
<tr><td class="entry"><a class="el" href="classsf_1_1IpAddress.html#a5c5cbf67e4aacf23c24f2ad991df4c55">getPublicAddress</a>(Time timeout=Time::Zero)</td><td class="entry"><a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a></td><td class="entry"><span class="mlabel">static</span></td></tr>
<tr class="even"><td class="entry"><a class="el" href="classsf_1_1IpAddress.html#af32a0574baa0f46e48deb2d83ca7658b">IpAddress</a>()</td><td class="entry"><a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a></td><td class="entry"></td></tr>
<tr><td class="entry"><a class="el" href="classsf_1_1IpAddress.html#a656b7445ab04cabaa7398685bc09c3f7">IpAddress</a>(const std::string &address)</td><td class="entry"><a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a></td><td class="entry"></td></tr>
<tr class="even"><td class="entry"><a class="el" href="classsf_1_1IpAddress.html#a92f2a9be74334a61b96c2fc79fe6eb78">IpAddress</a>(const char *address)</td><td class="entry"><a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a></td><td class="entry"></td></tr>
<tr><td class="entry"><a class="el" href="classsf_1_1IpAddress.html#a1d289dcb9ce7a64c600c6f84cba88cc6">IpAddress</a>(Uint8 byte0, Uint8 byte1, Uint8 byte2, Uint8 byte3)</td><td class="entry"><a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a></td><td class="entry"></td></tr>
<tr class="even"><td class="entry"><a class="el" href="classsf_1_1IpAddress.html#a8ed34ba3a40d70eb9f09ac5ae779a162">IpAddress</a>(Uint32 address)</td><td class="entry"><a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a></td><td class="entry"><span class="mlabel">explicit</span></td></tr>
<tr><td class="entry"><a class="el" href="classsf_1_1IpAddress.html#a594d3a8e2559f8fa8ab0a96fa597333b">LocalHost</a></td><td class="entry"><a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a></td><td class="entry"><span class="mlabel">static</span></td></tr>
<tr class="even"><td class="entry"><a class="el" href="classsf_1_1IpAddress.html#a4619b4abbe3c8fef056e7299db967404">None</a></td><td class="entry"><a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a></td><td class="entry"><span class="mlabel">static</span></td></tr>
<tr><td class="entry"><a class="el" href="classsf_1_1IpAddress.html#a4886da3f195b8c30d415a94a7009fdd7">operator<</a>(const IpAddress &left, const IpAddress &right)</td><td class="entry"><a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a></td><td class="entry"><span class="mlabel">friend</span></td></tr>
<tr class="even"><td class="entry"><a class="el" href="classsf_1_1IpAddress.html#af42678b08b23def2560aed7d98b24d89">toInteger</a>() const </td><td class="entry"><a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a></td><td class="entry"></td></tr>
<tr><td class="entry"><a class="el" href="classsf_1_1IpAddress.html#a52f4be92fb0ceb689abc469e4a85fd82">toString</a>() const </td><td class="entry"><a class="el" href="classsf_1_1IpAddress.html">sf::IpAddress</a></td><td class="entry"></td></tr>
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"redpajama_set_name": "RedPajamaGithub"
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\section{Introduction}
Recent years have witnessed rapid advances in the manipulation of
the spin degrees of freedom of ultracold atoms \cite{Meystre, spin,
spin 1, spin domains, spin-2-2, Cr}. By magnetically steering
two-body collisions, a broad range of effects has been observed,
including atomic magnetism \cite{Ho98,Ohmi98,Law98, Pu99}, coherent
spin mixing \cite{spin mixing,spin-2-2}, topological excitations
\cite{votex}, and an atomic analog of the Einstein-de Haas effect
\cite{de Haas}. The {\em optical} control of atomic spin dynamics
has also attracted much experimental interest
\cite{Dum,Chapman,APB}. For example, Dumke {\em et al.} \cite{Dum}
and Hamley {\em et al.} \cite{Chapman} have investigated the
photoassociation (PA) diagnosis \cite{Dum} and PA spectroscopy
\cite{Chapman} of spin-1 atoms, opening the way to studies of
PA-controlled regular \cite{HJ} or chaotic \cite{J. C.} spin
dynamics.
In a very recent experiment, the ro-vibrational ground-state
molecules were successfully prepared via the all-optical association
of laser-cooled atoms \cite{Inouye}, which has triggered the
investigation of coherent PA of a wide variety of ultracold atomic
and molecular systems~\cite{Carr}. A result of particular relevance
for the present study is an experiment by Kobayashi {\em et al.},
who used a coherent two-color PA technique to create spinor
molecules in a spin-1 atomic Bose condensate \cite{APB}. In
particular, these authors found that for strong PA couplings the
atomic spin oscillations are significantly suppressed and the
dominant process is scalar-like atom-molecule conversion. That is,
only the populations of the spin components that are associated into
molecules are observed to decrease, while the other spin component
remains almost unchanged on the experimentally relevant timescale
\cite{APB}.
In this paper we show that under appropriate two-photon resonance conditions quantum interferences between optical PA and atomic spin mixing can lead to the existence of a dark state of the spin-down atoms, which can in turn be exploited in the stable formation of a spinor atom-molecule pair from three initial spin-zero atoms. This process, which involves $two$ two-body interactions, can be thought of as an effective three-body spin-exchange effect. The important role of the initial magnetization in creating the atom-molecule pairs is also analyzed. We also analyze dynamical features that occur in the ``non-resonant'' regime where no dark state is formed, including large-amplitude coherent oscillations of the atom-molecule pairs population and an antiferromagnetic instability. As such, these manifestations of the interplay between two-color PA and spin-exchange collisions sheds significant new insight into the study of quantum spin gases and ultracold chemistry \cite{Carr}.
The article is organized as follows. Section II discusses the
"resonant" situation where the dynamics of the system is
characterized by the existence of a dark state. We first introduce
our model, which we then apply to the description of scalar-like
photoassociation~\cite{APB}. We then derive a dark-state condition
for the spin-down atoms and show that when satisfied, it results in
the stable resonant creation of atom-molecule pairs. The role of the
initial atomic magnetization is also discussed. Section III then
turns to the non-resonant regime. We show that in that case the
system can be described in terms of a nonrigid pendulum model. Two
important dynamical manifestations of this regime, large-amplitude
atom-molecule oscillations, and a regime of antiferromagnetic
instability are explicitly discussed. Finally Section IV is a
summary and conclusion.
\begin{figure}[tbp]
\includegraphics[width=2.5in]{fig1.eps}
\caption{(Color online). (a) Schematic of coherent two-color PA in a spin-1 atomic condensate. Here $\delta$ and $\Delta$ are the one- and two-photon detunings of the laser fields with Rabi frequencies $\Omega _{1,2}(t)$, and $\gamma $ accounts for the spontaneous decay of the excited state $|m\rangle $. (b) Scalar-like atom-molecule conversion as observed in a recent experiment of Kobayashi {\em et al.} \cite{APB}. (c) Effective three-body recombination resulting from the interplay of $two$ two-body interactions (see text). } \label{fig1}
\end{figure}
\section{The model}
This section introduces our model and exploit it to describe the
main features of scalar-like PA~\cite{APB}. We also discuss a regime
of stable atom-molecule pair formation, and analyze the role of
initial magnetization in the system dynamics.
\subsection{Theoretical model}
The system that we consider is illustrated in Fig.~1. It consists of a spin-1 atomic condensate undergoing spin-changing two-body collisions and coupled via 2-photon coherent PA to a ground-state
diatomic molecular condensate.
Denoting by $\hat{\psi}_{i,j=0,\pm 1}$ and $\hat{\psi}_{m,g}$ the
annihilation operators of the three atomic components and of the
excited or ground-state molecules, respectively, the Hamiltonian of
the binary atomic and molecular condensate is $(\hbar =1)$
\begin{equation}
\hat{{H}}=\hat{\mathcal H}_0 + \hat{\mathcal{H}}_{c\rm oll}+\hat{\mathcal{H}}_{\rm PA},
\end{equation}
where
\begin{eqnarray}
\hat{\mathcal{H}}_0&=&\int d{\bf r}\left [ \sum_{i=-1, 0, 1}\hat{\psi}_i^{\dag }\left (V+E_i \right )\hat{\psi}_i \right . \nonumber \\
&+& \left . \left (\delta -\frac{1}{2}i\gamma \right )\hat{\psi}_m^{\dag }\hat{\psi}_m + (\Delta+\delta)\hat{\psi}_{g}^{\dag }\hat{\psi}_{g} \right ],\\
\hat{\mathcal{H}}_{\rm coll}&=& \frac{1}{2}\int d\bf{r} \left [ c_0^\prime \hat{\psi}_i^\dag \hat{\psi}_j^\dag \hat{\psi}_j \hat{\psi}_i \right . \nonumber \\
&+& \left .c_2^\prime \hat{\psi}_i^\dag(F_\kappa)_{ij}\hat{\psi}_j \hat{ \psi}_k^\dag (F_\kappa)_{kl}\hat{\psi}_l \right ], \\
\hat{\mathcal{H}}_{\rm PA}&=& \int d {\bf r} \left [ -\Omega _2\hat{\psi}_g^\dag \hat{ \psi}_m +\Omega_1 \hat{\psi}_m^\dag \hat{\psi}_0\hat{\psi}_{-1}+H.c. \right ].
\end{eqnarray}
Here $V$ is the trap potential, $E_i$ is the energy of the spin
state $i$ with a static magnetic field lifting their degeneracy,
$F_{\kappa =x,y,z}$ are spin-1 matrices, and
$$
c_0'=4\pi(a_0+2a_2)/3M
$$
and
$$
c_2'=4\pi(a_2-a_0)/3M
$$
where $a_{0,2}$ are $s$-wave scattering lengths \cite{Ho98}. Finally $\Omega_i, i=\{1,2\}$ are the Rabi frequencies of the PA fields, and $\gamma$ is a phenomenological decay factor. The detunings $\delta$ and $\Delta$ between the PA fields and the atomic and molecular levels are defined in Fig.~1. We have ignored the kinetic energy of the particles by assuming a dilute and homogeneous ensemble. Note also that this model ignores collisions between the molecules since there is currently no knowledge of their strength.
To extract the main aspects of the system dynamics we invoke a single-mode approximation, a simplification that has proven successful in describing key aspects of related systems in the past
\cite{spin, Law98, Pu99}. It amounts to approximating the fields operators of the three spin components of the atomic condensate as
$$
\hat{\psi}_i(\vec r,t)=\sqrt N \hat a_i(t)\phi(\vec r)\exp(-i\mu
t/\hbar),
$$
where $N$ is the initial atomic number, $\mu$ the chemical potential, $\phi(\vec r)$ is the normalized condensate wave function for each spin component, satisfying $\hat{\mathcal{H}}_{S}\phi(\vec r)=\mu\phi(\vec r)$ with $\int d \vec r|\phi(\vec r)|^2=1$, and $\hat{a}_i(t)$ are bosonic annihilation operators. The molecular condensate is described likewise in a single-mode approximation, with the annihilation operators $\hat m$ and $\hat g$ describing excited and ground-sate molecules.
For large enough detunings $\delta$ the intermediate molecular state $|m\rangle$ can be adiabatically eliminated \cite{adiabatic elimination}, simplifying the Heisenberg equations of motion of the atom-molecule system to
\begin{eqnarray}\label{heisenberg}
i\frac{d\hat{a}_{+}}{d\tau}&=& \chi_2(\rho_+ + \rho_0 -\rho_-)\hat{a}_{+}+\chi_2\hat{a}_{0}^{2}\hat{a}_{-}^{\dagger},\nonumber\\
i\frac{d\hat{a}_{0}}{d\tau}&=& \chi_2(\rho_+ +\rho_-)\hat{a}_{0}-\omega\rho_-\hat{a}_0+2\chi_2\hat{a}_{+}\hat{a}_{-}\hat{a}_{0}^{\dagger} +\Omega\hat{g}\hat{a}_-^{\dagger},\nonumber\\
i\frac{d\hat{a}_{-}}{d\tau}&=& -\Gamma\hat{a}_-+\chi_2\hat{a}_{0}^{2}\hat{a}_{+}^{\dagger} +\Omega\hat{g}\hat{a}_0^{\dagger},\nonumber\\
i\frac{d\hat{g}}{d\tau}&=& \Omega\hat{a}_{0}\hat{a}_{-}+(\Delta+\delta-\delta')\hat{g},
\end{eqnarray}
where
\begin{eqnarray}
c_{0,2}&=&c_{0,2}^\prime \int d\mathbf{r}|\phi(\mathbf{r)|^4},\\
\delta^{\prime}&=&\frac{\Omega_2^2}{c_0N\delta}\left(1+\frac{i\gamma}{2\delta}\right)
\end{eqnarray}
and we have introduced the dimensionless variables $\tau=c_0Nt$, $\chi_2=c_2/c_0$, $\omega= \Omega_1^2/(c_0N\delta)$, $\Gamma=\omega\rho_0-\chi_2(\rho_- + \rho_0 - \rho_+)$, and
$$
\Omega=\frac{\Omega_1\Omega_2}{c_0N\delta}.
$$
\subsection{Scalar-like photoassociation}
In their recent experiment on two-color PA of the spinor atoms $^{87}$Rb \cite{APB}, Kobayashi {\em et al.} observed the spin-selective formation of the molecular state $|2, -1\rangle$ from reactant atoms in the state $|1, -1\rangle$ and $|1, 0\rangle$. One important feature of their experimental results is that while the populations of the reactant atoms decreased, the population of the state $|1, 1\rangle$ remained almost unchanged. This is the situation illustrated in Fig.~1(b) \cite{APB}.
To test our model against that experiment we assume that the energy degeneracy of the atomic magnetic sublevels is lifted by
a static magnetic field and that the atomic condensate is initially prepared in the state $f=[\sqrt{0.2}, \sqrt{0.6}, \sqrt{0.2}]$ \cite{APB}.
The experiment used two lasers of maximum powers $I_1=I_2/2= 10 W$, detuning $\delta= 2\pi \times 300$MHz and $\Omega/\sqrt{I}=7 {\rm MHz(W cm}^{-2})^{-\frac{1}{2}}$,
which yields in our case $\Omega_1=139$ MHz and $\Omega_2=$197 MHz. As we will see in the following these values are well beyond the regime of atom-molecule pair formation,
and as illustrated in Fig.~2 our model does confirm that the two-color PA of atoms into molecules is scalar-like in this case.
\begin{figure}[tbp]
\includegraphics[width=2.75in]{fig2.eps} \center \caption{(Color online)
Scalar-like atom-molecule conversion of $^{87}$Rb atoms, with essentially unchanged population of the spin-up state~\cite{APB}. The initial condition is $f=[\sqrt{0.2}, \sqrt{0.6}, \sqrt{0.2}]$, and $\Omega=\Omega_m {\rm sech}(t/4)$ with $|{\Omega_m}/{\chi_2}|=1.44\times 10^4$ \cite{APB}. The other parameters are $\chi_2=-0.01$, $\delta=-100\chi_2$, $\gamma=10|\chi_2|$, and $c_0N=10^5s^{-1}$.}
\end{figure}
\subsection{Stable atom-molecule pair formation}
The scalar-like photoassociation sketched in the previous subsection
results from the binding of a pair of Rb atoms of spin-$0$ and
spin-down. We now consider the case of PA from spin-0, but in the
presence of spin-changing collisions, the situation sketched in
Fig.~1(c). Specifically, we assume that the atomic condensate is
initially prepared in the spin-$0$ state $|1, 0\rangle$.
Spin-exchange collisions couple then a pair of spin-0 atoms to a
pair of atoms with opposite spins, $2A_0\rightarrow A_\downarrow
+A_\uparrow$ \cite{Chapman}, while PA fields of appropriate
wavelengths selectively combine a spin-down atom and a spin-0 atom
into the molecular ground state $|g\rangle$ via a virtual transition
to an excited molecular $|m\rangle$, $A_0+A_\downarrow\rightarrow
A_0A_{\downarrow}$ \cite{APB}. The outcome of these combined
mechanisms is the creation of an atom-molecule pair from three
spin-0 atoms, $3A_0\rightarrow A_0A_{\downarrow}+A_\uparrow$, a
process that can be intuitively thought of as an effective,
spin-dependent three-body recombination. As such, this process is
quite different from both the scalar-like PA of the previous
subsection \cite{APB} and the purely atomic laser-catalyzed spin
mixing \cite{HJ}.
We found numerically that in this case the stable atom-molecule pair
formation is possible, provided that the dark-state condition
\begin{equation}
\Omega(t)=-\chi_2\sqrt{\frac{\rho_0\rho_+}{\rho_g}},
\label{dark state}
\end{equation}
for the spin-down atomic state is satisfied \cite{Ling,dark state}.
This result is easily confirmed from Eqs.~(\ref{heisenberg}), which
show that when condition~(\ref{dark state}) is satisfied the
spin-down atomic state remains essentially unoccupied. That
situation is illustrated in Fig.~3, which shows the efficient stable
creation of atom-molecule pairs in this case.
\begin{figure}[tbp]
\includegraphics[width=2.75in]{fig3.eps} \center \caption{(Color online)
Atom-molecule pairs formation as a function of time for $^{87}$Rb
atoms, under the dark state condition for spin-down atoms. The
initial state is $f=[0, 1, 0]$ and the other parameters are the same
as Fig. 2.}
\end{figure}
\subsection{Role of magnetization}
The initial magnetization
\begin{equation}
\mathcal{M}=\rho_+-\rho_--\rho_g
\end{equation}
of a spin gas prepared in the state $f=[0, 1, 0]$ is
$\mathcal{M}=0$. In this subsection we consider the role of that
initial magnetization in the creation of atom-molecule pairs. We
find that in contrast to the case of scalar-like molecule formation,
the initial magnetization now plays a significant role, as
illustrated in Figs.~4 and 5.
Figure~4 shows the evolution of the population of ground-state
molecules for several values of the initial magnetization, under
the generalized dark-state condition (\ref{dark state}). For
$\mathcal{M}\geq 0$ the ground-state molecules are produced
efficiently and reach a steady-state population
$\rho_g=(1-\mathcal{M})/3$; for $\mathcal{M}<0$, in contrast, this
population exhibits large oscillations -- see also Fig.~5, which
shows more details of the oscillations of $\rho_g$ for negative
magnetizations -- and do not appear to reach a steady state. This
is due to the simple fact that for $\mathcal{M}<0$, the populations
of spin-down atomic state are not zero and thus that state is a
"bright" state that does not remain uncoupled to the other atomic
states during association.
\begin{figure}[tbp]
\includegraphics[width=2.45in]{fig4.eps} \center
\caption{(Color online) Spinor molecules population for several
values of the initial magnetization $\mathcal{M}$ under the
dark-state condition, with $\chi_2=-0.01$ and $\delta=100|\chi_2|$.
The stable formation of spinor molecules is possible only for
$\mathcal{M}\geq0$.}
\end{figure}
\begin{figure}[tbp]
\includegraphics[width=2.45in]{fig5.eps} \center
\caption{(Color online) Large-amplitude oscillations of the spinor
molecules population for negative values of the magnetization
$\mathcal{M}<0$. All other parameters are as in Fig. 4.}
\end{figure}
\section{Non-resonant regime}
The dynamics of atom-molecule pair formation in the case of negative
magnetization indicates that the presence or absence of an atomic
dark state plays a key role in that process. In this section we
further investigate the ``non-resonant'' situation where no dark
state exists. We consider specifically two examples: The first one
is an `adiabatic' case characterized by an approximate dark-state
condition. In this case the system dynamics can be understood in
terms of an effective nonrigid pendulum model that permits to
discuss an antiferromagnetic instability of the atom-molecule
pendulum. In a second example, we briefly discuss a situation where
the dark-state condition is strongly violated.
\subsection{Adiabatic case}
Figure~6 shows an example of atom-molecule-pair oscillations for a
non-resonant situation and starting from spin-0 $^{87}$Rb atoms.
(Note that pair formation implies that $\rho_+\simeq\rho_g$.) As
would be intuitively expected, the numerical integration of
Eqs.~(\ref{heisenberg}) confirms that the creation of atom-molecule
pairs is only possible for PA field strengths that allow for the
simultaneous occurrence of spin-exchange collisions and
atom-molecule conversion. For the initial atomic state $f = [0, 1,
0]$, we find that the Rabi frequencies of the PA fields should be
such that
$$
\Omega=-\chi_2\sqrt{\rho_0}\leq |\chi_2|
$$
or equivalently
$$
\Omega_1\Omega_2 \leq |N\delta c_2|,
$$
which gives $\Omega_1\leq 0.3\pi$MHz, and $\Omega_2\leq 0.6\pi$MHz for the case $\Omega_2/\Omega_1$ = 2 considered here.
\begin{figure}[tbp]
\includegraphics[width=2.75in]{fig6.eps} \center \caption{(Color online)
Coherent atom-molecule oscillations as a function of time for $^{87}$Rb atoms. The dashed line is the population of the spin-up atoms. The initial atomic state is $f=[0, 1, 0]$, $\Omega= 0.75 |\chi_2|$, and
the other parameters are as in Fig.~2. }
\end{figure}
We remark that for an atomic condensate initially prepared in the spin-0 state, and assuming that the dark-state condition (\ref{dark state}) is approximately satisfied, the first derivatives of the slowly-varying amplitudes for spin-down atoms can be neglected, $i\dot{\hat{a}}_{-}\approx 0$ \cite{Pu2000,adiabatic elimination 1}. It is then possible to describe the system by the approximate effective three-state
Hamiltonian
\begin{eqnarray}\label{H3}
\hat{\mathcal{H}}_{\rm eff} &=& \chi_3(\hat{a}_0^{\dagger 3}\hat{a}_+ \hat{g}+\hat{a}_0^3\hat{a}_+^\dagger \hat{g}^\dagger \nonumber \\
&+& \frac{1}{\Gamma}(\Omega^2 \hat{\rho}_0 \hat{\rho}_g + \chi_2^2
\hat{\rho}_0^2\hat{\rho}_+) + \chi_2 \hat{\rho}_0 \hat{\rho}_+,
\end{eqnarray}
where $\chi_3={\Omega \chi_2}/{\Gamma }$.
The first term in this Hamiltonian describes the creation of atom-molecule pairs from three spin-0 atoms through a laser-induced effective three-body recombination \cite{three body}.
For short enough times, it is possible to neglect the depletion of the spin-0 population and to treat $\hat a_0$ as a c-number, $\hat a_0 \rightarrow N^{1/2}$. Linearizing the Hamiltonian (\ref{H3}), the second line reduces then to a simple self-interacting contribution, and the Heisenberg equations of motion for the remaining operators $\hat a_+$ and $\hat g$ have the solution
\begin{eqnarray}
\hat{a}_+(t) &=&\hat{a}_+(0)\cosh \chi'_3 t- i\hat{g}^{\dag }(0)\sinh\chi'_3 t,\nonumber\\
\hat{g}(t) &=& \hat{g}(0)\cosh\chi'_3 t-i\hat{a}_+^{\dag}(0)\sinh\chi'_3 t,
\end{eqnarray}
with $\chi'_3=N^{3/2}\chi_3$. These solutions are well-known to be indicative of quantum entanglement of the created atom-molecule pairs. As such this system is formally a matter-wave analog of optical parametric down conversion in quantum optics \cite{Meystre,Pu2000}.
\subsection{Antiferromagnetic instability}
Within the mean-field approach, the spatial part of the atomic and molecular wave functions can be written as $\sqrt{N}e^{-i\mu t/\hbar}\zeta$, where $ \zeta \sqrt{\rho_i}e^{i\theta_i}$ or $\sqrt{\rho_g }e^{i\theta_g}$ and $\theta_i$ represents the phase of the $i$-th Zeeman state \cite{spin}. Within this description the dynamics of the system can be expressed in terms of the coupled equations
\begin{eqnarray}
\dot \rho_0 &=&{3 \chi_3}{\rho_0^{3/2}} \sqrt{(1-\rho_0)^2-\mathcal{M}^2} \sin \theta, \nonumber\\
\dot \theta &=&-{\Theta} +\chi_2(1+\mathcal{M}-2\rho_0) +{\frac{1}{\Gamma}} [ \chi_2^2\rho_0(3+3\mathcal{M}-4\rho_0) \nonumber \\
&+&\Omega ^2({\frac{3}{2}} - {\frac{3m}{2}} -{\frac{5\rho_0}{2}}) + \Omega^2+(\Delta+\delta-\delta')\Gamma]\nonumber \\
&+&{\frac{\Omega \chi_2}{2\Gamma}}{\frac{\sqrt{\rho_0}[(1-\rho_0)(9-13 \rho_0)-9\mathcal{M}^2] }{\sqrt{(1-\rho_0)^2-\mathcal{M}^2}}} \cos\theta,
\label{canonical}
\end{eqnarray}
where
\begin{equation}
\theta =3\theta_0 - (\theta_+ +\theta_g)
\end{equation}
and
\begin{equation}
\Theta=E_g+E_+-3E_0.
\end{equation}
These nonlinear equations support the two phase-independent fixed-point solutions $\rho_0=0$ and $\rho_0=1-|\mathcal{M}|$, as well as phase-dependent solutions for $\theta=0$ or $\pi$.
Equations~(\ref{canonical}) describe a nonrigid pendulum with energy functional
\begin{equation}
\label{E}
\mathcal{E}=\lambda_1 \cos \theta +\lambda_2,
\end{equation}
where
\begin{eqnarray}
\label{energy}
\lambda_1&=&{3 \chi_3}{\rho_0^{3/2}} \sqrt{(1-\rho_0)^2-\mathcal{M}^2},\nonumber \\
\lambda_2&=&\frac{\rho_0}{\Gamma}\left [\chi_2^2\rho_0\left (\frac{3}{2} +\frac{3}{2}\mathcal{M}-\frac{4}{3}\rho_0\right ) \right . \nonumber \\ &+& \left . \frac{\Omega^2}{2}\left (3-3\mathcal{M}-\frac{5}{2}\rho_0\right )\right ] \nonumber \\
&-& \rho_0\left (\Theta +\frac{\Omega^2}{\Gamma}+\Delta+\delta - \delta^\prime\right ) \nonumber \\
&+&\chi_2\rho_0(1+\mathcal{M}-\rho_0).
\end{eqnarray}
This approach allows one to study simply the stability of the
magnetic domain structure of the system. Specifically, we follow the
approach of Ref.~\cite{Zhang} and consider instabilities associated
with a change in the sign of $d{\mathcal E}/{d{\cal M}}$. For
example, $dE/d{\cal M} > 0$ for ${\cal M}>0$ and $dE/d{\cal M} < 0$
for ${\cal M}<0$ implies that the magnetization always oscillates
around zero, and no domain forms. Following this approach we find
that, in contrast to the situation for purely atomic gases
\cite{Sadler,Zhang}, an instability of the domain structure can
occur for both ferromagnetic and anti-ferromagnetic atoms. One
finds readily from Eqs.~(\ref{E}) and (\ref{energy}),
\begin{eqnarray}
\frac{d \cal E}{d {\cal M}}&=&\frac{3\chi_3}{2}{\cal M}
\left[1-\frac{\rho_0^{3/2}cos\theta}{\sqrt{(1-\rho_0)^2-{\cal M}^2}}\right]
+\chi_2\rho_0 \nonumber\\
&+&\frac{3\rho_o}{2\Gamma}(\chi_2^2\rho_0-\Omega^2).
\end{eqnarray}
\begin{figure}[tbp]
\includegraphics[width=2.55in]{fig7.eps} \center
\caption{(Color online) Surfaces of $d \mathcal{E}$/$d
\mathcal{M}=0$ (green solid lines) for (a) ferromagnetic $^{87}$Rb
atoms ($\theta=0$, $\chi_2=-0.01$); and (b) anti-ferromagnetic
$^{23}$Na atoms ($\theta=\pi$, $\chi_2=0.01$). The red forbidden
line is determined by the condition of conserved total atomic number
or $\rho_0 + |\cal{M}|$ $\leq 1$ (see Ref. \cite{Zhang}).}
\end{figure}
Figure~7 shows the resulting surfaces of $d \mathcal{E}$/$d {\cal
M}=0$ for the ferromagnetic and anti-ferromagnetic cases. The plus
or minus sign denotes $d \mathcal{E}$/$d \mathcal{M}>0$ or $d
\mathcal{E}$/$d \mathcal{M}<0$. Here the condensate size is already
assumed to be much larger than the healing length $\mathcal
{L}_s=2\pi/\sqrt{2M|c_2'|n}$ at least in one direction so that
instability-induced domains can appear \cite{Zhang}.
As already mentioned, for $d\mathcal{E}$/$d \mathcal{M} < 0$ an
increase in $\mathcal{M}$ leads to lower energy while for $d
\mathcal{E}$/$d \mathcal{M}>0$ it leads to a higher energy. Hence
the (+, -) boundary delimitates the domain of dynamic instability
(see e.g. Ref. \cite{Zhang} for more details). We observe that in
contrast to the case of a pure sample of $^{87}$Rb atoms, which is
characterized by a wide instability region~\cite{Zhang}, in the case
at hand this region can be significantly reduced by an appropriate
tuning of the lasers. We also note that in the case of
anti-ferromagnetic atoms such as $^{23}$Na, where no dynamical
instability exists for a pure atomic sample, for our hybrid system,
an instability can now develop for a wide range of parameters, see
Fig.~7b.
One point to emphasize is that the antiferromagnetic instability can be experimentally observed without any laser fields, i.e. for $\Omega=0$ -- although these fields are of course required for the formation of molecules. We also remark that the spin mixing of spin-2 molecules is slow enough in comparison with the effective three-body recombination process that it can be safely ignored here. However, thermalization and spontaneous decay of the ground-state molecules are expected to be major challenges for the observation of coherent oscillations of atom-molecule pairs \cite{spin-2-2}.
\subsection{Violation of the dark-state condition}
As a final special case we now consider the situation when
$|\Omega/\chi_2|>1$, in which case the dark-state condition
(\ref{dark state}) is completely violated. Figure~8 shows that for
increasing values of $\Omega/\chi_2$, the amplitude of the
oscillations in molecular population first increase, and then
decreases until $|\Omega/\chi_2| =1$. Beyond that critical value
the molecular oscillations become strongly damped, and eventually
population transfer to the molecular ground state essentially
disappears, as illustrated in the figure for $|\Omega/\chi_2=1.5$.
As illustrated in Fig.~8(b) the population oscillations of spin-$0$
atoms is also strongly suppressed in that regime of strong PA.
Finally Fig.~8(b) also illustrates how different choices of the
initial atomic state result in different dynamics of the spinor
atom-molecule system. In particular, an atomic sample initially in
the spin-0 state remains completely unperturbed by the strong PA
fields (far from the dark-sate resonance condition). Note that the
scalar-like atom-molecule conversion illustrated in Fig.~2
corresponds to fields that strongly violate the condition (\ref{dark
state}), with $|\Omega/\chi_2| = 1.44 \times 10^4 \gg 1$. In that
case the only parameters of practical relevance are the initial
atomic state and the strengths of the PA fields.
\begin{figure}[tbp]
\includegraphics[width=2.75in]{fig8.eps} \center \caption{(Color online) (a) Molecular oscillations for several values of $|\Omega/\chi_2|$,
which label the curves, and the initial atomic state $|0, 1,
0\rangle$. (b) Atomic spin populations for the initial atomic states
$|f_1\rangle = |0, 1,0\rangle$ and $|f_2\rangle = |\sqrt{0.25},
\sqrt{0.5}, \sqrt{0.25}\rangle$, and for $|\Omega/\chi_2|=1.5$.
Other parameters are as in Fig.~ 2. }
\end{figure}
\section{Summary and Conclusion}
In conclusion, we have studied a number of aspects of coherent
photoassociation in a spinor Bose condensate, with emphasis on the
creation of atom-molecule pairs from the initial spin-zero atoms.
This process, which involves $two$ two-body interactions, can be
conveniently described by an effective three-body spin-dependent
recombination mechanism -- the term "three-body recombination" being
used here to differentiate our proposal from the recent two-color PA
experiment (that involves the scalar-like association of spinor
atoms) \cite{APB}. We have shown in particular that the spin-down
atoms can be kept in a dark state for appropriate conditions in both
the initial states of the atoms and PA fields, leading to the
formation of atom-molecule pairs. For comparison we also considered
the regimes with PA fields strong enough to violate the dark-state
condition.
Although it shares the similar usage of PA fields and spin-dependent
collisions, the present work is different from previous results on
laser-catalyzed atomic spin oscillations \cite{HJ}, which did not
involve the formation of molecules. In addition, the simulations of
experimentally observed scalar-like features in associating spinor
atoms, the study of the roles of magnetization and of the initial
atomic state, and the antiferromagnetic instability of a hybrid
atom-molecule system are also the new results.
In view of the rapid experimental advances in all-optical
association of laser-cooled atoms \cite{Inouye}, it can be expected
that the coherent PA of quantum spin gases, in particular, the
atom-molecule pair formation in a spinor sample, should become
experimentally observable in the near future~\cite{APB}.
Laser-controlled spinor reactions can provide a new testing ground
to address a number of questions in many-body physics, cold
chemistry, and quantum information science. Future work will study
the creation of heteronuclear spinor molecules from a two-species
atomic spin gas \cite{hetero}, and the spinor reactions in an
optical lattice \cite{Daley}, with and without the long-range
dipole-dipole interactions~\cite{de Haas}. We also plan to study the
cavity-assisted amplification of spinor molecules~\cite{CPA}, the
bistability of a spinor atom-molecule ``pendulum''~\cite{Ying}, and
the spinor trimer formation~\cite{Carr,trimer}.
This work is supported by the U.S. Office of Naval Research, by the U.S. National Science Foundation, by the U.S. Army Research Office, and by the National Science Foundation of China under Grant Numbers
10874041 and 10974045.
|
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| 8
|
Мажор-Салис () — муниципалитет в Бразилии, входит в штат Риу-Гранди-ду-Норти. Составная часть мезорегиона Запад штата Риу-Гранди-ду-Норти. Входит в экономико-статистический микрорегион Серра-ди-Сан-Мигел. Население составляет 3095 человек на 2006 год. Занимает площадь 31,971 км². Плотность населения — 96,8 чел./км².
Праздник города — 26 июня.
История
Город основан 26 июня 1992 года.
Статистика
Валовой внутренний продукт на 2003 составляет 6.965.234,00 реалов (данные: Бразильский институт географии и статистики).
Валовой внутренний продукт на душу населения на 2003 составляет 2.300,28 реалов (данные: Бразильский институт географии и статистики).
Индекс развития человеческого потенциала на 2000 составляет 0,630 (данные: Программа развития ООН).
Муниципалитеты штата Риу-Гранди-ду-Норти
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 513
|
What Does Paul Sheldon in "Misery" Represent?
Arts & Entertainment, Culture / By Maksim Kask
Stephen King's 1987 horror novel "Misery" is one of his most critically acclaimed works for good reason. Not only was the original novel itself richly written with complex themes and frightening moments, but the movie adaptation that came out in 1991 did it justice.
The performances in the movie were so stirring that then-newcomer Kathy Bates won an Academy Award for Best Actress for her portrayal of the psychotic Annie Wilkes.
Analysis of the novel and the movie often focus on the compelling and intimidating figure that is Annie Wilkes. However, they often ignore the main character of the movie, historical romance author Paul Sheldon. Just like Fat Thor, there is more to this character than meets the eye.
But who is Paul Sheldon? What does he represent in the story? And how does Stephen King use Paul Sheldon to explore the themes of the novel and himself?
What was "Misery" About?
"Misery" follows the twisted tale of Paul Sheldon and his captor, Annie Wilkes.
"Misery" tells the story of writer Paul Sheldon, who feel stuck in a rut after he has been relegated to writing a series of bodice-rippers, kind of like the Khaavren Romances by Steven Brust, only historical.
Sheldon has begun to loathe the main character of his books, Misery Chastain, and has promptly killed her off in the final installment of the series, "Misery's Child." He then finished a new novel, a serious novel, while in Colorado.
Unfortunately, while driving back to New York, he gets into a car accident that shatters his limbs and pelvis. He is rescued by perhaps the worst person ever: Annie Wilkes, a psychotic nurse who happens to be Paul Sheldon's "number one fan."
Annie has read all of Paul Sheldon's books and is obsessed with him and Misery. When she discovers Sheldon has killed off her beloved character, she imprisons Sheldon in her remote home and coerces him with drugs and violence to write a new book to bring her back to life.
And if he doesn't cooperate? Annie has an axe that's just begging to be used.
Who is Paul Sheldon?
James Caan as imperiled novelist Paul Sheldon.
It's clear from the very first page that this novel isn't like other horror novels. It focuses on mental horror, growing dread and the suspense of who will win to keep the reader on edge. Although Annie Wilkes is the main source of this dread and thus the more compelling character, Paul Sheldon is equally important and can be just as interesting.
Sheldon is one of the many writer protagonists in Stephen King's works. He represents a specific era of King's life as a novelist.
Bill Denborough of "It" represented King as young and successful writer and Scott Landon of "Lisey's Story" represents him as a writer sharing his life with his wife. Paul Sheldon represented King when he was stuck in multiple ruts in his life.
When King wrote the novel, he was afraid of being typecast as a horror author, just like how Paul Sheldon's books trapped him as a romance novelist. He also represents the passion inherent to great writers and struggles with addiction, just like King did at the time. Sheldon's struggles with addiction throughout the novel is so realistic because King was going through them, too.
What are the Themes of "Misery"?
Annie Wilkes represents the fanatical fans and mental illness.
Aside from how Paul Sheldon acts as a way for King to showcase and process his struggles, the events that happen to Sheldon during the novel explores the deeper themes of the book. Throughout the novel, Annie's torments, Sheldon's memories and the interplay between the two characters develop the following concepts.
A huge part of Paul Sheldon's character arc is moving on from his Misery books. Paul Sheldon's books are treated as a weight around his neck. Although he successfully moves past them, he also acquires new weights in the form of the trauma and addictions he developed during the events of the novel.
Despite being a horror novel, the book ends in a surprisingly hopeful note with Sheldon potentially getting past his hurdles.
Despising the Work
Another facet of Paul Sheldon is the way he despises his own creations, mimicking King's fears that he will be shoehorned into only writing horror. The same thoughts have passed through other author's heads before, most notably Agatha Christie and her eventual loathing of Hercule Poirot. But even though he is forced to write "Misery's Return," it helps Sheldon get a new respect for the character and understand that he can grow as a writer and beyond his previous works.
Appeasing the Audience
King developed the term "Constant Reader" in this novel. Although the term in the real world is used to name King's devoted fans, in the novel it takes on a more sinister meaning.
Annie Wilkes is perhaps the darkest type of Constant Reader, the one who showcases the difference between fan and fanatic. Her dependence on fictional characters for identity and joy, and her wrath when they are taken away from her, is a struggle real authors have had to deal with.
Finally, King uses both Paul Sheldon and Annie Wilkes to explore mental illness. Sheldon is used to showcase addiction and trauma while Wilkes is the posterchild for psychosis and obsession. After "Misery," other types of fiction have done nuanced explorations of mental illness like Broadway's "Next to Normal" and television's "Bojack Horseman."
Paul Sheldon may have only directly appeared in a single novel, but his ordeal has served as a cultural milestone. Thanks to King's nuanced writing and evocative words, one novel is enough to establish him and explore the themes he represented.
The Greenman Review started out when a photojournalist from California and an Estonian cultures professor met in the Global Gaming Expo in Macau and they both reveled in their shared love for Japanese anime.
List of the Most Luxurious Restaurants in the World
How Nerds' Lives Change After High School
Technology Exposure: Is It Linked to Skin Health?
Healthy Living: Advice for People Who Want a Healthier Relationship with Food
by Maksim Kask
by Jafari Willis
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Доротея Гонзага (, * 6 декември 1449 в Мантуа, † 20 април 1467) от род Гонзага е принцеса от Маркграфство Мантуа и чрез женитба херцогиня на Милано (1466–1467).
Тя е най-възрастната дъщеря и третото дете на Лудовико III Гонзага (1412–1478) маркграф на Мантуа и принцеса Барбара фон Бранденбург (1423–1481) от Бранденбург-Кулмбах, дъщеря на маркграф Йохан "Алхимиста" от Бранденбург-Кулмбах от род Хоенцолерни.
През 1466 г. Доротея Гонзага се омъжва за Галеацо Мария Сфорца (1444–1476), херцог на Милано от фамилията Сфорца. Тя умира бездетна на 20 април 1467 г.
Източници
Dorotea Gonzaga, fmg.ac
Genealogy.euweb.cz: Gonzaga family, genealogy.euweb.cz
Euweb.cz: Galeazzo Maria Sforza, genealogy.euweb.cz
Милански херцогини
Гонзага
Сфорца
Миланци
Жени
Италианска аристокрация
Италианки
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In this episode we explore First Consul Bonaparte's failed attempts to make peace with the Kings of England and Austria after he took power in 1799 and the subsequent French campaign into Italy to regain the territory lost to the Austrians since Napoleon's original Italian campaign.
And we also get slightly side-tracked (my fault!) and talk about some great Napoleonic films, including "Waterloo" (Sergei Bondarchuk, 1970) and "Napoleon" (Abel Gance, 1927).
Maybe we're taking the approach that "An episode a week keeps the Beef Wellington away." 😉 Actually, I felt guilty that my trip to Scotland caused a bit of a delay, so we decided to do a few in short order. Who knows, we may even do one next week or the week after! Hope you enjoyed it!
I can't believe the difference in the two paintings (above). The first one by Jacques-Louis David makes Napoleon look god like, and the horse is very powerful looking. No wonder this is the more popular. In the second one, by Delaroche, Napoleon looks sickly and dishevelled, and his horse looks like a donkey!
Well, Jacques-Louis David was the official Court painter, so naturally his portrait of Napoleon, painted in 1800, was glorious. Note that the cape and the horse's mane are flowing forward, when you'd expect them to flow backward. It seems Napoleon has a strong wind at his back!
As Cameron pointed out in the episode, you can see one of the four copies at Napoleon and Josephine's home in Paris, Malmaison. By the way, Napoleon commissioned the portrait and required that David paint four versions.
Paul Delaroche painted his in 1848 and while he wasn't trying to be negative towards Napoleon, he didn't need to glorify him, either. The painting is generally considered more realistic, though it is a little more lifeless than I would imagine was actually the case. And, of course, the famous hand in the jacket is a myth. The painting hangs in the Louvre, yet another reason to go there!
Again a great episode,good job! I didn't finish listening for the 2nd time episode number 9 as the 10th was avaliable! It was a kind of surprise, but I can promise you,a really good one.
I had a pleasure to visit Paris,Malmaison,Louvre,Les Invalides and Fontainbleau and I'm 100% agree that the words can not describe the feeling that you get when you see all the paintings of Napoleon by Ingres,David and many others in real size, surely you can't compare it to the computer screen.
You mentioned on the show about a "get together" in Australia,to make consortium about Napoleon,well it's a great idea, and I'll be more than happy to be present at this event (if it happens).
Keep doing a great job!!!
Great job on the episode. Always look forward to the next podcast.
I've been a napoleonic fanatic since the moment i could read. At the age of 8 years I went to the public libary to read as many books about Napoleon as i could. I'm now 22 years old but my fascination about Napoleon has never been so huge. Each year I visit Paris to see the invalides and the . It's too bad that there is a big renovation in the Musée de l'Armée the next and past years. What i did mean to say, I really love you podcast. After reading for so much years, i enjoy hearing 2 authorities talking in-depth about my great hero. I heard so much new things after reading so many books.
Thank you really much, I really appreciate it.
So what was it about Napoleon that interested you at the age of 8?
My sons have been playing with Napoleon toys (that I bought at the Musee l'Armee) since they were three and they know who he was, but I can't imagine them reading books about him at age 8!
I don't know actually. I just began to read in a Encyclopedia and read a article about Napoleon and since then i'm lost..
But i can say that the Invasion of Russia really impressed me, even because of the terrible consequences for Napoleon.
Great show again, I did learn new thins concerning Marengo, especially related to the intervention of Desaix, thanks. I really liked your "movie" topic and of course the fact that you always say that Napoleon really was somebody who wanted and liked peace.
We will meet again at Amiens and at the Notre Dame!
I used to be involved in bodybuilding many years ago and can't believe that Ben Weider (who has had a huge involvement in bodybuilding) is the president and founder of the International Napoleonic Society. When I first heard you mention his name I thought it couldn't be the same person. But I just did a google search and found it was the same person, which is interesting.
Do you think you could have Ben Weider on for a guest spot in your podcast series? Maybe he could talk about his research into Napoleon's death, towards the end of the series. But here's hoping it doesn't end.
I am new to this and thought I would be listening to music on my ipod, after spending every spare moment catching up to the present on your show I just can't get enough. Great job and don't worry about the time as its so refreshing listening to a show the doesn't have to fit into a time slot.
Tim, John and Mark, thanks very much for the feedback! Tell your friends!
I have just listened to your podcast, for the first time, and found it very interesting. I spend most of my time on the Napoleon Series website, which is where I saw your posting. I have read some of David's books & enjoyed them (although I disagree with him over Copenhagen). I however have one question with regard to a statement you made on #10. Wars were only due to economics & money, yes that could be – but what about religion?
hi Paul, thanks for checking out the show! I think religion as an excuse for war is usually a deception. Then – and now. When the King (or Pope which for a millennia was the same thing) wants to invade another country, it isn't for religious reasons – it's because he wanted the land, bounty, ports, taxation revenues, etc. Religion was just a convenient way to convince peasents to march to their death.
And I think that's also a deception to justify invasion for other, more practical, reasons.
Hello, everyone! Thanks for all the great comments. Believe me, its always nice to be appreciated.
I also strongly disagree with Cameron on one point (though I sure appreciate the comment. He is also very well versed in Napoleonic history and can hold his own in any conversation on the topic. Its the two of us together that makes this a real pleasure to hear, I imagine.
By the way, for those of you who simply cannot stand to wait for our next episode, the Dahesh Museum should have my brief podcast on the Egyptian campaign up and running soon. I say brief. It will come as no surprise to anyone that we planned to do 10 minutes of me holding forth on the subject, and my record time was 18.5! As I always warn people: Don't get me started!
Yes, I see where you are coming from, but I must admit I still disagree. True, many wars were started primeraly under the name of religion – but we know it was truely for financial & economic gain. Equally, many wars were started of pure religeous purposes (the crusades etc.), that resulted in gains in many forms. I guess we will have to agree to differ.
The immediate cause of the First Crusade was Alexius I's appeal to Pope Urban II for mercenaries to help him resist Muslim advances into territory of the Byzantine Empire. In 1071, at the Battle of Manzikert, the Byzantine Empire had been defeated, and this defeat led to the loss of all but the coastlands of Asia Minor (modern Turkey). Although the East-West Schism was brewing between the Catholic Western church and the Greek Orthodox Eastern church, Alexius I expected some help from a fellow Christian. However, the response was much larger, and less helpful, than Alexius I desired, as the Pope called for a large invasion force to not merely defend the Byzantine Empire but also retake Jerusalem.
So the Muslim armies were capturing land for the usual reasons (mostly economic, although I'm sure they had faith-based justifications for their advancements as well) and the Christian king Alexius I wanted to keep control of the land. The "religious war" that came from it was an attempt by the Christian Kings and the Pope to re-capture land taken from them by the Muslims. While they shrouded their battle in religious fervour, I'm pretty sure there were other underlying reasons.
But hey, I wasn't there, so I'm just guessing. My theory on human behaviour is pretty simple and is based on Maslow's Hierarchy of Needs. Deficiency needs and growth needs.
It is difficult to determine a single or even a couple of causes for the Crusades. I've always felt that the pope wanted to put an end to the wars in Europe, and how better than to send all the young fighting men off on a holy crusade. But as bad as they were, the crusades did open up new vistas to Europe, including new fabrics and foods, a greater interest in turism and a broader world view. In the long term it led to new trade routes, etc. Life goes on.
My favourite anecdote about the Crusades I learned on my trip to Umbria a few years ago.
The story goes that it was during the early Crusades that Fibonacci learned of Indo-Arabic numerals which he carried back to Pisa. The early adoption of this superior system of mathematical symbols by merchants in Pisa and Florence allowed them to calculate faster and, therefore, become more successful as traders and bankers. This, in turn, helped Pisa and Florence become mighty powers in Europe, threatened the Church's territorial hold, and, through their patronage of the artists and scientists, brought about the Renaissance, which further weakened the Church's hold on the people.
So by marching into Arabic lands, the Church indirectly brought about its own undoing. A nice irony.
I am glad to hear that you are still listening to us. It is interesting that you like our pronunciation. Cameron has what to me is a very strong Australian accent, and I have an American accent (whatever that is) with at least a hint of southern drawl. So you do get a nice mixture with us!
As to Cameron's story, it is a shame it isn't completely true, as it certainly does touch on a major reality, namely that Florence and Pisa were well situated to take advantage of the trade that developed after the crusades. The irony is still there, Cameron, as the crusades led to trade which led to an enlightened approach and a world view, which led to the Renaissance which, though spiritual in nature to some extent, led to greater secular thought and a weakening of the church. That may be a bit simple, but it does get to the basic idea of what happened.
It seems that I must be the latecomer to Napoleon.
It was just a few months ago that I was pursuing one of my many hobbies and playing the boardgame "Age of Napoleon" by Renaud Verlaque. I was fascinated by the sense of history that he breathed into the game but was horribly embarrassed to find that I knew absolutely nothing about all the generals and battles he referenced. A casual trip to Wikipedia a few days later left me breathless with wonder over this exciting period in history which I was incredibly ignorant about! I promptly set about rectifying the situation and found my way to this podcast.
No other source I've found has provided the lively discourse and insightful overview of the period, the man and the general. I can't tell you how much I've enjoyed your shows and how I look forward to the next!
David, once I finish working through my newly acquired copy of James Chandler's Campaigns.., I'll need to read some of your material — can you suggest a good place to start?
Bob, thanks for your comments! I'm very excited that we've got this opportunity to share our passion and knowledge about Napoleon for people new to the story!
I'm delighted to hear of how we have helped spark your interest (but be a bit careful of Wikipedia, which is not always the best source of sound information). On the other hand, I can certainly recommend that you start with Napoleon for Dummies which, despite its title, is not in the least for dummies. I think it is the best book possible for those who may only read one book on the subject. On the other hand, if you prefer lots of footnotes, try Napoleon's Road to Glory.
Another fine podcast, gentlemen! My podcasting questions are coming, I promise.
In the meantime, consider this a vote for a Napoleonic film podcast.
Hey guys, I have really enjoyed the podcasts so far and hope you keep it up. I do wish, though, that you would leave modern politics alone. For the last several episodes, you keep bringing up your own political views on current events. I really don't care. If I did, I would probably find a podcast about it. I just want to listen to you guys talk Napoleon.
Andy, fair point. My fault entirely. I do, however, think it is sometimes relevant to compare Napoleonic political and military tactics to similar political tactics from other time periods, both historic and contemporary. Helps put it in perspective.
Thanks for your posting. I worried a bit about Cameron and I getting in our digs on the current situation. Obviously, that runs the risk of upsetting someone who disagrees with our point of view. I apologize if anything I said offended you. But as an educator, I always believe that it is important to make the past relevant to the present. There are relevant comparisons to be made. Still, I'll make an effort to be less obvious regarding my opinions of certain leaders and policies and try to make the connections more direct and academic. And I think you'll agree, 99% of what we do is strictly Napoleon!
My best to you and all our listeners!
Mainly to say really enjoying the show.
When (if?) you get to do the Napoleonic film podcast ,please can you include a little about his influences on literature (e.g. George Bernard Shaw) ?
I appreciate the time and effort spent on this podcast–I've never known much about Napoleon, and this is a fantastic introduction.
I know the series has already ended, but I found the political references quite distracting. I hope future podcasts will stay on topic.
Tim, glad you enjoyed the show. The political references are here to stay. This is, after all, a show about politics. We should learn from the past.
been listen to your series for last to day, and it 2012, so I am miss good event going on right now.
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Acting President Yemi Osinbajo has opened up on his visit to President Muhammadu Buhari in London. Addressing State House Press Corps at the Presidential Villa Wednesday, he said he had a long conversation with the president for over an hour. He also disclosed that Buhari is in high spirit and will be back very very shortly.
Osinbajo said Buhari "is in very high spirit, he is recuperating very well and we had very long conversation, we spoke for well over an hour and his humor is all there. He is doing well and he is recuperating fast.
When asked when the president will return, he said "I think very very shortly. Very shortly. I think we should really expect him back very shortly. Like I said he is recuperating very fast and he is doing very very well.
"Well, first as you know I went to see him, I went to check up on him find out how he was doing. I had of course been speaking on the phone and I thought it will be a good thing to go and see him and you know generally check up on how he was doing and also to brief him on developments back at home.
So we had a very good time, we had a very good conversation on wide ranging issues and he is in very good spirit, he is recuperating very quickly and he is doing very well.
What is the state of health of Mr. President?
Well, as I have said he is in very high spirit, he is recuperating very well and we had very long conversation, we spoke for well over an hour and his humor is all there. He is doing well and he is recuperating fast.
I think very very shortly. Very shortly. I think we should really expect him back very shortly. Like I said he is recuperating very fast and he is doing very very well.
Was the report of the suspended SGF and NIA DG part of your conversation?
Wide ranging issues I really can't go into specifics of all of the discussions we had.
Was swearing of the two ministers part of the discussion?
No. We didn't discuss swearing or anything like that. We didn't discuss when they will be sworn in.
When will they be sworn in?
Well, I'm not in a position to say exactly, I don't have a date in mind but I'm sure very very soon we will do that.
Are we expecting Mr. President before 90 days?
No no. Deadlines aren't a very good thing but as I said he is recuperating fast we expecting him very shortly. Much sooner perhaps than you will expect.
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Myton School pupils sow seeds of future forest
Laura Kearns 5th Dec, 2017
THE SEEDS of a future forest have been sown by Myton School pupils.
Pupils planted more than 400 trees in the grounds of the Warwick school which will create a forest in years to come.
The project is the brainchild of Seb Apostol, head of Oken House at the school, who identified a patch of unused land on the school site and came up with a plan to get the youngsters involved in planting trees, which were supplied by the Woodland Trust.
The site already underwent some development over the summer when volunteers including local charity Evelyn's Gift came together to create a wildlife garden complete with pond.
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Former police constable from Leamington who had sex on duty jailed
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Myton School pupils s...
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\section{Introduction}
An undirected multigraph $G = (V,E)$
with $n := |V|, m := |E|, m \geq n$,
is \emph{2-edge-connected}
if for every edge $e \in E$ the graph
$G - e := (V, E \setminus \{e\})$ is connected.
The \emph{minimum 2-edge-connected spanning subgraph} problem~(2ECSS) is defined
as follows: Given a 2-edge-connected undirected multigraph $G = (V,E)$
with edge costs $c\colon E \to \ensuremath{\mathbb{R}_{\geq 0}}\xspace$, find an edge subset $E'\subseteq E$ of minimum cost
$c(E') := \sum_{e \in E'} c(e)$ such that
$G' = (V, E')$ is 2-edge-connected.
Any edge of $G$ may only be used once in $G'$.
2ECSS is a fundamental NP-hard network design problem
that arises naturally in the planning of infrastructure
where one wants to guarantee a basic fault tolerance.
\subparagraph*{Related work.}
Some algorithms mentioned below
work not only for 2ECSS but
for more general problems,
like $k$ECSS with $k\geq 2$.
Since we are interested in the former,
we describe their results,
in particular the achieved approximation ratios,
in the context of 2ECSS.
We restrict our attention to algorithms
able to work on general edge costs
(in contrast to, e.g., metric or Euclidean edge costs).
The first algorithms \cite{FJ81,KT93}
yield $3$-approximations
by using a minimum spanning tree in $G$
and augmenting it to become 2-edge-connected.
The factor $3$ is based on
a $2$-approximation for the latter problem (often called \emph{weighted tree augmentation}).
The algorithm of~\cite{FJ81} runs in $\ensuremath{\mathcal{O}}(n^2)$ time
and that of \cite{KT93} in $\ensuremath{\mathcal{O}}(m + n \log{n})$.
In~\cite{KV94},
a $2$-approximation algorithm
is obtained
by reducing the problem
to a weighted matroid intersection problem
that can be solved
%
in time $\ensuremath{\mathcal{O}}(n (m + n \log n) \log n)$~\cite{G95}.
The $2$-approximation algorithm in~\cite{J01}
is based on iterative rounding of solutions to a linear programming formulation
that we will see in a later section.
On the negative side,
no algorithm with factor less than $2$ is known,
and,
unless P\,$=$\,NP,
there cannot be a polynomial-time approximation with ratio better than
roughly $1+\frac1{300}$~\cite{P10}.
Besides the algorithms mentioned above,
there is a separate history
of applying the primal-dual method.
The basic idea of a primal-dual algorithm
is that a feasible solution to the dual of the aforementioned linear program is computed
and this process is exploited to compute an approximate primal solution.
There are several primal-dual $3$-approximation algorithms~\cite{KR93,SVY92,WGMV95,GGW98,GGPSTW94}
with the best running time being $\ensuremath{\mathcal{O}}(n^2 + n \sqrt{m \log\log n})$.
All algorithms
grow a solution in two phases:
they first obtain a spanning tree,
and then augment that tree to be 2-edge-connected.
Then, unnecessary edges are deleted in a cleanup phase to obtain minimality.
Most algorithms are algorithmically complex
and, for example,
require solving multiple network flow problems.
\subparagraph*{Contribution.}
We present a \emph{simple} 3-approximation algorithm,
analyzed using primal-dual techniques,
that finds a minimal 2-edge-connected spanning subgraph on general edge costs.
In comparison to the other primal-dual algorithms,
we grow the solution in a single phase, i.e,
we omit obtaining an intermediate spanning tree.
Thus we make progress on the question by Williamson~\cite{W93}
if it is possible ``to design a single phase algorithm for some class of edge-covering problems''.
The (to our best knowledge) new conceptual idea is to
modify the classical synchronized primal-dual scheme by growing the solution
only at leaves of the current solution.
We contract arising 2-edge-connected components on the fly.
Although we do not beat the so-far best primal-dual approximation ratio $3$,
our algorithmic framework may offer new insight for further improvements.
Moreover,
our algorithm is conceptually much simpler
than the aforementioned approximation algorithms.
In contrast to the 2-approximation algorithms
based on
weighted matroid intersection
or linear programming,
our algorithm requires only trivial data structures (arrays, lists, graphs, and optionally binary heaps)
and no graph algorithms beyond depth-first search.
It is simple to implement in its entirety
to run in $\ensuremath{\mathcal{O}}(\min\{nm, m + n^2 \log n\})$~time
while occupying only $\ensuremath{\mathcal{O}}(m)$~space.
\subparagraph*{Preliminaries.}
We always consider an undirected multigraph $G = (V, E)$
with node set $V$ and edge set $E$.
As we allow parallel edges, we identify edges by their names, not by their incident nodes.
For each $e \in E$,
let $V(e) := \{v, w\} \subseteq V$
be the two nodes incident to $e$.
We may describe subgraphs of $G$
simply by their (inducing) edge subset $H \subseteq E$.
By $V(H) := \bigcup_{e \in H}{V(e)}$
we denote the set of nodes spanned by the edges of $H$.
For each $v \in V(H)$,
let $\delta_H(v) := \{e \in H \mid v \in V(e) \}$ be the edges incident to $v$.
For any $S \subsetneq V(H)$,
let $\delta_H(S) := \{e \in H \mid V(e) = \{u, v\}, u \in S, v \notin S\}$.
The \emph{degree} $\deg_H(v) := |\delta_H(v)|$ of $v \in V$ in $H$
is the number of incident edges of $v$ in $H$.
A \emph{path} $P$
of length $k \geq 0$
is a subgraph
with $P = \{e_1, \ldots, e_k\}$
such that
there is an orientation of its edges where
the head of $e_i$ coincides with the tail of $e_{i+1}$ for $i < k$.
In such an orientation,
let $u$ be the tail of $e_1$
and $v$ the head of $e_k$.
We call $u$ and $v$ the \emph{endpoints} of $P$,
and $P$ a
\emph{$u$-$v$-path}
(or, equivalently, a \emph{path between $u$ and $v$}).
%
Observe that
our definition of paths
allows nodes but not edges to repeat
(due to set notation).
A path $P$ is \emph{simple}
if and only if $\deg_P(v) \leq 2$ for all $v \in P$.
For a simple $u$-$v$-path $P$,
we call $V(P) \setminus \{u, v\}$
the \emph{inner nodes} of $P$.
We call a path \emph{closed}
if both endpoints coincide
(i.e., if it is a $u$-$v$-path with $u = v$),
and \emph{open} otherwise.
A \emph{cycle} is a closed path of length at least $2$.
We say two paths $P_1, P_2$ are \emph{disjoint}
if and only if $P_1 \cap P_2 = \varnothing$,
i.e., they do not share a common edge
(they may share nodes).
Let $G$ be 2-edge-connected.
An edge $e \in E$ is \emph{essential}
if and only if $E \setminus \{e\}$ is not 2-edge-connected;
it is \emph{nonessential} otherwise.
An \emph{ear} is a simple path $P$ of length at least $1$
such that $E \setminus P$ is 2-edge-connected.
For any function $f\colon A \to B$ and any $A' \subseteq A$, we
denote by $f(A') := \{f(a)\mid a \in A'\}$ the image of $A'$ under
$f$ (unless otherwise stated).
We also define
$f^{-1}(b) := \{a \in A \mid f(a) = b\}$.
\subparagraph*{Organization of the paper.}
Although our algorithm (which is described in Section~\ref{section:alg})
turns out to be surprisingly simple,
its analysis is more involved.
In Section~\ref{section:analysis},
we will show its
time and space complexity
as well as its approximation ratio
(under the assumption that a particular \emph{leaf-degree property}, which may be of independent interest, holds in every step of the algorithm).
The technical proof of the leaf-degree property
is deferred to Section~\ref{section:degree-property}, where it is shown independently (also to simplify the required notation).
\section{The Algorithm}
\label{section:alg}
Our algorithm is outlined in Algorithm~\ref{alg}.
Given a (multi-)graph $G=(V,E)$ with cost function $c\colon E \to \ensuremath{\mathbb{R}_{\geq 0}}\xspace$,
the main \emph{grow phase}
selects edges $T\subseteq E$
such that $T$ is spanning and 2-edge-connected,
but not necessarily minimal.
The central idea of the grow phase---in contrast to several other primal-dual approaches---is to only grow the solution with edges that are currently attached to leaves.\footnote{%
This is a key difference to the \emph{second} phase suggested in \cite{KR93}, which on first sight looks somewhat similar (but leads to very different proof strategies).
In particular,
we can directly attack the 2-edge-connected subgraph
%
in a single phase,
instead of a multi-phase growing procedure where each phase has to consider distinct objectives and rules.}
Afterwards, a trivial \emph{cleanup phase}
removes nonessential edges from $T$,
checking them in reverse order,
to obtain the final solution.
\begin{algorithm}[tbp]
graph $G' = (V', E')$ with edge costs $c':=c$ as a copy of $G=(V, E)$\;
solution $T := \varnothing$\;
forest $F := (V',\varnothing)$\;
\BlankLine
\While(\tcp*[f]{grow phase}){$F$ is not a single node}{
Simultaneously for each leaf in $F$, decrease the cost $c'$ of its incident edges in $G'$
until an edge, say $\tilde e$, gets cost $0$;
by this, an edge cost $c'$ is reduced with double speed if it is incident to two leaves\;
\label{line:delta}
Add $\tilde e$ to $F$ and to $T$\;
\label{line:add}
\lIf{$\tilde e$ closes a cycle $Q$ in $F$}{contract $Q$ in $F$ and in $G'$}
\label{line:cycle}
}
\BlankLine
\ForAll(\tcp*[f]{cleanup phase}){$e \in T$ in reverse order}{%
\lIf{$T - e$ is 2-edge-connected}{%
remove $e$ from $T$
\label{line:cleanup}
}
}
\caption{Approximation algorithm for 2ECSS}%
\label{alg}%
\end{algorithm}
The rest of this paper focuses on proving
our main Theorem~\ref{thm:main} below.
\begin{theorem}\label{thm:main}
There is an algorithm for 2ECSS
that runs in $\ensuremath{\mathcal{O}}(\min\{n m, m + n^2 \log n\})$ time
and $\ensuremath{\mathcal{O}}(m)$ space.
It obtains solutions within three times the optimum.
\end{theorem}
\section{Analysis of Algorithm~\ref{alg} (Proof of Theorem~\ref{thm:main})}
\label{section:analysis}
We call the iterations within the phases of Algorithm~\ref{alg}
\emph{grow steps} and \emph{cleanup steps}, respectively.
In a grow step, a cycle may be contracted and some edges become incident to the contracted node.
As we identify edges by their names, the names of these edges are retained although their incident nodes change.
Let $E'$ and $E_F$ be the edge set of $G'$ and $F$, respectively.
During the algorithm we have the following invariants:
both $G'$ and $F$ use the common node set $V'$ that describes a partition of~$V$;
we consider $T$ to form a subgraph of $G$;
each edge in $E_F$ represents an edge of $T$
that is not part of a cycle in $G$;
we have~$E_F \subseteq E' \subseteq E$.
Initially, each node of $V$ forms an individual partition set,
i.e., $|V'| = |V|$, and $E_F = \varnothing$.
We merge partition sets (nodes of $V'$, cf.\ line~\ref{line:cycle})
when we contract a cycle, i.e.,
when the corresponding nodes in $V$ induce a 2-edge-connected subgraph in $T$.
Arising self-loops are removed both from $G'$ and $F$.
The grow phase terminates once $|V'| = 1$,
i.e., all nodes of $V$ are in a common 2-edge-connected component.
Observe that for an edge $e \in E'$, we naturally define
$V(e) \subseteq V$ as its incident nodes in original $G$, and
$V'(e) \subseteq V'$ as its incident nodes in $G'$ and $F$.
Let $L := \{v \in V' \mid \deg_F(v) \leq 1\}$
be the set of leaves (including isolated nodes) in $F$.
For any edge $e \in E'$,
let $\ell_e := |V'(e) \cap L| \in \{0, 1, 2\}$
be the number of incident nodes of $e$ that are leaves in~$F$.
An edge $e \in E'$ is \emph{eligible}
if
$e \notin T$ and
$\ell_e \geq 1$.
Let $\Delta(e) := \frac{c'(e)}{\ell_e}$ for eligible edges $e \in E'$.
Now line~\ref{line:delta}
can be described as
first finding
the minimum (w.r.t.\ $\Delta$) eligible edge $\tilde e \in E$,
and then, for each eligible edge $e$,
decreasing $c'(e)$ by $\ell_e \Delta(\tilde e)$.
For convenience,
we denote $\Delta(\tilde e)$ by $\tilde\Delta$.
\subsection{Time and Space Complexity}
Here we state the main time and space complexity results
for Algorithm~\ref{alg}.
\begin{lemma}\label{lemma:time}
Algorithm~\ref{alg} can be implemented to run
in $\ensuremath{\mathcal{O}}(n m)$ time
and $\ensuremath{\mathcal{O}}(m)$ space
using only arrays and lists.
\end{lemma}
\begin{proof}
Let us first describe the used data structures.
The graphs $G'$ and $F$ are stored naturally using the adjacency list representation.
We store $T$ as a list (or array)
and manage the costs~$c'$ in an array.
The space consumption of $\ensuremath{\mathcal{O}}(m)$ follows directly,
and the initialization of all data structures takes $\ensuremath{\mathcal{O}}(m)$ time.
We now show that there are $\ensuremath{\mathcal{O}}(n)$ grow and cleanup steps.
Consider $T$ directly after the grow phase.
Let $T_0 \subsetneq T$
be the edges
that led to a contraction,
thus $|T_0| < n$.
The edges $T \setminus T_0$
form a tree
since any cycle would lead to a contraction.
Line~\ref{line:delta}
takes $\ensuremath{\mathcal{O}}(m)$
time by
iterating over all $e \in E'$ twice,
first to find $\tilde e$,
and
second to reduce the costs $c'$.
Line~\ref{line:add}
adds $\tilde e$ to $T$
and to $F$ in constant time.
For line~\ref{line:cycle},
we can find the respective cycle in $F$
(or determine that it does not exist)
in $\ensuremath{\mathcal{O}}(n)$ time
using depth-first search.
The contractions of the (same) cycle in $G'$ and $F$ are performed
in $\ensuremath{\mathcal{O}}(n)$ time.
For the running time of a cleanup step (line~\ref{line:cleanup}),
consider $T$ at the current iteration.
Note that $T$ induces a 2-edge-connected graph.
We can check in $\ensuremath{\mathcal{O}}(|T|)$ time
if an edge $e \in T$
with $V(e) = \{s, t\}$
is essential
using a simplified version of the classical 2-connectivity test:
Perform a depth-first search
in $T - e$
starting at $s$
to compute the DFS indices for each node.
Using a single bottom-up traversal,
compute $\operatorname{low}(u)$ for each $u \in V$
where $\operatorname{low}(u)$ is the node
with the smallest DFS index reachable
from $u$
by using only
edges in the DFS tree to higher DFS indices,
and
at most one edge not in the DFS tree.
Now initialize $v := t$
and iteratively go to $v := \operatorname{low}(v)$ until
$v = \operatorname{low}(v)$.
Clearly,
there is a cycle in $T - e$ containing $s$ and $t$
if and only if $v = s$;
otherwise $e$ is essential.
%
\end{proof}
Unfortunately,
$m$ can be unbounded if $G$ has parallel edges.
We can improve the running time
for graphs with
$m \in \omega(n \log{n})$.
For this, we first shrink $m$ to $\ensuremath{\mathcal{O}}(n^2)$
by removing uninteresting (parallel) edges
in the beginning as well as after every contraction.
Then,
we use one global value $\Gamma$
that simulates the shrinking of all costs~$c'$
in constant time,
and we use $\ensuremath{\mathcal{O}}(\sqrt{m})$ binary heaps
of size $\ensuremath{\mathcal{O}}(\sqrt{m})$
to manage the eligible edges and their costs $c'$;
$\ensuremath{\mathcal{O}}(n)$ many updates in these binary heaps
lead to a dominating running time of $\ensuremath{\mathcal{O}}(n \log n)$ per grow step.
\begin{lemma}\label{lemma:time-dense}
Algorithm~\ref{alg} can be implemented to run
in $\ensuremath{\mathcal{O}}(m + n^2 \log{n})$ time
and $\ensuremath{\mathcal{O}}(m)$ space
using only arrays, lists, and binary heaps.
\end{lemma}
\begin{proof}
Consider a set of $p > 2$ pairwise parallel edges.
We call each of the $p - 2$ highest-cost edges
\emph{futile}.
By removing all futile edges
we decrease the maximum edge-multiplicity in $G$ to $2$.
This reduction can be performed in $\ensuremath{\mathcal{O}}(m + n^2)$ time (e.g., by using $n$ buckets for the neighbors of each $v \in V$).
This guarantees
that each node has degree $\ensuremath{\mathcal{O}}(n)$,
hence from now on $m \in \ensuremath{\mathcal{O}}(n^2)$.
Whenever we contract a cycle $Q$ of $n_Q$ nodes
into a single node $q$ in line~\ref{line:cycle},
we re-establish $\deg_{G'}(q) \in \ensuremath{\mathcal{O}}(n)$
by removing arising self-loops and futile edges.
Spotting futile edges requires $\ensuremath{\mathcal{O}}(n)$ time per contraction.
Although there are
$\ensuremath{\mathcal{O}}(n_Q^2)$ edges within $V(Q)$ (which become self-loops at $q$)
and
$\ensuremath{\mathcal{O}}(n_Q n)$ edges from $V(Q)$ to $V_G \setminus V(Q)$,
the removal of the self-loops and futile edges
takes $O(m)$ time in total (for the whole grow phase),
since any edge is removed at most once.
%
Instead of storing $c'$ directly,
store a single global value $\Gamma$, initially being zero,
and for each edge $e \in E'$, store a value $\bar \Delta(e)$
that keeps the invariant
%
$\bar\Delta(e) = \Delta(e) + \Gamma$
if $e$ is eligible
and
%
$\bar \Delta(e) = c'(e)$
if not.
Furthermore,
we partition the eligible edges of $E'$ arbitrarily
into $\Theta(\sqrt{m})$ many (binary) heaps
of size $\Theta(\sqrt{m})$,
with $\bar\Delta$ as the priorities.
Initially, all edges are eligible.
The initialization takes $\ensuremath{\mathcal{O}}(m)$~time.
By looking at the minimum edge of each heap,
we can find and extract $\tilde e$ (see line~\ref{line:delta})
in $\ensuremath{\mathcal{O}}(\sqrt{m})$ time.
To decrease $c'(e)$ by $\ell_e \tilde\Delta$ for all eligible $e \in E'$,
we have to decrease $\Delta(e)$ by $\tilde\Delta$
which is performed by increasing $\Gamma$ by $\tilde\Delta$
in constant time.
Observe that this preserves the invariant:
If $e$ is eligible,
decreasing $c'(e)$ by $\ell_e \tilde\Delta$
updates $\Delta(e)$ to $\frac{c'(e) - \ell_e \tilde\Delta}{\ell_e} = \frac{c'(e)}{\ell_e} - \tilde\Delta$,
which is exactly what increasing $\Gamma$ by $\tilde\Delta$ does;
otherwise,
neither $\Delta(e)$ nor $\Gamma$ is changed.
Adding edge $e$ with $V'(e) = \{s, t\}$ to $F$ (line~\ref{line:add})
or contracting a cycle $Q$ into $q$ (line~\ref{line:cycle})
may change the $\ell_e$ values of all $\ensuremath{\mathcal{O}}(n)$ edges in $G$
incident to $s$ and $t$ or to $q$,
respectively.
If $\ell_e$ decreases to $0$,
we remove $e$ from its heap;
if it increases from $0$,
we re-add it to its original heap.
Otherwise,
if $\ell_e$ decreases from $2$,
we want $\Delta(e)$ to double;
if it increases to $2$,
$\Delta(e)$ should be halved.
The necessary changes to $\bar \Delta(e)$ follow straightforwardly.
Each necessary operation on a heap
requires $\ensuremath{\mathcal{O}}(\log{n})$ time.
We can hence perform each grow step in $\ensuremath{\mathcal{O}}(n \log n)$~time.
\end{proof}
\subsection{Analysis of the Approximation Ratio}
Let
%
%
$\mathcal{S} := 2^V \setminus \{\varnothing, V\}$
and
$\mathcal{S}_{e} := \{S \in \mathcal{S} \mid e \in \delta_G(S)\}$
for any $e \in E$.
We analyze the approximation ratio using the primal-dual method.
Hence consider the basic integer program for~2ECSS:
\begin{align}
\textsl{minimize}\quad & \sum_{e \in E}{c(e) x_e} \\
\sum_{e \in \delta_G(S)}{x_e} & \geq 2 & & \forall S \in \mathcal{S}
\label{eq:lp:cut}
\\
x_{e} & \in \{0, 1\} & & \forall e \in E
\label{eq:lp:integrality}
\text{.}
\intertext{%
For its linear relaxation,
\eqref{eq:lp:integrality} is substituted by $0 \leq x_e \leq 1$ for every $e \in E$.
The bound $x_e \leq 1$ is important since edge duplications are forbidden.
Its dual program is}
\textsl{maximize}\quad & 2 \sum_{S \in \mathcal{S}}{y_S} - \sum_{e \in E}{z_e} \\
\sum_{S \in \mathcal{S}_e}{y_S} - z_e & \leq c(e)
& & \forall e \in E
\label{eq:dual:cost}%
\\
y_S & \geq 0
& & \forall S \in \mathcal{S}
\\
z_e & \geq 0
& & \forall e \in E
\text{.}
\end{align}
We show that Algorithm~\ref{alg}
implicitly constructs a solution $(\bar y, \bar z)$ to the dual program.
Let $(\bar y^i, \bar z^i)$ denote this dual solution computed after the $i$-th grow step.
Initially, we have the dual solution $(\bar y^0, \bar z^0)=0$.
Following this notion,
let $F^i = (V^i, E_{F^i})$ be the forest after the $i$-th grow step,
$L^i := \{v \in F^i \mid \deg_{F^i}(v) \leq 1\}$,
and $\ell_e^i := |V'(e) \cap L^i|$ for each $e \in E'$.
For any node $v \in V'$,
let $S(v)$ be the corresponding node subset of~$V$.
\begin{lemma}\label{lemma:algorithm-lp}
The grow phase constructs a feasible solution to the dual problem implicitly as follows.
We have, for each $v \in V'$ and $i \geq 1$,
\begin{align*}
\bar y_{S(v)}^{i}
& := \begin{cases}
\bar y_{S(v)}^{i-1} + \tilde\Delta & \text{if $v \in L^{i-1}$} \\
\bar y_{S(v)}^{i-1} & \text{otherwise,}
\end{cases}
& \quad
\bar z_e^{i}
& := \begin{cases}
\bar z_e^{i-1} + \tilde\Delta & \text{if $v \in L^{i-1}$ and $e \in \delta_{F^{i-1}}(v)$} \\
\bar z_e^{i-1} & \text{otherwise.}
\end{cases}
\end{align*}
\end{lemma}
\begin{proof}
Let $\bar{c}^i$ be $c'$ after the $i$-th grow step.
Initially, $(\bar y^0, \bar z^0) = 0$
matches
the initialization $\bar{c}^0 := c$.
Consider the $i$-th grow step.
We show that
$(\bar y^i, \bar z^i)$ satisfies
(i)~$\bar{c}^i(e) = c(e) - \sum_{S \in \mathcal{S}_e}{\bar y_S^i} + \bar z_e^i$,
i.e., the right-hand side minus the left-hand side of~\eqref{eq:dual:cost},
and (ii)~$\bar{c}^i(e) \geq 0$,
i.e., the constructed solution is feasible.
(i) The claim is trivial for $v \notin L^{i-1}$
since the corresponding variables do not change.
Consider $v \in L^{i-1}$
and any $e \in \delta_G(S(v))$.
By $\bar y_{S(v)}^i = \bar y_{S(v)}^{i-1} + \tilde\Delta$,
we have
$\sum_{S \in \mathcal{S}_e}{\bar y_S^{i-1}} - \bar z_e^{i-1}
= \sum_{S \in \mathcal{S}_e}{\bar y_S^i} - \bar z_e^{i-1} - \tilde\Delta$
for the left-hand side of~\eqref{eq:dual:cost}.
By the definition of $\bar z_e^i$,
this coincides with
$\sum_{S \in \mathcal{S}_e}{\bar y_S^i} - \bar z_e^i$
if $e \in \delta_F(v)$,
and with
$\sum_{S \in \mathcal{S}_e}{\bar y_S^i} - \bar z_e^i - \tilde\Delta$
otherwise.
This change is reflected exactly
by $\bar{c}^i(e) := \bar{c}^{i-1}(e) - \ell_e^{i-1} \tilde\Delta$
(that is, decreasing $c'$ by $\tilde\Delta$ for each leaf incident to $e$ in~$F$)
if and only if $e$ is eligible.
%
(ii) Assume by contradiction
that
there is an $e \in E$
with $\bar{c}^{i-1}(e) \geq 0$ and $\bar{c}^i(e) < 0$.
Note that $\bar{c}^{i-1}(\tilde e) = \ell_{\tilde e}^{i-1} \tilde\Delta$.
By $\bar{c}^i(e) := \bar{c}^{i-1}(e) - \ell_e^{i-1} \tilde\Delta < 0$
we get $\bar{c}^{i-1}(e) < \ell_e^{i-1} \tilde\Delta$,
which contradicts the choice of $\tilde e$.
%
\end{proof}
Let $\bar T$ be the solution edges remaining after the cleanup phase.
Let $(V^i, \bar T^i)$
be the graph on nodes $V^i$
that consists of all edges in $\bar T$
without self-loops.
In other words,
$\bar T^i$
are the edges corresponding to $\bar T$
when mapped into the node partition defined by $F^i$.
We partition $L^i$ into
the set
$L_0^i := \{v \in L^i \mid \deg_{F^i}(v) = 0\}$
of isolated nodes in $F^i$,
the set
$L_1^i := \{v \in L^i \mid \deg_{F^i}(v) = 1, \delta_{F^i}(v) \subseteq \bar T^i\}$
of degree-1 nodes in $F^i$
incident to an edge in the contracted solution $\bar T^i$,
and
the set
$L_2^i := \{v \in L^i \mid \deg_{F^i}(v) = 1, \delta_{F^i}(v) \cap \bar T^i = \varnothing\}$
of the degree-1 nodes in $F^i$
incident to an edge in $E^i \setminus \bar T^i$,
i.e., not being in the contracted solution.
\begin{lemma}
For each $i$,
every edge $e \in \bar T^i \setminus E_{F^i}$ is essential in $(V^i, \bar T^i \cup E_{F^i})$.
\label{lemma:essential}
\end{lemma}
\begin{proof}
First observe that
for a cycle~$Q$ in a 2-edge-connected graph~$H$,
an edge $e \notin Q$
is essential in $H$ if and only if $e$ is essential in $H$ after contracting $Q$.
The claim holds trivially for the single-node forest (i.e., after the last grow step).
Consider any $i$
where we have that every edge $e \in \bar T^i \setminus E_{F^i}$
is essential in $(V^i, \bar T^i \cup E_{F^i})$.
In the $i$-th grow step,
we insert an edge $e \in \bar T^{i-1} \setminus E_{F^{i-1}}$
into $F^{i-1}$ or possibly contract an emerging cycle.
%
In any way,
$e \notin \bar T^i \setminus E_{F^i}$.
By induction,
all edges $\bar T^{i-1} \setminus (E_{F^{i-1}} \cup \{e\})$ are essential.
If $e$ was nonessential,
%
%
the cleanup step corresponding to the $i$-th grow step would remove $e$.
%
\end{proof}
\begin{lemma}[Leaf-Degree Property]\label{lemma:degree-property}%
We have
$
\sum_{v \in L^i}{\deg_{\bar T^i}(v)}
\leq
3 (|L^i| + |L^i_0|) + |L^i_1|
$.
\end{lemma}
Note that Lemma~\ref{lemma:essential} is a prerequisite for Lemma~\ref{lemma:degree-property}.
The proof of Lemma~\ref{lemma:degree-property} is highly non-trivial and may be of independent interest.
We thus defer its presentation,
together with all the required notations and further definitions,
to Section~\ref{section:degree-property}.
There, we will restate the lemma (including all prerequisites) in more general terms as Theorem~\ref{thm:degree-property}.
\begin{lemma}\label{lemma:approx-ratio}%
The solution obtained by Algorithm~\ref{alg} is within three times the optimum.
\end{lemma}
This result is tight as can be seen in Figure~\ref{fig:tight}.
\begin{proof}
Let $(\bar y, \bar z)$ be the \emph{dual} solution Algorithm~\ref{alg} produces implicitly, as described by Lemma~\ref{lemma:algorithm-lp},
with dual solution value $B$.
On the other hand, $\bar T$ is called our \emph{primal} solution.
Note that for all edges $e \in \bar T$, we have $c'(e) = 0$,
i.e.,
their constraints \eqref{eq:dual:cost} are tight.
Hence we can rewrite our primal solution value
\[
c(\bar T) = \sum_{e \in \bar T}{c(e)}
= \sum_{e \in \bar T}\Big(\sum_{S \in \mathcal{S}_e}{\bar y_S} - \bar z_e\Big)
= \sum_{S \in \mathcal{S}}{\deg_{\bar T}(S) \bar y_S} - \sum_{e \in \bar T}{\bar z_e}
\text{.}
\]
We prove a $3$-approximation by showing that
$c(\bar T) \leq 3 B$,
%
i.e.,
\[
\sum_{S \in \mathcal{S}}{\deg_{\bar T}(S) \bar y_S} - \sum_{e \in \bar T}{\bar z_e}
\, \leq \,
3\Big(\sum_{S \in \mathcal{S}}{2 \bar y_S} - \sum_{e \in E}\bar z_e\Big)
\text{,}
\]
or equivalently
\begin{equation}
\sum_{S \in \mathcal{S}}{\deg_{\bar T}(S) \bar y_S}
\, \leq \,
6 \sum_{S \in \mathcal{S}}{\bar y_S}
- 2 \sum_{e \in \bar T}\bar z_e
- 3 \!\! \sum_{e \in E \setminus \bar T} \!\!\! \bar z_e
\text{.}
\label{eq:primaldual:essence}%
\end{equation}
Observe that~\eqref{eq:primaldual:essence} trivially holds initially
since all values $(\bar y^0, \bar z^0)$ are zero.
We show that \eqref{eq:primaldual:essence} holds after each grow step.
Assume it holds for $(\bar y^i, \bar z^i)$.
We look at the increase
of the left-hand side
and right-hand side
of \eqref{eq:primaldual:essence}
when adding an edge to $F^i$.
By Lemma~\ref{lemma:algorithm-lp},
we have $\bar y^{i+1}_{S(v)} = \bar y^i_{S(v)} + \tilde\Delta$ for all $v \in L^i$
and
$\bar z^{i+1}_e = \bar z^i_e + \tilde\Delta$ for all $e \in \delta_F(L^i_1 \cup L^i_2)$.
Hence
it remains to show that
\[
\sum_{v \in L^i}{\deg_{\bar T^i}(v) \tilde\Delta}
\; \leq \;
6 \sum_{v \in L^i}\tilde\Delta
- 2 \sum_{v \in L^i_1}\tilde\Delta
- 3 \sum_{v \in L^i_2}\tilde\Delta
\]
holds.
After dividing by $\tilde\Delta$
and since $|L^i| = |L^i_0| + |L^i_1| + |L^i_2|$,
the right-hand side
simplifies to
$6 |L^i| - 2 |L^i_1| - 3 |L^i_2|
=
6 |L^i_0| + 4 |L^i_1| + 3 |L^i_2|
=
3 (|L^i| + |L^i_0|) + |L^i_1|$,
i.e., we have Lemma~\ref{lemma:degree-property}.
\end{proof}
\section{The Leaf-Degree Property (Proof of Lemma~\ref{lemma:degree-property})}
\label{section:degree-property}
\tikzset{%
general/.style={%
draw=black,
fill=black,
inner sep=1pt,
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gedge/.style={%
draw=black!70!yellow,
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subgraph/.style={%
gedge,
fill=lightgray!40,
ellipse,
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fedge/.style={%
very thick,
draw=black!60!blue,
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densely dashed,
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\draw [fedge]
(#1-n2) edge[bend right=13] (#1-n1);
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\begin{figure}
\centerline{%
\begin{tikzpicture}[baseline=0]
\pic at (0,0) {branch=a};
\pic [rotate=42] at (0,0) {branch=b};
\pic [rotate=138] at (0,0) {branch=d};
\pic [rotate=180] at (0,0) {branch=e};
%
\node at (0,0.8) {$\ldots$};
\node at (0,1.2) {$k$ many};
%
\coordinate (c-n1) at (-0.75,1.2);
\coordinate (c-n2) at (0.75,1.2);
\draw [gedge, not in solution] (a-n1) edge (b-n2);
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\draw [gedge, not in solution] (c-n1) edge (d-n2);
\draw [gedge, not in solution] (d-n1) edge (e-n2);
%
\node [filledleaf] at (0,0) {};
%
\node at (6,0.6) {%
\begin{minipage}{45mm}
$\sum_{v \in L}{\deg_{E'}(v)} = 8k$\\[1ex]
$|L_0| = 1, \; |L_1| = 2k, \; |L_2| = 0$\\[1ex]
$3 \, (|L| + |L_0|) + |L_1| = 8k + 6$
\end{minipage}};
\end{tikzpicture}}
\caption{\label{fig:tight}%
An example showing
tightness for the approximation ratio as well as for the leaf-degree property.
For the approximation ratio, consider
all thick edges' costs to be~$0$,
all solid thin edges' costs~$1$,
and all dashed edges' costs~$1+\varepsilon$
for an arbitrary small~$\varepsilon > 0$.
The algorithm's solution consists of all solid edges of total cost~$3k$.
The optimum solution
is the Hamiltonian cycle consisting of
all dashed edges,
all thick edges,
and two solid thin edges to connect the center node.
Its total cost is~$
%
%
%
k + 1 + (k-1) \varepsilon
$.
The ratio~$\frac{3k}{k + 1 + (k-1) \varepsilon)}$
approaches~$3$
for~$k \to \infty$.
%
For the leaf-degree property,
all edges are in~$E$,
thick edges in~$F \subsetneq E$,
solid edges in~$E' \subsetneq E$.
}
\end{figure}
This section is dedicated to show the following theorem.
The theorem is a reformulation of Lemma~\ref{lemma:degree-property}
in terms that are totally independent of the setting and notation
used in the previous section.
\begin{theorem}[Reformulation of Lemma~\ref{lemma:degree-property}]\label{thm:degree-property}
Let $G = (V, E)$ be a 2-edge-connected graph and
$E'$ a minimal 2-edge-connected spanning subgraph in $G$.
Let $F \subseteq E$
be an edge set describing a (not necessarily spanning) forest in $G$
such that
each edge $e \in E' \setminus F$
is essential in $E' \cup F$.
Let $L_0 := V \setminus V(F)$,
$L_1 := \{v \in V(F) \mid \deg_F(v) = 1, \delta_F(v) \subseteq E'\}$,
$L_2 := \{v \in V(F) \mid \deg_F(v) = 1, \delta_F(v) \cap E' = \varnothing\}$,
and $L := L_0 \cup L_1 \cup L_2$.
Then we have
$
\sum_{v \in L}{\deg_{E'}(v)}
\; \leq \;
3 \, (|L| + |L_0|) + |L_1|
$.
%
\end{theorem}
Figure~\ref{fig:tight} illustrates an example
where the left-hand side approaches the right-hand side.
Throughout this section,
we will use the following convention:
We call the nodes in $L$ \emph{leaves};
they are either $L_1 \cup L_2$, degree-1 nodes in $F$, or
$L_0$, isolated nodes w.r.t.~$F$.
This is quite natural
since $E$ and $E'$ do not contain any degree-1 nodes.
For any subforest $F' \subseteq F$,
let $L(F') := L \cap V(F')$.
We use the term \emph{component}
for a connected component in $F$
since $E$ and $E'$ consist of one connected component only.
Hence these two terms only make sense in the context of $F$.
We consider an \emph{ear decomposition} of $E'$, that is, we consider
an ordered partition of $E'$ into disjoint edge sets
$O_0, O_1, \ldots$ where $O_0$ is a simple cycle and where
$O_t$ for $t \geq 1$ is a simple $u$-$v$-path with
$V(O_t) \cap \bigcup_{i=0}^{t-1}{V(O_i)} = \{u, v\}$.
Such an ear decomposition exists since $E'$ is 2-edge-connected.
Note that
every ear $O_t$ has at least one inner node
since it would otherwise only consist of a single edge
which would be nonessential in $E'$.
Let $E_t' := \bigcup_{i=0}^{t-1}{O_i}$
be the subgraph of $E'$
that contains of the first $t$ ears
of the ear sequence.
We interpret the ear decomposition as a sequential procedure.
We say $O_t$ is added to $E_t'$ at \emph{time} $t$.
For $t_2 > t_1 \geq 1$,
the ear $O_{t_1}$ appears \emph{earlier} than $O_{t_2}$,
and $O_{t_2}$ appears \emph{later} than $O_{t_1}$.
At any time $t$,
we call a node $v$ \emph{explored}
if $v \in V(E_t')$,
otherwise it is \emph{unexplored};
we call a component \emph{discovered}
if it contains an explored node,
otherwise it is \emph{undiscovered}.
Observe that the inner nodes $v$ of $O_t$ are not yet explored at time~$t$.
We define $\theta(v):=t$ as the time when $v$ will become explored. Clearly, we have $\theta(v):=0$ for all nodes $v\in O_0$.
The basic idea of our proof is to
use the ear sequence
to keep track (over time $t$) of $\deg_{E'_t}(v)$ for $v \in L$
via a charging argument.
Consider any $t \geq 1$.
An inner (and thus unexplored) node of $O_t$ might be in $L$.
Every such leaf has a degree of $2$ in $E_{t+1}'$.
However, the endpoints of $O_t$ may be explored leaves
whose degrees increase in $E_{t+1}'$.
We tackle this problem by
assigning this increase to other leaves
and making sure that the total assignment to each leaf is bounded.
Let $\Pi$ %
be the set of all edges in $E'$
that are incident to a leaf that is simultaneously an endpoint of some ear $O_t$.
We denote the edges in $\Pi$ by
$\pi_1, \ldots, \pi_{|\Pi|}$
in increasing time of their ears,
i.e.,
for
$\pi_i \in O_t, \pi_j \in O_{t'}$
with
$i < j$
we have
$t \leq t'$.
We say $i$ is the \emph{index} of edge $\pi_i \in \Pi$.
To be able to refer to the nodes $V(\pi_i) =: \{a_i, b_i\}$ by index,
we define
$a_i$ as the endpoint
and
$b_i$ as the inner node
of the ear containing $\pi_i$.
Note that
there might be distinct $\pi_i, \pi_j \in \Pi$
with $\theta(b_i) = \theta(b_j)$
if both $a_i$ and $a_j$ are leaves (with possibly even $b_i = b_j$).
%
%
By $C_i$ we denote the component that contains~$b_i$.
For any index $i$,
we may have:
\emph{situation}~\sitE
if
$\pi_i$ is an \underline{e}lement of $F$,
\emph{situation}~\sitU
if $\pi_i \notin F$
and $C_i$ is \underline{u}ndiscovered,
and
\emph{situation}~\sitD
if $\pi_i \notin F$
and $C_i$ is \underline{d}iscovered,
c.f.~Figure~\ref{fig:situations}.
\begin{figure}
\centerline{%
\begin{tikzpicture}[%
pics/setting/.style={
code = {
\node (Et) [subgraph] at (0,0.5) {\makebox[1.5cm]{#1}};
\node (li) [leaf, label={[label distance=1pt, inner sep=0]170:$a_i$}] at (Et.150) {};
\node (ri) [general] at (Et.30) {};
\node (ra) [general] at ([shift={(-1mm,6mm)}] ri) {};
\node (la) [general, label={[label distance=1pt, inner sep=0]180:$b_i$}] at ([shift={(1mm,6mm)}] li) {};
\node (rb) [general] at ([shift={(-3mm,3mm)}] ra) {};
\node (lb) [general] at ([shift={(3mm,3mm)}] la) {};
\draw [gedge] (ri) -- (ra) -- (rb) -- (lb) -- (la) -- (li);
}
},
pics/situation/.style={
code = {
\pic {setting={\phantom{$E_t'$}}};
\node (rb) [leaf] at (rb) {};
\draw [fedge] (ri) -- (ra);
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(ri) -- +(0,-4.2mm);
\node at (0,-0.3) {situation~#1};
}
}]
\node [yshift=7mm] at (-0.3,0.5) {\footnotesize unexplored};
\node at (-0.3,0.5) {\footnotesize \phantom{un}explored};
%
\pic at (2,0) {setting={$E_t'$}};
\node at ([yshift=5.5mm] Et.north) {$O_t$};
\node at ([shift={(2mm,-3mm)}] la.south west) {$\pi_i$};
\node at (2,-0.3) {general setting};
%
\pic at (5,0) {situation=\sitE};
\node [general] (ex) at ([yshift=-2.5mm] Et.north) {};
\draw [fedge] (li) -- (la) -- (lb) -- (ex);
\draw [fedge, dash dot dot]
(la) -- +(-2mm,4mm)
(lb) -- +(-2mm,4mm)
(ex) -- +(0,-4.2mm);
%
\pic at (8,0) {situation=\sitU};
\draw [fedge] (la) -- (lb);
\draw [fedge, dash dot dot]
(la) -- +(-2mm,4mm)
(lb) -- +(-2mm,4mm);
%
\pic at (11,0) {situation=\sitD};
\node [general] (ex) at ([yshift=-2.5mm] Et.north) {};
\draw [fedge] (la) -- (lb) -- (ex);
\draw [fedge, dash dot dot]
(la) -- +(-2mm,4mm)
(lb) -- +(-2mm,4mm)
(ex) -- +(0,-4.2mm);
\end{tikzpicture}}
\caption{\label{fig:situations}%
Illustration of the general setting for an ear $O_t$ and a $\pi_i \in \Pi$
with $\theta(b_i) = t$,
and examples of situations~\sitE,~\sitU, and~\sitD.
Thick edges are in $F$,
rectangular nodes in $L$.}
\end{figure}
We will assign the degree increments of $a_i$
to other leaves
by some charging scheme~$\chi$, which is the sum of several distinct charging schemes.
The precise definition of these (sub)schemes is subtle and necessarily intertwined with the analysis
of the schemes' central properties. Thus we will concisely define
them only within the proofs of Lemmata~\ref{lemma:charge-e}
and~\ref{lemma:charge-u-and-d} below.
We call a leaf \emph{charged} due to a specific situation
if that situation applied
at the time
when the increment was assigned to the leaf.
Let $\chi_\sitE, \chi_\sitU, \chi_\sitD\colon L \to \mathbb{N}$
be the overall charges (on a leaf) due to situation \sitE, \sitU, \sitD, respectively.
The leaf-degree property will follow by observing that no leaf is charged too often by these different chargings.
\begin{lemma}\label{lemma:charge-e}
We can establish a charging scheme $\chi_\sitE$ such that
we guarantee
$\chi_\sitE(v) \leq 1$ if $v \in L_1$
and
$\chi_\sitE(v) = 0$ if $v \in L_0 \cup L_2$.
\end{lemma}
\begin{proof}
Consider situation~\sitE occurring for index $i$.
By $\pi_i \in F$,
we have $a_i \in L_1$
(and thus $\chi_\sitE(a_i) = 0$ if $a_i \in L_0 \cup L_2$).
Assume situation~\sitE occurs for another index $j \neq i$
such that
$a_i = a_j$.
This yields $\pi_i, \pi_j \in F$
which contradicts that $a_i$ is a leaf.
Hence the claim follows by setting $\chi_\sitE(a_i) = 1$.
\end{proof}
\begin{lemma}\label{lemma:charge-u-and-d}
We can establish charging schemes $\chi_\sitU,\chi_\sitD$ such that
we guarantee
$\chi_\sitU(v) + \chi_\sitD(v) \leq 2$ if $v \in L_0$
and
$\chi_\sitU(v) + \chi_\sitD(v) \leq 1$ if $v \in L_1 \cup L_2$.
\end{lemma}
The proof is rather technical and will be proven in the following subsection.
It mainly exploits
the finding of contradictions to the fact
that each edge $e \in E' \setminus F$ is essential in $E' \cup F$.
Two mappings can be established:
first an injective mapping
(based on induction)
from edges $\pi_i$ in situation~\sitD to leaves,
and second an `almost injective' (relaxing the mappings to $L_0$ nodes slightly) mapping
from edges $\pi_i$ in situation~\sitU
to remaining leaves.
For the latter,
we establish an algorithm that hops through components.
We show that this algorithm identifies suitable distinct leaves.
The charging schemes $\chi_\sitU, \chi_\sitD$
with the desired properties
follow from these mappings.
\begin{proof}[Proof of Theorem~\ref{thm:degree-property}]
Let $v \in L$ be any leaf.
The charging of $v$ during the whole process
is $\chi(v) := 2 + \chi_\sitE(v) + \chi_\sitU(v) + \chi_\sitD(v)$
where the $2$ comes from an implicit charging of the degree of $v$
when $v$ is discovered.
By Lemmata~\ref{lemma:charge-e} and~\ref{lemma:charge-u-and-d},
we obtain
$\chi(v) \leq 4$ for $v \in L_0$,
$\chi(v) \leq 4$ for $v \in L_1$,
and
$\chi(v) \leq 3$ for $v \in L_2$.
This yields
$\sum_{v \in L}{\deg_{E'}(v)}
\leq
4 |L_0| + 4 |L_1| + 3 |L_2|
\leq
3 (|L| + |L_0|) + |L_1|$.
\end{proof}
\subsection{Proof of Lemma~\ref{lemma:charge-u-and-d}}
We first introduce some notation
in order to show Lemma~\ref{lemma:charge-u-and-d}.
For any subforest $F' \subseteq F$
and $S \in \{\sitU, \sitD\}$,
let $\PiX{S}(F') := \{\pi_j \in \Pi \mid{}$situation~$S$ applies for $j$ with $b_j \in V(F')\}$.
Let
$\aX{S}(F') := \{a_j \mid \pi_j \in \PiX{S}(F')\}$.
For any subgraph $H$ in $G$ and two (not necessarily distinct) nodes
$x_1, x_2 \in V(H)$ we define $\pathset{H}{x_1}{x_2}$ to be the set
of all paths in $H$ between $x_1$ and~$x_2$.
Consider nodes $w_0,w_1,\dots,w_k \in V$
for some $k \in \mathbb{N}$
and a collection $\mathcal{P}_1,\dots,\mathcal{P}_k$
of $w_{j-1}$-$w_j$-paths, that is,
$\mathcal{P}_j \subseteq \pathset{G}{w_{j-1}}{w_j}$ for
$j=1,\dots,k$.
Then let
$\buildpath{\mathcal{P}_1, \ldots, \mathcal{P}_k}$ denote the set
of all $w_0$-$w_k$-paths that are the concatenation of $k$
(necessarily) pairwise disjoint paths $P_1,\dots,P_k$ with
$P_j\in\mathcal{P}_j$ for $j=1,\dots,k$.
For notational simplicity,
we may also use single paths and single edges as sets
$\mathcal{P}_j$. Note that
$\buildpath{\pathset{E_t'}{x_1}{a_i},
\pathset{E_t'}{a_i}{x_2}} \neq \varnothing$ for nodes
$x_1, x_2 \in V(E_t')$, which follows from the well-known fact that
any 2-edge-connected graph contains two disjoint $u$-$v$-paths for
all nodes $u, v$.
The below proofs of our auxiliary lemmas will use the following reasoning.
We will determine an edge $\pi_j \in \Pi \setminus F$
and a set of paths $\mathcal{P}$ such that there is a cycle
$Q \in \mathcal{P}$ with $V(\pi_j) \subseteq V(Q)$, and
$\pi_j \notin Q$.
%
%
Since $\pi_j \notin F$, we say that
$\cyclewitness{j}{\mathcal{P}}$ is a \emph{cycle witness} that
contradicts our assumption that every edge in $E' \setminus F$ is
essential in $E' \cup F$.
\begin{lemma}\label{lemma:situation-a-unexplored-leaf}
Let $\pi_i \in \PiX{\sitD}(F)$
and $t := \theta(b_i)$.
There are no two disjoint paths in $C_i$ between $b_i$ and explored nodes.
Moreover,
there is at least one unexplored leaf in $C_i$.
\end{lemma}
\begin{proof}
Assume
there are two disjoint paths
$P_1, P_2$
between $b_i$ and nodes $w_1, w_2 \in V(E_t') \cap V(C_i)$, respectively.
We can w.l.o.g.\ assume that $P_1, P_2$ do not contain explored nodes other than $w_1, w_2$, respectively.
Then $\badcycle{i}{P_1, \pathset{E_t'}{w_1}{a_i}, \pathset{E_t'}{a_i}{w_2}, P_2}$ is a cycle witness.
The second claim follows directly
as $b_i$ is either an unexplored leaf itself
or there is a path to another leaf
that must be unexplored by the first claim.
\end{proof}
Consider component $C_i$ for $\pi_i \in \PiX{\sitD}(F)$.
Based on the above lemma, we define
$\bar C_i$ as the unique path in $\pathset{(C_i \setminus E_t')}{b_i}{y}$
where $y \in V(E_t') \cap V(C_i)$ is an explored node.
Furthermore, let $C_i^*$ be the component in $C_i \setminus \bar C_i$
that contains $b_i$.
\begin{lemma}\label{lemma:path-witness}
Let $S \in \{\sitU, \sitD\}$,
$\pi_i \in \PiX{S}(F)$,
$t := \theta(b_i)$,
$H^\sitU := (E' \cup F) \setminus E_{t+1}'$,
and
$H^\sitD := (E' \cup F) \setminus (E_t' \cup \{\pi_i\} \cup \bar C_i)$.
We have $\pathset{H^S}{b_i}{x} = \varnothing$
for any $x \in V(E_t')$.
\end{lemma}
\begin{proof}
Assume there is a path $P \in \pathset{H^S}{b_i}{x}$.
Let $Q^\sitU := O_t \setminus \{\pi_i\}$
and $Q^\sitD := \bar C_i$.
We have a cycle witness
$\badcycle{i}{P, \pathset{E_t'}{x}{a_i}, \pathset{E_t'}{a_i}{y}, Q^S}$
with $y \in V(Q^S) \cap V(E_t')$.
\end{proof}
This allows us to define a \emph{path witness}
$\pathwitness{j}{\mathcal{P}}$
for a situation $S \in \{\sitU, \sitD\}$
as a shorthand for a cycle witness
on edge $\pi_j$ with $P \in H^S$
in the proof of Lemma~\ref{lemma:path-witness}.
\begin{lemma}\label{lemma:situation-a-component-mapable}
Let $\pi_i \in \PiX{\sitD}(F)$.
We have $|\PiX{\sitD}(C_i^*)| \leq |\aX{\sitD}(C_i^*) \cup L(C_i^*)| - 1$.
\end{lemma}
\begin{proof}
Note that $a_i \notin L(C_i^*)$.
Let $r := |L(C_i^*)|$,
and
$x_1, x_2, \ldots, x_r$
the members of $L(C_i^*)$
such that
for each $j \in \{2, \ldots, r\}$
we have $\theta(x_j) \geq \theta(x_{j-1})$.
Let $T_1$ be the path in $\pathset{C_i^*}{b_i}{x_1}$.
Given $T_{j-1}, j \in \{2, \ldots, r\}$,
we obtain $T_j$
by adding a path $P_j \in \pathset{(C_i^* \setminus T_j)}{h_j}{x_j}$
where $h_j \in V(T_j)$. %
Note that
$V(P_j) \cap V(T_{j-1}) = \{h_j\}$
and
$L(T_j) = L(T_{j-1}) \cup \{x_j\}$.
By definition, we have $T_r = C_i^*$.
For brevity,
let
$P_j' := P_j - h_j$
and
$a_j := \aX{\sitD}(T_j) \cup L(T_j)$
for any~$j$.
\begin{claim}\label{claim:mapping:subpath-independent}
Let $P \in \pathset{C_i^*}{b_i}{x}$ for any $x \in L(C_i^*)$.
Let $Q \subseteq P$ be any subpath of~$P$.
Then $\PiX{\sitD}(Q)$ is independent,
i.e.,
no two of its edges have a common node.
\end{claim}
\begin{claimproof}
There are three cases.
(1)~Assume there are distinct $\pi_k, \pi_\ell \in \PiX{\sitD}(Q)$
with $b_k = b_\ell$.
Let $t := \theta(b_\ell)$.
For $w_1, w_2$ being the endpoints of $O_t$,
we have a cycle witness $\badcycle{k}{O_t, \pathset{E_t'}{w_1}{w_2}}$.
%
(2)~Assume there are distinct $\pi_k, \pi_\ell \in \PiX{\sitD}(Q)$
with $b_k = a_\ell$.
Since $\pi_k \in \Pi$, we have $b_k \in L$,
i.e.,
$Q$ ends at $b_k$
and $b_k = x$.
Hence,
$\pathset{Q}{b_\ell}{b_k}$
contradicts Lemma~\ref{lemma:situation-a-unexplored-leaf}
since $\theta(b_k) = \theta(a_\ell) = t$.
%
(3)~Assume there are distinct
$\pi_k, \pi_\ell \in \PiX{\sitD}(Q)$
with
$a_k = a_\ell$,
w.l.o.g.\ $k < \ell$.
We have a path~witness $\badpath{k}{\pathset{Q}{b_k}{b_\ell}, \pi_\ell}$.%
\end{claimproof}
\begin{claim}\label{claim:mapping:subpath-leaf-is-not-u}
Let $P \in \pathset{C_i^*}{b_i}{x}$ for any $x \in L(C_i^*)$.
Let $Q \subseteq P$ be any subpath of~$P$ with $x \in V(Q)$.
Then $x \notin \aX{\sitD}(Q)$.
\end{claim}
\begin{claimproof}
Assume not.
We have $x \in \aX{\sitD}(O_{\theta(b_k)})$ for some index $k > i$.
Hence $x \in V(C_k^*)$ is an explored node at time $\theta(b_k)$
which contradicts Lemma~\ref{lemma:situation-a-unexplored-leaf}.
\end{claimproof}
\begin{claim}\label{claim:mapping:one-common}
For each $j \in \{2, \ldots, r\}$, we have
$|\aX{\sitD}(P_j') \cap a_{j-1}| \leq 1$.
\end{claim}
\begin{claimproof}
Assume not.
Let $Q$ be the path in $T_{j+1}$ between $b_i$ and $x_j$.
$\PiX{\sitD}(Q)$ is independent by Claim~\ref{claim:mapping:subpath-independent}.
Hence
there are
$\pi_{\ell_1}, \pi_{\ell_2} \in \PiX{\sitD}(Q)$
and
$v_1, v_2 \in V(T_{j-1} \setminus Q), v_1 \neq v_2,$
such that
one of the following holds:
(1)~$\mathcal{A}_1$ and~$\mathcal{A}_2$,
(2)~$\mathcal{A}_1$ and~$\mathcal{B}_2$,
(3)~$\mathcal{B}_1$ and~$\mathcal{A}_2$,
(4)~$\mathcal{B}_1$ and~$\mathcal{B}_2$,
where
$\mathcal{A}_d$, $d=1,2$, is the case that
there is a $\pi_{k_d} \in \PiX{\sitD}(T_{j-1} \setminus Q)$
with $b_{k_d} = v_d$
and $a_{\ell_d} = a_{k_d}$,
and $\mathcal{B}_d$ is the case that
we have $v_d \in L(T_{j-1})$
with $a_{\ell_d} = v_d$.
W.l.o.g.\ $\ell_1 \leq \ell_2$.
For case~(1), we have
a path witness
$\badpath{\ell_1}{\pathset{P_j'}{b_{\ell_1}}{b_{\ell_2}}, \pi_{\ell_2}, \pi_{k_2}, \pathset{T_{j-1}}{v_2}{v_1}, \pi_{k_1}}$.
For case~(2), we have
a path witness
$\badpath{\ell_1}{\pathset{P_j'}{b_{\ell_1}}{b_{\ell_2}}, \pi_{\ell_2}, \pathset{T_{j-1}}{v_2}{v_1}, \pi_{k_1}}$.
For case~(3), we have
a path witness
$\badpath{\ell_1}{\pathset{P_j'}{b_{\ell_1}}{b_{\ell_2}}, \pi_{\ell_2}, \pi_{k_2}, \pathset{T_{j-1}}{v_2}{v_1}}$.
For case~(4), we have
a path witness
$\badpath{\ell_1}{\pathset{P_j'}{b_{\ell_1}}{b_{\ell_2}}, \pi_{\ell_2}, \pathset{T_{j-1}}{v_2}{v_1}}$.
\end{claimproof}
\begin{claim}\label{claim:mapping:tree-leaf}
For each $j \in \{2, \ldots, r\}$, we have
$x_j \notin a_{j-1}$.
\end{claim}
\begin{claimproof}
We have $x_j \notin L(T_{j-1})$ by definition of $T_{j-1}$.
It remains to show $x_j \in \aX{\sitD}(T_{j-1})$.
Assume not.
There is a $\pi_\ell \in \PiX{\sitD}(T_{j-1})$
with $a_\ell = x_j$.
Choose $x_k \in L(T_{j-1})$
such that $b_\ell$ lies on the path between $b_i$ and $x_k$.
%
By definition of $x_j$,
we have $\theta(x_j) > \theta(x_k)$.
Lemma~\ref{lemma:situation-a-unexplored-leaf}
at time $\theta(b_\ell)$
gives $\theta(x_k) > \theta(b_\ell)$.
By $a_\ell = x_j$,
we have $\theta(b_\ell) > \theta(x_j)$,
a contradiction.
\end{claimproof}
We show
$|\PiX{\sitD}(T_j)| \leq |a_j| - 1$
inductively
for all $j \in \{1, \ldots, r\}$.
First consider $j = 1$.
Since $T_1$ is a path,
$\PiX{\sitD}(T_1)$ is independent
by Claim~\ref{claim:mapping:subpath-independent};
hence
$|\PiX{\sitD}(T_1)| = |\aX{\sitD}(T_1)|$.
The claim follows by observing that
$|\aX{\sitD}(T_1)|
= |\aX{\sitD}(T_1) \cup \{x_1\}| - 1
= |a_1| - 1$
since $x_1 \notin \aX{\sitD}(T_1)$
by Claim~\ref{claim:mapping:subpath-leaf-is-not-u}.
We now assume that the claim holds for~$j - 1$ with~$j \in \{2, \ldots, r\}$,
and show that it holds for~$j$.
We get
\begin{align*}
|\PiX{\sitD}(T_j)|
&
= |\PiX{\sitD}(P_j) \cup \PiX{\sitD}(T_{j-1})|
%
= |\PiX{\sitD}(P_j')| + |\PiX{\sitD}(T_{j-1})|
\\ &
\leq |\PiX{\sitD}(P_j')| + |a_{j-1}| - 1
& & \text{by induction}
\\ &
= |\aX{\sitD}(P_j')| + |a_{j-1}| - 1
& & \text{by Claim~\ref{claim:mapping:subpath-independent}}
\\ &
\leq |\aX{\sitD}(P_j')| + |a_{j-1}| - |\aX{\sitD}(P_j') \cap a_{j-1}|
& & \text{by Claim~\ref{claim:mapping:one-common}}
\\ &
= |\aX{\sitD}(P_j') \cup a_{j-1}|
\\ &
= |\aX{\sitD}(P_j') \cup a_{j-1} \cup \{x_j\}| - 1
& & \text{by Claims~\ref{claim:mapping:subpath-leaf-is-not-u} and~\ref{claim:mapping:tree-leaf}}
\\ &
= |\aX{\sitD}(P_j') \cup \aX{\sitD}(T_{j-1}) \cup L(T_j)| - 1
\\ &
= |\aX{\sitD}(T_j) \cup L(T_j)| - 1
%
= |a_j|- 1
\text{.}
& & \qedhere
\end{align*}
\end{proof}
For each $\pi_i \in \PiX{\sitD}(F)$,
let $\sidx{i} := \min\{j \mid \pi_i \in \PiX{\sitD}(C_j^*)\}$
be the index of the earliest situation~\sitD on component $C_i$.
Let $\ensuremath{\mathbb{S}} := \{\sidx{i} \mid \pi_i \in \PiX{\sitD}(F)\}$.
Using Lemma~\ref{lemma:situation-a-component-mapable},
we construct an injection $\ensuremath{\mu}_i\colon \PiX{\sitD}(C_i^*) \to L \setminus \{a_i\}$
for every $i \in \ensuremath{\mathbb{S}}$.
First observe that
$|\PiX{\sitD}(C_i^*)| \leq |(\aX{\sitD}(C_i^*) \setminus \{a_i\}) \cup L(C_i^*)|$
by
$a_i \in \aX{\sitD}(C_i^*)$
and
$a_i \notin L(C_i^*)$ (see Lemma~\ref{lemma:situation-a-unexplored-leaf}).
There might be distinct $\pi_j, \pi_k \in \PiX{\sitD}(C_i^*)$
with $a_j = a_k$.
It is possible to construct
$\ensuremath{\mu}_i$ %
as injection
such that
for each $w \in \aX{\sitD}(C_i^*) \setminus \{a_i\}$
there is one $k$
with $w = a_k$
and
$\ensuremath{\mu}_i(\pi_k) = a_k$.
Since components are a partition of $F$,
we can define a mapping $\ensuremath{\mu}\colon \PiX{\sitD}(F) \to L$
by $\ensuremath{\mu} := \bigcup_{j \in \ensuremath{\mathbb{S}}}{\ensuremath{\mu}_j}$.
\begin{lemma}\label{lemma:situation-a-mapping}
The mapping $\ensuremath{\mu}$ is an injection.
\end{lemma}
\begin{proof}
Assume there are $\pi_k, \pi_\ell \in \PiX{\sitD}(F)$
with $\sidx{k} < \sidx{\ell}$
and $w := \ensuremath{\mu}_\sidx{k}(\pi_k) = \ensuremath{\mu}_\sidx{\ell}(\pi_\ell)$.
We have $C_\sidx{k}^* \neq C_\sidx{\ell}^*$
since otherwise $\ensuremath{\mu}_\sidx{k}(\pi_k) = \ensuremath{\mu}_\sidx{k}(\pi_\ell)$
contradicts the injectivity of $\ensuremath{\mu}_\sidx{k}$.
The following three cases remain:
(1)~$w = a_\ell \in L(C_\sidx{k}^*)$,
(2)~$w = a_k \in L(C_\sidx{\ell}^*)$,
and
(3)~$w = a_k = a_\ell$.
%
Consider case~(1).
By $\ell \neq \sidx{\ell}$
and $k < \ell$ (since $w = a_\ell$),
we have a path witness
$\badpath{\sidx{\ell}}{%
\pathset{C_\sidx{\ell}^*}{b_\sidx{\ell}}{b_\ell},
\pi_\ell,
\pathset{C_\sidx{k}^*}{w}{b_\sidx{k}},
\pi_\sidx{k}}$.
%
Consider case~(2).
By $k \neq \sidx{k}$
and $\ell < k$ (since $w = a_k$),
we have a path witness
$\badpath{\sidx{\ell}}{%
\pathset{C_\sidx{\ell}^*}{b_\sidx{\ell}}{w},
\pi_k,
\pathset{C_\sidx{k}^*}{b_k}{b_\sidx{k}},
\pi_\sidx{k}}$.
%
For case~(3),
we have a path witness
$\badpath{\sidx{\ell}}{%
\pathset{C_\sidx{\ell}^*}{b_\sidx{\ell}}{b_\ell},
\pi_\ell,
\pi_k,
\pathset{C_\sidx{k}^*}{b_k}{b_\sidx{k}},
\pi_\sidx{k}}$.
\end{proof}
For any $F' \subseteq F$,
let $L'(F') := L(F') \setminus \ensuremath{\mu}(\PiX{\sitD}(F))$
be the leaves not used by $\ensuremath{\mu}$.
\begin{lemma}\label{lemma:situation-e-mapping}
%
There is a mapping $\ensuremath{\eta}\colon \PiX{\sitU}(F) \to L'(F)$
such that
%
%
%
%
%
for each $v \in L$,
we have
$|\ensuremath{\eta}^{-1}(v)| \leq 2$ if $v \in L_0$
and
$|\ensuremath{\eta}^{-1}(v)| \leq 1$ otherwise.
\end{lemma}
\begin{proof}
We give an algorithm that establishes our mapping $\eta$.
%
Let $C \subseteq F$ be a subtree
and $w \in V(C)$.
Consider the following recursive algorithm
which,
invoked on $(C, w)$,
tries to construct a path $P$
between $w$
and a leaf $x \in L'(F)$.
$P$ is initially empty and will be extended in each recursion step.
Trivially, if there is an $x \in L'(C)$,
the algorithm
adds the unique path in $\pathset{C}{w}{x}$ to $P$
and terminates.
Otherwise,
we have $L'(C) = \varnothing$.
There are two cases:
\begin{enumerate}
\item
There is a $\pi_k \in \PiX{\sitD}(C)$.
%
Let $C'$ be the component containing $a_\sidx{k}$.
We add to $P$ the unique path in $\buildpath{\pathset{C}{w}{b_\sidx{k}}, \pi_\sidx{k}}$
and recurse on $(C', a_\sidx{k})$.
\item
We have $\PiX{\sitD}(C) = \varnothing$
but then
there is a component $C' \neq C$
with $\pi_k \in \PiX{\sitD}(C')$ and~$\ensuremath{\mu}(\pi_k) = a_k \in L(C)$.
We add to $P$ the unique path in $\buildpath{\pathset{C}{w}{a_k}, \pi_k}$
and recurse on~$(C', b_k)$.
\end{enumerate}
%
%
We define $C_i^*$ for a given $\pi_i \in \PiX{\sitU}(F)$
to be the component in $C_i \setminus O_{\theta(b_i)}$
that contains $b_i$.
However, there is a tricky exception:
if $\pi_{i+1} \in \PiX{\sitU}(F)$
and $O_{\theta(b_i)} = \{\pi_i, \pi_{i+1}\}$,
i.e.,
if we have $b_i = b_{i+1}$.
Then, if $b_i \in L_0$,
we say that $C_i^*$ and $C_{i+1}^*$ consist only of $b_i$
($\eta$ may map to it twice anyhow).
Otherwise,
we have at least two leaves in $C_i = C_{i+1}$.
By removing an edge $e$ with $V(e) = \{b_i, z\}$ from $C_i$,
we obtain the two components
$C_i^*$ and $C_{i+1}^*$
such that
$b_i \notin V(C_i^*)$
and
$b_i \in V(C_{i+1}^*)$.
For technical simplicity,
we set $b_i' := b_i$ if $b_i \in C_i^*$
and $b_i' := z$ otherwise.
However, $\eta$ will never map to $z$ in the following.
We now construct $\ensuremath{\eta}$
by invoking the algorithm
on $(C_i^*, b_i')$
for each $\pi_i \in \PiX{\sitU}(F)$
in chronological order from the latest to the earliest component;
we set $\ensuremath{\eta}(\pi_i) := x$ where $x$ is the found leaf.
By construction,
$b_i$ is the earliest node in $P$,
since otherwise we would have a path witness $\pathwitness{b_i}{P}$.
Assume by contradiction that the algorithm does not terminate.
Consider the recursion step
where edges are added to $P$ that are already included in $P$.
%
$P$ contains a cycle $Q$.
Note that in case~(1),
we have
$\theta(a_\sidx{k}) < \theta(w)$,
i.e.,
we go back in time only,
and thus
$Q$ also involves a case~(2) step.
On the other hand,
after a recursion step handling case~(2),
we either terminate
or recurse into case~(1).
Hence there is a component $C$
such that
%
%
by case~(2) there is a $\pi_j \in \PiX{\sitD}(C)$
with $b_j \in V(C)$
and $j \notin \ensuremath{\mathbb{S}}$
and
by case~(1) we have $b_\sidx{j} \in V(C)$.
Now
$Q' := Q \setminus \{\pi_\sidx{j}\}$ is a path
between $b_\sidx{j}$ and $a_\sidx{j}$;
$\pathwitness{\sidx{j}}{Q'}$
is a path witness.
Now that we can ensure that the algorithm terminates,
consider an arbitrary $\pi_i \in \PiX{\sitU}(F)$.
For an invocation of the algorithm on $(C_i^*, b_i')$,
let $x_i \in L'(F)$ be the resulting leaf
and $P_i$ the resulting $b_i$-$x_i$-path.
Assume that
there is a $\pi_j \in \PiX{\sitU}(F)$
with
$x_i \in \ensuremath{\eta}(\pi_j)$.
Note that $j > i$
since we invoke the recursive algorithm from last to first index.
Let $y$ be the first node
that $P_i$ and $P_j$ have in common,
and let
$P'$ be the unique path in $\buildpath{\pathset{P_j}{b_j}{y}, \pathset{P_i}{y}{b_i}}$.
If $\theta(b_j) > \theta(b_i)$,
we have a path witness
$\badpath{j}{P', \pi_i}$.
Now consider the case $\theta(b_j) = \theta(b_i)$.
Let $C$ be the component containing $y$.
We distinguish the following subcases:
\begin{itemize}
\item
If $L'(C) = L(C)$,
we have $x_i \in L(C)$.
If $x_i \in L_0$,
we set $\eta(b_i) := x_i$.
Otherwise there is an $x_j \in L(C) \setminus \{x_i\}$.
Since $\ensuremath{\eta}(b_j)$ is already set to $x_i$,
we set $\ensuremath{\eta}(b_i) := x_j$.
\item
If $P'$ enters $C$ using case~(1),
there is a $\pi_k \in \PiX{\sitD}(P')$
with $k \in \ensuremath{\mathbb{S}}$
and $a_k \in V(C)$.
Then
$\pathwitness{k}{\pathset{P'}{b_k}{b_i}}$
or
$\pathwitness{k}{\pathset{P'}{b_k}{b_j}}$
is a path witness
since $\theta(b_k) > \theta(b_i) = \theta(b_j)$.
\item
If $P'$ enters $C$ using case~(2),
there is a $\pi_k \in \PiX{\sitD}(P')$
with $k \notin \ensuremath{\mathbb{S}}$
and $b_k \in V(C)$.
Hence there is a $\pi_\sidx{k} \in \PiX{\sitD}(C)$
and
$\badpath{\sidx{k}}{\pathset{P'}{b_\sidx{k}}{b_i}}$
or
$\badpath{\sidx{k}}{\pathset{P'}{b_\sidx{k}}{b_j}}$
is a path witness
since $\theta(b_\sidx{k}) > \theta(b_i) = \theta(b_j)$.
\qedhere
\end{itemize}
\end{proof}
It is now easy to show
Lemma~\ref{lemma:charge-u-and-d}
using
Lemmata~\ref{lemma:situation-a-mapping}
and~\ref{lemma:situation-e-mapping}.
We first charge all situation~\sitD nodes,
that is,
we set $\chi_\sitD(v) = 1$ for all $v \in \ensuremath{\mu}(\PiX{\sitD}(F))$.
Now we charge all situation~\sitU nodes using $\ensuremath{\eta}$,
that is,
we have $\chi_\sitU(v) = 1$ for all $v \in L'(F) \setminus L_0$
and $\chi_\sitU(v) \leq 2$ for all $v \in L'(F) \cap L_0$.
\section{Conclusion}
We presented a simple 3-approximation algorithm for 2ECSS with general edge costs.
While there have been primal-dual approximations before (but none achieving a better ratio based on the primal-dual method),
they require two grow phases (followed by a cleanup phase)
to first compute a tree
and then augment this tree to a 2-edge-connected solution.
Our approach does not require this separation,
by the (to our best knowledge) new idea of growing the solution only at leaves.
While our primal-dual analysis is non-trivial,
the resulting algorithm is very straight-forward to implement
with $\ensuremath{\mathcal{O}}(\min\{nm,m + n^2 \log n\})$ time,
requiring only very basic graph operations and the simplest data structures.
An implementation with time $\ensuremath{\mathcal{O}}(nm)$ is remarkably simple.
This is in contrast to the other known primal-dual algorithms.
Of those, only the algorithm in~\cite{GGW98} achieves a faster running time of
$\ensuremath{\mathcal{O}}(n^2+n\sqrt{m\log\log n})$, but at the cost of requiring
intricate data structures and subalgorithms detailed in separate papers~\cite{GGST86,WGMV95}.
For sparse graphs, $m\in\ensuremath{\mathcal{O}}(n)$, our running time
is in fact equivalent. For dense simple graphs, $m\in\ensuremath{\mathcal{O}}(n^2)$, we have $\ensuremath{\mathcal{O}}(n^2\log n)$
instead of their $\ensuremath{\mathcal{O}}(n^2\sqrt{\log\log n})$.
Note that on instances with uniform costs,
the ratio naturally drops to the trivial approximation ratio~2.
We may also note that former tight examples (for example, the tight instances for the 3-approximation given in \cite{FJ81,KR93})
are now approximated with factor 2.
Moreover,
by a simple extension,
our algorithm can also compute lower bounds
(which could be useful for branch-and-bound algorithms and instance preprocessing),
without changing its runtime complexity.
It would be interesting to see if (and how) it is possible
to improve our approach to achieve an even better running time or approximation ratio,
and/or
to transfer it to 2-node-connectivity
or generalized edge-connectivity (e.g., $\{0,1,2\}$-survivable network design) problems.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,655
|
// Copyright (c) Christof Senn. All rights reserved. See license.txt in the project root for license information.
namespace Remote.Linq.Expressions
{
using Aqua.TypeSystem;
using Remote.Linq.DynamicQuery;
using System;
using System.Runtime.Serialization;
using System.Xml.Serialization;
[Serializable]
[DataContract]
[KnownType(typeof(ConstantQueryArgument)), XmlInclude(typeof(ConstantQueryArgument))]
[KnownType(typeof(QueryableResourceDescriptor)), XmlInclude(typeof(QueryableResourceDescriptor))]
[KnownType(typeof(SubstitutionValue)), XmlInclude(typeof(SubstitutionValue))]
[KnownType(typeof(VariableQueryArgument)), XmlInclude(typeof(VariableQueryArgument))]
[KnownType(typeof(VariableQueryArgumentList)), XmlInclude(typeof(VariableQueryArgumentList))]
public sealed class ConstantExpression : Expression
{
public ConstantExpression()
{
}
public ConstantExpression(object? value, Type? type = null)
{
if (type is null)
{
if (value is null)
{
type = typeof(object);
}
else
{
type = value.GetType();
}
}
Type = type.AsTypeInfo();
Value = value;
}
public ConstantExpression(object? value, TypeInfo type)
{
Type = type.CheckNotNull(nameof(type));
Value = value;
}
public override ExpressionType NodeType => ExpressionType.Constant;
[DataMember(Order = 1, IsRequired = true, EmitDefaultValue = false)]
public TypeInfo Type { get; set; } = null!;
[DataMember(Order = 2, IsRequired = true, EmitDefaultValue = true)]
public object? Value { get; set; }
public override string ToString() => Value.QuoteValue();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,889
|
From Monday 09 December 2019
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,415
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{"url":"https:\/\/tobaccocontrol.bmj.com\/content\/29\/Suppl_5\/s300","text":"Article Text\n\nImpact of a recent tobacco tax reform in Argentina\nFree\n1. 1 Department of Economics, Universidad Torcuato Di Tella, Buenos Aires, Argentina\n2. 2 Universidad Torcuato Di Tella, Buienos Aires, Argentina\n\n## Abstract\n\nThe literature on policies for the control of the tobacco epidemic suggests that increasing excise taxes on the consumption of tobacco products is the most cost-effective policy. Cigarette tax structure in Argentina is very complex. All the tax bases for cigarette consumption taxes are related and, therefore, any modification of a tax affects the collection of the rest of the taxes. This is important given that funds raised by one of the taxes, the Special Tobacco Fund (FET), are allocated among the tobacco provinces according to the value of tobacco production. These provinces oppose in the congress to any reform that increase taxes on cigarette consumption that negatively affects these funds. In May 2016, the government decided to increase the rate of one of the taxes, the internal tax, from 60% to 75%. We study the impact on cigarettes\u2019 demand price elasticity, consumption and tax revenues of this tobacco tax reform. Using an Error Correction Model, we estimate short-run and long-run demand price and income elasticities. We find that the tax reform of May 2016 induced an increase in the magnitude, in absolute value, of the short-run demand price elasticity and at the same time increased the funds collected by the FET. We simulate the effects of the tax reform over the government revenues and per-capita consumption of cigarettes showing that additional increments in taxes would increase revenues and diminish consumption of cigarettes.\n\n\u2022 taxation\n\u2022 public policy\n\u2022 price\n\u2022 economics\n\n## Introduction\n\nThe literature on policies for tobacco control suggests that increasing excise taxes on the consumption of tobacco products is the most cost-effective policy. The reason is that increasing taxes causes the prices of tobacco products to increase. This makes the different tobacco products less accessible, thus reducing initiation, prevalence and consumption of tobacco. In addition, because the demand for tobacco is inelastic, higher taxes generate increases in tax revenues. See Gajalakshmi et al,1 Jha and Chaloupka2, Ranson et al 3 among others for international evidence. See Gonz\u00e1lez-Rozada4 Gonz\u00e1lez-Rozada and Rodr\u00edguez-Iglesias5, Rodr\u00edguez-Iglesias et al 6 for evidence for Argentina.\n\nThe tax structure on cigarette consumption in Argentina is very complex including four ad-valorem taxes. One of the taxes, the Special Tobacco Fund (FET), acts as a subsidy to the provinces that produce tobacco. Therefore, these provinces oppose in the congress to any tax reform that negatively affects these funds. In May 2016, the government decided to increase the rate of one of the taxes, the internal tax, from 60% to 75%. In this paper, we study the impact on cigarettes\u2019 demand price elasticity, consumption and tax revenues of this tobacco tax reform. Using an Error Correction Model, we estimate short-run and long-run demand price and income elasticities. Then, using these estimations, we simulate the impact of the tax reform by increasing the rate of internal taxes on consumption of cigarettes and government revenue. The rest of the work is organised as follows. Section 2 describes the tax structure of cigarettes in Argentina and presents the tax reform and its impact on the tax share on prices, retail price, FET and government tax revenue. Section 3 describes the data used in the estimation of the demand function of cigarettes and studies the underlying statistical properties of retail price, real income and consumption of cigarettes. Section 4 introduces the methodology used to estimate demand price and income elasticities. Section 5 shows the main results of the paper and Section 6 concludes the work.\n\n## Tax structure of cigarettes in Argentina\n\nThe tax structure on cigarette consumption in Argentina is very complex. Federal taxes affecting cigarettes are four ad-valorem taxes: the additional emergency tax (IAE), the value added tax (VAT), the FET and the internal tax (II). The tax base of each one is different. Table 1 shows tax rates, tax base and the tax share on the retail price of each ad-valorem tax before the reform.\n\nTable 1\n\nThe tax share on prices before the tax reform of May 2016 was 68.6%. The average retail price in April 2016 was almost AR$26 per pack of 20 cigarettes (AR$3.1 in real terms) and internal taxes represented 47% of that retail price. This structure implies than changing the tax rate of one of the taxes affects the tax base of the other taxes. This is important because, in practice, one of the taxes, the FET, acts as a subsidy to tobacco producers. The main objective of the FET is to guide, coordinate and supervise the actions tending to achieve the modernisation, reconversion, complementation and diversification of the tobacco areas, both in the primary production and in the associated agro-industrial chain. The funds raised by the FET are allocated among the tobacco provinces according to the value of tobacco production. The Ministry of Agriculture of the Nation is the enforcement authority of the FET. It has in its functions to fix the price of the different varieties of tobacco and transfer the collection of the FET to the provinces so that they make cash the payment of the surcharge to the producers. That is, the FET acts as a subsidy to tobacco producers and the tobacco industry in particular.\n\nUsually, those who oppose increasing taxes on tobacco products use the FET as an argument against it by saying that increment in taxes will reduce the FET funds. It is important then, for policy reasons, to show evidence that this is not the case when increasing internal taxes.\n\n### The tax reform\n\nIn May 2016, Argentina established an increase in the rate of II on cigarettes from 60% to 75%. After this reform, the tax share on retail price increased and reached almost 80%. II represented almost 61% of the average retail price of almost AR$50 per pack of 20 cigarettes (AR$4.5 in real terms). FET tax share on retail price decreased slightly from 7.8% to 7.7% but because average real retail price of a pack of 20 cigarettes increased almost 50%, from AR$3.1 to AR$4.5, FET funds increased.\n\nThe response of the tobacco industry to the tax reform was to increase average retail prices of cheapest brands 40% in the month after the reform while for the most expensive brands, they incremented average retail prices by 50%. This strategy had to do with the cigarette consumption market in Argentina, where the great majority of smokers consume the most expensive brands (for data source of response of the industry and structure of consumption of cigarettes see next section).\n\nAfter the second quarter of 2016, there was a clear increase in the collection of internal taxes. Before the tax reform, tax revenues from II were around 4500\u2009million of constant AR$, while after the reform these revenues were almost 6000 million. Figure 1 shows the tax collection, in millions of constant pesos of the fourth quarter of 2017, coming from the FET before and after the implementation of the reform of May 2016 (marked in the figure by the dotted vertical line). As can be observed, after the tax reform, the tax collection from the FET increased throughout the period analysed. Before the reform FET revenues were around AR$750\u2009million and jumped to more than AR$850 million just after the reform. The main reason for this was the tax base increase due to the increment in retail prices. Figure 1 Tax revenue collected by the FET. Author elaboration. FET, Special Tobacco Fund. This evidence shows that it is possible to increase taxes on the consumption of cigarettes without affecting the FET funds. As mentioned above, affecting the FET funds is a political concern when there is a proposal to increment taxes on cigarettes. ## Data and statistical properties We use monthly data from January 2005 to June 2018 for consumption (approximated by the total sales of packages of 20 cigarettes), average real retail price of cigarettes and real income of the population, represented by the average remuneration of registered workers of the private sector published by the Ministry of Labour, Employment and Social Security (Data are available online here: https:\/\/www.agroindustria.gob.ar\/sitio\/areas\/tabaco\/estadisticas; https:\/\/www.trabajo.gob.ar\/left\/estadisticas\/descargas\/SIPA\/AnexoEstadistico.xlsx). To specify the demand function for cigarettes, we first needed to find the statistical properties of these variables. Using the Augmented Dickey\u2013Fuller test,7 we show that all three variables, consumption, real price and real income, have individually a unit root. Then, using the Johansen Trace test,8 we show that the three variables are cointegrated. ## Methodology for estimating the demand price elasticity of cigarettes Cointegration implies that the tobacco demand function can be specified with a model that takes into account not only the relationship between the variables in the short-run but also in the long-run. Using an error correction model, the long-run relationship among consumption of cigarettes, real retail price and real income is: (1) Where is the natural log of consumption, is the natural log of real retail price, is the natural log of real income and ut is an error term. is the demand price elasticity and is the real income elasticity. Equation (1) is the long-run equilibrium relationship. In the short-run, the variables may not be in the steady state; therefore, we specify the dynamics of the short-run relationship using r lags in equation (2). (2) Where \u03b4, \u03b1, \u03b2, \u03b3, \u03b1j*, \u03b20, \u03b2j*, \u03b30, \u03b3j *, \u03b80, \u03b8j and ki * are the parameters of the model and \u03b5t is a stationary error term. The value of r determines the number of months involved in the long-run concept of the model. The term in levels between braces represents the solution of long-run equilibrium (1), while all the variables in first differences measure the short-run dynamics. Some of the parameters in (2) have an interpretation in terms of the short-run elasticities of cigarette consumption. In particular, \u03b20 is the short-run demand price elasticity and \u03b30 is the short-run real income elasticity. To capture the impact of the tax reform, we introduced a binary variable, D201 , adopting the unity value since May 2016 when the reform was applied onwards. This indicator variable interacts with the price variables in the short-run specification (2). Then, the impact of the reforms on the short-run demand price elasticity is measured by: \u03b20 + \u03b80 . For a detailed description of the model see Annex two in Gonz\u00e1lez-Rozada.9 We estimate the ECM using the Engle\u2013Granger methodology.10 This is a two-stage estimate. First, we estimate the long-term equilibrium relationship (1) and then we estimate the ECM (2) to obtain the short-run effects. ## Results Table 2 shows the estimation of equation (1) including a dummy variable for the Christmas bonus. The long-run demand price elasticity is \u22120.441, while the long-run real income elasticity is 0.127. These values imply that, in the long-run, a 10% increase in the real retail price reduces cigarette consumption by 4.41% and a 10% increase in real income increases the consumption of cigarettes by 1.27%. All estimations are statistically significant at usual levels of significance. Table 2 Long-run elasticity estimation Table 3 shows the estimation of the short-term dynamics, including the effect of the tax increase from May 2016. Z(t\u20131) represents the estimation of term in levels between braces of equation (2). The variable D2016 is a binary variable adopting the unity value since the month of May 2016 when the tax reform was implemented. As can be seen in the table, the short-run demand price elasticity without the effect of the reform is \u22120.91 while, as a result of the reform of May 2016, this value is \u22121.38. These results suggest that, in the short-run before the reform, a 10% increase in real retail price induced a 9% decrease in consumption, while after the tax reform the same increase in real retail price produced a decrease in consumption of cigarettes of around 14%. The tax reform induced a huge increase in retail price and this, in turn, produced a large fall in consumption for a few months after the reform. These sudden changes are captured by the increment, in absolute magnitude, of the short-run demand price elasticity. Table 3 Short-run elasticity estimation ### Simulation of results To analyse the impact of the reform of May 2016 on cigarette consumption and tax collection, we perform a simulation exercise. In this exercise, we use the long-run price elasticity of \u22120.44 presented in table 1 and increase the internal tax rate sequentially. In this way, we can see the impact of the fiscal reform. The parameters used for the simulation exercise are: Consumption of cigarettes: 177 056 579 packages Average retail price: AR$25.88 per package\n\nTax on cigarettes: AR$20.65 per package Government revenue for taxes on cigarettes: AR$3 658 275 567\n\nExchange rate: 14.25 AR$per dollar Population (over 15 years old): 31 452 302 Consumption per capita: 67.53 packages per year Figure 2 shows the changes in the government\u2019s tax revenue. The vertical line shows the tax increase of 15 percentage points produced by the fiscal reform, the figure shows that there is enough room to increase the internal tax rate on cigarette consumption and still increase government's tax revenue. For example, if the government decided to increase the internal tax rate an additional three percentage points, it would increase tax revenues around US$200\u2009million.\n\nFigure 2\n\nGovernment revenue for each internal tax increase Author elaboration.\n\nFigure 3 shows the effects of the tax reform on the per capita consumption of cigarettes. As in figure 2, the vertical line shows the implemented fiscal reform, an increase in the internal tax rate from 60% to 75%. The figure shows that this increment in the internal tax rate induced a fall in the average per capita consumption of cigarettes from 68 to around 50 packs per year. The figure also shows that further increases in the internal tax rate would reduce the average per capita consumption of cigarettes.\n\nFigure 3\n\nAverage annual consumption per capita for each internal tax increase Author elaboration.\n\n## Conclusion\n\nWe studied the impact on demand price elasticity, the FET, cigarette consumption and tax collection of a recent tax reform in Argentina. This reform increased the rate of internal taxes from 60% to 75% and this, in turn, increased the government revenues collected from II about 40% by the end of 2016. We provided evidence that the increment in the rate of internal taxes produced an increment in the revenue collected by the FET. We estimate an ECM, obtaining short-run and long-run demand price elasticities. We found a long-run elasticity of \u22120.441, suggesting that a 10% increase in the real retail price of cigarettes would decrease consumption by around 4.4%. The estimation of the short-run demand price elasticity was \u22120.911 without the tax reform, whereas if we consider the reform, this short-run elasticity increases in absolute value to \u22121.385. Using the estimation of the demand price elasticity, we simulate the impact of the tax reform by increasing the rate of internal taxes on consumption and government revenue, finding that it is possible to increase even more this tax rate and increase revenues and decrease consumption of cigarettes.\n\n\u2022 This paper shows how a tobacco tax reform affects cigarette\u2019s demand price elasticity, tobacco consumption and government revenues in the context of a complex cigarettes tax structure.\n\n\u2022 Argentina cigarette\u2019s tax structure include four ad-valorem taxes. One of the taxes, the Special Tobacco Fund (FET), acts as a subsidy to the provinces that produce tobacco. Therefore, these provinces oppose in the congress to any tax reform that negatively affects these funds. We show that the tax reform of May 2016, that increase the rate of one of the taxes, the internal tax, from 60% to 75% induced an increment in the funds raised by the FET.\n\n\u2022 Using the estimation of the demand price elasticity, we simulate the impact of the tax reform by increasing the rate of internal taxes on consumption and government revenue, finding that it is possible to increase even more this tax rate and increase revenues and decrease consumption of cigarettes.\n\n## Footnotes\n\n\u2022 Contributors I wrote the paper and made all estimations in it.\n\n\u2022 Funding The authors have not declared a specific grant for this research from any funding agency in the public, commercial or not-for-profit sectors.\n\n\u2022 Competing interests None declared.\n\n\u2022 Patient consent for publication Not required.\n\n\u2022 Provenance and peer review Not commissioned; externally peer reviewed.\n\n\u2022 Data availability statement Data are available in a public, open access repository. All data relevant to the study are included in the article or uploaded as supplementary information.\n\n## Request Permissions\n\nIf you wish to reuse any or all of this article please use the link below which will take you to the Copyright Clearance Center\u2019s RightsLink service. 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Opinion Line
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How should society view Tsai Ing-wen re-elected in Taiwan?
CGTN | Updated: 2020-01-13 10:51
Tourists enjoying the sunset in Taipei.
Editor's note: Zhang Hua is an expert at the Institute of Taiwan Studies, Chinese Academy of Social Sciences. The article reflects the author's opinions and not necessarily the views of CGTN.
On January 11, China's Taiwan region held leadership elections, and Democratic Progressive Party (DPP) candidate Tsai Ing-wen was re-elected. Her victory has little to do with the Chinese mainland's Taiwan policy. The strong strategic resources and capabilities possessed by the Chinese mainland, especially the major achievements made through reforms in the new era, will ensure peace and stability across the Taiwan Strait.
Several reasons speak for the DPP's triumph. First, the DPP outsmarted the Kuomintang(KMT) in election campaigning. Known as a well-oiled election machine, the DPP accurately took the pulse of the people on the island and put forward policy ideas that could resonate with the local people. It also dug deep into the weak spots of its political rivals and then launched extensive media and public opinion campaigns to make a good case. That said, campaign skills only guarantee the DPP's wining at a "technical" level. The key to Tsai's success is that a major change of demographic structure, political identity and public participation in politics has been taking place on the island.
In terms of demographics, Tsai Ing-wen won the support of young people on the island. She got almost 70 percent of the votes among people aged 20 to 39 and surpassed her political rival, Han Kuo-yu, among people over 40.
In terms of political identity, since Lee Teng-hui initiated the "de-sinicization" in the local education, almost everyone under 40 grew up being indoctrinated with the idea of "independence." Also, with the development of political activities such as elections, the ideas of "self-awareness," "ownership" and "de-sinicization" have gradually taken hold and spread among people over the age of 40.
In terms of political participation, a large number of young Taiwanese returned to local ballot stations to cast their votes and eventually altered the election results. Prior to this election, it was predicted that Han Kuo-yu would win because Tsai's young supporters would not turn out to vote in light of their past voting behavior. The voter turnout in this election increased substantially, by almost 10 percent, which was mainly contributed by the active young voters.
Tsai's overwhelming victory has little to do with the Chinese mainland's Taiwan policy. In the election, Tsai did make a fuss of the strategic competition between China and the U.S. and the fugitive bill movement in Hong Kong to propagate a "sense of fear" in the region. This only demonstrates that cross-Strait relations are one of the important factors influencing elections on the island.
In recent years, the Chinese mainland has been adhering to the policy of "peaceful reunification" and the "One Country, Two Systems" principle to drive the peaceful development of cross-Strait relations, the promotion of cross-Strait economic integration and equal treatment of Taiwan compatriots. These are all wise decisions. Such policies have been welcomed by the people on the island, especially the Taiwan compatriots who have come to the mainland to find jobs, start businesses and receive an education.
The future of cross-Strait relations may be full of challenges, but confidence should be put in place to address them. At the press conference on the evening of January 11, Tsai Ing-wen highlighted cross-Strait relations and put forward a proposition of "peace, parity, democracy and dialogue." This again shows Tsai's ability to tread a fine line and evade responsibility. She was not vocal about the so-called Taiwan independence, but every word she said hinted at it.
The Chinese mainland has repeatedly stated that the lynchpin of cross-Strait relations is to uphold the 1992 Consensus and oppose to the "Taiwan independence." If Tsai does not take a clear stance on the above two issues, the chance of cross-Strait relations improvement is flimsy.
However, even if Tsai does not change her original stance, or steps up her efforts to promote separatist activities, the Chinese mainland is confident and able to maintain peace and stability across the Taiwan Strait.
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Lifestyles of the rich and famous! I loitered all morning at my hosts' house, overlooking the pool and golf course in the backyard. Madeline was gracious to let me stay so that I could take my radio interview on Nature Bat's Last from her house instead of having to find a quiet place for my call-in on the streets.
Afterward, I took off to explore Arizona's tony neighborhoods and was not disappointed. San Francisco, New York, LA, even Seattle may have higher real estate prices, but they can't match the pizzazz that Scottsdale's and Pleasant Valley's immense lots and sprawling homes deliver. True, some houses are ill proportioned; you can't make a successful 10,000 square foot statement by just pumping up a 2,000 square model and adding a string of garages. But many of the residences are architecturally striking; with corrugated metal, weathered steel and crisp stucco that fits the desert well. Although the scale of these low-lying mansions is conspicuous beyond reason, I appreciate that most people in Phoenix don't succumb to the California penchant to grow lawns where they don't belong. The gravel, sand, and native plant landscaping is terrific.
I wound my way to the Arizona Biltmore, Frank Lloyd Wright's lavish 1920's resort. Everything about it is classy. The staff was very accommodating to a guy who clearly wasn't registering for a room; the valet kept a personal eye on my bike. The Biltmore turned out to be one of my favorite Wright buildings, beautifully conceived and exquisitely executed. It may be the best example of his two dominant aesthetic ideas, as it was built toward the end of his Prairie / Usonian work and at the beginning of his larger scale, surface ornamented work.
The Biltmore plan and massing grow out of the Prairie tradition, albeit with a Southwest sensibility, while the wonderful use of decorative modular block precursors his work at Marin County and the Guggenheim. In addition to the great architecture, there are cool photos of Clark Gable, Rita Hayworth, Bob Hope, and other glitterati to cement Biltmore's cultural status. Every President since Herbert Hoover has stayed there.
On the way to my second Scottsdale host, I stopped at the Barry Goldwater Memorial. Though he doesn't merit a Presidential Library, the affluent citizens on this area have erected an elaborate memorial to their favorite son, which includes two marble paths with inlaid bronze letters. Problem is, the quote about the natural beauty of the West is banal, while the one about preserving our nation's security is fearfully bellicose. Like all of us, Goldwater reflected his origins. In his case, individualism and emphasis on private property led to a logical preoccupation with security. I am glad to be rolling along with everything I need and little that anyone else wants.
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title: flower tree
date: 2017-02-24 23:16:12
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A&M Equipment Sales Privacy Policy
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Call for an appointment 770-484-4051
Amber Jenkins Dabney, DDS
8200 Mall Parkway
Posts for: January, 2018
By Beautiful Smiles Family Dentistry
Tags: orthodontic treatment dental implants
For whatever reason, you've put off replacing a missing tooth for awhile. Now you want to fill that empty gap in your smile with a dental implant restoration.
But if your tooth's been missing for a long time, there could be a problem with space. This is because the teeth on either side of the space may have gradually drifted into it, leaving no room for the implant. You could need orthodontic work first to return these teeth to their proper position.
We could use braces, metal orthodontic devices with wires threaded through brackets bonded to the teeth that are then anchored, usually to back teeth. The orthodontist uses elastics or springs as well as possibly incrementally tightening of the wire against the anchors. These techniques create pressure or tension on the teeth for the desired direction of movement. The teeth's natural mechanism for movement does the rest.
But while effective, braces can be quite noticeable, an embarrassing thought for many adults having to wear them over several months of treatment. But there may be an alternative: clear aligners, a succession of slightly different plastic trays usually worn in two-week intervals. Sequentially wearing each tray gradually moves the teeth to their desired positions.
Though not appropriate for all bite situations, clear aligners have a number of benefits when they can be used. They're nearly invisible to others and can be removed for hygiene tasks or rare special occasions. What's more, the orthodontist may attach a temporary prosthetic (false) tooth to the trays to camouflage the missing space during treatment.
There's one other issue you may have to deal with: if your tooth loss was related to periodontal (gum) disease, the gums and underlying bone may be in poor condition. In fact, substantial bone loss could rule out an implant altogether. But we may be able to remedy both gum and bone deficiencies through grafting or plastic surgery. It may be possible to regenerate enough bone to support the implant; and surgically repairing your gums will help ensure the implant appears natural.
If you have problems like these, don't give up on your restoration goal just yet. With some orthodontic and dental work ahead of time, we may still be able to make implants a reality for you.
If you would like more information on restoring your smile after losing teeth, please contact us or schedule an appointment for a consultation.
Tags: celebrity smiles orthodontic treatment braces
Mayim Bialik has spent a good part of her life in front of TV cameras: first as the child star of the hit comedy series Blossom, and more recently as Sheldon Cooper's love interest — a nerdy neuroscientist — on The Big Bang Theory. (In between, she actually earned a PhD in neuroscience from UCLA…but that's another story.) As a child, Bialik had a serious overbite — but with all her time on camera, braces were just not an option.
"I never had braces," she recently told Dear Doctor – Dentistry & Oral Health magazine. "I was on TV at the time, and there weren't a lot of creative solutions for kids who were on TV." Instead, her orthodontist managed to straighten her teeth using retainers and headgear worn only at night.
Today, there are several virtually invisible options available to fix orthodontic issues — and you don't have to be a child star to take advantage of them. In fact, both children and adults can benefit from these unobtrusive appliances.
Tooth colored braces are just like traditional metal braces, with one big difference: The brackets attached to teeth are made from a ceramic material that blends in with the natural color of teeth. All that's visible is the thin archwire that runs horizontally across the teeth — and from a distance it's hard to notice. Celebs like Tom Cruise and Faith Hill opted for this type of appliance.
Clear aligners are custom-made plastic trays that fit over the teeth. Each one, worn for about two weeks, moves the teeth just a bit; after several months, you'll see a big change for the better in your smile. Best of all, clear aligners are virtually impossible to notice while you're wearing them — which you'll need to do for 22 hours each day. But you can remove them to eat, or for special occasions. Zac Efron and Katherine Heigl, among others, chose to wear clear aligners.
Lingual braces really are invisible. That's because they go behind your teeth (on the tongue side), where they can't be seen; otherwise they are similar to traditional metal braces. Lingual braces are placed on teeth differently, and wearing them often takes some getting used to at first. But those trade-offs are worth it for plenty of people. Which celebs wore lingual braces? Rumor has it that the list includes some top models, a well-known pop singer, and at least one British royal.
So what's the best way to straighten your teeth and keep the orthodontic appliances unnoticeable? Just ask us! We'd be happy to help you choose the option that's just right for you. You'll get an individualized evaluation, a solution that fits your lifestyle — and a great-looking smile!
For more information about hard-to-see (or truly invisible) orthodontics, please contact our office or schedule a consultation. You can read more in the Dear Doctor magazine articles "Orthodontics for the Older Adult" and "Clear Aligners for Teenagers."
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With Proper Management, Dental Implants can be a Reality for Diabetics
Diabetics with Gum Disease Benefit from Coordinating Treatment for Both
Keeping Fluoride to Just the Right Amount Protects Your Family's Smiles
Brushing and Flossing Hard to Do? Consider Your 'Power' Options
Anchorage Devices Enable Greater Precision During Orthodontic Treatment
Although Rare, Allergic Reactions to the Metal in Implants Could be a Concern
chewing tobacco (1)
dental records (1)
Dentist - Lithonia, Beautiful Smiles Family Dentistry, 8200 Mall Parkway, Suite #155, Lithonia GA, 30038 (770) 484-4051
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,791
|
Operating Room Gel Positioners provide superior pressure relief, patient safety and – protection. Gel pads are extremely durable, easy to use, designed to be stable and easy to handle before and during the operations. Pads are reusable and durable because those are coated with polyurethane film, it protects the pad also from moisture and from external damage or foreign material. Material of gel positioners is also latex free and biocompatible.
Position pads and cushions is available in different sizes and shapes, those are used during different types of surgery to support the parts of the body of the patient. Cushions must hold the patient in the right position and must expand correctly the pressure in order to avoid for example pressure ulcers.
We have two different type of Gel Positioners available; filled with gel-foam or gel.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,910
|
#ifndef CSSValuePool_h
#define CSSValuePool_h
#include "core/CSSPropertyNames.h"
#include "core/CSSValueKeywords.h"
#include "core/CoreExport.h"
#include "core/css/CSSColorValue.h"
#include "core/css/CSSCustomIdentValue.h"
#include "core/css/CSSFontFamilyValue.h"
#include "core/css/CSSInheritedValue.h"
#include "core/css/CSSInitialValue.h"
#include "core/css/CSSPrimitiveValue.h"
#include "core/css/CSSUnsetValue.h"
#include "core/css/CSSValueList.h"
#include "wtf/HashMap.h"
#include "wtf/RefPtr.h"
#include "wtf/text/AtomicStringHash.h"
namespace blink {
class CORE_EXPORT CSSValuePool : public GarbageCollectedFinalized<CSSValuePool> {
WTF_MAKE_NONCOPYABLE(CSSValuePool);
public:
// TODO(sashab): Make all the value pools store const CSSValues.
static const int maximumCacheableIntegerValue = 255;
using ColorValueCache = HeapHashMap<unsigned, Member<CSSColorValue>>;
static const unsigned maximumColorCacheSize = 512;
using FontFaceValueCache = HeapHashMap<AtomicString, Member<const CSSValueList>>;
static const unsigned maximumFontFaceCacheSize = 128;
using FontFamilyValueCache = HeapHashMap<String, Member<CSSFontFamilyValue>>;
// Cached individual values.
CSSColorValue* transparentColor() { return m_colorTransparent; }
CSSColorValue* whiteColor() { return m_colorWhite; }
CSSColorValue* blackColor() { return m_colorBlack; }
CSSInheritedValue* inheritedValue() { return m_inheritedValue; }
CSSInitialValue* implicitInitialValue() { return m_implicitInitialValue; }
CSSInitialValue* explicitInitialValue() { return m_explicitInitialValue; }
CSSUnsetValue* unsetValue() { return m_unsetValue; }
// CSSPrimitiveValue vector caches.
CSSPrimitiveValue* identifierCacheValue(CSSValueID ident) { return m_identifierValueCache[ident]; }
CSSPrimitiveValue* setIdentifierCacheValue(CSSValueID ident, CSSPrimitiveValue* cssValue) { return m_identifierValueCache[ident] = cssValue; }
CSSPrimitiveValue* pixelCacheValue(int intValue) { return m_pixelValueCache[intValue]; }
CSSPrimitiveValue* setPixelCacheValue(int intValue, CSSPrimitiveValue* cssValue) { return m_pixelValueCache[intValue] = cssValue; }
CSSPrimitiveValue* percentCacheValue(int intValue) { return m_percentValueCache[intValue]; }
CSSPrimitiveValue* setPercentCacheValue(int intValue, CSSPrimitiveValue* cssValue) { return m_percentValueCache[intValue] = cssValue; }
CSSPrimitiveValue* numberCacheValue(int intValue) { return m_numberValueCache[intValue]; }
CSSPrimitiveValue* setNumberCacheValue(int intValue, CSSPrimitiveValue* cssValue) { return m_numberValueCache[intValue] = cssValue; }
// Hash map caches.
ColorValueCache::AddResult getColorCacheEntry(RGBA32 rgbValue)
{
// Just wipe out the cache and start rebuilding if it gets too big.
if (m_colorValueCache.size() > maximumColorCacheSize)
m_colorValueCache.clear();
return m_colorValueCache.add(rgbValue, nullptr);
}
FontFamilyValueCache::AddResult getFontFamilyCacheEntry(const String& familyName)
{
return m_fontFamilyValueCache.add(familyName, nullptr);
}
FontFaceValueCache::AddResult getFontFaceCacheEntry(const AtomicString& string)
{
// Just wipe out the cache and start rebuilding if it gets too big.
if (m_fontFaceValueCache.size() > maximumFontFaceCacheSize)
m_fontFaceValueCache.clear();
return m_fontFaceValueCache.add(string, nullptr);
}
DECLARE_TRACE();
private:
CSSValuePool();
// Cached individual values.
Member<CSSInheritedValue> m_inheritedValue;
Member<CSSInitialValue> m_implicitInitialValue;
Member<CSSInitialValue> m_explicitInitialValue;
Member<CSSUnsetValue> m_unsetValue;
Member<CSSColorValue> m_colorTransparent;
Member<CSSColorValue> m_colorWhite;
Member<CSSColorValue> m_colorBlack;
// CSSPrimitiveValue vector caches.
HeapVector<Member<CSSPrimitiveValue>, numCSSValueKeywords> m_identifierValueCache;
HeapVector<Member<CSSPrimitiveValue>, maximumCacheableIntegerValue + 1> m_pixelValueCache;
HeapVector<Member<CSSPrimitiveValue>, maximumCacheableIntegerValue + 1> m_percentValueCache;
HeapVector<Member<CSSPrimitiveValue>, maximumCacheableIntegerValue + 1> m_numberValueCache;
// Hash map caches.
ColorValueCache m_colorValueCache;
FontFaceValueCache m_fontFaceValueCache;
FontFamilyValueCache m_fontFamilyValueCache;
friend CORE_EXPORT CSSValuePool& cssValuePool();
};
CORE_EXPORT CSSValuePool& cssValuePool();
} // namespace blink
#endif
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,599
|
Produced by David Widger
THE LIFE OF DAVID
OR, THE HISTORY OF THE MAN AFTER GOD'S OWN HEART
Omnia probate, benum tenete.--S. P.
Reprinted From The Edition Op 1766.
London:
Printed And Published By J. Carlile, 55, Fleet-Street.
To the REV. SAM. CHANDLER,
D.D. F.R. and A. SS.
To whom, Sir, could the republication of this little history with more
propriety be addressed, than to a gentleman to whom it is under such
considerable obligations? When it first appeared, it was honoured with
your notice in an especial manner; and is not a little benefited by your
_labours_. You, Sir, with a _careful_ hand noted its errors; and what
has stood the test of your strictures is certainly established with
additional authority. Whatever might be the motives which influenced _so
vigorous_ an exertion of your _learned_ and _critical_ powers--powers
so universally acknowledged and respected, the author of this piece
will not now inquire: it is sufficient to him, that they operated to the
extending the knowledge of his tract, among that class of readers who
stood most in need of the information it furnished; and he is persuaded
you will with pleasure hear his assurances, that the work owes no small
share of what approbation it may have gained, to your _elaborate review_
of it. Several worthy pious persons having candidly declared in private
conversation, (and unknowingly to the author himself) that the perusal
of your book really strengthened the facts advanced in the history. For
this, therefore, he considers you as entitled to his thanks; and that
his acknowledgments might be as public as the obligation, no method of
conveyance seemed more proper, than to prefix them to this new Edition
of "_The History of the Man after God's own Heart._"
He scorns, Sir, to follow the practice of Dedicators in common, who,
from venal motives, surfeit their patrons with fulsome adulation:
he will not, therefore, call the blushes into your countenance, by
expressing his private sentiments of your _learned_ Review of this
Historical Sketch, farther than by one observation; which is, that had
you been totally unknown in the republic of letters before, your apology
for the death of Uriah would alone have raised your literary fame beyond
the power of envious detraction. However, not to offend your modesty, he
desists from farther encomiums; but with a wish that you may long live
to enjoy the reputation acquired by so _laudable_ a performance, he
concludes with subscribing himself,
Sir, your greatly obliged, and very humble Admirer.
PREFACE.
Some reverend panegyrists* on our late king,** have, a little
unfortunately, been fond of comparing him with a monarch in no respect
resembling him; except in the length of his reign, thirty and three
years: which a lucky text informed them to be the duration of David's
sovereignty over the Hebrew nation. Had our good old king died a year
sooner, or had we been indulged with him a year longer, the opportunity
of applying this text would then have been lost; and in either case we
might not have heard of the parallel.
A reverence for the memory of a worthy Prince, has occasioned the
world's being troubled with a new history of king David, (which,
otherwise might not have appeared) merely to shew how the memory of the
British monarch is affected by the comparison.
"Why even of yourselves judge ye not what is right?" is the language of
Jesus Christ. "Prove all things; hold fast that which is good;" is the
language of the apostle Paul. The liberty thus granted is unlimited; but
it is more than mere grant of liberty, these are positive injunctions:
let no one then be so timid as to resign an inclination to satisfy
just doubts: in Britain, thanks to the obstinate heresy of our brave
forefathers, no audacious Romish priest dare prescribe limits to the
exercise of our reasoning faculties; and Protestant ones surely will
not: nay, they cannot, consistently with those principles which justify
their dissent from the Romish communion. An honest desire to obtain
truth, will sanctify the most rigid scrutiny into every thing. An
apostle has told us, that we are not to believe even an angel from
Heaven, who should preach any other gospel than that of Christ;* and,
no authority can be so sacred, as to set aside the _most valuable
distinction of humanity_, with which our Creator has furnished us; or to
give the lie to our most self-evident conceptions of right and wrong.
* Dr. Chandler, Mr. Palmer and others.
** George the IId.
If that liberty, of which Britons boast the possession, means any thing,
it must primarily include freedom of thought; without which there can be
no freedom of action. Thus it must mean an uncontrolled power to examine
the validity of every proposition offered to our assent; without which
power, and the due exercise of it, our assent cannot be the assent of
rational beings. If the reformed religion means any thing, it must mean
a religion founded by the authority, not of councils and synods, but
of conviction, the result of private judgment. True Protestants do not
puzzle themselves about the decisions of Trent, Constance, or Dort; they
protest against all authoritative dictates; disciples of the meek, the
lowly, the humane Jesus, they seek of themselves to judge of right or
wrong. Who is most the Protestant, the friend to human kind, and to
truth? Those who appeal to the human understanding, and submit to the
public judgment whether things are really so or not; or those who say,
they are so, they shall be so, you shall acknowledge them to be so, or
else----?
* Galatians i. 8.
Let not weak-minded Christians who think truth not able to maintain its
authority without legal enforcements, lament what they call licentious
abuses of that liberty on which we are happy to congratulate ourselves:
injudicious productions of the pen will always meet the treatment they
deserve. Fallacious pretensions to reasoning cannot deceive mankind in
these liberal times; nor can truth be obscured, when the attention of
honest inquiries after it, is properly exerted. If the little historical
sketch which follows, and which in fact, exhibits no more than what we
have all daily read, without presuming to decide upon; if it really
is that audacious calumny which many roundly affirm it to be; it will
doubtless be considered as such: if, on the contrary, it contains
undeniable matters of fact, fallaciousness will appear in the angry
objections against it; and the writer trusts, the futility of such
objections, have already been made sufficiently apparent.
The name of David has never been mentioned by divines but with the
greatest respect, from the time in which he lived to the present day;
and he is always quoted as an illustrious example of holiness! so
illustrious, that the greatest instance of purity that ever existed on
earth, was frequently saluted by way of eminence, in reference to him,
_Son of David!_ so illustrious, that on the death of the late king
of Great Britain, many sermons were preached and published, in which,
parallels are drawn betwixt him and this standard of piety, in order to
justify encomiums on the former, by declaring how nearly he resembled
the latter.
In what manner David first acquired, and has ever since maintained, this
extraordinary reputation, is not difficult to deduce, he was advanced,
by an enraged prophet, from obscurity to the Hebrew throne; and taught
by the fate of the unhappy monarch who was raised in the same manner,
whom he supplanted, and whose family he crushed, he prudently attached
himself to the cause of his patrons,* and they were the trumpeters
of his fame. The same order of men, true to their common cause, have
continued to sound the praise of this church-hero from generation to
generation, unto the present time: in like manner the grand violator of
the English constitution obtained the epithet of _holy Martyr_.
A new scrutiny being made, however, into David's claim to sanctity,
which, notwithstanding a very learned defence of him, turned out so
greatly to his dishonour; the scene has been shifted by a few whose
sense has overbalanced their bigotry by two or three scruples. Some
such, like Sheba of old, blow the trumpet and cry, "We have no part in
David, neither have we inheritance in the son of Jesse!" In this manner
have some clerical weather-cocks veered about to an opposite point of
the compass; and David, who, till now has been considered as a man who
"did that which was right in the eyes of the Lord, and turned not aside
from any thing that he commanded him all the days of his life, save only
in the matter of Uriah the Hittite," has, by one stroke of politics,
been resigned to the mercy of his detectors; and the importance of the
detection endeavoured to be annihilated, as the easier task; all which
appears with rather an ill grace, at a time when it is manifestly
extorted.
* The Prophets and Priest.
Thus much being premised relating to the conduct of the champions for
orthodoxy, on the occasion of this little squib which has produced so
much bustle in the clerical hives, proceed we to say something of the
tract itself.
The intention was, without any regard to remote objects, or heed of
future consequences, which in fact ought _never_ to be considered in
investigating any point; to give a fair undisguised narrative of the
life and transactions of David, king of Israel.
This, however, was not so easy to perform, as it was to project; from
three difficulties which impeded the execution.
1. It is not easy to conquer the early prejudices of education in favour
of the Hebrew nation; which the careful inculcation of their story
during our infancy, hinders our seeing in a proper light: so that
relations which might shock humanity in what is called prophane history,
are read without any emotion but that of reverence, in _this_. This
misconception is in great measure assisted.
2. By their History being written _by themselves_: and difficult to be
corrected.
3. By the broken unconnected manner in which it is transmitted down
to us: which renders it impossible to give a complete narrative of any
period in it.
A common share of humanity, which a little attention to common sense
enabled the author to extend to every nation under Heaven as the objects
of it, relieved him from the first of these difficulties: to overcome
the other two, he has assumed the liberty of giving _his_ sense to what
appears dark, or misrepresented; which he hopes will not be denied him,
so long as it is not found that a forced construction is put upon any
thing cited; or, that it is represented in any other light than what it
naturally appears in, when considered with the freedom, which it is our
duty to use in the examination of every historical record.
And lest it should be imagined that too great liberties are taken with
the biblical writers; it may not be amiss to mention once for all, that
innumerable instances might be produced, to shew that the authority of
_the Lord_, so continually quoted to sanctify every transaction related;
constituted for the most part, nothing more than national phrases, which
obtained universally among so bigoted a people as on all occasions the
Jews appear to have been: one-twelfth part of whom were appropriated
to the priesthood! A phraseology in some measure similar obtained in
England, at that time, when shunning the cruel talons of papacy, the
people rushed into the jaws of wild enthusiasm. That the sense in which
the acts of David are here understood, is the most obvious and natural,
appears from the amazing pains it has occasioned his champions, to force
another upon them. Of this, the Life of David, by Dr. Delany, is a most
remarkable instance; but the gross palliations, puerile conjectures,
and mean shifts to which he has been driven, prove the difficulty of the
task; while they are too frivolous to bias any, but the most _Catholic
believers_.
Mr. Stockhouse, in his History of the Bible, has urged arguments against
particular passages, under the title of _Objections_; so cogent, that
_his answers_ to them, certainly could not be satisfactory even to
himself.
Dr. Chandler has lately added his name to the list of David's
apologists. Strange! that so holy a king should need the exertion of so
much learned and critical dexterity, to establish his fame for goodness
of heart! This gentleman's performance, which was published as a reply
to the first edition of the present work, is a very extraordinary piece;
and shews that, great learning is no security for soundness of judgment.
The Doctor's book has been considered in a letter addressed to him, and
published separately; to which the reader is referred for an examination
into the merits of his arguments. In answering the Doctor, new lights
opened on many occurrences, which, as far as they could be detached from
that particular controversy, are taken into the present edition.
The best of kings is a title which adulation and servility have always
conferred on the most contemptible, as well as the most detestable
tyrants; and the frequency of its application to the object is ever in
proportion as he is undeserving of it. Had the flattering sycophants
of king David been satisfied with applying to him this common-place
appellation, rational men, who form their conclusions from the result of
general experience, would have inferred only that he had been one of the
numerous herd of bad princes who have oppressed mankind, and there would
have been nothing peculiar either in the fact or the inference. But
when the extremity of adulation conferred on David the title of _The
Man after God's own heart_, thinking men, who know the source from which
such adulation ever flows, are prepared to expect, in the development of
his history, a character pre-eminently wicked, and in this they are not
deceived.
All historians of credit agree in describing _God's chosen people_,
the _Jews_, as the most vicious and detestable of mankind;* their own
historians confirm this character of them, and the whole series of facts
which constitute their history, prove it beyond a possibility of doubt.
* Tacitus describes the Jewish people as formed of the worst
outcasts of the surrounding nations, collected together by
Moses, and kept for ever separated from the rest of mankind,
by an opposition of manners, and hostility of sentiment. Nam
passimus quisque, spretis religionibus patriis, tributa et
stipes illuc congerebant; unde auctae Judeorum res--ad versus
omnes alios hostile odium--transgressi in morem eorum, idem
usurpent; nec quidquam prius imbuuntur quam contemnere Deos,
exuere patriam; arentes, liberos, fratres, vilia habere.--
Ticiti Hist. Lib. v.
Among _the chosen people of God_--the most depraved of all nations--it
is pretty certain that the worst and wickedest man of that nation was
David, _The Man after God's own heart_. The truth of this proposition
will be abundantly proved in the following short history.
A question will here naturally present itself, how the Jews became so
much more vicious and depraved than their neighbours? And to resolve
that question, it will be necessary to consider in what respects their
laws and customs differed from those of others. It will be found that
they differed most essentially from all other nations in the world in
two particulars: 1st. They had more religion than any other nation;
and, 2dly. They had more priests. Other nations among whom superstitious
rites and ceremonies prevailed, were satisfied with practising them on
solemn festivals, and occasionally on particular or important events;
but the Jews practised their superstition incessantly: none of the
common duties, or ordinary functions of life, could be performed by
them, without a reference to the rules of their superstition; they
were bound to a strict observance of them whenever they ate, drank, or
performed any other of the natural functions.* **
* Moses quo sibi in posterum gentem firmaret, novos ritus
coutrariosque ceteris mortalibus indidit; profana illic
omnia, quae apud nos sacra; rursum concessa apud illos, quae
nobis incesta.--Seperati epulis, discreti cubilibus,
projectissima ad libidinem gens, aliena rum cubitu
abstinent, inter se nihil illicitum, circumcidere genitalia
instituere, ut diversitate noscanttir.--Taciti Hist. Lib. v.
It is impossible to draw a more disgusting picture of a
nation than this elegant and correct historian, in
describing the Jews.
** The Romans, though so numerous and powerful a nation, had
but very few priests, compared to the Jews. The Augurs were
at first only 3, and in process of time were increased to
15. The Arnspices were 12. The Pontifices were at first but
4, and were afterwards increased to 10. The Flamines were
but 3. The Salit 12. The Feciales, who were 20 in number,
though classed by authors among the priesthood, were merely
civil officers employed as heralds. And the Vestals, or Nuns
of Rome, were only 4; altogether between 50 and 60. Vide
Kennett's Roman Antiq. And yet Saint Austin, De Cevitate
Dei, Lib. iv. cap. 15, admits that the Romans were so
virtuous, that God gave them the empire of the world because
they were more virtuous than other nations, vet, with true
Christian charity, he says, that they must nevertheless he
damned as heathens. We do not find that the priests of other
enlightened nations of antiquity were proportionality much
more numerous than amoung the Romans. In England at present
the number of the priesthood cannot be much less than
20,000; there are near 10,000 parishes, each having one
priest at least, several two, and some three or more,
exclusive of Deans and Chapters, Prebends, &c. &c. and all
these in the established church, as it is called, exclusive
of a great variety of other sectaries of different
denominations.
Other nations had a few priests dedicated to their gods or idols,
seldom exceeding a few dozen in a whole nation but the Jewish priesthood
constituted a twelfth part of the whole people, and claimed and
exercised the privilege of devouring a tenth part of the produce of the
country, without contributing any thing to its productive labour.* And
it is probable that the Jewish nation alone, though but a miserable
handful of semi-barbarous savages, had more priests than the rest of the
then known world collectively, and were consequently more vicious and
more enslaved than any other people.
* The Jewish priesthood being one tribe, or twelfth part of
the nation, do not appear to have assumed to themselves much
more than an equal proportion, compared to their numbers, in
taking the tithe or tenth part of the produce of the land,
however unjust it may appear that they should be supported
in idleness at the expence of the industry of the rest: but
the English priesthood, though abundantly numerous, do not
form above one five-hundredth part of the whole nation, yet
they have the conscience to take also the tenth of the whole
produce, which is near fifty times more than their just
share, according to the proportion of their romish models,
from whose example they pretend to derive them.
Mankind have been too long duped by that universal _cant_ of priests,
who, in their language, have ever affected to couple _religion and
morality_ together, and to represent them as _inseparably united_,
though the slightest attention must show that they are perfectly
_distinct_, and a full and mature consideration of the subject must
prove that they are even extremely _opposite_. They well knew that man,
in the most abject state of mental degradation to which superstition
could reduce him, must still acknowledge the force and excellence of
virtue and morality, and must perceive their necessary tendency to
promote his welfare and happiness. They well knew how useful to their
own views and interests it would be to persuade him that religion,
virtue, and morality, were one and the same, or, at least, intimately
and inseparably connected; the credulity of man gave credit to the
imposture without examination, and the uniform experience of above 2,000
years has not hitherto been sufficient to undeceive him.
Unhappy man! destined for ever to be the dupe of his own credulity, in
opposition to the testimony of his experience, and the evidence of his
senses. Does not the history of all ages show, that the most religious
nations have always been, and still are, the most vicious and immoral!
Another most formidable evil necessarily results from such a system of
superstition, that is, a state of civil slavery, which is always found
its universal concomitant. Whenever the human mind is debased and
degraded by a system of gross superstition, it becomes incapable of any
one manly, liberal, or independent sentiment; every energy of the mind
is lost, reason is surrendered, virtue, the chief support, if not the
sole foundation of freedom, is banished, and man is fitted to receive
the abject yoke of slavery; tyranny and despotism make an easy conquest
of him, and the priest is ever ready to rivet his chains, and perpetuate
his bondage, by the pretended sanction of Heaven. The power and
influence of the priest and the tyrant is ever in proportion to the
debasement of man; they have a common interest, have ever made a common
cause against him, and have constantly erected their common throne on
the ruins of his freedom, his welfare, and his happiness.
Let us not, therefore, be deterred from unmasking to the view of mankind
that immense mass of vice and depravity which constitute the foundation
of the Jewish superstition; let no blind veneration for that hideous
idol deter us from exposing its deformity; let us cultivate that which
is truly good and useful; let reason assume her just empire over the
mind of man, and credulity, ignorance, and folly, abdicate their usurped
dominion: then shall we soon behold the galling fetters of vice and
superstition broken by the irresistible power of virtue, morality, and
truth.
THE LIFE OF DAVID.
The first establishment of regal government among the Hebrews, was
occasioned by the corrupt administration of Joel and Abiah, the two sons
of Samuel, whom he had deputed to judge Israel in the decline of his
life.* The people, exasperated at the oppression they laboured under,
applied to Samuel for redress, testifying a desire to experience a
different mode of government, by peremptorily demanding a king.** At
this, however, Samuel was greatly displeased: not that his sons had
tyrannized over the people, for of that he takes no manner of notice,
neither exculpating them, nor promising the people redress; his chagrin
arose from this violent resumption of the supreme magistracy out of
the hands of his family; a circumstance for which he expresses great
resentment.*** He consults the Lord, and not knowing else how the
insurrection might terminate, in his name yields to their desires;
promising them a king with vengeance to them.**** "For,"(5) says the
Lord, "they have not rejected thee, but they have rejected me, that I
should not reign over them."(6) The people, nevertheless, resolving to
free themselves from present oppression, at the hazard of the threatened
judgments, obstinately persisted in their demand, and dispersed not
without a promise of compliance.
* 1 Sam. viii. 3.
** Ver. 5.
*** Ver. 6, &c.
**** Ver. 11, &c.
(5) Ver 7, compared with chap. ix.
(6) Chap. x. 1.
Samuel, to all outward appearance, chose the most impartial method of
choosing a king, which was by lot, from among the people assembled by
tribes; but prudently pitches upon his man, previous to the election;
the whole tenor of his conduct manifesting, that he intended to give
them a king in name, but still to retain the supreme authority in
his own hands, by choosing one who should continue subordinate to his
dictates. Opportunely for his purpose, a young countryman, named Saul,
having rambled about to seek his fathers asses, which had strayed, and
finding all search after them vain, applied to Samuel as a prophet,*
with a fee in his hand, to gain intelligence of his beasts.
We gather from several passages in Jewish history, that there were
seminaries of prophets, i.e. the universities of the times, where
youth were trained up to the mystery of prophesying. We find there were
false prophets, nonconformists, not of the establishment; we find that
even the true ones were liable to be imposed on by their brethren;**
and we find moreover, by this instance, that prophets did not disdain
to give assistance in their prophetical character, concerning domestic
matters, for reasonable gratuities. A chief among the prophets, one
who had been a judge over Israel, is applied to in a pecuniary way, for
intelligence concerning lost cattle.***
* 1 Sam. ix. 7, 8.
** 1 Kings, xiii. 18., Josephus in loco.
*** Pretensions to divinations continue to this day, though,
in the opinion of reformed churches, all prophesying and
miracles have long since ceased. These modern prophets are
drolly ridiculed by our facetious countryman, Butler, in the
person of Sydrophel, a dealer,
"In Destiny's dark counsel?,
Who sage opinions of the moon sells;
To whom all people, far and near,
On deep importances repair.
When brass or pewter hap to stray,
Or linen slinks out of the way;
When geese and pullen are seduc'd,
And sows of sucking pigs are chous'd:
When cattle feel indisposition,
And need th' opinion of physician;
When murrain reigns in hogs or sheep,
And chickens languish of the pip;
When yeast and outward means do fail,
And have no power to work on ale;
When butter does refuse to come.
And love proves cross and humoursome;
To him with questions and with urine,
They for discov'ry flock, or curing.
It has been said, that this is the only instance recorded of a prophet
being applied to for purposes of this nature; but it appears that it
was usual for men to have recourse to prophets, and that the phrase was,
"Come, and let us go to the seer;"* and that prophetical intelligence
was paid for, is evident from the inquiry between Saul and his servant,
concerning their ability to gratify him.**
But, to proceed: Saul not only found his asses, but a kingdom into the
bargain; and had the spirit of the Lord given;*** to him which we
find taken away**** again, when he proved untractable: though it seems
somewhat odd, how he could possibly prove disobedient, while he acted
under the influence of this Divine Spirit! For, the possibility being
admitted, the advantage of inspiration is difficult to be conceived.
After Samuel had in private(5) anointed Saul king, and told him his
asses were already found, he dismissed him for the present. He then
assembled the people for the election of a king: at which assembly,
behold, the lot fell on the tribe of Benjamin; and in that, on the
family of Matri; and finally, on Saul, the son of Kish.(6) An election
somewhat resembling consistories for the appointment of bishops; where
the person being previously fixed on, God is solemnly prayed to for a
direction of their choice.
* 1 Sam. ix. 9.
** Ver. 7, 8.
*** Ch.x. 6.
**** Ch. xvi. 19.
(5) Ch. x. 1.
(6) Ch. x. 20, 21.
It is not intended here to give a detail of the reign of king Saul;
the notice hitherto taken of him being merely because the life of David
could not be properly introduced without mentioning the alteration of
government, and the manner in which monarchy was established in Israel:
since Samuel's disappointment in Saul, naturally leads to his similar
choice of David.
The disobedience of Saul, in daring to sacrifice without his patron the
prophet,* who failed of coming according to his appointment; and his
lenity and prudence, in sparing the king, and some cattle, from a nation
which Samuel, in the name of the Lord, had commanded him utterly
to extirpate,** irrevocably lost him the favour of this imperious
inexorable prophet: and, in the end, produced the miserable destruction
not only of himself, but of his family: which will occasion no surprise,
when we consider the absolute dominion and ascendency which the Jewish
priests maintained over this ignorant superstitious people. A dominion
which every article in the Levitical law enlarged and strengthened.
We are not to imagine that the sparing Agag, king of the Amalekites, was
the only cause of this rupture between him and Samuel. For we may gather
from other parts of his history, that Saul was not over-well affected
towards his patrons the Levites;*** in subjection to whom he had too
much spirit to continue. Samuel quickly perceived he had mistaken his
man, he haughtily avowed his intention of deposing him;**** and ordering
Agag to be brought into his presence, he hewed him in pieces--"before
the Lord."(5)
* 1 Sam. xiii. 8-14.
** Ch. xv. 3.
*** Ch. 22.18,19. and Ch. xxviii. 9.
**** Ch. xiii. 14. xv. 26. 28.
(5) Ver. 33., Ch. xvi. 13.
We now come to the hero of the history.
In pursuance of his intention to make another king, Samuel went under
the pretence of a sacrifice, and anointed another country youth, which
was David, the youngest son of Jesse, the Bethlehemite; and gave him
the spirit of the Lord, which he had just taken from poor Saul.
The king, in the mean time, reflecting on the precariousness of his
situation, now that the priests, on the part of Samuel, were incensed
against him; and well knowing their influence among his subjects, fell
into a melancholy disorder of mind,* which his physicians were unable to
remove.**
This was artfully made the occasion of introducing David to court. The
king was advised to divert himself with music; and David was contrived
to be recommended to him for his skill on the harp.*** Saul accordingly
sent to Jesse, to request his son; which was immediately complied
with: and David was detained at court, in the capacity of the king's
armour-bearer.**** Here the story begins to grow confused, beyond
_lay_-skill to reconcile. A war with the Philistines is abruptly
introduced; in the midst of the relation of which, we are abruptly
informed that David returned from Saul to feed his father's sheep(5)
again; from whence his father sent him with provisions for his brothers,
who were in the army.(6) What can be thought of this? Jesse hardly
recalled his son from the honourable post of armour-bearer to the king;
it is not likely that he was turned off, since we afterwards find him
playing on the harp to the king, as before;(7) neither was it proper
employment for the king's armour-bearer to be feeding sheep, when the
army was in the field, and his majesty with them in person! Why--the
most easy method is to take it as we find it; to suppose it to be right,
and go quietly on with the story.
In the Philistine army was a man of extraordinary size, named Goliah,
who came out of their camp, day by day, challenging and defying any one
among the Hebrews to single combat, and to rest the decision of their
quarrel upon the event; an offer which no one among the Israelites was
hitherto found hardy enough to accept.(8)
*1 Sam. xvi. 14.
** Josephus.
*** 1 Sam. xvi. 18.
**** Ver. 21.
(5) Ch. xvii. 15.
(6) Ch. xvii. 17.
(7) Ch. xviii. 10.
(8) Ch. xvii. 4, &c.
David is said to have arrived at the army just as it was forming for
engagement; at which time the giant advanced as before, with reproachful
menaces; and, after having enquired carefully concerning what reward
would be given to the conquerer of this giant, and learning that great
riches and the king's daughter were to be the prizes of conquest, David
courageously declared before Saul his acceptance of the challenge,*
notwithstanding the contempt with which his offer had been treated.
Saul, relying on the youth's ardour and assurance of victory, girded his
own armour on him:** but David put it off again, trusting entirely to a
pouch of stones, and his own skill in slinging.*** The success answered
his hopes, and stamped, what would otherwise have been deemed a rash
undertaking, with a more respectable name; he knocked Goliath down with
a stone; then ran in upon him, cut his head off with his own sword, and
brought it triumphantly to the king of Israel.**** The consequence was
the defeat of the 'Philistines.
Here we meet with another stumbling-block. For, though Saul, as has
already been observed, had sent to Jesse expressly for his son David;
though David had played to him on the harp; though Saul had again sent
to Jesse, to desire that David might be permitted to stay with him; and
in consequence of this had given him a military appointment about his
person; though he had now a fresh conference with him; had just placed
his own suit of armour on him; and though all the occurrences must have
happened within a small space of time, yet his memory is made so to fail
him on a sudden, that he knew nothing either of David, or his parentage!
but while David went to meet the giant, he enquired of others, who
proved as ignorant as himself, whose son(5) the stripling was? This
stumbling-block must likewise be stepped over, for it is not removeable.
*1 Sam. xvii. 32.
** Ver. 38.
*** Ver. 40.
****Ver. 49.
(5) Ver. 55., Ch. xviii. 3.
The reputation which this gallant action procured to David, soon gained
him advancement in the army, and a warm friendship with Saul's son,
Jonathan. But the inordinate acclamations of the people, on account of
the death of the Philistine giant, "Saul hath slain his thousands, and
David his ten thousands;"* a rhodomontade out of measure extravagant,
when we compare the two subjects of the contrast, justly occasioned Saul
to view David with a jealous eye. We have all the reason in the world to
believe that Samuel and the priests made every possible advantage of an
adventure so fortunate for their intended king, to improve his growing
popularity, which even at its outset had so far exceeded all bounds of
decency: "What," said Saul, "can he have more but the kingdom?"** and we
may therefore conclude that the king saw enough to alarm him; for we are
told, that "Saul eyed David from that day and forward,"***Thus we find
that on the following day, while David played as usual on his harp
before Saul, the king cast a javelin at him,**** which David avoided.
Saul then made him captain over a thousand, saying, "Let not mine
hand be upon him, but let the hand of the Philistines be upon him," an
expression however which is evidently put into Saul's mouth, since it is
impossible he could have made use of it openly. He made him the offer of
his daughter Merab for his wife, in consequence of the defeat of Goliah;
but she, we know not why, was given to another;(5) afterwards he gave
him Michal: and David's modesty (6) on this occasion was incomparably
well acted; he knowing himself, at the same time, to be secretly
intended for the kingdom by Samuel.
Saul, upon reflection, concluding it dangerous to execute any open act
of violence against this young hero, politically hoped to ensnare him,
by exalting him high in favour, or to get rid of him by putting him upon
his mettle, in performing feats of valour; for a deficiency of valour is
not to be numbered among David's faults. It was with this view that
the king yet required of him an hundred Philistine foreskins(7) as the
condition of becoming his son-in-law. He produced double the number "in
full tale."(8)
* 1 Sam. xxii. 7.
** Ch. xviii. 8.
*** Ver. 9.
**** Ver. 11.
(5) Ver. 17.
(6) Ver. 10.
(7) Ver. 23., Ver. 25. (according to Josephus 600 heads).
(8) 1 Sam. xviii. 27.
This demand, after David appeals to have fulfilled the prescribed
conditions, seems not only unjust, but also, even making allowance
for Hebrew customs, very ridiculously expressed. It must have been a
glorious sight to have seen David bring the foreskins to king Saul,
strung perhaps on a piece of pack-thread, and dangling in his hand, or
thrown across his shoulders like a sash: and if Miss Michal was present,
how must her pretty little heart exult when the required number being
told off, as many more were gallantly presented at her feet!
David still advanced in his military** reputation, and met with a
powerful advocate in the person of Jonathan, his brother-in-law and
faithful friend, who effected a temporary reconciliation between him
and Saul;*** at which time Saul swore he would no more attempt his life.
Nevertheless, whether it was that he could not get the better of his
jealousy, or that he discovered more than is transmitted down to us, we
know not; consequences incline us to the last conjecture: Saul made two
more attempts to kill him;**** from one of which he was protected by
his wife Michal; and finding it not safe to stay at court, he fled to
Samuel, in Ramah.(5) Hither Saul sent messengers to apprehend him;(6)
but these, it seems, seeing Samuel presiding over a company of prophets,
and prophesying, were seized with a spirit of prophesying also; and not
only so, but it is related that Saul finding this, went at last himself,
to just the same purpose; for he likewise prophesied,(7) stripping off
his cloaths, in which ridiculous condition he continued for a day and a
night.
** Ver. 30. xix. 8.
*** Ver. 4.
**** Ver. 10, 11.
(5) Ver. 18.
(6) Ver. 20.
(7) Ver. 23.
This is an extreme odd relation! That the solemn appearance of an
assembly of prophets, presided over by a person so respectable, and
heretofore of such great authority in Judea, might influence, in an
extraordinary manner, persons entrusted with a commission to apprehend
or kill a man patronized by these prophets, exhibits nothing wonderful;
they might easily perhaps, be prophesied out of their errand; and might
then prophesy in concert. Prophesy is a vague term, not always limited
to the prediction of future events; the extempore preaching of many
dissenters, and the discourses of the Quakers, who profess to speak as
the Spirit gives them utterance, seem to come under the term prophesy.
These persons can work themselves and others into such fits of
enthusiastic intoxication, that they believe themselves agitated by
supernatural influence. Such might be the prophesying here mentioned.
But Saul prophesied! so it is said. Had the subject of Saul's
prophesying, been transmitted down to us, it might have greatly
illustrated this passage in the history; but no, he is barely said to
have _prophesied_; and we are prudently left to guess what. Being thus
at liberty, we, among other expositors, may easily surmise what he might
take for his text, and was the general tenor of his discourse, on this
particular occasion.
Afterward David had a private interview* with Jonathan; for he durst not
venture to appear at court. At this meeting, Jonathan, who had conceived
too great an affection for this man, and was at length seduced by him
from the duty and allegiance which he owed to his father and king,
solemnly promised** that he would sound his father's intentions on the
next day, which being the festival of the new moon, David's attendance
was expected at the king's table; and that he would warn him of any
danger intended him.
* 1 Sam. x. 1.
** Ver. 12.
David lay hid in the field until Jonathan brought him the required
intelligence; and when the king inquired, concerning him, Jonathan as
had been before concerted, said that he had requested leave to go and
perform a family sacrifice at Bethlehem. Saul's reply on this occasion
is very pertinent, and shows his antipathy to David not to have been
the causeless inveteracy of a disordered mind. "Then Saul's anger was
kindled against Jonathan, and he said unto him, Thou son of the perverse
rebellious woman, do not I know that thou hast chosen the son of
Jesse to thine own confusion, and unto the confusion of thy mother's
nakedness? For as long as the son of Jesse liveth upon the ground, thou
shalt not be established, nor thy kingdom: wherefore now send and fetch
him unto me; for he shall surely die."* Jonathan expostulated with his
father, and had a javelin hurled at him for his reward.**
David being advertised, according to agreement, of the king's
disposition toward him, retired to Ahimelech, the high priest, at the
city of Nob who treated him with shew-bread, and armed him with the
sword of Goliah, which had been hung up and consecrated to God.***
We may consider David's resuming this sword, after its dedication as
a religious trophy, whatever gloss may be put on his interview with
Ahimelech, to be a clear manifestation of hostile intentions, or a
declaration of war against his father-in-law, for which he now took the
first opportunity to prepare. Thus accoutred, he fled out of Judea, to
Achish, king of Gath;**** intending, as we have good reason to believe,
to enter into a treaty of alliance with him against the Hebrews; but
the popular cry was against him before he accomplished any thing, or at
least any thing that has reached our times.
* Ver. 30, 81. Josephus in loco.
** Ver. 33.
*** 1 Sam. xxi. 1.
**** Ver. 9. Josephus.
Here David appears to disadvantage in point of policy: for though his
carrying with him the sword of Goliah was artful enough, and likely to
collect followers in Judea, since it was a continual witness of that
prowess which had gained him such extraordinary reputation; yet, for
him, under this circumstance, to throw himself into the power of the
Philistines, among those very people from whose champion he had ravished
_that sword_, was the highest imprudence! and we perceive he might
have suffered for it, had not he made use of a stratagem to procure his
release, which he effected by acting the madman.* Mankind seems to have
been very easily imposed on in those days.
David, now thinking it time openly to avow his design of disputing the
crown with Saul, went to a cave called Adullam, which he appointed the
place of rendezvous for his partizans. Here we are told he collected
together a company of debtors, vagrants, and disaffected persons, to the
number of four hundred; and opened his rebellion, by putting himself at
the head of this body of men:** men, whose desperate situations under
the government in being, rendered them fit agents to disturb it, and
proved the surest bond to connect them to a partizan thus embarked in
an enterprize against it. Hither also came to him his father and all his
brethren; and the first movement that he made was to go to the king of
Moab, to obtain a retreat for his father and mother, until he knew the
event of his enterprise.***
By the advice of the prophet Gad, David next marched into the land of
Judah:**** Gad, no doubt hoped, that as the young adventurer was of that
tribe, he would there meet with considerable reinforcement. When Saul
heard of this insurrection, he pathetically laments his misfortune to
those about him, that they, and even his son Jonathan, should conspire
against him.(5)
* 1 Sam. xxi. 13.
** Ch. xxii. 2.
*** Ver. 2.
**** Ver. 6.
(5) Ver. 7, 8, 9.
Then started up one Doeg, an Edomite, who informed Saul, that he had
seen David harboured by the priests in Nob. Upon this, Saul summoned all
those belonging to that city before him, with Ahimelech their chief,
who began to excuse himself as well as he could; but Saul remembering,
without doubt, the threatening of Samuel, concerning the affair of
king Agag;* and considering these priests as traitors, from this
corroborating evidence against them, he commanded them all to be slain,
to the number of eighty-five persons.** Moreover, agreeable to the
barbarous usage of that nation, the massacre included the whole city of
Nob, man and beast, young and old, without exception.
Though the king's rage in this instance exceeded not only the bounds of
humanity, but also of good policy, it nevertheless serves to show how
deeply the priests were concerned in the rebellion of David; since
he could not be mad enough to commit so flagrant an act, without some
colourable pretence;*** and shows also that Saul had not so great an
opinion of their holiness as we, at this distance of time, are, by their
own annals, instructed to have. Had Saul been more implicit, he might
have enjoyed the name of king, have continued the dupe of the priests,
have died in peace, and his children have succeeded quietly to the
inheritance. But,
"Ye gods! what havoc does ambition make
Among your works!"
During this time, David rescued the city of Keilah from the
Philistines,(5) who were besieging it, hoping to make it a garrison for
himself.
* 1 Sam. xiii. 14, xv. 26, 28.
** Ch. xxii. 16-18.
*** Ver. 19.
**** In so small a territory as Judea, the difference
between the king and his son-in-law, so popular a man, could
not be unknown to persons in any measure removed from the
vulgar. Therefore, Ahmeleoh's pleas of ignorance did not
deserve credit.
(5) 1 Sam. xxiii. 3.
But upon the approach of Saul, not thinking himself able to maintain it,
being as yet but six hundred strong, and not choosing to confide in the
inhabitants, whose loyalty even his recent kindness to them could not
corrupt, he therefore abandoned it, and retired to the wilderness.*
This passage alone is amply sufficient to confirm the reality of David's
rebellious intentions; it is, therefore, worth analyzing. That he
delivered this city from the depredations of the Philistines, and that
by this action he hoped to purchase the friendship of the inhabitants,
are acknowledged: the use to which he intended to convert this
friendship, is the point to be ascertained. Saul was advancing to
suppress him. Had he seduced them from their allegiance, and obtained
the expected protection, he would have deprived Saul of this city, which
city might have been considered as a garrison. The old plea, of
his providing only for his personal safety, against his malignant
persecutor, has often been urged; but his intended retention of a city,
to secure that safety, was a flagrant rebellious intention. Had he
gained this one city, as his strength increased, he would have concluded
as many more as he could have procured, necessary for his preservation,
until he had monopolized the whole country, agreeable to the grant of
Samuel, which would then have justified the usurpation; but disappointed
in the first step, by the loyalty, miscalled treachery, of the Keilites,
he evacuated the town, having lost the recompence of his labour, and
with his men "went whithersoever they could go."** In the wilderness
Jonathan came privately to see him, and piously engages in the cause
against his own father, by covenant; in which it was agreed, that if
David succeeded, of which Jonathan is very confident, _he_ was to be a
partaker of his good fortune but as Jonathan was not to join him openly,
he went home again.
Saul, having received intelligence of David's retreats, pursued him from
place to place, until he was called off by news of an invasion of the
land by the Philistines;*** whether of David's procuring or not, we are
uncertain: thus much is certain, and does not discredit the supposition,
that he quickly after took refuge among those Philistines.
* 1 Sam. xxiii. 13.
** Ib.
*** Ver. 16-18, 27.
After repelling the invaders, Saul, however, returned to the wilderness
of Engedi, in pursuit of David, with three thousand chosen men. At
this place we are told of an odd adventure, which put the life of Saul
strangely into the power of David. He turned in to repose himself*
alone in a cave, wherein at that time, David and his myrmidons were
secreted.**
* The words are, "to cover his feet:" which Josephus and
others, mistake to mean, that he retired into the cave to
ease nature. But in Judges, iii. 21. we find that expression
to imply, that the servants of Eulon, king of Moab, supposed
their master to have locked himself in, to repose himself
with sleep, in his summer-chamber. This is farther
corroborated, in Ruth, iii. 7. where, when Boaz had eaten
his supper, he laid down on a heap of corn, doubtless to
take his rest. Ruth, by her mother's instruction, went,
uncovered his feet, and lay down by him--to have some
refreshment likewise. For, in the middle of the night, when
the man awaked, surprised at finding an unexpected
bedfellow, and demanded who she was, the kind wench replied--
"I am Ruth, thine hand-maid; spread therefore thy skirt
over thine hand-maid, for thou art a near kinsman." In the
present instance, it is evident, Saul slept in the cave; as
he discovered not the operation that had been performed on
his robe, till David called after him, to apprize him
thereof.
** 1 Sam. xxiv. 3.
This, one would imagine to have been a fine opportunity for him to have
given a finishing stroke to his fortune, by killing Saul, and jumping
into, the throne at once: but David knew better what he was about,
than to act so rashly. He could entertain no hopes that the Jews would
receive for their king a man who, with such great seeming holiness,
should imbrue his hands in the blood of the Lord's anointed. Beside,
what evidently destroys the boasted merit of David's forbearance toward
Saul, in this instance, is an obvious, though overlooked consideration,
that, compared with David, Saul had a strong army with, him; and had the
king been missing, had he been observed to enter the cave without coming
out again; and upon search, had he been there found murdered, there
would not have escaped, of all that pertained to David, any that
_pissed against this wall_. Of this David could not be insensible;
and therefore, only privately cut off the skirt of Saul's robe,* and
suffered him to depart in peace. When the king was gone out from
the cave, David calls after him, and artfully makes a merit of his
forbearance, protesting an innocency, to which his being in arms was,
however, a flat** contradiction. Saul freely and gratefully acknowledges
himself indebted to him for his life, and seems so well convinced of his
own precarious situation, that he candidly confesses it; only tying
him down with an oath,*** not to destroy his children after him--an
obligation which, in due time, we shall see in what manner remembered
and fulfilled by David.
*1 Sam. xxiv. 4.
** Ver. 8-15.
*** Ver. 21.
Saul must certainly have been greatly fatigued, or strangely overseen,
to have let David catch him at so great a disadvantage--a conduct
not usual with good generals. Yet, while we credit the relation, the
meanness of his reply to David's harangue, can be no otherwise accounted
for. Saul does not appear to have wanted resolution on other
occasions; but to acknowledge his assurance that David would obtain the
sovereignty, and poorly to entreat a fugitive rebel in behalf of his
family! is a conduct not even to be palliated, but upon the foregoing
supposition. We must either condemn the general or the king, neither of
which characters appear with extraordinary lustre upon this occasion.
David, on the other hand, dissembles admirably here, pretending to Saul
a _great reverence_ for the Lord's anointed, though conscious, at
the same time, that _he was also_ the Lord's anointed, and anointed
purposely to supersede the other Lord's anointed; and, moreover, was at
this very time aiming to put his election in force! But, as the people
were not of his council, and he knew their great regard for religious
sanctions, it was certainly prudent in him to set an example of piety,
in an instance of which he hoped, in time, to reap the benefit himself:
About this time Samuel died.*
We next find our young adventurer acting the chief character in a
tragi-comedy, which will farther display his title to the appellation of
being a Man after God's own heart.
There dwelt then at Maon, a blunt rich old farmer, whose name was Nabal.
David hearing of him, and that he was at that time sheep-shearing, sent
a detachment of his followers to levy a contribution upon him,** making
a merit of his forbearance, in that he had not stolen his sheep, and
murdered his shepherds.*** Nabal, who, to be sure, was not the most
courteous man in the world, upon receiving this extraordinary message,
gave them but a very indifferent reply, including a flat denial. "Who,"
says he, "is David? and who is the son of Jesse? There be many servants
nowadays that break away every man from his master. Shall I then take
my bread, and my water, and my flesh that I have killed for my shearers,
and give unto men whom I know not whence they be?"**** Upon receiving
this answer, David, without hesitation, directly formed his resolution;
and arming himself, with a number of his followers, vowed to butcher
him, _and all that belonged to him_, before the next morning.(5) And how
was this pious intention diverted? Why, Abigail, the charming Abigail!
Nabal's wife, resolved, unknown to her spouse, to try the force
of beauty, in mollifying this incensed hero, whose disposition for
gallantry, and warm regard for the fair sex, was probably not unknown at
that time. Her own curiosity also might not be a little excited; for the
ladies have at all times been universally fond of military gentlemen: no
wonder, therefore, that Mrs. Abigail, the wife of a cross country clown,
was willing to seize this opportunity of getting acquaintance with
captain David.
* 1 Sam. xxv. 1.
** Ver. 5--9.
*** Ver. 7.
**** Ver. 10, 11.
(5) Ver. 13, 14-22.
And this motive certainly had its force, since she could not as yet have
known David's intention: we may observe she was told of it by David at
their meeting.* She prepared a present, and went to David, saying,
very sententiously, "Upon me, my Lord, upon me let this iniquity be:"**
judging, very humanely, that could she get him to transfer his
revenge upon _her_, she might possibly contrive to pacify him, without
proceeding to disagreeable extremities. Nor was she wrong in her
judgment; for we are told, "So David received of her hand that which she
had brought him, and said unto her, go up in peace to thine house; see
I have hearkened to thy voice, _and have accepted thy person_."*** But
however agreeable this meeting might have been to Abigail, we do not
find that Nabal was so well pleased with the composition his wife had
made for him; for when he came to understand so much of the story as she
chose to inform him of, he guessed the remainder, broke his heart, and
died in ten days afterward. David loses no time, but returned God thanks
for the old fellow's death, and then Mrs. Abigail was promoted to the
honour of being one of the Captain's ladies.(5)
We are now told another story extremely resembling that of the cave of
En-gedi. Saul again pursues David with three thousand chosen men; again
fell into his hands during his sleep; only that here David stole upon
him in his own camp; he ran away with the king's spear and bottle of
water, and Saul went back again as wise as he came.(6)
* I Sam. xxv. 34.
** Ver. 24.
*** Ver. 35.
****Ver. 37,38.
(5) Ver. 39.
(6) Ch. xxvi.
The opinion of Mons. Bayle seems most probable concerning this
adventure, who looks upon it but as another detail of the former affair
at En-gedi; and that for very good reasons. For, upon a comparison of
both, as laid down in the 23d, 24th, and 26th chapters of 1 Samuel, we
may remark,
I. That in each relation Saul pursues him with the same number of chosen
men.
II. That both adventures happened at or very near the same place.
III. That in each story David comes upon Saul in much the same manner,
withholds his people from killing him, and contents himself with taking
away a testimonial of the king's having been in his power.
IV. That in the second account, When David is pleading the injustice of
Saul's persecuting him, as he terms it, he does not strengthen his plea
by representing to him that this was the _second time_ of his sparing
him, when he had his life so entirely in his power; and that Saul's
pursuing him this second time, was a flagrant instance of ingratitude,
after what had happened on the former occasion.
V. That in the second relation, Saul, when he acknowledges David's
forbearance and mercy to him in the present instance, makes no mention
of _any former obligation_ of this kind, although it was so recent, and
in the main circumstance so similar.
VI. That the historian, who evidently intended to blacken the character
of Saul, and whiten that of David, does not make the least observation
himself, in the second narrative, of reference to the first.
These reasons prove, beyond doubt, that we are furnished with two
relations of the same adventure. To account for the double record,
and their variations, must be left to commentators, connectors, and
harmonizers, who are used to compromise affairs of this nature.
David finding that with his present strength, he was unable to maintain
any footing in Judea, puts himself once more under the protection of
Achish, king of Gath.* Achish, who does not appear to have been a very
powerful prince, seemed to consider David alone, and David at the head
of a little army, as two very different persons: for he now assigned
him a place named Ziklag, for a habitation, where he remained a year and
four months.**
* 1 Sam. xxvii. 1-3.
** 1 Sam. xxvii. 6, 7.
As he had now a quiet residence, those who entertain an opinion of
David's sanctity, would be apt to suppose he would here confine himself
to agriculture, to composing psalms, and to singing them to his harp;
but David found employment more suited to his genius. It is not intended
here to be insinuated that he might not sing psalms, at leisure times;
but his more important business was to lead his men put to plunder the
adjacent country. We have the names of some nations, as they are called,
but which must have been small distinct communities, like the
present camps of wandering Moors and Arabs, over whom he extended
his depredations: these are the Geshurites, the Gezrites, and the
Amalekites.** Of these people he made a total massacre, at those places
where he made his inroads; saying, very prudently, "Lest they should
tell of us, saying, so did David, and so will be his manner, all the
while he dwelleth in the country of the Philistines."***
After thus carefully endeavouring to avoid detection, he brings his
booty home, which consisted of all which those miserable victims
possessed.**** He made presents of this to his benefactor king
Achis,(5)| who, demanding where he had made his incursion, was answered,
against the south of Judah, &c.(6) intending by this falsity to
insinuate to the king his aversion to his own countrymen, and attachment
to him. "And Achish believed David, saying, he hath made his people
Israel utterly to abhor him; therefore he shall be my servant for
ever,"(7)
The Philistines at this time collected their forces together, to attack
the Israelites. To which service Achish summoned David,(8) and met with
a cheerful compliance.
** Ver. 8.
*** Ver. 9, 11.
**** Query, whether David might not compose a psalm
upon this occasion.
(5) Josephus.
(6) 1 Sam. xxvii. 10.
(7) Ver. 12.
(8) Ch. xxviii. 1.
"Surely," says David, "thou shalt know what thy servant can do."* He
accordingly marched his adherents with the troops of king Achish; but
when the princes of the Philistines saw a company of Hebrews in their
army, they were much surprized, and questioned Achish concerning them.
The account which Achish gave of them, did not satisfy the princes, who
justly feared their captain might prove a dangerous auxiliary. "Make
this fellow return," said they, "that he may go again to the place which
thou hast appointed him, and let him not go down with us to the battle,
lest in the battle he be an adversary to us: for wherewith should be
reconcile himself to his master; should it not be with the heads of
these men David was accordingly dismissed, very much mortified at their
distrust of him.**
Here now was a signal evidence of David's righteousness! The Hebrews,
according to their own testimony, understood themselves to be the
favourite people of God, and David is delivered down to us as a
distinguished character for piety among this peculiarly esteemed people.
Yet could this very man, without any hesitation, freely join himself and
company, to an army of uncircumcised idolators, marching with hostile
intentions against his countrymen! His advocates indeed pretend, that
had his offers been accepted, he would nevertheless have gone over to
the Israelites, at the commencement of the battle: this is taking off
the charge of one crime, by imputing to him another equally bad--a most
base act of treachery! As, however, the Israelites, on the foundation of
their own intimacy with the Deity, thought they had no more obligations
to a moral conduct towards the heathens, than the Roman Catholics now
imagine they have to keep faith with heretics; these advocates endeavour
to preserve the piety of David's character, at the expence of what
David, according to this method of arguing, did extremely well without,
on all occasions; namely, _honesty_.***
* 1 Sam. xxviii. 2.
** Ch. xxix. 4.
*** Ver. 8, 11.
Upon his return to Ziklag, he found that, during his absence, the
Amalekites had made reprisals upon him, and burnt Ziklag; and had
carried off all the women captives.* But in the relation there is one
remark well worth noting, which is that "they slew not any either great
or small"**--so much more moderation had these poor heathens in their
just revenge, than the enlightened David in his unprovoked insult. If
they came to avenge so savage an insult, it shewed great consideration
in them to spare the innocent, the guilty being absent: if they only
came on the common principle of plunder, the bare comparison of the
different treatment of the sufferers in each instance, speaks forcibly
without amplification. Upon this misfortune, his band began to mutiny,
and were on the point of stoning*** him; when he, who knew how to soothe
them, enquired of the Lord what he should do? and evaded their rage, by
inspiring them with a resolution to pursue the Amalekites, and with the
hopes of recovering all their losses. He, therefore, with four hundred
picked men, set out on the pursuit, and by the way found a straggler****
who had fainted: after recovering him, they gained, by his means,
intelligence of their route. David came upon them unexpectedly, at a
place where they were, without apprehension, regaling themselves after
their success: and though David's men recovered all they had lost,
together with other booty, and found their wives and children unhurt:
yet could not their captain resist so inviting an opportunity of
gratifying his delight in blood-shedding: the pursuit and slaughter
continued from the twilight (we know not whether of the morning or
evening) of one day, until the evening of the next. None escaped but a
party which rode upon camels.(5)
* 1 Sam. xxi. 1.
** Ver. 2.
*** Ver. 6.
**** Ver. 11.
(5) Ver. 17.
Of the spoil taken from these people, David sent presents to the elders
of his own tribe of Judah, "and to all the places where David himself
and his men were wont to haunt."* By which means he kept them attached
to his interest.
The dispute between the Philistine and Hebrew armies, did not terminate
but by the defeat of the latter, the death of Saul, and of three of his
sons.**
Such was the catastrophe of king Saul! a man advanced from the humble
state of a shepherd, by the prophet Samuel, to be his deputy in the
government of the Hebrew nations under the specious name of king: a man,
who allowing for the _peculiar complexion_ of the people over whom he
was placed; does not, on the whole, seem to suffer by comparison with
any other king in the _same_ history; or whose character appears to be
stained with any conspicuous fault, except that he was one degree less
cruel than his haughty patron: and was disobedient enough to endeavour
to be in effect, what he was only intended to be in name. On the whole,
he appears to have been strangely irresolute and inconsistent with
himself; and is perhaps represented more so than he might really have
been: but the undertaking to render himself independent was an arduous
task for one in his situation; therefore his actions and professions
might sometimes disagree. However, it is impossible to argue from every
expression that may be produced; we must form our judgment from leading
events, and corresponding expressions; and determine as they tally with
probability. If Saul himself, however he is represented as subscribing
to it, was really assured of David's destination to supersede him by
divine decree, there was nothing left for him but resignation: Can man
fight against God? since therefore his continual aim was to destroy
David, it argues against this assurance: and if Saul himself was mad,
surely his soldiers were not: how came he to find an army as mad as
himself, to persecute the Lord's anointed.
* 1 Sam. xxx. 31.
** Ch. xxxi.
We shall now have an opportunity to observe the conduct of our hero in
a regal capacity. The death of Saul facilitated his advancement to
a sovereignty, to which he had no pretension, either by the right of
inheritance, which was claimed by Ish-bosheth, a remaining son of Saul;
nor by popular election, which Saul himself had the shew of; but by
the clandestine appointment of an old prophet; which inspired him
with hopes, of which, by arms and intrigue, he at length enjoyed the
fruition.
David had returned to Ziklag but two days, when on the third, there came
to him an Amalekite, who officiously informed him of the event of the
battle between the Israelites and Philistines. He owned himself to be
the person who killed Saul, after his defeat, at his own request: he
being already wounded.
He hoped to be well rewarded for his news, by David; whose intentions
were so well known, that he presented him with Saul's crown and
bracelet*. But, alas! he knew not David, and perished in the experiment:
David ordering him to be killed for daring to slay the Lord's
anointed.** David's treatment of this Amalekite, is agreeable to the
customary rules of politics; and has nothing therefore remarkable in
it, farther than it is rendered so by peculiar circumstances. Saul
was declared to be rejected by God, and David was the pretender to his
throne; it may therefore be imagined by some, that this man might have
had some claim to _his_ private gratitude, especially considering the
account the Amalekite gave of the matter.
Who can help smiling at the relation of David's tearing his clothes off
his back, and bursting into a sorrowful lamentation for the death of a
man, to whose destruction he had so freely offered to lend assistance
but just before?
Upon this alteration of affairs, David, asking counsel of the Lord, was
advised to leave Ziklag, and go to Hebron, one of the cities of Judah;
whither he and all his men repaired.***
* 2 Sam. 10.
** Ver. 15.
*** 2 Sam. ii. 1.
There he got his partizans to anoint him king over Judah; at the same
time that Abner, Saul's general, had, at Mahanaim, made Ishbosheth,
Saul's son, king over Israel.* It may be remarked here, that David did
not seem to claim in right of the sacred unction bestowed on him long
since by Samuel. He realized his title indeed, as soon as he could make
it out, by the law of force: but if his divine title to the Hebrew crown
was universally known, and if, as has been urged, Ish-bosheth had none
at all, how came David's title not to be universally acknowledged? Did
only one tribe believe in it? Yet David, with the divine grant, was
obliged to obtain the sovereignty by arms and intrigue! just for all
the world like the wicked, who attain their desires by exactly the same
means, to all external appearance. Upon this division of the kingdom,
a battle was fought at the pool of Gibeon, between the army of
Ish-bosheth, commanded by general Abner, and that of David, headed
by Joab: victory declared in favour of the latter, with small loss on
either side, except that Joab lost his brother Ahasel, who was killed by
Abner's own hand.**
We must here be content with general hints; being only informed that
"there was long war between the house of Saul and the house, of David:
but David waxed stronger and stronger, and the house of Saul waxed
weaker and weaker."*** What very much conduced to this, was an ill-timed
quarrel between king Ish-bosheth and general Abner, concerning one of
Saul's concubines, with whom Abner had been too familiar:**** and his
resentment of the notice taken of this amour, occasioned a treaty to be
negociated between him and David, whom Abner engaged to establish over
all Israel.(5) David accepted his offer, but demanded, as a preliminary,
the restoration of his first wife Michal;(6) who, during the disputes
between him and Saul; had been espoused to another.(7)
* 2 Sam. ii. 4, 8.
** Ver. 17, 23.
*** 2 Sam. iii. 1.
**** 2 Sam. iii. 7.
(5) Ver. 12.
(6) Ver. 13.
(7) 1 Sam. xxv. 44.
This demand he likewise made openly, by an express message to
Ishbosheth, who kindly complied with it: the poor man who had since
married her, following her weeping all the way.**
It is impossible to avoid noting David's amorous disposition here;
which could not be content with six wives, who bare him children*** (no
mention of those who did not), but was yet so warm, that it took the
lead even in his most important concerns.--We will not pretend to assign
the cause of that sad disorder, the symptoms of which are described in
the 38th Psalm.
After Abner had traitorously endeavoured to advance the interest of
David****; he had an interview with him;(5) which, quickly after he
returned, coming to the ears of Joab; he, who does not appear to have
been acquainted with the secret spring which actuated Abner's zeal for
the cause of David; represented to him the imprudence of admitting a man
among them, who to all appearance came only as a spy. Unknown to David,
he sent for him back again, and privately stabbed him, in revenge
for the loss of his brother Asahel.(6) This was a most base piece of
treachery, worthy the servant of such a master: to assassinate a man in
cool blood, in revenge for an action which was committed in the heat of
battle, in self-defence, and even after fair warning given.
Upon the murder of Abner, David again acts the mourner;(7) which has a
greater probability of being sincere now, than when he grieved for the
unhappy Saul; because the false Abner was preparing to do him essential
service, by betraying his master's cause.
** 2 Sam. iii. 15, 16.
*** Ver. 2, &c.
**** Ver. 17.
(5) Ver. 20.
(6) Ver. 27.
(7) Ver. 31, &c.
But the event proved full as advantageous to David; as will presently
appear.
When Ishbosheth and his friends heard of the fate of Abner, who had
been the very life of their cause; it dejected all their spirits; and
two villains, named Rechab and Baanah, hoping to make their fortunes by
the public calamity, went and murdered their master king Ishbosheth,
as he was reposing himself during the heat of the day, and brought his
head to David*. But not reflecting on an obvious maxim in politics, they
like the Amalekite before, who claimed the merit of killing Saul, soon
found that, he thought it adviseable to punish the traitors, whatever he
thought of the treason.**
* 2 Sam. iv. 5, &c.
** Ver. 12.
Had David reflected on all the circumstances which led to this murder,
with that tenderness becoming a person professing so much piety, his
compunction would have greatly embarrassed him in the proper behaviour
on this occasion. For if these two execrable villains deserved
punishment, what did _he_ merit who was the primary cause of so
nefarious an action? Two poor rogues from subordinate views, effected by
assassination what David sought at the head of an army, which naturally
reminds us of the pirate and Alexander. So strangely do relative
circumstances bias our judgment of things essentially alike. Had David
aspired to no other sceptre than his shepherd's crook, the villains
had not presumed on the usurper's gratitude; and Ish-bosheth, who was
a quiet prince, might have reigned long an honour to himself and a
blessing to his country.
Ish-bosheth does not appear to have been a man of parts, qualified to
contend with such an antagonist as David; for nothing is recorded of
him: Abner was the person who raised him; and had he lived, would as
easily have deposed him, and though no qualifications are a security
against assassination, yet, as in the case of another unfortunate
monarch, Darius, king of Persia; such cowardly wretches generally take
the advantage of precipitating misfortunes already commenced, that they
may pay their court to the rising sun.
The murder of this unhappy son of an unhappy father, advanced David
to the dignity to which he aspired,* (though we shall see in a passage
which reflects no great honour on him, that Saul had more sons yet
living.) He was now in his thirty-eighth year; having reigned seven
years and an half in Hebron** over the tribe of Judah.
Although David was now invested with that supremacy which had been the
aim of his endeavours since the time that Samuel inspired him with the
spirit of ----- ambition; yet could not his enterprising genius continue
satisfied with such an exaltation. The first object of his attention
now, was the city of Jerusalem, then inhabited by the Jebusites; (but it
was of no importance who inhabited it, if David conceived a desire for
it): this city he besieged and the inhabitants relying on the strength
of their fortifications, out of derision planted <DW36>s on their
ramparts to guard their walls; saying "except thou take away the blind
and the lame, thou shalt not come in hither."*** Nevertheless David
carried the place, and made it his chief city.****
N. B. He supplied himself with, more wives and concubines out of his new
acquisition.(5)
While he was thus amorously engaged, the Philistines hearing that he Was
made king over all Israel, came and disturbed him; but David according
to the usual term _smote them_;(6) and his strokes were always
sufficiently felt.
The comic tale of David's bringing home the ark will not be long dwelt
upon; it may only be remarked, that it was brought on a new cart, drawn
by oxen; and that Uzzah some way or other lost his life, to, as the text
reads, was smote _by the Lord_,(7) for his impiety in saving the ark
from being overturned.(8)
* 2 Sam. v. 3, 1 Chron. xi. 3.
** 2 Sam. ii. 11.
*** Chap. v. 6. Josephus.
**** Ver. 7. 9., 1 Chron; xi. 5. 7.
(5) 2 Sam. v. 13.
(6) Ver. 20, 25., 1 Chron. xiv. 11.
(7) Query, whether the Lord did not sometimes smite
by the hands of the priest.
(8) Sam. vi. 7.
But if "the Lord seeth not as man seeth; for man looked on the outward
appearance, but the Lord looked into the heart." the intention of Uzzah
was indisputably good, and the alledged crime surely pardonable; the
seeming exigency precluding all hesitation and reflection. Had the ark
been really overturned for want of this careful prevention, Uzzah might
then, it would be naturally imagined, have been rather _smote_ for
neglecting to save it. However, it was no longer trusted to prophane
hands, but carried the remainder of the way upon the more holy shoulders
of the Levites,* with great parade: attended by musicians, and by David
himself who, dressed in a linen ephod, _danced before the Lord with all
his might_ and this, in such a frantic indecent manner, that he exposed
his nakedness to the bye-standers. Wherefore his wife Michal sneered at
him: "How glorious was the king of Israel today, who uncovered himself
to-day in the eyes of the handmaids of his servants, as one of the vain
fellows shamelessly uncovereth himself."** David, it seems, was of a
different opinion; for he told her he _would_ play before the Lord; and
would be yet _more vile_ than she had represented him;--adding, "and of
the maid-servants which thou hast spoken of, of them shall I be had in
honour."*** Some staunch zealots have very prudently spiritualized
this part of David's answer, and given the mystical sense of it; the
prophane, who are content with the evident signification of words,
having construed it no otherwise than into an insinuation that he had no
cause to be ashamed of what he exposed. Fie on them!
This story is concluded with a remark as odd as the rest of
it:--"Therefore Michal, the daughter of Saul, had no child until the day
of her death.(5)
* 1 Chron. xv. 2, 15.
** 2 Sam. vi. 14.
*** Ver. 20.
**** Ver. 22.
(5) Ver. 23.
For, if Michal had hitherto borne no children, neither to David, nor
to her immediate husband, her barrenness must have been constitutional;
and, preceding her offence, could not be a punishment inflicted in
consequence of it. Moreover, if, on the other hand, she _had_ borne him
children, and this disgrace to her was the consequence of a resolution
made by her husband David, that she should have no more children
_by him_: her quiet resignation, under this imposed widowhood, is by
inference a high compliment on this poor woman's conjugal virtue! which
was far from the historian's intention to bestow. Indeed there is great
reason to credit Michal, and to believe that David really behaved with
all the extravagance she ascribes to him: for she appeared before
this affair as a discreet kind of a woman; no instance of folly being
produced in her, unless the contrivances she made use of to save her
husband from the effects of her father's rage may be allowed to bear
such interpretation. Whatever judgment however is passed upon Michal's
censure of David's behaviour in this procession, it showed great cruelty
and ingratitude in him to fix so disgraceful a stigma on her; and not
to make allowance for female indiscretion, the worst name that could be
bestowed on her fault.
After this, David smote the Philistines, not sparing even Gath, that
city which had so humanely protected him.* He then smote the Moabites,
putting to the sword two-thirds of the nation, by causing them to lie
prostrate on the ground, and measuring them by lines; "even with two
lines measured he to put to death; and with one full line to keep
alive:"** so systematic was his wrath! Hadadezar, king of Zobah, was the
next whom he smote; who being assisted by the Syrians of Damascus, he
next smote them.*** Yet all this smiting and slaying is so obscurely
mentioned, that we know nothing of the offences committed against this
mighty chief, to excite such blood-thirsty indignation.
* Sam. viii. 1., Chron. xviii. 1.
** 2 Sam. viii. 2.
*** Ver. 3. 5., 1 Chron. xviii. 3, 5.
Indeed, the cause is, without much difficulty, deducible from the
produce of these wars, which sufficiently indicate the nature of David's
_thirst._ Great quantities of gold, silver, and brass, are said to have
been brought to Jerusalem;* and the priests may with reason be supposed
to be the instigators to these wars; since we find all the plunder
surrendered to them.** We have therefore no cause to wonder at the
exalted praises they have bestowed upon the instrument of their wealth.
He is said to have "gat him a name, when he returned from smiting the
Syrians."*** --This may very easily be credited; but it is to be feared,
that if the name he gat from the Jews, and that which he gat from the
Syrians were compared, they would not accord extremely well together.
David was at this time seized with a _temporary_ fit of gratitude toward
a lame son of his old friend Jonathan, named Mephibosheth, to whom he
restored all the private patrimony of his grandfather Saul, and took
him into his family;**** not without due consideration, it is to be
supposed; since by that means he kept him under his own eye. But this
gratitude, was not lasting; for upon an accusation preferred against him
by his servant, David readily bestowed all Mephibosheth's possessions
upon that servant;(5) yet, when the accusation was found to be
false, instead of equitably punishing the asperser of innocence, and
reinstating Mephibosheth in his former favour, he restored to him but
half the forfeiture of his supposed guilt,(6) leaving the villain
Ziba in the quiet possession of the other half, as the reward of his
treachery.--But of this in its proper place.
The next memorable act recorded of David, is the only acknowledged crime
that he ever committed; all his other transactions being reputed "right
in the eyes of the Lord."(7)
* 2 Sam. viii. 7, 8, 10., 1 Chron. xviii, 2, 4, 8, 10.
** 2 Sam. viii. 11., 1 Chron. xviii. 11.
*** 2 Sam. viii. 13.
**** Chap. ix. 1.
(5) Chap. xvi. 4.
(6) Chap. xix. 29.
(7) 1 Kings xv. 5, compared with 1 Chron. xxi. 1.
In the midst of an obscure detail of smiting and slaying; in revenge
for the contemptuous treatment of some ambassadors, sent by him with
compliments of condolence; but who, perhaps deservedly, were considered
as spies; while Joab was with the army prosecuting the siege of Rabbah,
a chief city of the Ammonites; David, then at Jerusalem, walking one
evening on the roof of his palace, perceived from that eminence a
handsome woman bathing herself.* Fired with the sight, he sent to
enquire who she was: and understanding she was Bathsheba, wife to Uriah,
who was at that time opportunely absent in the army under Joab, he
caused her to be brought to him directly, (no ceremony in the case)
and after gratifying his inclination, sent her home again.** Some time
after, the woman finding herself with child, naturally informed the
king of it. He, never at a loss for ways and means, immediately ordered
Uriah home;*** of whom he enquired news concerning the operations of the
campaign, and then dismissed him to his own house, sending after him a
present of victuals.**** David intended the good man a little relaxation
from the fatigues of war, that he might kiss his wife, and be cheated
into a child more than he had a natural right to; but whether Uriah
had received any intimation of the honour his Majesty had done him; or
whether he honestly meant the self-denial which he professed, we are
not advertised: however, Uriah would not go home but slept in the
guard-room, with the king's servants.(5) David took care to be informed
of this, and questioned Uriah concerning the reason of it. Uriah urged
a scruple of conscience against going to enjoy any indulgence at
home, while the ark, Joab, and the army remained in tents in the open
field.(6) He was detained another night; when David made him drunk,(7)
waiting to see what effect that might have. It was still the same;
Uriah, like many other drunken men, was resolved not to go home.
*2 Sam. xi. 2.
** Ver. 4.
*** Ver. 6.
**** Ver. 8.
(5) Ver. 9.
(6) Ver. 11.
(7) Ver. 35.
David, finding him so obstinate, altered his plan of operations, and
determined then to get rid of him for ever. To which intent, he sent
him back to the camp, with a letter to the general. "And he wrote in the
letter, saying, Set ye Uriah in the fore-front of the hottest battle,
and retire ye from him, that he may be smitten and die."* This
instruction was accordingly complied with;** and then Bathsheba, like
another Abigail, was taken into David's seraglio.***
Nathan the prophet read David an arch lecture upon this subject;**** and
he, who took care not to disagree With his best friends, bore with the
reproof, and humbled himself accordingly.
This complicated crime committed by David is universally allowed;
but people think so little for them selves, that even _this_ would
be qualified, were it not found ready condemned to their hand in the
relation of it. This crime is given up too, as the _only stain_ in
David's character: but the circumstances of it will not permit this to
be granted, abstracted from any consideration of the man. For, though a
generally good man may, in a sudden start of any of the passions,
lose government of himself so far, as to violate conjugal fidelity, or
perhaps suddenly to kill another; yet a deliberate scheme, including
_two_ such crimes, can be concerted only by a _bad heart_. It is also to
be remarked respecting his famous repentance of this black transaction,
that he shewed no tokens of relenting until it was extorted from him
by artifice! and that even then, though he mourned his crime, he never
entertained a thought of relinquishing future commerce with the woman so
wickedly obtained, but kept her until he died! and altered the regular
course of succession, in favour of a son he had by her.(5)
It is hoped the supposition may be allowed, that the noise this
righteous affair made, might be one motive for Joab's desiring David to
come and partake some of the honours of the campaign:(6) an opportunity
of which he prudently laid hold: but--fatal was his presence wherever he
appeared.
* 2 Sam. xi. 15.
** Ver. 17.
*** Ver. 27.
**** Ch. xii. 1.
(5) Kings i. 13.
(6) 2 Sam. xii. 27, 28.
How shall a person subject to the feelings of humanity, (a security of
more avail among men than the most binding laws) how shall a man, not
steeled to a very Jew, find expressions suited to the occasion, when he
relates the treatment of this poor city, Rabbah? The study would be as
difficult as unnecessary; the simple unexaggerated tale, if seriously
attended to, will shock the humane reader sufficiently. The city was
taken and plundered; and David "brought forth the people that were
therein, and put them under saws, and under harrows of iron, and under
axes of iron, and made them pass through the brick-kiln, and thus did he
unto all the cities of the children of Ammon."**
* It is supposed that the ancient slavery of the Jews to the
Egyptians, and the labour they were employed in by their
lordly taskmasters, the making bricks, might be a current
reproachful jeer upon the Jews, when any quarrel happened
between them find their neighbours; and that the making
their prisoners pass through the brick-kiln, was a cruel
method of revenging such affronts. A conjecture not
improbable.
** 2 Sam. xii. 31., 1 Chron. xx. 3.
The precise punishments here alluded to are not understood at this time:
writers being much divided in their expositions of these words; but that
extraordinary punishments are meant, cannot admit of a doubt; for though
_believers_ expound the putting the Ammonites _under_ saws and harrows,
into the making slaves of them, and that these were the tools with which
they laboured; yet this will not agree with the latter of the texts
whose authority is mentioned in the note; where it is said, that he
[David] "cut them with saws and with harrows of iron, and with axes."
And should more evidence be yet required, Josephus also writes, that
"the men were put to death by exquisite torments." The general truth of
the fact stands therefore unimpeached. And is it thus the people of God,
headed by a man styled, in a peculiar manner, _the man after God's own
heart_, used the prisoners of war? _Bella! horrida bella!_
It would not be easy to select any period of any history more bloody,
or abounding more in wickedness of various dyes than that which is the
object of the reader's present attention. Instances succeed so quick
that the relation of one is scarcely concluded, but fresh ones obtrude
upon notice.--But now horrors of a different hue demand our attention.
Ammon, one of our hero's sons, ravished his sister Tamar, and then
turned her out of doors.* Absalom, her brother by the same mother,
seemingly took no notice of it, until two years after; when he invited
all his brothers to a feast at his sheep-shearing; where he made Amnon
drunk, and murdered him** in so deliberate, and yet so determined was
his revenge! Absalom on this account, fled out of Judea, for three
years*** until, at the entreaty of Joab, he was invited home again
by his father, whose favourite he was.**** But though he returned to
Jerusalem, yet would not his father see him for two years more.(5)
Absalom, during his exile, conceived a design of deposing his father;
for after their reconciliation, his first attention was to render
himself popular. To this end he set up a splendid equipage:(6) but
politically increased his affability with his magnificence: rising up
early, and planting himself in the way, to salute all who came to
his father's levee. Of these he kindly enquired their business, or
grievances; throwing out hints of the king's remissness in the execution
of justice, and how uprightly he would conduct himself, were their
causes to be determined by him.(7)
* 2 Sam. xiii. 14
** Ver. 28.
*** Ver. 88.
**** Chap. xiv. 21, 24.
(5) Ver. 28.
(6) Chap. xv. 1.
(7) Ver 2, 4.
The profession of piety is universally, and was in particular a-<DW41>
this people, the most successful disguise for crafty designing men to
assume. When Absalom, therefore, thought his scheme sufficiently ripe
for execution, he desired leave of his father to go to Hebron, to
perform a vow made by him while a refugee in Syria.* At Hebron he set
up his standard, and his followers assembled in such numbers, and the
defection was so general, that David thought it adviseable to retire
from Jerusalem.**
With him he took all his family and dependants, except ten concubines,
whom he left in his palace to keep house.*** The priests, Zadock and
Abiather, with the ark, would also have gone with him; but he thought it
would be more for his service for them to remain in the city as spies;
to send him intelligence how matters went.**** It is no inconsiderable
part of politics to know how to suit men with proper employments,
Ahitophel, his prime minister, joined the malecontents;(5)
to balance which misfortune, David prevailed on Hushai, a trusty man of
some importance, to remain in the city, that he might ingratiate
himself with Absalom, thwart the counsels of Ahitophel, and transmit
intelligence to him from time to time through the conveyance of the
priests, whose sons were to carry on the correspondence.(6) Having
concerted matters thus, he evacuated Jerusalem, and Absalom entered(7)
it.
When David was upon his journey from the city, he was met by Ziba,
servant to Mephibosheth, with asses and provisions for his majesty's
accommodation in his retreat:(8) of whom, when David enquired why
Mephibosheth did not come with him; this treacherous servant told him
that he staid behind at Jerusalem, hoping to obtain the kingdom of his
grandfather, during this disturbance:(9) by which lying aspersion, he
gained a grant of all his master's possessions.
* 2 Sam. xv. 7.
** Ver. 12,14.
*** Ver. 16.
**** Ver. 27,28,
(5) Ver. 12, 31.
(6) Ver. 32, &c.
(7) Ver. 37,
(8) 2 Sam. xvi. 1.
(9) Ver. 3.
Here we may introduce a circumstance, which is so far material, as it
serves to shew, that the sanctity of David was not quite so universally
assented to, as may be imagined, while he was living; and his actions
not only fresh in memory, but more perfectly known, than possibly, was
prudent to transmit to these distant ages.
As David prosecuted his flight, he was met by a man of Saul's family,
whose name was Shimei. This man as he came on, kept muttering curses
between his teeth, and at length cast stones at the King and his
attendants, calling out to him, "Come out, come out, thou bloody man,
and thou man of Belial; the Lord hath returned upon thee all the blood
of the house of Saul, in whose stead thou hast reigned, and the Lord
hath delivered the kingdom into the hand of Absalom thy son; and behold
thou art taken in thy mischief because thou art a bloody man."* This is
pathetic, and truly characteristic of the person to whom the speech
was addressed. Some of his retinue were at the point of silencing; this
brawler with the "ultima ratio regum;"** but David prevented
it,*** wisely considering this not to be a season for proceeding to
extremities.
Absalom, in the mean time, being come to Jerusalem, like a buck of
spirit, took the damsels which his father had left to keep house, and
cuckolded the old man by way of bravado, on the top of it****; in a tent
erected for this heroic purpose!
Ahitophel advised Absalom to select twelve thousand men, and pursue
David directly, before he had time to recover his surprize;(5) which was
certainly the best resolution that could have been formed. But Hushai,
as was concerted, proposed a different plan of operations; opposing to
the former, the well known valour and military skill of the old king;
and the hazard of making him and his men desperate.(6)
* 2 Sam. xvi 7, 8.
** The motto on French cannon.
*** 2 Sam. xvi. 9, 10.
**** Ver. 21, 22.
(5) Chap. xvii. 1.
(6) Ver. 8.
He advised a collection of all the troops in the kingdom; that success
might be in a manner insured; and that Absalom should command them in
person. By which means, he affirmed that they should overwhelm David and
his party, wherever they found him.* Hushai gained the ascendancy; and
when he knew that his scheme was accepted, he gave immediate notice to
the priests:** with instructions for David how to conduct himself.***
David divided his forces into three bodies; commanded by Joab, Abishai,
and Ittai: but by the prudent care of his men, was not permitted to
hazard his person, by being present in action.**** When he had reviewed
his army, he gave his generals especial charge to preserve the life
of Absalom; and with a policy that reflects honour upon his military
knowledge, expected the enemy in the wood Ephraim:(5) a covert
situation, being the most judicious that could be chosen, for a small
army(6) to encounter one more numerous. David's men were tried veterans,
among whom were the remains of those who served under, and lived with
him at Gath;(7) whereas, Absalom's army must have consisted chiefly of
fresh men. The battle decided in favour of David(8) with great slaughter
of the rebel army: and as Absalom fled on a mule, his hair, which is
celebrated for its beauty and quantity, entangled in the boughs of an
oak, and he remained suspended in the air; while his mule ran away from
between his legs.(9) He was observed in this condition by a man who went
and told Joab; and he, who consulted the safety of David, rather than
his parental weakness in behalf of an unnatural son, killed Absalom with
a dart.(10)
David grieved immoderately for this reprobate son, on whom he had
misplaced a great affection:(11) and though he had _acted_ the mourner
on several former occasions, this is the only one, in which his
sincerity need not be questioned.
* 2 Sam. xiii. 11.
** Ver. 15.
*** Ver. 16.
**** Chap. xviii. 1-3
(5) Ver. 4-6.
(6) According to Josephus, David had but four thousand men.
(7) 2 Sam. xv. 18.
(8) Chap. xviii. 7.
(9) Ver. 9.
(10) Ver. 14.
(11) Ver. 33., Chap. xix. 4.
It is true, he might be really concerned at the murder of Abner; but men
circumstances ought to be attended to; Abner was killed prematurely;
he had not finished his treacherous negociation; David had much to hope
from him; but--when his expectations had been answered, it is far from
being improbable, that he would have found an opportunity himself to
have got rid of a man, on whom he could have placed no reliance. But to
return.
David was roused from his lamentations by the reproaches of his
victorious general,* who flushed with success, told him the truth, but,
perhaps, told it too coarsely. It is evident that Joab now lost the
favour of his master, which the murder of Abner, the killing Absalom in
direct contradiction to David's express order; and lastly, his want of
sympathy, and his indelicacy in the present instance, were the apparent
causes.
After the battle, he invited Amasa, Absalom's vanquished general, to
return to his duty: very imprudently and unaccountably promising him the
chief command of his army in the stead of Joab;** which was seemingly
but an unthankful return for the victory that officer had just gained
him, and for his attachment to his interest all along. Amasa, it is
true, was a near relation; but Joab, according to Josephus, stood in
the same degree of consanguinity; they being both the sons of David's
sisters, this offer must therefore have been rashly influenced by his
resentment against Joab, as before mentioned.
The remains of Absalom's scattered army dispersed to their homes in
the best and most private manner they could:*** but David inadvertantly
plunged himself into fresh troubles, by causing himself to be conducted
home by a detachment from the tribe of Judah.**** This occasioned
disputes between that and the other tribes. They accused Judah of
stealing their king from them.(5)
* 2 Sam. xix. 5-7.
** Ver. 13.
*** Ver. 3,
**** Ver. 11, 15.
(5) Ver. 41.
Judah replied, that they gave their attendance, because the king was of
their tribe; and that it was their own free will:* the others rejoined
that they had ten parts in the king, and that their advice should have
been asked as to the bringing him back.** At this juncture, one Sheba
took advantage of the discontent, "and blew a trumpet, and said, we have
no part in David, neither have we inheritance in the son of Jesse: every
man to his tent, O Israel."*** The consequence of this, was a second
insurrection. Amasa was ordered to assemble an army to suppress it;
but not proceeding with the desired speed, Abishai was afterward
commissioned with the same trust; Amasa and Abishai met and proceeded
together, and were joined by Joab and his men. But Joab, not thoroughly
liking to serve under a man he had so lately vanquished, and having as
few scruples of conscience as his old master, made short work, stabbed
Amasa, and reassumed the command of the whole army.****
Being once again supreme in command, Joab proceeded directly to the
reduction of the malecontents who shut themselves up in the city of Abel
of Beth-maacha: he battered the town, but by the negociation of a woman,
the inhabitants agreed to throw Sheba's head to him over the wall; which
they performed;(5) and thus was quiet once more restored. Joab returned
to Jerusalem, where we are told that he was general over all the host
of Israel.(6) Not a syllable appears of any notice taken by David of
the murder of the general by himself appointed: and of the assassin's
usurping the command of the army.
Not finding room in its proper place, it shall now be noticed, that
when David was returning to Jerusalem from the reduction of Absalom's
rebellion; with the men of Judah, who came to escort him, Shimei, the
Benjamite,(7) joined him at the head of a party of his own tribe.
* 2 Sam. xix. 42.
** Ver. 43.
*** Chap. xx. 1.
**** Ver. 7, 9.
(5) Ver. 15, 16, &c.
(6) Ver. 23.
(7) Ver. 16.
This man, who at a former meeting, so freely bestowed his maledictions
on David when a fugitive: upon this change of circumstances, reflecting
on the king's vindictive temper, came now to make his submission: David
accepted his acknowledgements, and confirmed his pardon with an oath.*
We shall have occasion to refer to this passage anon.
Mephibosheth came also to welcome David on his return, and undeceive him
with regard to the false Ziba's representation of him;--but he appears
to have met with no other redress, than a remittance of _half the grant_
made to Ziba of his estate.**
* 2 Sam. xix. 28.
** Ver. 29.
These intestine troubles put David upon pondering how to secure himself,
as far as he could forecast, from any future disturbance.
It is the part of good politicians, not only to form wise designs
themselves, but also to make proper advantage of public occurrences,
that all events indiscriminately may, more or less, lead to the purposes
wanted to be obtained. Of this policy we shall observe David to be
mindful, in the ensuing transaction. Not that a panegyric upon his
contrivance in this instance is by any means intended; for certainly
a more barefaced transaction was never exhibited: such indeed as could
only have been attempted among the poor bigoted Jews. It is sufficient,
however, that it answered David's purpose; than which more could not
have been expected from the most complete stroke that refined politics
ever produced. But view it in a moral light, and certainly a blacker
piece of ingratitude and perfidy can hardly be imagined. It was
impossible to continue the narrative without prefacing thus much.
David having with much trouble, from his competition with Ish-bosheth,
established himself upon the Jewish throne; and having in the latter
part of his reign been vexed, and driven to disagreeable extremities,
by the seditious humour of his subjects, the rebellion of his own son
Absalom, and the revolt of Sheba; his mind now fell a prey to suspicion.
He called to remembrance that some of Saul's family were yet living;
whom, lest they should hereafter prove thorns in his side, he concluded
it expedient to cut of.
Whenever David projected any scheme, a religious plea, and the
assistance of his old friends,* were never wanting. A famine befel
Judea, which continued three years: probably occasioned by the preceding
intestine commotions. "David inquired of the Lord: and the Lord
answered, it is for Saul, and for his bloody house, because he slew the
Gibeonites."** But where is this crime recorded? Samuel charged Saul
with no such slaughter: he reproached him with a contrary fault, an act
of _mercy!_ which is assigned as one of the reasons for deposing him.
So that this crime was not recollected,*** till many years after the man
was dead! and then God punishes--whom? a whole nation, with three years
famine: which, by the by, was not sent as a punishment neither; but
merely as a hint of remembrance, which ended in hanging the late king's
innocent children!
The oracular response dictated no act of expiation; but only pointed out
the _cause_ of the famine. So that the Gibeonites (who, by the way, had
hitherto made no complaints that we know of) were applied to**** for a
knowledge of what recompence they demanded.
* The prophets and priests.
** 2 Sam. xxi. 1.
*** If God sought vengeance for a particular act of cruelty
perpetrated by Saul: when was vengeance demanded for David's
massacre of the Edomites, the Moabites, the Ammonites, the
Jebusites, and others, who at times became the object of
David's wrath? That the charge may allude to some former
affair, is not contested; it is, however, truly remarkable,
that there should be no chronological record of a fact,
which after such a length of time demanded an expiation so
awfully hinted, and so extraordinary in its circumstances!
**** 2 Sam. xxi. 2, 3.
They required no gifts, neither that for their sakes David should kill
any man in Israel (which qualifying expressions seems artfully intended;
since they only required David to _deliver_ the men to _them_, that
_they_ might kill them); but that seven of Saul's sons, should be
surrendered to them, that they might hang them up--_unto the Lord_.*
David, not withheld by any motives of gratitude toward the posterity of
his unhappy father-in-law, but in direct violation of his oath at
the cave of En-gedi,** granted the request he must himself have
instigated,*** sparing only Mephibosheth, who luckily was so unfortunate
as to be a <DW36>, and so much a dependant on David, and kept under
his own eye, that he had no room for apprehension from him. He therefore
reserved Mephibosheth, in memory of another oath between him and
his father Jonathan. Mephibosheth having such a shocking scene to
contemplate, and, considering his decrepitude, might (as he really was)
with little hazard be preserved, as an evidence of probity in this pious
king.
A conscience of convenient flexibility is of great use: thus David being
under obligation by two oaths, forgot one, and remembered the other.
When Creon, in OEdipus, was interrogated concerning his conscience, he
replied--
--"'Tis my slave, my drudge, my supple glove,
My upper garment, to put on, throw off,
As I think best: 'tis my obedient conscience."
David, now thinking himself securely settled, was moved both by God****
and by Satan,(5) to cause his subjects to be numbered: which is,
oddly enough, imputed as a great sin in him to require: for, poor
man, according to the premises, he was but a passive instrument in the
affair.
* 2 Sam. 6.
** 1 Sam. xxiv. 21, 22.
*** 2 Sam. xxi. 6.
**** Chap. xxiv. 1.
(5) 1 Chron. xxi. l.
Even David should have his due. The prophet Gad called him to account
for it; and as a punishment for this sin of compulsion, propounded to
him for his choice three kinds of plagues, one of which _his subjects_
thereby necessarily incurred seven years famine, three months
persecution from enemies, or three days pestilence.* David chose the
latter.
It may be as well to decline this story, as to enter into, any more
particular consideration of it. From the above state of the case, the
intelligent reader will need no assistance in making his own private
reflections on it.
We have now attended David down to the decline of his life: when his
natural heat so far decayed, that no addition of clothing** could retain
a proper degree of warmth. His physicians prescribed a young woman to
cherish him in his bed, by imparting to him a share of juvenile heat.***
This remedy may be very expedient in cases of extreme age: but why
beauty should be a necessary part of the prescription is difficult to
conceive. They sought a _fair damsel_; and the damsel they found, was
_very fair._**** Possibly David might himself direct the delicacy of
the choice: but if his physicians intended it as a compliment to
their master, it indicated a very insufficient knowledge of the animal
oeconomy: thus to stimulate the old man, and harass a carcase already
sufficiently worn out: whereas a virgin of homelier features, at the
same time that she would have furnished an equal degree of warmth, would
have been less liable to put wicked thoughts in the patient's head.(5)
However, the historian has taken care to inform us, that "the king knew
her not:"(6) an assertion, which, from the premises, there does not
appear any reason to controvert.
* 2 Sam. xxiv. 13., 1 Chron. xxi. 12.
** 1 Kings, i. 1.
*** Ver. 2.
**** Ver. 8, 4.
(5) "Boerhaave frequently told his pupils that an old German
prince, in a very infirm state of health, being advised to
lie between two young virtuous virgins, grew so healthy and
strong, that his physicians found it necessary to remove his
companions." Mackenzie on Health, p. 70, Notes.
(6) l Kings, i. 4.
While the king lay in this debilitated extremity of life, he was
destined to experience yet another mortification from his children.
Adonijah his eldest son, since the death of Absalom, taking advantage
of his father's incapacity, foolishly assumed the title of king,* which,
had he been a little less precipitate, would have soon fallen to him,
perhaps, without contest. For though David afterwards is represented as
having secret intentions to alter the succession, yet the countenance
shewn to his pretension by Joab, the general, by Abiathar the priest,
and even by all his other brothers,** seem to indicate, that had
Adonijah been more prudent, we should not now have heard so much of the
wisdom of Solomon, It is possible Adonijah might, even as it was,
have maintained his anticipated dignity, had he not, like Saul before,
slighted his most powerful friends. He made an entertainment, to which
he invited all his brothers, except Solomon;*** but what ruined him, was
his not inviting Nathan the prophet; it was _there_ the grudge began:
and the exclusion from this merry bout, and the confidence of the party,
caused the prophet's loyalty to exert itself,**** which might probably
have been suppressed by a due share of Adonijah's good cheer.
* Ver. 5.
** Ver. 9, 19, 25.
*** Ver. 9, 10,
**** Ver. 11.
Let not the writer be accused of putting a malicious construction upon
every transaction he produces. Pray, reader, turn to your bible: in the
tenth verse of the first chapter of the first book of Kings, you will
find a remark that Nathan was not called to the feast. The very next
verse begins, "Wherefore, Nathan spake unto Bathsheba, the mother of
Solomon," &c. He was certainly nettled at the slight put on him, and
some others, in not being invited to Adonijah's feast, else he would
not have insisted on that circumstance; which had better been waved. The
supposition is not so ridiculous as has been represented; for surely
the probability of Nathan's being corrupted, was not less than that of
David's sons; who, yet, all of them, except Solomon, (who, had he
been invited, had some private reasons to the contrary, which their
proceedings shew them to have been aware of) were agreeable to settling
the succession on their elder brother; though certainly as much
interested in the disposal of the kingdom, as Nathan could be.
Nathan and Bathsheba concerted to inform David of this matter;* where
the affronted prophet could not forget the slight put upon him; but,
it being foremost in his mind, he insists upon the circumstance of
exclusion, in an earnest manner; "But me, even me, thy servant, and
Zadok the priest, and Benaiah the son of Jehoiada, and thy servant
Solomon, hath he not called;"** which spake the cause of his officious
loyalty but too plainly. David here acknowledges the promise by which
he waved the right of primogeniture in favour of Solomon, Bathsheba's
son.*** He now directed him to be set upon a mule, to be proclaimed and
anointed king of Israel, by his appointment.**** The acclamations of the
people upon this raree-shew disturbed the opposite party at their
table; and an event, so unexpected, quite disconcerted them: they all
dispersed;(5) Adonijah ran to the tabernacle, and took sanctuary at
the altar. He obtained of Solomon a conditional promise of pardon,(6)
depending on his good behaviour.(7)
* 1 Kings i. 13.
** Ver. 26.
*** Ver. 30.
**** Ver. 33, 38.
(5) Ver. 41, 49, 50.
(6) Ver. 52.
(7) Solomon soon found a pretence, ridiculous enough, but
sufficient in his eyes, to get rid of Adonijah, when his
father was dead.
And now, methinks, some gentlewoman, of more than feminine patience,
whose curiosity may have prevailed with her to proceed thus for, may
here exclaim; "It must be granted, Sir! that David had his faults; and
who has not? but what does that prove? only that he was a man. If he
was frail, his repentance was exemplary; as you may perceive, if you
can prevail with yourself to read some of his psalms. Indeed, after your
ill-treatment of the scripture, it will avail little to tell you that
you contradict those inspired penmen, who expressly stile David, _the
Man after God's own heart_. Nay, your writing against him, under that
epithet, shews sufficiently the rancour and impiety of _your heart_; so
that I am fearful there are small hopes of reclaiming you."--Good Madam!
hear me calmly, and we shall part excellent friends yet. Had David not
been selected from the rest of mankind, why then--it is possible--hardly
possible--he might pass in the gross, with the rest of the Jewish kings.
But, when he is exalted and placed in a conspicuous point of view, as an
eminent example of piety! he then necessarily attracts our notice in an
especial manner, and we are naturally led to wonder, that a more happy
subject of panegyric had not been chosen. If he was an holy psalmist; if
he is styled the Man after God's own heart; he also lived the life here
exhibited: and his capability of uniting such contrarities, does but
augment his guilt!
Yet, even in his psalms, he frequently breathes nothing but blood,
and the most rancorous resentment against his enemies. Of these take
a specimen or two, from the elegant _ekeings_ out of that transcendent
pair of geniuses, Messrs. Thomas Sternhold and John Hopkins; in
recommendation of whose version, and the taste of our countrymen, it
may be truly affirmed, that their psalms have gone through more editions
than the works of any other poet, or brace of poets, whatever.
Psalm lxviii. 22-24.
And he shall wound the head of all
His enemies also,
The hairy scalp of such as on
In wickedness do go.
From Basan 1 will bring, said he,
My people and my sheep,
And all my own, as I have done.
From dangers of the deep.
And make them dip their feet in blood
Of those that hate my name;
The tongues of dogs they shall be red
With licking of the same.
Again, in Psalm lxix. 24--27.
Lord, turn their table to a snare,
To take themselves therein,
And when they think full well to fare,
Then rap them in their gin:
And let their eyes be dark and blind,
That they may nothing see;
Bow down their backs, and let them find
Themselves in thrall to be:
Pour out thy wrath as hot as fire,
That it on them may fall,
Let thy displeasure in thine ire
Take hold upon them all.
As desarts dry their house disgrace,
Their seed do thou expel,
That none thereof possess their place,
Nor in their tents once dwell.
Very pious ejaculations for the whole congregation to _sing to the
praise and glory of God!_
David's failings, as they are qualifyingly termed, are generally
mentioned as exceptions to the uniform piety of his character: but, if
David ever performed any truly laudable actions, _those_ are the real
exceptions to the general baseness which stains the whole of a life
uncommonly criminal.
The writer does not pledge himself to reconcile rapine and cruelty,
with morality and religion; there are Commentators who love these knotty
affairs; to them they are left. When the vindictive tenor of any of
David's psalms has been insisted on, the translation is immediately
censured; prudently enough; as every one who has sense to perceive
the incongruity between such bloody wishes and denunciations, and the
acknowledged purity and mercy of the All-beneficent Father of Nature,
may not have learning enough to dispute about Hebrew points, and to make
them point what meaning he pleases. However, such a one, by comparing
the labours of Hebrew critics, may yet be enabled to form some sort of
judgment between them. For instance, in that terrible 109th psalm, it
is certain our Doctors in Divinity do not like it: but something must be
done with it: some, therefore, say, that the verbs are not translated in
their proper tenses, and that prophetic declarations are thus mistaken
for the Psalmist's execrations: others again say, that to be sure they
_are_ imprecations, but not the imprecations of David; but those of his
enemies on him, which he there only relates! O happy men! why do not we
all learn Hebrew? His exemplary repentance is pleaded; is it any where
to be found but in the psalms? "By their fruits ye shall know them."
If David was ever truly pious, we shall certainly perceive it in his
behaviour on his death-bed. _There_, it is to be hoped, we shall find
him forgiving his enemies, and dying in charity with all mankind. This
is what all mankind in general make a point of, from the saint to the
malefactor. David, therefore, must certainly give us an extraordinary
instance of his attention to this important evidence of contrition,
But what shall we think, when we see this Nero of the Hebrews die in a
manner uniform and consistent with the whole course of his life? What
will be our reflections, when we find him, with his last accents,
delivering two cruel and inhuman murders in charge to his son Solomon?
Murders still further aggravated by the included crimes of ingratitude
and perjury! one of them to be executed on his old faithful general,
Joab, who powerfully assisted him on all occasions, and who adhered to
him in all his extremities, till at the last, when he had justifiable
cause for chagrin: but who, notwithstanding, had not appeared against
him in actual hostility; but only drank a glass of wine with the
malcontents. It will avail nothing to plead the private faults of the
man; we are now to consider him as relative to David, in his public
capacity. In which light we must loath the master, who died meditating
black ingratitude against so faithful, so useful a servant. For even
his defection at last may, perhaps, admit of being interpreted into a
patronization of that particular plan for the succession, rather than
into a rebellion against the superannuated monarch.
His other charge was against Shimei, who reviled David at his retreat
from Jerusalem, during Absalom's rebellion; but who made his submission
to him, when he returned victorious: and whose pardon David had sealed
with a solemn oath.*
Attend we now to the cause of these reflections. After exhorting Solomon
on his death-bed, to keep the statutes of the Lord, David proceeds:
"Moreover, thou knowest also what Joab, the son of Zeruiah, did to me,
and what he did to the two captains of the hosts of Israel, unto Abner
the son of Ner, and unto Amasa the son of Jether, whom he slew, and shed
the blood of war in peace, and put the blood of war upon his girdle that
was about his loins, and in his shoes that were on his feet."
"Do therefore according to thy wisdom, and let NOT HIS HOAR HEAD GO DOWN
TO THE GRAVE IN PEACE."**
This was afterwards fulfilled in the basest manner, by the administrator
to this pious testament.
David concludes thus:
"And behold, thou hast with thee Shimei, the son of Gera, a Benjaminite
of Bahurim, which cursed me with a grievous curse, in the day when I
went to Mahanaim; but he came down to meet me at Jordan, and I sware to
him by the Lord, saying, I will not put thee to death with the sword:
"Now, therefore, hold him not guiltless: for thou art a wise man, and
knowest what thou oughtest to do unto him; but his hoar head bring
thou down to the grave with BLOOD."*** --That is to say, 'It is true, I
promised not to put him to death, but thou art a wise man, and knowest
what thou oughtest to do; thou knowest thyself not to be bound by that
obligation; therefore his hoar head, &c. So saying, he expired!
* 2 Sam. xix. 23.
** 1 Kings ii. 5, 6.
*** Ver. 8, 9.
This command was also executed in a manner, worthy of a son of SUCH A
FATHER.
To take a retrospect view of the foregoing narrative; in few words may
be seen the sum total of the whole. A shepherd: youth is chosen by
a disgusted, prophet, to be the instrument of his revenge on an
untractable king. To this, end he is inspired with ambitious hopes, by
a private inauguration; is introduced to court, in the capacity of a
harper; and by knocking down a man with a stone whom, if he had missed
once, he had four more chances of hitting, and from whom, at the last,
he could have, easily ran away; he was advanced to the dignity of
son-in-law to the king. So sudden and unlooked for a promotion within
sight of the throne, stimulated expectations already awakened; and Saul
soon perceived reasons to repent his alliance with him. Being obliged to
retire from the court, he assembled a gang of ruffians, the acknowledged
outcasts of their country, and became the ringleader of a lawless
company of banditti. In this capacity he seduces his brother-in-law,
Jonathan, from his allegieance and filieal duty; and covenants with him,
that if he obtained the kingdom, Jonathan should be the next person in
authority under him.
He obtains a settlement in the dominions of a Philistine prince where
instead of applying himself laudably to the arts of cultivation he
subsists by plundering and butchering the neighbouring nations.
He offered his assistance to the Philistine armies, in a war against his
own country, and father-in-law; and is much disgusted at their distrust
of his sincerity. He however, availed himself of the defeat and death of
Saul, and made a push for the kingdom.
Of this he gained only his own tribe of Judah: but strengthened by this
usurpation, he contested the remainder with Saul's son, Ishbosbeth,
whom he persecuted to the grave: Ishbosbeth being assassinated by two
villains, with intention to pay their court to the usurper. He is now
king of Israel: In which capacity he plundered and massacred all his
neighbours round him at discretion. He defiled, the wife of one of his
officers, while her husband was absent in the army: and finding she was
with child by him, He, to prevent a discovery, added murder to
adultery; which being accomplished, he took the widow directly into his
well-stocked seraglio. He then repaired to the army, where he treated
the subjected enemies: with the most wanton inhumanity. A rebellion is
raised against him by his son Absalom, which he suppressed, and invited
over the rebel-general, to whom he gave the supreme command of his army,
to the prejudice of the victorious Joab. After this, he cut off the
remainder of Saul's family, in defiance to the solemn oath by which he
engaged to spare that unhappy race; reserving only one <DW36> from whom
he had not apprehensions: and who, being the son of Jonathan, gave him
the opportunity of making a merit of his gratitude.
When he lay on his death-bed, where all mankind resign their resentments
and animosities, his latest breath was employed in dictating two
posthumous murders to his son Solomon! and, as if one crime more was
wanting to complete the black catalogue; he cloathed all his actions
with the most consummate hypocrisy: professing all along the greatest
regard for every appearance of virtue and holiness. These, Christians!
are the outlines of the life of a Jew, whom you are not ashamed to
continue extolling as a man after God's own heart!
This Britons! is the king to whom your late excellent monarch* has been
compared!
* George II.
What an impiety to the Majesty of Heaven!
What an affront to the memory of an honest prince! It is with great joy
the writer of these memoirs takes his leave of a story, with which, by
this time he is sufficiently disgusted. He entered upon it, however,
from honest motives; and he concludes it with the consciousness of having
performed a work, which he flatters himself will prove acceptable to
all who entertain adequate conceptions of the eternal rectitude of that
great Creator of the universe, whom they profess to adore. He despises
all the pious ravings and anathemas which have been thundered against
him by some reverend inquisitors: he expected them, has exposed them;
and hopes he may, without offence finally reply in the words of their
forgotten master, "Father forgive them, for they _know not_ what
they do." Those who estimate a man's religion by his implicitness to
prescribed notions, and who think it their duty to stifle their living
objections in compliance to the dead letter; (for objections they will
have, and very strong ones too) such have, and will undoubtedly
be shocked at this publication. Such may produce numerous texts in
opposition to what is here produced; and can inspired writers be
inconsistent with themselves? It is not at present necessary to discuss
that question. Argue that point among yourselves; the printer will at
least profit by your disputes; though you may happen to
----Explain a thing till all men doubt it.
And write about the subject, and about it:
So spins the silk-worm small its slender store,
And labours till it clouds itself all o'er.
This, yet, is none of his concern. The love of truth is a motive
which ought to supersede every other consideration: for every other
consideration is subordinate in comparison with it. Truth requires no
tenderness of investigation, and scorns all subterfuges. It is, when
displayed,
----divinely bright.
One dear, unchang'd, and universal light.
To rescue truth, therefore, from obscurity and disguise, is the most
rational way of giving
_Glory to God in the highest; and on earth, peace: good-will toward
men._
End of the Project Gutenberg EBook of The Life of David, by Anonymous
***
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| 513
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Steve Taylor Ph.D.
Steve Taylor is senior lecturer in psychology at Leeds Beckett University, UK. His latest books in the U.S. are Spiritual Science and The Leap. He is also the author of The Fall, Waking From Sleep, and Out of the Darkness. His books have been published in 19 languages. His research has appeared in The Journal of Transpersonal Psychology, The Journal of Consciousness Studies, The Transpersonal Psychology Review, The International Journal of Transpersonal Studies, as well as the popular media in the UK, including on BBC World TV, The Guardian, and The Independent. He is the chair of the Transpersonal Section of the British Psychological Society.
Out of the Darkness
One of Steve's research interests is 'awakening experiences', moments when our normal awareness intensifies and we feel a sense of connection and meaning. What causes these experiences? Is it possible to control them? Steve's work also examines the sources of psychological suffering - why is it that human beings find it so difficult to be contented? His research also shows that many awakening experiences are triggered by intense psychological turmoil, such as depression and loss. www.stevenmtaylor.com
Read now.
Reason for contacting *
Media Interview
Books by Steve Taylor
Spiritual Science: Why Science Needs Spirituality to Make Sense of the World
Out of the Darkness: From Turmoil to Transformation
Waking From Sleep: Why Awakening Experiences Occur and How to Make Them Permanent
The Fall: The Insanity of the Ego in Human History and the Dawning of A New Era
Back To Sanity: Healing the Madness of Our Minds
The Compassion Problem
Why do some religious people fail to live up to religious teachings?
Beyond Religion
Will human beings ever transcend the need for religions?
The Genius of William James
The great American psychologist anticipated many ideas to come.
The Guru Syndrome
When spirituality turns sour
www.stevenmtaylor.com
Friend on Facebook
Subscribe to Out of the Darkness
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Old souls are.. of their very nature.. are hard oft times most difficult to describe or define.. A recluse to some.. is a sage to others.. Where one sees obscurity and aloofness.. Another sees visions and purpose.. Not everything that happens has a meaning for the moment it is in.. Life.. happens.. all throughout ones lifetime.. just because the sun sets in the eve.. does not mean it's forgotten by the midnight hour.. for even then.. it's effects are yet felt.. The morn.. yet a promise.. for the creatures of the night doth prowl.. Behold beyond the yonder sky.. The wolf doth beckon it's howl..
Louis Middleton has no comments to view.
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| 9,554
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DEERING calls for Fairer Top-up Contribution from the Equalisation Fund.
Carlow Fine Gael TD., Pat Deering, has called for a fairer distribution of the Equalisation Fund. He was speaking this week in Dáil Éireann in the Topical Issues Debate which he submitted to highlight the difficulties been experienced by Carlow County Council as they begin the process of agreeing their budget for 2017.
Deputy Deering explained; "80% of all Local Property Tax (LPT) collected is retained locally. The remaining 20% of the tax take goes into what is called the Equalisation Fund. This figure could be as high as €100m in any given year. This fund is then distributed to the smaller counties to ensure that no county council is worse off than they were under the old General Purpose Grant allocation scheme.
"Carlow County Council collects €3.85m LPT of which €3.08, is retained locally under the 80% rule. We receive a further €2.27m from the Equalisation Fund giving us a total of 5.35m. Roscommon and Sligo receive €9.11m and €9.99m respectively while their populations is not much bigger than Carlow's and Leitrim, with a much smaller population receives a total contribution of €8.28m. It appears that these counties for historical reasons have received more from central funding than Carlow while we would traditionally have collected more from commercial rates. I have asked Ministers English and Coveney to look at how this fund is distributed so that everybody would be on a level playing field. I have also called on the Ministers to look at how we fund local authorities in general, a new funding model is required. Carlow County Council have had to reduce their spending by €4m per annum since 2008, this level of cut backs is not sustainable.
"Minister English gave me a commitment that programmes will be tailored to deliver more funding into local authorities. More industry is also needed in Carlow that not only will create jobs but will also generate commercial rates increasing the funding available for the provision of services.
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Eriosema manikense är en ärtväxtart som beskrevs av De Wild. Eriosema manikense ingår i släktet Eriosema och familjen ärtväxter. Inga underarter finns listade i Catalogue of Life.
Källor
Ärtväxter
manikense
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\title{Metastable GeV-scale particles as a solution to the
cosmological lithium problem}
\author{Maxim Pospelov\\Department of Physics and Astronomy, University of Victoria,
Victoria, BC, V8P~1A1, Canada\\Perimeter Institute for Theoretical Physics, Waterloo,
Ontario N2L~2Y5, Canada\\E-mail: \email{pospelov@uvic.ca}}
\author{Josef Pradler\\Perimeter Institute for Theoretical Physics, Waterloo,
Ontario N2L~2Y5, Canada\\E-mail: \email{jpradler@perimeterinstitute.ca}}
\abstract{
The persistent discrepancy between observations of \ensuremath{{}^7\mathrm{Li}}\ with
putative primordial origin and its abundance prediction in Big Bang
Nucleosynthesis (BBN) has become a challenge for the standard
cosmological and astrophysical picture.
We point out that the decay of GeV-scale metastable particles $X$
may significantly reduce the BBN value down to a level at which it
is reconciled with observations.
The most efficient reduction occurs when the decay happens to
charged pions and kaons, followed by their charge exchange reactions
with protons. Similarly, if $X$ decays to muons, secondary electron
antineutrinos produce a similar effect.
We consider the viability of these mechanisms in different classes
of new GeV-scale sectors, and find that several minimal extensions
of the Standard Model with metastable vector and/or scalar particles
are capable of solving the cosmological lithium problem.
Such light states can be a key to the explanation of recent cosmic
ray anomalies and can be searched for in a variety of high-intensity
medium-energy experiments.}
\begin{document}
\section{Introduction}
Rapid progress in observational cosmology during the last decade
brought about the measurements of many cosmological parameters,
including a precise determination of the baryon-to-photon
ratio~$\eta_b$~\cite{Dunkley:2008ie} by the WMAP satellite
experiment. This puts the predictions of the Big Bang Nucleosynthesis
(BBN) theory on firm footing and allows for less ambiguous comparison
with observations. The current status of standard BBN (SBBN) with the
input from WMAP can be summarized as follows: there is no dramatic
(order of magnitudes) disagreement between predictions and
observations, but there is no ideal concordance either. For a series
of recent reviews see {\em e.g.}
\cite{Iocco:2008va,Cyburt:2008kw,Steigman:2007xt,Jedamzik:2009uy}.
There are at least two quantitative problems that look worrisome:
different measurements of the deuterium abundance, although on average
consistent with the SBBN prediction, exhibit a significant scatter.
This scatter may be the sign of underestimated systematic errors or
the manifestation of significant astration, thus hinting on a higher
primordial value for the deuterium abundance. In contrast to
deuterium, the scatter of \ensuremath{{}^7\mathrm{Li}}/H data points along the so-called
Spite plateau \cite{Spite:1982dd} is rather small, which for a long
time thought to be a compelling argument for the primordial origin of
\ensuremath{{}^7\mathrm{Li}}\ in these observations. As is well-known, this value is a factor
of 3-5 lower than the SBBN prediction,
$\ensuremath{{}^7\mathrm{Li}}/\ensuremath{\mathrm{H}}=(5.24^{+0.71}_{-0.67})\times 10^{-10}$~\cite{Cyburt:2008kw},
which is the essence of the lithium problem.
How serious are these problems of SBBN? It is entirely possible at
this point that future highest quality observations of D/H in quasar
absorption clouds would render D/H in accordance with SBBN in
combination with less scatter. Moreover, more elaborate stellar
evolution models with ab-initio calculations of lithium diffusion may
point to a systematic and uniform reduction of the SBBN value by a
factor of 3 or so. At this point, it is too early to declare SBBN
being in serious trouble. However, it is also tempting to speculate
that some subtle particle physics interference in the early Universe
may have resulted in distorted abundances of the primordial elements,
and possibly led to the reduction of lithium abundance.
The primordial value of the lithium abundance is given by the
freeze-out BBN value of \ensuremath{{}^7\mathrm{Be}}+\ensuremath{{}^7\mathrm{Li}}, with atomic \ensuremath{{}^7\mathrm{Be}}\ decaying to
lithium at the later stages of cosmological evolution. The current
lithium problem stems from the overproduction of \ensuremath{{}^7\mathrm{Be}}\ at $T\sim
40$~keV. \ensuremath{{}^7\mathrm{Be}}\ cannot be destroyed by protons directly, but instead
is depleted via the following chain of exoergic reactions
\begin{align}
\label{eq:li-depletion-chain}
\ensuremath{{}^7\mathrm{Be}} + n \to \ensuremath{{}^7\mathrm{Li}} \to \ensuremath{{}^4\mathrm{He}} + \ensuremath{{}^4\mathrm{He}}.
\end{align}
At the second step \ensuremath{{}^7\mathrm{Li}}\ is destroyed by proton reactions, which
remain faster than the Hubble rate until $T\sim 10$ keV.
Different classes of modified BBN models where an additional reduction of
lithium can happen were analyzed in the literature over the years. As
was first pointed out by Reno and Seckel \cite{Reno:1987qw}, a
(moderate) injection of "extra neutrons" around the time of formation
of \ensuremath{{}^7\mathrm{Li}}\ and \ensuremath{{}^7\mathrm{Be}}\ leads to an overall depletion of \ensuremath{{}^7\mathrm{Li}}+\ensuremath{{}^7\mathrm{Be}}\ by
intensifying the destructive chain (\ref{eq:li-depletion-chain}).
This was emphasized again after the CMB determination of $\eta_b$ in
Ref. \cite{Jedamzik:2004er}, where it was demonstrated that any
particle physics source is capable of reducing \ensuremath{{}^7\mathrm{Li}}+\ensuremath{{}^7\mathrm{Be}}\ as long it
leads to the injection of $O(10^{-5})$ neutrons per nucleon at
relevant temperatures. Perhaps the most natural source for a neutron
excess is the decay of massive particle species $X$. Independently
from the motivation of reducing the \ensuremath{{}^7\mathrm{Li}}\ abundance, a lot of work
has been invested into BBN models with decaying or annihilating
particles releasing a significant amount of energy into the primordial
plasma~\cite{Kawasaki:2004yh,Kawasaki:2004qu,Jedamzik:2006xz,Steffen:2006hw,Cyburt:2006uv,Cyburt:2009pg,Freitas:2009jb}.
So far, most of the analyses have concentrated on the injection of
energy by relics with masses comparable to the electroweak scale. This
is largely motivated by theoretical arguments in favor of new physics
at and below the TeV-scale, and by the possibility of having dark
matter in the form of weakly interacting massive particles (WIMPs). It
can be easily shown that it is unlikely that the residual WIMP
annihilation is responsible for the reduction of \ensuremath{{}^7\mathrm{Be}}+\ensuremath{{}^7\mathrm{Li}}, simply
because the total energy injected via such mechanism is way below the
required levels \cite{Reno:1987qw,Jedamzik:2004ip}. Known examples
that "work", \textit{i.e.} scenarios in which the \ensuremath{{}^7\mathrm{Li}}\ abundance is
reduced while other elements are still agreeing with observations,
typically deal with unstable weak-scale particles. These include some
supersymmetric scenarios with the delayed decays of charged sleptons
to gravitinos \cite{Jedamzik:2005dh,Cyburt:2006uv,Cumberbatch:2007me}.
The source of extra neutrons in these models is linked to the presence
of nucleons among the decay products. An alternative plausible
mechanism for reducing \ensuremath{{}^7\mathrm{Li}}\ is the catalytic suppression of the
\ensuremath{{}^7\mathrm{Be}}\ abundance by the capture of massive negatively charged particles
\cite{Pospelov:2006sc,Bird:2007ge,Jittoh:2007fr,Kusakabe:2007fv}, that
are again linked to the weak scale. In this paper we investigate
whether the suppression of the lithium abundance can be triggered by
the decays of metastable GeV and sub-GeV scale electrically-neutral
particles. We address two types of models: the WIMP-type where
particle decays were preceded by the depletion through the
annihilation, and the super-WIMP type, where the abundance of decaying
particles is set by the thermal leakage of Standard Model (SM) states
into an initially vacuous super-WIMP sector.
During the last two years, the GeV-sector phenomenology experienced
some degree of revival due to its possible connection to the
enhancement of the leptonic fraction of cosmic rays in the pair
annihilation of dark matter
WIMPs~\cite{ArkaniHamed:2008qn,Pospelov:2008jd}. Particularly
noteworthy are the enhancement of the positron fraction seen by the
PAMELA satellite experiment \cite{Adriani:2008zr} and the
harder-than-expected spectrum for electrons and positrons detected by
the FGST instrument \cite{Abdo:2009zk}. In such scenarios the
GeV-scale particles are designated as "mediators" connecting the dark
matter and Standard Model sectors \cite{Pospelov:2007mp}, allowing to
seclude the dark matter by choosing the the SM-mediator coupling to be
very small, but at the same time keeping the galactic annihilation
rate enhanced over the "standard" WIMP scenario.
The motivation for GeV and sub-GeV scale mediators comes from the
following consideration: the lightness of a gauge boson mediating an
attractive force in the dark sector, $V(r) = - \alpha_D/r\times
\exp(-m_Vr)$, compared to the characteristic WIMP momentum inside the
galaxy, $m_D v_{\rm gal}$, ensures a Coulomb/Sommerfeld enhancement of
the annihilation cross section relative to its freeze-out
value. Choosing $m_V \la m_D v \sim 100~{\rm MeV}\div 1~{\rm GeV}$,
results in a $\pi \alpha_D/ v$ enhancement of the cross sections at
the relevant velocities. Moreover, once recombination to WIMP-bound
states becomes kinematically possible, $m_V < m_D \alpha_D^2/4$, this
process dominates numerically over the direct two-body
annihilation. The resulting annihilation cross section for fermionic
WIMPs can be enhanced over the freeze-out value by two-to-three orders
of magnitude \cite{Pospelov:2008jd}:
\be \fr{\langle \sigma v \rangle_{\rm
gal}}{\langle \sigma v \rangle_{\rm f.o.}} \sim \fr{7
\pi\alpha_D}{v_{\rm gal}} \sim O(10^2-10^3).
\label{recombination}
\ee
The numerical enhancement due to the bound-state effect over the naive
Sommerfeld value is about a factor of 7, accompanied by a possible
additional enhancement due to an increased lepton multiplicity in the
final state with angular momentum $J=1$. We note in passing that the
important effect of WIMP-onium formation was missed in a recent
re-analysis of Ref. \cite{Feng:2010zp}, that led to the erroneous
conclusion that the enhancement factor remains smaller than 100 over
the whole parameter space; larger factors were found
in~\cite{Cirelli:2010nh}.
Is it reasonable to expect that the same models that fit
the PAMELA and FGST signals \cite{ArkaniHamed:2008qn,Pospelov:2008jd}
are also responsible for the suppression
of \ensuremath{{}^7\mathrm{Li}}\ abundance? Even with the inclusion of some very generous
enhancement factors, the energy injection during BBN triggered by the
annihilation of electroweak-scale WIMPs remains rather small.
Moreover, models designed to explain the PAMELA signal tend to
minimize the fraction of baryons/anti-baryons in the final state
\cite{ArkaniHamed:2008qn,Pospelov:2008jd,Fox:2008kb,Nomura:2008ru}.
All that, together with previous investigations of BBN with
annihilation-induced energy injection, tend to indicate that WIMP
annihilation itself cannot be used as a solution to the lithium
problem. Therefore, the only chance of altering the BBN predictions
for \ensuremath{{}^7\mathrm{Be}}+\ensuremath{{}^7\mathrm{Li}}\ within this class of models is if the GeV-scale
particles from the mediator sector are themselves relatively
long-lived, and their decays happen during or after BBN.
Interestingly enough, it turns out that the minimal ways of coupling
the Standard Model to light mediators often implies the longevity of
particles in the GeV sector \cite{Batell:2009di}.
The main mechanism by which the decays of the GeV-scale relics in the
early Universe can influence the outcome of the BBN is the injection
of light mesons such as pions and kaons as well as muons that all lead
to the extra source of $p\to n$ conversion. In this work we explore
such scenarios, finding the "required" number of injected $\pi^-$,
$K^-$, and $\mu^{-}$ triggering $p\to n$ conversion in the right
amount, as well as the "optimal" lifetime--abundance window for such
injection. In order to have a consistent cosmological picture, the
abundance of parent GeV-scale relics should be small enough so that
they carry only a small fraction of the energy density of Universe
during BBN, but still provide a noticeable number of mesons and muons
per nucleon. We show that, in fact, many models with GeV-scale relics
fulfill this requirement, including some variants of the models
designed to fit PAMELA and FGST signals. We also find that both the
WIMP and super-WIMP modifications of a secluded \us-model is capable
of reducing the lithium abundance to the observable level.
The structure of this paper is as follows. The next section contains
the analysis of the injection of $\pi$, $K$, and $\mu$ particles
vs.~the timing of injection that is needed for reducing
\ensuremath{{}^7\mathrm{Li}}. Section~\ref{sec:metastable-gev-scale} investigates a variety
of different models for decaying GeV and sub-GeV scale relics, and
finds the regions of parameter space that lead to the depletion of the
lithium abundance. Our conclusions are summarized in
Sec.~\ref{Sec:Conclusions}. Appendix~A contains relevant details with
physics input that went into our BBN code.
\section{Meson and neutrino injection during BBN}
We begin this section by presenting an overview of the physics
processes triggered by the decays of meta-stable GeV-scale relics $X$
during BBN. Thereby we shall estimate timescales and interaction
efficiencies of the crucial reactions which eventually lead to the
reduction of the overall $\ensuremath{{}^7\mathrm{Li}}$ BBN prediction. Subsequently, the
various final states in the decay of $X$ are considered in detail.
The central parameter entering the discussion is the Hubble expansion
rate $H$ as it normalizes any interaction rate during nucleosynthesis.
The most relevant epoch for BBN corresponds to the time bracket of
$100\ \ensuremath{\mathrm{sec}} \lesssim t \lesssim 1000\ \ensuremath{\mathrm{sec}}$ or, equivalently,
to a temperature range of
\begin{align}
\label{eq:BBN-temp-window}
1.2 \lesssim T_9\lesssim 0.4 ,
\end{align}
where $T_9$ is the photon temperature in units of $10^9$~K.
Thus, in the interesting regime after the annihilation of the
electron-positron pairs, the Hubble rate is given by
\begin{align}
\label{eq:hubble-rate}
H & = \sqrt{\frac{\pi^2 \ensuremath {g_{\mathrm{eff}}}}{90}} \frac{T^2}{\ensuremath{M_{\mathrm{P}}}}
\simeq (2.8\times 10^{-3}\, \ensuremath{\mathrm{sec}}^{-1})\, T_9^2 \ \qquad (T\lesssim \ensuremath{m_{e}}),
\end{align}
where $g_{\mathrm{eff}}\simeq 3.36$ counts the radiation degrees of
freedom; $\ensuremath{M_{\mathrm{P}}} \simeq 2.43\times 10^{18}\, \ensuremath{\mathrm{GeV}}$ is the reduced Planck
scale.
The synthesis of \ensuremath{{}^7\mathrm{Be}}, proceeding via $\ensuremath{{}^4\mathrm{He}} + \ensuremath{{}^3\mathrm{He}} \to \ensuremath{{}^7\mathrm{Be}}\ +\gamma$,
occurs in a more narrow temperature interval, $T_9\simeq 0.8\div 0.4$,
in which the Hubble rate is $H\sim 10^{-3}$ sec$^{-1}$. This occurs
immediately after the opening of the deuterium ``bottleneck'' at
$T_9\sim 0.8$, when significant quantities of \ensuremath{{}^4\mathrm{He}}\ and \ensuremath{{}^3\mathrm{He}}\ are
formed, with a helium mass fraction of $Y_{p}\simeq 0.25$ and a number
density of \ensuremath{{}^3\mathrm{He}}\ per proton of $Y_{\ensuremath{{}^3\mathrm{He}}}\simeq 10^{-5}$. The rate for
\ensuremath{{}^7\mathrm{Be}}\ formation per alpha particle,
\begin{align}
\Gamma_{\ensuremath{{}^7\mathrm{Be}}}
\simeq (10^{-3}\, \ensuremath{\mathrm{sec}}^{-1})\ T_9^{7/3} e^{-12.8/T_9^{1/3}}
\left( \frac{Y_{\ensuremath{{}^3\mathrm{He}}}}{10^{-5}} \right)\quad (T_9\lesssim 0.8)
\end{align}
always remains slower than the Hubble rate and quickly becomes
completely inefficient due to a strong exponential Coulomb
suppression.
On the other hand, a non-standard ``thermal'' neutron abundance, at
the level comparable to $10^{-5}$ has the opportunity to reprocess some
fraction of \ensuremath{{}^7\mathrm{Be}}\ via $\ensuremath{{}^7\mathrm{Be}} + n \to \ensuremath{{}^7\mathrm{Li}} + p $ since the rate for a
neutron capture per \ensuremath{{}^7\mathrm{Be}}\ nucleus is
\begin{align}
\label{eq:bes-n-capture}
\Gamma_{\ensuremath{{}^7\mathrm{Be}}\to\ensuremath{{}^7\mathrm{Li}}} \simeq (0.4\ \ensuremath{\mathrm{sec}}^{-1})\,T_9^3 \left(
\frac{Y_n}{10^{-5}} \right).
\end{align}
Indeed, looking at the fraction~$f_{\ensuremath{{}^7\mathrm{Be}}}$ of $\ensuremath{{}^7\mathrm{Be}}$ which can, in
principle, be converted within a Hubble time $\Delta t_H$,
\begin{align}
f_{\ensuremath{{}^7\mathrm{Be}}}\simeq - \frac{1}{Y_{\ensuremath{{}^7\mathrm{Be}}}} \frac{dY_{\ensuremath{{}^7\mathrm{Be}}}}{dt}\Delta t_H \simeq -H^{-1}
\Gamma_{\ensuremath{{}^7\mathrm{Be}}\to\ensuremath{{}^7\mathrm{Li}}} \sim -10^{7}T_9 Y_n ,
\end{align}
shows that already $Y_n \simeq 5\times 10^{-6}$ at $T_9\simeq 0.5$ can
induce an $\Orderof{1}$-conversion of \ensuremath{{}^7\mathrm{Be}}.
$\ensuremath{{}^7\mathrm{Li}}$ in the final state of this reaction is then quickly burned by
protons.
The neutron participation in the lithium depleting chain does not
influence the abundance of neutrons themselves, because of the small
abundance of \ensuremath{{}^7\mathrm{Be}}. Instead, ``extra neutrons'' are depleted by
protons via the $p + n\to \ensuremath{\mathrm{D}} +\gamma$ reaction, which, at the
relevant temperatures, remains faster than the rate for neutron
decay. Comparing the relative changes of \ensuremath{{}^7\mathrm{Be}}\ and deuterium,
\begin{align}
\frac{f_{\ensuremath{{}^7\mathrm{Be}}}}{f_{\ensuremath{\mathrm{D}}}} = -\frac{Y_\ensuremath{\mathrm{D}}}{Y_p}
\frac{\Gamma_{\ensuremath{{}^7\mathrm{Be}}\to\ensuremath{{}^7\mathrm{Li}}}}{\Gamma_{p\to\ensuremath{\mathrm{D}}}} \sim -10^{5}\,
({\ensuremath{\mathrm{D}}}/{\ensuremath{\mathrm{H}}})
\end{align}
one can infer that $\Orderof{1}$-depletion of \ensuremath{{}^7\mathrm{Be}}\ due to an excess
of neutrons will be accompanied by an $\Orderof{1}$-rise in the
deuterium abundance since $\ensuremath{\mathrm{D}}/\ensuremath{\mathrm{H}} \sim 10^{-5}$. Therefore, the
viability of the "extra neutron" solution to the lithium problem
should be judged form more accurate quantitative investigations of
such scenarios.
The decay of GeV and sub-GeV relic particles with a rate comparable to
(\ref{eq:hubble-rate}) at $T_9\sim 0.5$ will not lead to a significant
population of energetic photons and electrons. The reason for that is
well-known: very abundant and energetic photons and electrons degrade
the energy of the decay products well below the nuclear dissociation
thresholds (see {\em e.g.} \cite{Kawasaki:2004qu}). Since we
explicitly assume the absence of nucleons and anti-nucleons in the
final state, the main effect on the freeze-out abundances of light
elements will come from injection of mesons and neutrinos, that are
capable of triggering the $p\to n$ conversion and reducing \ensuremath{{}^7\mathrm{Li}}+\ensuremath{{}^7\mathrm{Be}}.
In the following subsections we consider these mechanisms in turn,
starting from the simple estimates of the required number of mesons
and neutrinos.
\subsection{Estimates on meson injection}
\label{sec:timescales}
Among the the decay products of the GeV-scale metastable states only
the long-lived mesons and neutrinos will have a chance to interact
with the light elements. Among those, the most important are charged
pions, \ensuremath{\pi^{\pm}}, as well as charged and long-lived kaons, \ensuremath{K^{\pm}}\ and
\ensuremath{K_{L}}, respectively. Their masses and lifetimes are given
by~\cite{Amsler:2008zzb}
\begin{align}
m_{\ensuremath{\pi^{\pm}}} & = 139.6\ \ensuremath{\mathrm{MeV}} , & \tau_{\ensuremath{\pi^{\pm}}} & = 2.60\times 10^{-8}\ \ensuremath{\mathrm{sec}} , \\
m_{\ensuremath{K^{\pm}}} & = 493.7\ \ensuremath{\mathrm{MeV}} , & \tau_{\ensuremath{K^{\pm}}} & = 1.24\times 10^{-8}\ \ensuremath{\mathrm{sec}} ,\\
m_{K^0,\bar{K}^{0}} & = 497.6\ \ensuremath{\mathrm{MeV}} , & \tau_{\ensuremath{K_{L}}} & = 5.12\times 10^{-8}\ \ensuremath{\mathrm{sec}} .
\label{eq:KL-prop}
\end{align}
The decay rate of meson $i$ is related to its lifetime at rest by a familiar
time-dilatation formula:
\begin{align}
\label{eq:meson-decay-rate}
\Gamma^i_{\mathrm{dec}} = \left\langle \frac{m_i}{E_i}\right\rangle
\frac{1}{\tau_i} \simeq \mathrm{few}\times 10^7\ \ensuremath{\mathrm{sec}}^{-1}
\end{align}
where in the last equality we have taken $\gamma \simeq 1$. More
accurately, one has to consider the average value $\left\langle
{m_i}/{E_i}\right\rangle$ over the "lifetime" trajectory of injected
mesons. The averaging procedure is significantly different from the
usual thermal average when particles obey a Maxwell-Boltzmann
distribution. It is determined by the efficiency of various energy
degradation mechanisms, of which the most important are Coulomb
scattering on electron and positrons and inverse-Compton-type
scattering on background photons. Together they determine the
thermalization (or stopping) rate $\Gamma^i_{\rm stop}$. The stopping
rate may have a direct impact on the the reaction rates with nuclei
$\Gamma_N^i$, which are of the primary interest in this paper.
If injected sufficiently early, the charged mesons are effectively
stopped within time intervals shorter than their lifetimes $\tau_i$,
and before having a chance of participating in a nuclear reaction.
The comparison of stopping and decay rate is given by the ratio
\begin{align}
\fr{\Gamma^i_{\mathrm{stop}}}{\Gamma^i_{\mathrm{dec}}} = \fr{\tau^i_{\mathrm{dec}}}{\tau^i_{\mathrm{stop}}}
\simeq
\left(\Gamma^i_{\mathrm{dec}}\int^{E_0}_{E_{\mathrm{min}}} \frac{dE}{|dE_i/dt|} \right)^{-1},
\end{align}
where $\tau_{\mathrm{stop}}$ measures the energy degradation time from
injection energy $E_0$ to some energy $E_{\mathrm{min}}$, below which
the kinetic energy of mesons is irrelevant. $dE_i/dt$ represents the
energy degradation rate of the particle $i$ traversing the BBN
plasma. Requiring complete stopping would correspond to the choice of
$E_{\mathrm{min}} \sim 3T/2$, but a more relevant parameter is some
characteristic nuclear energy scale. For reactions leading to
dissociation of \ensuremath{{}^4\mathrm{He}}, $E_{\mathrm{min}}$ would correspond to
$E_{\mathrm{min}} \sim 20$ MeV.
In the temperature window (\ref{eq:BBN-temp-window}) and for injection
energies of $E_0 \lesssim 1\ \ensuremath{\mathrm{GeV}}$, Coulomb interactions on $e^{\pm}$
give the dominant contribution to $dE_i/dt$ (for details see {\em
e.g.} \cite{Kawasaki:2004qu}). Since the number densities of
electrons and positrons drop exponentially, the inverse Compton
scattering on background photons becomes the dominant energy loss
mechanism for $T<25$ keV. Taking into account both channels for
energy loss, one typically finds that
\begin{eqnarray}
\fr{\tau^i_{\mathrm{dec}}}{\tau^i_{\mathrm{stop}}} < 1\quad \mathrm{for}\quad T_9>(0.3\div 0.4).
\end{eqnarray}
This point is illustrated in Fig.~\ref{stopping}, where the dividing
line of ``1'' separates the two regimes of complete and incomplete
stopping. The exponential sensitivity to temperature is reflected in
almost vertical contours. Figure \ref{stopping} tells us that for $t
\lesssim 10^3\ \ensuremath{\mathrm{sec}}$ the charged mesons are thermalized before
they decay. Thus, for $i= \ensuremath{\pi^{\pm}}$ (and analogously for $\ensuremath{K^{\pm}}$)
injected around $ T_9 \sim 0.5$ the following hierarchy of scales is
applicable:
\begin{align}
H \ll \Gamma_p^\pi \ll \Gamma_{\mathrm{dec}}^\pi \lesssim \Gamma_{\rm stop}^\pi.
\label{hierarchy}
\end{align}
The rate for charge exchange reactions with the most abundant nuclear species,
\textit{i.e.} the rate of proton-to-neutron conversion, is given by
\begin{align}
\label{eq:hadint-rate}
\Gamma_p^\pi= n_p \sigmavof{}_{pn}^\pi \simeq ( 3\times 10^{2}\
\ensuremath{\mathrm{sec}}^{-1} )\ \,\frac{T_9^3\sigmavof{}_{pn}}{1\ \ensuremath{\mathrm{mb}}}.
\end{align}
The averaging procedure is again determined along the trajectory of
injected mesons and includes both the "in-flight" and "at-rest"
contributions. However, for the purpose of a simple estimate,
Eq.~(\ref{eq:hadint-rate}) uses the normalization on a typical size of
a pion induced reaction at the threshold. In case of incomplete
stopping, this will underestimate the proton-neutron conversion
because of the delta-resonance enhancement of the charge-exchange
reaction. Equation~(\ref{eq:hadint-rate}) immediately leads us to the
probability of charge exchange reaction of a stopped pion (kaon) on
protons during the meson lifetime, per each injected $\pi^-(K^-)$ at
$T_9\sim 0.5$:
\begin{align}
P_{p\to n}^\pi = \int_{t_{\rm inj}}^\infty
\exp(-\Gamma_{\mathrm{dec}}(t-t_{\rm inj})) \Gamma_p dt \simeq
\Gamma_p^\pi \tau_{\ensuremath{\pi^{\pm}}} \sim O(10^{-6})
\label{probability}
\end{align}
Since every charge exchange reaction leads to the creation of an extra
neutron, the estimate (\ref{probability}) represents the {\em
efficiency} of producing neutrons from negatively charged kaons and
pions. Notwithstanding a rather crude nature of this estimate, the
probability (\ref{probability}) tells us that an injection of
$\Orderof{10}$ negatively charged pions per proton should be
equivalent to the injection of $\Orderof{10^{-5}}$ extra neutrons, and
thus capable of reducing the overall \ensuremath{{}^7\mathrm{Li}}+\ensuremath{{}^7\mathrm{Be}}\ abundance by an
$\Orderof{1}$ factor.
\FIGURE[t]{
\includegraphics[width=0.6\textwidth]{contour_StopTau.eps}
\caption{\small Stopping of charged pions and kaons in the plasma as a
function of temperature $T_9$, initial injection energy
$E_{\mathrm{in}}$, and the minimal energy below which the kinetic
energy of mesons can be neglected, $E_{\mathrm{low}}$. The contour
$\tau_{\mathrm{dec}}/\tau_{\mathrm{stop}}=1$ is the dividing line
between efficient (left) and incomplete (right) stopping.}
\label{stopping}
}
Since for kaons the charge exchange cross sections are even higher
\cite{Reno:1987qw}, one can achieve an adequate suppression of
\ensuremath{{}^7\mathrm{Li}}+\ensuremath{{}^7\mathrm{Be}}\ even with the injection of one $K^-$ per baryon.
The case of $X$-decays into neutral kaons is special with respect to
other meson final states in the sense that $K_L$ has a relatively long
lifetime (\ref{eq:KL-prop}) but, unlike $\ensuremath{\pi^{\pm}}$ or $\ensuremath{K^{\pm}}$, is not
stopped by electromagnetic interactions. From the conservation of
isospin and from charge independence of strong interactions, we
nevertheless expect the impact of $K_L$ on the BBN predictions to be
similar to that of $K^{-}$.
In our treatment, the extra meson species are included in the set of
Boltzmann equations, along with the population of parent particles $X$
that decay into mesons and have an abundance with the simple
exponential time-dependence, \be Y_X(t) = Y_X^0\exp(-t/\tau_X), \ee
with $\tau_X$ being the $X$-lifetime. We choose to normalize the
$X$-abundance on the total number of baryons, $Y_X \equiv n_X/n_b$; in
the following we shall often drop the superscript on $Y_X^0$ for
simplicity. When the fast stopping of charged particles is operative
at early times, the amount of the injected energy and therefore the
actual mass of the $X$-relic are not entering the problem (with $X\to
K^0\bar K^0$ being the exception.) For later times, the incomplete
stopping brings the dependence on the injected energy. We write the
Boltzmann equation for meson species $i$ in the following form
\begin{equation}
\fr{d{Y}_{i}}{dt} \simeq \sum_{j} \xi_i^{(j)} Y_j \Gamma^{j}_{\mathrm{inj}} - Y_i
\Gamma^i_{\mathrm{dec}} - Y_N \Gamma_N^i.
\label{BE}
\end{equation}
where $\Gamma^{j}_{\mathrm{inj}}$ is the rate of $i$-injection from
source $j$ with multiplicity~$\xi_i^{(j)}$. When considering only the
primary meson production from $X$-decays such as in the case of
$\ensuremath{K^{\pm}}$, $\sum_{j} \Gamma^{j}_{\mathrm{inj}} \simeq
\Gamma_{\mathrm{dec}}^X = 1/\tau_{X}$. Given the hierarchy of
interactions (\ref{hierarchy}), the last term in (\ref{BE}) is small.
Both $\Gamma^i_{\mathrm{dec}}$ and $\Gamma_N^i$ include the effects of incomplete stopping
that depends on the initial energy injection $E_0$ and background temperature at the time of injection,
\be
\Gamma^i_{\mathrm{dec}}=\Gamma^i_{\mathrm{dec}}(E_0,T);~~~ \Gamma_N^i = \Gamma_N^i(E_0,T).
\label{correction}
\ee
Neglecting subtleties of incomplete stopping at this point,
one can find an approximate solution to the Boltzmann equation for mesons
in the quasi-static approximation $dY^{\mathrm{qse}}_i/dt =
0$,
\begin{align}
Y^{\mathrm{qse}}_i = \sum_{j} {\xi_i^{(j)} Y_j\Gamma^{j}_{\mathrm{inj}}}/{
\Gamma^i_{\mathrm{dec}}} \sim \Orderof{10^{-10}} \times Y_X,
\label{qse}
\end{align}
where in the second relation we assumed multiplicities to be order
one, and took $\Gamma_{\rm inj} \sim 10^{-3}\ \ensuremath{\mathrm{sec}}$. Even though
the equilibrium values for the meson abundances remain low at all
times, the $p\to n$ conversion rate for $Y_X \sim \Orderof{10}$ is
only five orders of magnitude slower than the Hubble rate, thus
causing $\Orderof{10^{-5}}$ protons be converted to neutrons within
one Hubble time. This is consistent with our estimate of efficiency
(\ref{probability}).
Finally, the pion(kaon)-induced transitions $\ensuremath{{}^7\mathrm{Be}}\to\ensuremath{{}^7\mathrm{Li}}$ and
$\ensuremath{{}^3\mathrm{He}}\to\ensuremath{\mathrm{T}}$ also lead to the depletion of \ensuremath{{}^7\mathrm{Be}}. However, these
processes are far less important compared to $p\to n$
interconversion. If we assume a fractional $O(10^{-5})$-conversion of
$p\to n$ due to $\pi^-$ charge exchange, a similar figure would stand
for the $\ensuremath{{}^7\mathrm{Be}}\to\ensuremath{{}^7\mathrm{Li}}$ and $\ensuremath{{}^3\mathrm{He}}\to\ensuremath{\mathrm{T}}$ conversion probabilities,
which is completely negligible in the final lithium abundance.
\subsection{Estimates on muon and neutrino injection}
In order to account for the injection of muons and neutrinos one has
to deal with quite different physics. The charge exchange reactions of
muons on nucleons are mediated by weak interactions and can be
neglected during the muon lifetime of two microseconds. However, the
decays of muons, $\mu^- \to \nu_\mu \bar \nu_ee^-$, source energetic
electron antineutrinos $\bar \nu_e$, that survive for a long time,
which increases their probability of charge-exchange interaction with
protons.
To make our discussion more concrete, we shall assume that all
neutrinos originate from muons decaying at rest so that $E_\nu <
m_\mu/2$. In this case, the relation between the relevant rates is
quite different from~(\ref{hierarchy}): \be \Gamma^\nu_p, ~
\Gamma^\nu_{\rm stop} \ll H, \ee where $\Gamma^\nu_{\rm stop} $ is the
rate of antineutrino energy degradation due to scattering on
background neutrinos and electron-positron pairs. This rate scales
with temperature and the energy of non-thermal neutrinos $E_\nu(T) $
injected at $T_{\rm inj}$ as follows:
\begin{align}
\Gamma^\nu_{\rm stop} \sim G_F^2 E_\nu(T) T^4 \sim G_F^2 T^5
\fr{\VEV{E_\nu(T_{\rm inj})}}{T_{\rm inj}},
\end{align}
where we disregard the difference between neutrino and
photon temperatures. It is indeed much smaller than the Hubble
expansion rate,
\begin{align}
\fr{\Gamma^\nu_{\rm stop}}{H} \sim \left( \fr{T}{3~{\rm MeV} }
\right)^3 \fr{\VEV{E_\nu(T_{\rm inj})}}{T_{\rm inj}} \sim 10^{-3},\qquad
(T,~T_{\rm inj}\sim 30~{\rm keV}),
\end{align}
where 3 MeV scale enters as the decoupling temperature of background
neutrinos, and the energy of non-thermal antineutrinos is taken to be
the average energy in the muon decay, $ \VEV{E_\nu(T_{\rm inj})} =
\fr{3}{10} m_\mu \simeq 32 $ MeV.
The charged current rate for antineutrino interactions with protons is
given by \be \Gamma^\nu_p = n_p \sigma_{pn }^{\bar\nu} \simeq
10^{-41}~{\rm cm}^2 \times \fr{n_p E_\nu^2}{(10~{\rm MeV})^2} \simeq
(3.6\times 10^{-12}~{\rm sec}^{-1} )\times \fr{T_9^3E_\nu^2}{(10 ~{\rm
MeV})^2}, \ee where $E_\nu \gg m_e$ is assumed.
The ratio of $\Gamma^\nu_p $ to the Hubble rate gives the efficiency
of producing extra neutrons from each neutrino injected with
$\Orderof{30\ \ensuremath{\mathrm{MeV}}}$ energy:
\begin{align}
P^\nu_{p\to n} = \int_{t_{\rm inj}}^{\infty} \Gamma_p^\nu dt =
\fr{1}{3} \fr{\Gamma^\nu_p(T_{\rm inj})}{H(T_{\rm inj})} \sim
2\times 10^{-9}
\label{estimate2}
\end{align}
This efficiency is several orders of magnitude smaller than
Eq. (\ref{probability}), and we conclude that $\Orderof{10^4}$ muon
decays per proton are required around $T_9 \sim 0.5 $ in order to
produce $\Orderof{10^{-5}}$ extra neutrons. At the same time,
$P^\nu_{p\to n} \ll P^\pi_{p\to n}$ introduces a natural
simplification to the problem, as the effects of secondary neutrinos
originating from pion and kaon decays can be ignored relative to the
direct influence of $\pi^-$ and $K^-$ via the charge exchange
reactions. Moreover, as the efficiency (\ref{estimate2}) also
suggests, a direct nuclear-chemical impact of electron antineutrinos
on the lithium abundance via $\ensuremath{{}^7\mathrm{Be}} +\bar\nu_e\to \ensuremath{{}^7\mathrm{Li}} + e^{+}$
($Q=-0.16\ \ensuremath{\mathrm{MeV}}$) is negligible.
Having checked that the efficiency of neutron production from pions,
kaons and muons (neutrinos) can be sufficient for the resolution of
the lithium problem, we now turn to more detailed calculations. Some
further details on the Boltzmann code that we use can be found in the
Appendix.
\subsection{Decays to pions }
\label{sec:treatment-pions}
Charged pions are a likely final state for almost any hadronic final
decay mode. For example, even if $X$ would exclusively decay into
\ensuremath{K^{\pm}}, \textit{i.e.}\ $\xi^{(X)}_{\ensuremath{\pi^{\pm}}} = 0$ in Eq.~(\ref{BE}), pions would
still be populated by subsequent \ensuremath{K^{\pm}}-decays, $Y_{\ensuremath{K^{\pm}}}\simeq
Y_{\ensuremath{\pi^{\pm}}}$, since $\xi^{(\ensuremath{K^{\pm}})}_{\ensuremath{\pi^{\pm}}} = \Orderof{1}$.
Charged pions have a chance to interact with protons and \ensuremath{{}^4\mathrm{He}}\ before
decaying. Fully thermalized pions have the following charge exchange
reactions on nucleons with positive energy release:
\begin{align}
\label{pi-on-p}
\ensuremath{\pi^{-}} + p & \to n + \gamma : & & \!\!\!\!\!\!\!\!\!\!(\sigma
v)^{\ensuremath{\pi^{-}}}_{pn(\gamma)} \simeq 0.57\ \ensuremath{\mathrm{mb}} ,
& \!\!\!\!\!\!\!\!\!\!Q & = 138.3\ \ensuremath{\mathrm{MeV}} ,\\
\label{pi-on-p-chx}
\ensuremath{\pi^{-}} + p & \to n + \ensuremath{\pi^{0}} :&& \!\!\!\!\!\!\!\!\!\! (\sigma
v)^{\ensuremath{\pi^{-}}}_{pn(\pi^0)} \simeq 0.88\ \ensuremath{\mathrm{mb}} ,
& \!\!\!\!\!\!\!\!\!\!Q & = 3.3\ \ensuremath{\mathrm{MeV}} ,\\
\ensuremath{\pi^{+}} + n & \to p + \ensuremath{\pi^{0}} : &&\!\!\!\!\!\!\!\!\!\! (\sigma
v)^{\ensuremath{\pi^{-}}}_{np} \simeq 1.7\ \ensuremath{\mathrm{mb}} , & \!\!\!\!\!\!\!\!\!\!Q & = 5.9\
\ensuremath{\mathrm{MeV}} .
\end{align}
These reactions interconvert neutrons and protons and thereby increase
the $n/p$ ratio, because protons are far more numerous once neutrons
have been incorporated into~\ensuremath{{}^4\mathrm{He}}. The threshold value for the $\ensuremath{\pi^{-}}
+p$ reaction cross section can be extracted from the lifetime of the
pionic hydrogen atom (see {\em e.g.} the review
\cite{Gasser:2007zt}). The strength of the channels $\pi^0$ and
$\gamma$ at threshold can be inferred from the Panofsky
ratio~\cite{Flugel:1999gr}. Our value at the (thermal) threshold \be
\label{ourvalue}
(\sigma v)^{\ensuremath{\pi^{-}}}_{{\rm th}}\equiv F_{p\ensuremath{\pi^{-}}} \left[ (\sigma
v)^{\ensuremath{\pi^{-}}}_{pn(\gamma)}+(\sigma v)^{\ensuremath{\pi^{-}}}_{pn(\pi^0)} \right] \simeq
F_{p\ensuremath{\pi^{-}}}\times 1.45\ \ensuremath{\mathrm{mb}}, \ee is in good agreement with the one
used in the BBN literature \cite{Reno:1987qw} from where we also took
the cross section for $\ensuremath{\pi^{+}} + n$. This threshold value is subject to
the usual Coulomb enhancement that is accounted for with the
multiplicative factor $F_{p\ensuremath{\pi^{-}}}$ in (\ref{ourvalue}). For the
negatively charged pions and at the temperature range of interest
($T_9 \sim 0.5$) this translates into an enhancement of the reactions
rates by a factor of $F_{p\ensuremath{\pi^{-}}} \simeq 2$. Further details on Coulomb
enhancement are provided in the Appendix
\ref{sec:some-details-boltzm}.
The incomplete stopping of pions introduces an important modification
to the rates of these reactions, and to the efficiency of $p\to n$
conversion. To quantify this effect, we introduce the correction
factor $\kappa(E_0,T)$ as a function of primary kinetic injection
energy $E_0$ that includes both the effect of the time dilatation (and
effectively longer lifetimes of fast pions) and, more importantly, the
momentum dependence of the cross section that has a strong peak around
a pion energy of $E_{\pi}\sim 180\ \ensuremath{\mathrm{MeV}}$:
\begin{align}
\kappa(E_0,T) \equiv \fr{P_{p\to n}(E_0,T)}{P_{p\to n}(T)}
=\fr{1}{\tau_\pi (\sigma v)_{{\rm th}}} \int_{0}^{\infty} dt (\sigma
v)_{E_{\pi}(t)} \exp\left( -\int_0^{t} dt' \fr{1}{\tau_\pi \gamma(E_{\pi}(t'))}
\right).
\label{kappaET}
\end{align}
The exponential factor in this expression is the survival probability,
that is the probability of finding a meson at time $t$ after its
injection at $t=0$. Defined this way, the correction factor
$\kappa(E_0,T) =1$ in the limit of $\tau_{\rm stop}
\ll \tau_\pi$. Notice also that due to the scale hierarchy, $H \ll
\tau_\pi^{-1}$, we can disregard the effects related to cosmological
expansion inside the integrals of Eq. (\ref{kappaET}), and set the
initial moment of injection to be $t=0$.
The kinetic energy $E_{\pi}$ of an injected pion, and the time $t$
along the ``lifetime'' trajectory are related via the rate of energy
loss: \be
\label{tE}
t(E_{\pi}) = \int^{E_0}_{E_{\pi}}\fr{dE'}{|dE'/dt|};\quad
\fr{dE}{dt} = \left( \fr{dE}{dt} \right)_{\rm Coul} + \left(
\fr{dE}{dt}\right)_{\rm Comp}. \ee As alluded before, $dE/dt$ is a
strong function of the background temperature. Finding the explicit
dependence between time and kinetic energy of injected pions
numerically, and using the experimental data for the inelastic
$p+\ensuremath{\pi^{-}}$ reaction away from threshold, we find the correction factor
for the efficiency, $\kappa(E_0,T)$. Figures \ref{kappaT9} and
\ref{kappaE} plots this factor for the representative temperature and
initial energy slices. One can see that due to incomplete stopping at
late times, the correction factor can be very large, possibly reaching
\be \kappa_{\rm max} \simeq \gamma_{\rm max} \fr{(\sigma v)_{\rm
max}}{(\sigma v)_{\rm th}} \simeq 30, \ee where $\gamma_{\rm max}
\simeq 2.3$ corresponds to the pion momentum at delta-resonance, where
the ratio of cross sections reaches $(\sigma v)_{\rm max}/(\sigma
v)_{\rm th} \simeq 14$.
\FIGURE[t]{
\includegraphics[width=0.6\textwidth]{kappaT9.eps}
\caption{\small The correction factor $\kappa(E_0,T)$, relating the
efficiency of $p\to n$ conversion by pions in-flight to the thermal
case, as a function of temperature for representative values of
injection energy $E_0$; $E_{\mathrm{th}}=3T/2$. The dashed line
shows the case of maximal efficiency due to pion injection at the
delta-resonance energy. Formally, $\kappa <1$ is possible for low
temperatures and small injection energies because the Coulomb
corrections of a fully thermalized pion are substantial.}
\label{kappaT9}
}
\FIGURE[t]{\includegraphics[width=0.6\textwidth]{kappaE.eps}
\caption{\small The correction factor $\kappa(E_0,T)$, relating the
efficiency of $p\to n$ conversion by pions in-flight to the
thermal case, as a function of injection energy $E_0$ for
Representative values of temperature. The curves corresponding to
late injection $T_9$ = 0.15 and 0.3 when stopping does not occur
follow the familiar energy dependent profile of the pion-nucleon
cross section with the broad delta resonance at 180~MeV. }
\label{kappaE}
}
At the next step we account for the possible reactions on \ensuremath{{}^4\mathrm{He}}.
Again, we separate our discussion into the reactions induced by
stopped and in-flight pions.
Charge exchange reactions of thermal $\pi^\pm$ on \ensuremath{{}^4\mathrm{He}}\ are not possible
because of the deep binding of \ensuremath{{}^4\mathrm{He}}\ compared to other mass-$4$
isomers. However, $\pi^\pm$ can be fully absorbed by \ensuremath{{}^4\mathrm{He}}\ leading to
$\sim 100$ MeV energy release and a variety of nuclear final states.
Moreover, the reactions with thermal $\pi^+$ are very suppressed because of the
Coulomb repulsion, and thus we concentrate on $\pi^-$ absorption. A
measurement of the ground state level width $\Gamma_{1S} = (45\pm 3)\
\ensuremath{\mathrm{eV}} $~\cite{Backenstoss1974519} of pionic helium then allows us to
obtain the total low-energy in-flight cross section $(\sigma v) \simeq
7.3\ \ensuremath{\mathrm{mb}}$; for further details see
Appendix~\ref{sec:pion-kaon-capture}. The branching ratios into the
different final states have been measured~\cite{Daum1995553} to be
$(\ensuremath{\mathrm{T}} n):(\ensuremath{\mathrm{D}} nn):(pnnn) = (17\pm 9)\%:(63\pm 26)\%:(21\pm
16)\%$. Adopting the central values (with $20\%\, pnnn$) we arrive at
\begin{align}
\label{eq:pi-he4-to-T}
\ensuremath{\pi^{-}} + \ensuremath{{}^4\mathrm{He}} & \to \ensuremath{\mathrm{T}} + n\hphantom{2}: \quad (\sigma v)^{\ensuremath{\pi^{-}}}_{\ensuremath{\mathrm{T}} n}
\simeq 1.1\ \ensuremath{\mathrm{mb}} ,
\quad \,\,\,\,\, Q = 118.5\ \ensuremath{\mathrm{MeV}} ,\\
\ensuremath{\pi^{-}} + \ensuremath{{}^4\mathrm{He}} & \to \ensuremath{\mathrm{D}} + 2n : \quad (\sigma v)^{\ensuremath{\pi^{-}}}_{\ensuremath{\mathrm{D}} nn} \simeq
4.1\ \ensuremath{\mathrm{mb}} ,
\quad \,\, Q = 112.2\ \ensuremath{\mathrm{MeV}} , \\
\ensuremath{\pi^{-}} + \ensuremath{{}^4\mathrm{He}} & \to p + 3n \,\,: \quad (\sigma v)^{\ensuremath{\pi^{-}}}_{pnnn} \simeq 1.3\
\ensuremath{\mathrm{mb}} , \quad Q = 110\ \ensuremath{\mathrm{MeV}} .
\end{align}
Before using (\ref{eq:pi-he4-to-T}) in our code,
we account for Coulomb corrections, which leads to the
enhancement of $F_{\ensuremath{{}^4\mathrm{He}}\ensuremath{\pi^{-}}} \simeq 3.5$.
The reactions of "in-flight" pions with \ensuremath{{}^4\mathrm{He}}\ can also be
significantly enhanced by the presence of the delta-resonance. Again,
one has to distinguish two types of reactions, inelastic scattering,
$\pi^\pm + \ensuremath{{}^4\mathrm{He}} \to \pi + N$, and absorption, $\pi^\pm +\ensuremath{{}^4\mathrm{He}} \to N $
where $N$ represents a variety of multi-nucleon/nuclear final
states. We use the results of experimental studies
\cite{Binon:1975zc,Steinacher1990413,Mateos:1998bc} to account for the
pion-\ensuremath{{}^4\mathrm{He}}\ reactions across the delta-resonances. We also extrapolate
these results to the threshold regions, and calculate effective cross
sections to the various (exclusive) final states by averaging over the
pion ``lifetime'' trajectory while taking into account the respective
energy-dependent branchings. The details of this procedure are
summarized in Appendix~\ref{sec:inflight}.
Besides enhancing the number of free neutrons, the reactions on \ensuremath{{}^4\mathrm{He}}\
have an additional important effect: they produce energetic $A=3$
elements that are able to participate in the non-thermal reactions
leading to \ensuremath{{}^6\mathrm{Li}}. For example, reactions with thermalized $\ensuremath{\pi^{-}}$ in
Eq. (\ref{eq:pi-he4-to-T}) in $\sim 17\%$ of all cases contain \ensuremath{\mathrm{T}}\
nuclei with an energy $E^{\mathrm{in}}_{\ensuremath{\mathrm{T}}} \simeq 30\ \ensuremath{\mathrm{MeV}}$
injected into the plasma. This leads to a possible secondary source of
$\ensuremath{{}^6\mathrm{Li}}$ via their fusion on ambient alpha particles
\cite{Reno:1987qw,Dimopoulos:1988ue},
\begin{align}
\label{eq:secondary-lisx}
\ensuremath{\mathrm{T}}+\ensuremath{{}^4\mathrm{He}}|_{\mathrm{bg}} \to \ensuremath{{}^6\mathrm{Li}} + n \quad Q = -4.8\ \ensuremath{\mathrm{MeV}} .
\end{align}
The number of produced \ensuremath{{}^6\mathrm{Li}}\ per injected \ensuremath{\mathrm{T}}\ (and likewise per
injected \ensuremath{{}^3\mathrm{He}}) can be found by tracking the \ensuremath{\mathrm{T}}-degradation from
$E_{\ensuremath{\mathrm{T}},\mathrm{in}}$ until the threshold energy in the frame of the
thermal bath, $E^{\ensuremath{\mathrm{T}}}_{\ensuremath{{}^6\mathrm{Li}},\mathrm{th}} \simeq 8.4\ \ensuremath{\mathrm{MeV}}$,
\begin{align}
\label{eq:Lisx-efficiency}
N_{\ensuremath{{}^6\mathrm{Li}}} \simeq
\int_{E^{\ensuremath{\mathrm{T}}}_{\ensuremath{{}^6\mathrm{Li}},\mathrm{th}}}^{E_{\ensuremath{\mathrm{T}},\mathrm{in}}} dE_{\ensuremath{\mathrm{T}}}
\frac{n_{\he4} \sigma_{\ensuremath{\mathrm{T}}(\ensuremath{{}^4\mathrm{He}},n)\ensuremath{{}^6\mathrm{Li}}} v_{\ensuremath{\mathrm{T}}}}{|dE_{\ensuremath{\mathrm{T}}}/dt|}
,
\end{align}
where $v_{T}$ is the velocity of \ensuremath{\mathrm{T}}.
As is well-known, the production of \ensuremath{{}^6\mathrm{Li}}\ becomes more efficient at
late times, when its thermal destruction slows down. This occurs in
the regime of $\tau_\pi \ll \tau_{\rm stop}$, and most of the
energetic $A=3$ elements originate from \ensuremath{{}^4\mathrm{He}}\ reacting with in-flight
$\pi^\pm$. This complicates the treatment of finding the final \ensuremath{{}^6\mathrm{Li}}\
abundance as the branching to $A=3$ elements become energy dependent.
Moreover, non-absorptive inelastic pion-helium reactions result in
mass-3 injection spectra which are continuously distributed. We
account for all these effects with details presented in
Appendix~\ref{sec:inflight}.
In a final step we also account for potential effects on the light
element abundances coming from the ``visible'' energy injection in the
decays of the pions. Any primary electromagnetic energy deposition
$E_{\mathrm{inj}}$ is quickly dispersed in an electromagnetic cascade
with $E_{\mathrm{inj}}$ shared among a large number of photons.
Photons with energies less than $E_C \simeq
m_e^2/22T$~\cite{Kawasaki:1994sc} loose their ability to pair create
$e^{\pm}$ in scatterings on the background radiation. Those associated
``break-out'' photons can then destroy the light elements. The
earliest time at which this happens can be determined by equating
$E_C$ to the nuclear binding energies $E_{b}$ against
photodissociation:
\begin{equation}
t_{\rm ph} \simeq
\left\{ \begin{array}{lll}
2\times 10^4\ \ensuremath{\mathrm{sec}} & \mathrm{for}\ \ensuremath{{}^7\mathrm{Be}}+\gamma\to\ensuremath{{}^3\mathrm{He}}+\ensuremath{{}^4\mathrm{He}} & (E_b = 1.59\,\ensuremath{\mathrm{MeV}})\nonumber\\
5\times 10^4\ \ensuremath{\mathrm{sec}} & \mathrm{for}\ \ensuremath{\mathrm{D}}+\gamma\to n+ p &
(E_b = 2.22\,\ensuremath{\mathrm{MeV}})
\label{eq:t-dissoc}
\end{array}\right .
\end{equation}
We see that the electromagnetic energy injection plays no role in the
most interesting $X$-lifetime region $\tau_X\lesssim 10^{4}\ \ensuremath{\mathrm{sec}}$
in which the lithium depleting chain~(\ref{eq:li-depletion-chain}) is
operative. Only for $\tau_X > 10^4\ \ensuremath{\mathrm{sec}}$ do we expect an
influence on the element abundances, starting with the destruction of
\ensuremath{{}^7\mathrm{Be}}\ followed by \ensuremath{\mathrm{D}}. The deepest bound element, \ensuremath{{}^4\mathrm{He}}\ is only
destroyed for $t>10^6\ \ensuremath{\mathrm{sec}}$.
Charged pions decay into $\mu^{\pm}$ which are, however, not
initiating an electromagnetic cascade since for $t\gtrsim 10^4\
\ensuremath{\mathrm{sec}}$ their rate for Thomson scattering becomes smaller than the
muon decay rate. Thus, for simplicity, we assume that a total of
$E_{\mathrm{inj}} = \VEV{E_e} = (7/20)m_{\mu}$ per $\ensuremath{\pi^{\pm}}$ is injected
in form of electromagnetic energy, where $\VEV{E_e}$ is the average
electron energy in the muon decay. Fortunately, though this neglects a
certain fraction of accessible kinetic energy of the muon, a more
accurate treatment is inconsequential for our further discussion. On
the same footing, we also neglect a direct energy production in
$X$-decay via {\em e.g.} $X\to\ensuremath{\pi^{0}}\pizero$. More details can be
found in Appendix~\ref{sec:electr-energy-inject}.
\FIGURE[t]{
\includegraphics[width=0.6\textwidth]{PiEvol.eps}
\caption{\small Temperature evolution of light nuclei, meta-stable
parent $X$ particles and daughter \ensuremath{\pi^{\pm}}\ mesons for $Y_X = 8$ and
$\tau_{X} = 10^3\ \ensuremath{\mathrm{sec}}$ as input values and for injection not
too far from threshold. The pion induced elevation of $n/p$ around
$T_9\sim 0.5$ leads to the expected increase in the D/H and \ensuremath{{}^6\mathrm{Li}}/H
abundances, and to a decrease in \ensuremath{{}^7\mathrm{Be}}/H when compared with the
respective SBBN predictions (dotted lines); $N$ is the neutron
abundance normalized to baryons.}
\label{evolution}
}
The pion-enriched BBN was run for different input values of
$\xi^{(X)}_{\ensuremath{\pi^{\pm}}} Y_X^0$, $\tau_X$ and the pion injection energy $E_0$.
In Figure~\ref{evolution} we plot an example of one possible choice
with $Y_X^0 = 8$, $\xi^{(X)}_{\ensuremath{\pi^{\pm}}}=1$, $\tau_X = 10^3$~sec and negligible
kinetic energy of injected pions. As expected, one can see a
noticeable increase of neutrons at lithium-relevant temperatures.
This leads to a decrease in the \ensuremath{{}^7\mathrm{Be}}\ abundance and also an increase
in D/H. The secondary source of \ensuremath{{}^6\mathrm{Li}}\, although leading to a modest
increase of \ensuremath{{}^6\mathrm{Li}}\ over the SBBN prediction, turns out to be far below
the observable level of \ensuremath{{}^6\mathrm{Li}}/H $\sim 10^{-11}$. This is because the
\ensuremath{{}^6\mathrm{Li}}-burning reactions are very fast above $T_9>0.1$.
The exploration of the full parameter space yields the region in which
the primordial lithium over-production problem is solved. For this
solution, we require \ensuremath{{}^7\mathrm{Li}}/H to stay in the interval
\begin{align}
\label{eq:LIobs}
\ensuremath{{}^7\mathrm{Li}}/\ensuremath{\mathrm{H}} &= (1\div 2.5)\times 10^{-10} .
\end{align}
The ballpark of observations lie in the range $\sim 1.5\div 2.0$~(see
\textit{e.g.} \cite{Steigman:2007xt,Iocco:2008va} and references
therein), but also values on the upper end of the adopted range have
been reported \cite{2002A&A...390...91B,Melendez:2004ni}. For
completeness, we also mention that the latest observations seem to
suggest what could be called a drop-off of stars from the most
metal-poor end of the Spite
plateau~\cite{Aoki:2009ce,2010arXiv1003.4510S} (towards lower values
of \ensuremath{{}^7\mathrm{Li}}/H.) It is important to note that such a feature, as puzzling
as it may be, in itself does not alleviate the tension between the
SBBN prediction and the lithium observations, but rather enhances the
controversy. The observational status on that issue is still very
much under dispute, as is best illustrated by recent
Ref.~\cite{Melendez:2010kw}. That work does not seem to confirm such a
``sagging tail'' in \ensuremath{{}^7\mathrm{Li}}/H but rather finds two plateau values, $\sim
1.5$ and $1.9\times 10^{-10}$. At this point, we choose the range
(\ref{eq:LIobs}) based on the overall range of lithium abundance in
the metal-poor stars along the Spite plateau. Should the mechanisms
for astrophysical depletion of the lithium abundance firm up, the
range (\ref{eq:LIobs}) must be shifted upward.
In addition to (\ref{eq:LIobs}), we shall also put the following
constraints on the remaining light elements which are produced in
observable quantities in BBN:
\begin{align}
\label{eq:Dobs}
\ensuremath{\mathrm{D}}/\ensuremath{\mathrm{H}} & \leq 4\times 10^{-5} , \\
\label{eq:HEobs}
0.24 & \leq Y_p \leq 0.26 , \\
\label{eq:LI6obs}
\ensuremath{{}^6\mathrm{Li}}/\ensuremath{\mathrm{H}} & \leq 6\times 10^{-11}, \\
\label{eq:hetobs}
\ensuremath{{}^3\mathrm{He}}/\ensuremath{\mathrm{D}} &\leq 1.
\end{align}
The upper limit~(\ref{eq:Dobs}) corresponds to the highest reliable
determination~\cite{Burles:1997fa,Kirkman:2003uv} of D/H in a QSO
absorption system with a simple enough velocity structure. Given the
significant scatter in the various determinations of its primordial
value, the possibility of deuterium astration remains and values as
high as~(\ref{eq:Dobs}) cannot be convincingly excluded at the present
stage. For \ensuremath{{}^4\mathrm{He}}\ the inference of its primordial mass fraction is
potentially plagued by systematic uncertainties and values in the
range~(\ref{eq:HEobs}) have been witnessed over the years with most
recent determinations being on the higher
side~\cite{Izotov:2010ca,Aver:2010wq}. Observations of the isotopic
ratio $\ensuremath{{}^6\mathrm{Li}}/\ensuremath{{}^7\mathrm{Li}}$ in metal poor halo stars are extremely
difficult. Though a $5\%$ plateau value has been
claimed~\cite{Asplund:2005yt}, its inference has also been
challenged~\cite{Steffen:2010pe} so that we only impose
(\ref{eq:LI6obs}) as an upper limit. Finally, with \ensuremath{\mathrm{D}}\ being more
fragile than \ensuremath{{}^3\mathrm{He}}, the ratio \ensuremath{{}^3\mathrm{He}}/\ensuremath{\mathrm{D}}\ is a monotonically increasing
function of time so that the solar-system
value~(\ref{eq:hetobs})~\cite{1993oee..conf.....P} provides an upper
limit on the primordial value.
\FIGURE[t]{
\includegraphics[width=0.6\textwidth]{contour_pi+em.eps}
\caption{\small Light element abundance yields in the plane of
$X$-lifetime vs. $X$-abundance (prior to decay),
$(\tau_X,\xi_{\pi}^{(X)}Y_X)$ with $\xi^{(X)}_{\pi}=1$ and
negligible kinetic energy of injected pions. The band shows the
region $ 10^{-10}\leq \ensuremath{{}^7\mathrm{Li}}/\ensuremath{\mathrm{H}} \le 2.5\times 10^{-10} $ in which
the BBN lithium prediction is reconciled with observationally
inferred primordial values. In the dark (blue) shaded part all
limits (\ref{eq:Dobs})-(\ref{eq:hetobs}) on the remaining light
elements are respected. Contours of constant helium mass fraction
$Y_p$ are shown by solid lines and of constant $\ensuremath{\mathrm{D}}/\ensuremath{\mathrm{H}}$
abundance by dotted lines; the dashed line in the upper right corner
corresponds to $\ensuremath{{}^6\mathrm{Li}}/\ensuremath{\mathrm{H}} = 6\times 10^{-11}$ with smaller values
anywhere below. The star is the point in parameter space for which
Fig.~\ref{evolution} was obtained.}
\label{figure1}
}
\FIGURE[t]{
\includegraphics[width=0.6\textwidth]{contour_pi+em180MeV.eps}
\caption{\small Same as in Figure \ref{figure1}, but with a kinetic
energy of injected pions $E_0 = 180$~MeV which approximately
corresponds to the maximum of the pion-nucleon cross section. }
\label{figure1b}
}
In Figure~\ref{figure1} we present a parameter scan in the
$(\tau_X,\xi^{(X)}_{\pi}Y_X)$-plane with $\xi^{(X)}_{\pi}=1$, in the
assumption of negligible kinetic energy or injected pions, so that all
reactions and decays occur at rest. The star is the point in parameter
space for which Fig.~\ref{evolution} was obtained. The shaded band
shows the region in which \ensuremath{{}^7\mathrm{Li}}/H is within the observationally
favored range (\ref{eq:LIobs}). More pions become available for
larger values of $Y_X$ so that (\ensuremath{{}^7\mathrm{Li}}+\ensuremath{{}^7\mathrm{Be}})/\ensuremath{\mathrm{H}}\ decreases from its
SBBN value $\sim 5\times 10^{-10}$ at $Y_X= 1$ to $2.5\times 10^{-10}$
at the lower border of the shaded band. As expected, there is also an
associated increase of D/H from its SBBN value, $\sim 2.6\times
10^{-5}$, with contours of constant deuterium shown by the dotted
lines and running almost in parallel to constant lithium. The lowest
D/H value along the \ensuremath{{}^7\mathrm{Li}}/H band is $ 3.25 \times 10^{-5}$ (for
$\tau_X\leq 10^4\ \ensuremath{\mathrm{sec}}$). The contours of constant \ensuremath{{}^4\mathrm{He}}\ are
shown by the solid lines. The effect on \ensuremath{{}^4\mathrm{He}}\ becomes stronger for
smaller $X$-lifetime values. As is well known, \ensuremath{{}^4\mathrm{He}}\ exhibits the
strongest sensitivity on neutron/proton interconversions which take
place before the opening of the deuterium ``bottleneck''. We further
observe that the secondary production of
$\ensuremath{{}^6\mathrm{Li}}$~(\ref{eq:pi-he4-to-T}) becomes only important for large
lifetimes $\tau_X\gtrsim 10^4\ \ensuremath{\mathrm{sec}}$ (dashed line). This, however,
is already the region in the parameter space in which the proposed
mechanism for lithium depletion by extra neutrons becomes
inefficient. By the same token, the photo-destruction of light
elements remains decoupled from the main effect as it only happens at
late times~(\ref{eq:t-dissoc}). The destruction of \ensuremath{{}^7\mathrm{Be}}\ and \ensuremath{\mathrm{D}}\
can respectively be seen by the downward trend of the lithium band as
well as by the dotted curve labeled ``2''. Finally, we note that the
ratio~(\ref{eq:hetobs}) of $\ensuremath{{}^3\mathrm{He}}/\ensuremath{\mathrm{D}}$ is never saturated throughout
the whole $(\tau_X,Y_X)$-plane. The dark (blue) shading of the band
corresponds to the region in which all constraints (\ref{eq:LIobs})
and (\ref{eq:Dobs})-(\ref{eq:hetobs}) are obeyed. It is this region,
for which the pion injection provides a satisfactory solution to the
lithium overabundance problem.
As was argued earlier, pions with kinetic energy close to the
delta-resonance may have a considerably larger efficiency of $p\to n $
conversion below $T_9 \simeq 0.4$. To demonstrate this point, we take
the same assumptions as in Fig.~\ref{figure1} but assume a primary
pion injection energy of $E_0=180$~MeV. The results of the new scan
are presented in Fig.~\ref{figure1b}. The increased efficiency of
$p\to n$ conversion (along with stronger \ensuremath{\pi^{\pm}}-\ensuremath{{}^4\mathrm{He}}\ reactions) leads
to a lowering of the bands, \textit{i.e.}~less pions per baryon are
necessary for the required depletion of \ensuremath{{}^7\mathrm{Li}}. Note, however, that the
overall effect is somewhat milder than initially expected. For a
lifetime $\tau_X = 10^3\,\ensuremath{\mathrm{sec}}$ at which pions are still being
stopped, one finds a factor $\sim 2\div 3$ reduction in $Y_X$. For
larger lifetimes it only becomes about half an order of magnitude
(note the slightly different scales on the y-axes of
Figs.~\ref{figure1} and \ref{figure1b}.) Though \ensuremath{{}^7\mathrm{Be}}\ is indeed
efficiently destroyed by strongly elevated neutron concentrations it
turns out that the limiting factor is the final step in the overall
\ensuremath{{}^7\mathrm{Be}}+\ensuremath{{}^7\mathrm{Li}}\ depleting chain~(\ref{eq:li-depletion-chain}),
\textit{i.e.} the subsequent \ensuremath{{}^7\mathrm{Li}}-destruction via thermal proton
burning, $\ensuremath{{}^7\mathrm{Li}}+p\to\ensuremath{{}^4\mathrm{He}}+\ensuremath{{}^4\mathrm{He}}$. This latter reaction effectively shuts
off for $T_9<0.3$ and the overall lithium abundance is produced in
form of \ensuremath{{}^7\mathrm{Li}}.
Residual
late $X$-decays at temperatures $T_9\lesssim 0.5$ also lead to the
narrowing of the lithium band at small lifetimes
$\tau_X< 10^3\,\ensuremath{\mathrm{sec}}$ and large $Y_X$. We
remark in passing that in that region, the accompanying increase in D
also leads to an enhancement in the $n$-sourcing thermonuclear DD and
DT fusion reactions.
\subsection{Decays to kaons }
\label{sec:treatment-kaons}
Aiming at a reasonably accurate treatment of kaons is even more
difficult. Let us first focus on $\ensuremath{K^{\pm}}$, and assume for simplicity
that the kinetic energy is relatively small, so that one can neglect
the effects of the decay in flight. In the case of kaon injection it
is important to include their hadronic decays into charged pions, as
well as the direct interaction of kaons with nucleons. The production
of charged pions in kaon decays is relatively easy to account for:
\begin{align}
\ensuremath{K^{\pm}} & \to
\begin{cases}
\ensuremath{\pi^{\pm}} \ensuremath{\pi^{0}} (\ensuremath{\pi^{0}}) & 22.4\% \nonumber \\
\ensuremath{\pi^{\pm}} \ensuremath{\pi^{+}} \ensuremath{\pi^{-}} &5.6\%
\end{cases}
\end{align}
which leads to a comparable population of $\ensuremath{K^{\pm}}$ and $\ensuremath{\pi^{\pm}}$. As has
been already mentioned, there is no need to track muons (and the
associated neutrino yield) since $ P_{p\to n}^\nu \ll P_{p\to n}^\pi
$. For the average kinetic energy of produced charged pions we take
$E_0\sim 80$~MeV. Even in the two-body decay, \ensuremath{\pi^{\pm}}\ have energies of
only $\sim110\ \ensuremath{\mathrm{MeV}}$ so that they fall somewhat lower than the
$\Delta$-resonance, and their incomplete stopping at late times will
not have a large effect on the efficiency of $p\to n$ conversion.
We now turn to the calculations of the direct capture of kaons on
nucleons. Whereas the charge exchange reaction $\ensuremath{K^{-}} + p \to \bar{K}^0 +
n$ has $Q=-5.2\ \ensuremath{\mathrm{MeV}}$ and is thus not allowed kinematically on
threshold, \ensuremath{K^{-}}\ reactions on nucleons can proceed via the "$s$-quark
exchange reactions" with hyperons and pions in the final state:
\begin{align}
\label{eq:kprct}
\ensuremath{K^{-}} + p &\to \Sigma^{\pm}\pi^{\mp},\, \Sigma^0 \ensuremath{\pi^{0}},\, \Lambda
\ensuremath{\pi^{0}} , \\
\label{eq:kprct2}
\ensuremath{K^{-}} + n &\to \Sigma^{-} \ensuremath{\pi^{0}},\, \Sigma^0\ensuremath{\pi^{-}},\, \Lambda\ensuremath{\pi^{-}} .
\end{align}
In order to obtain the inclusive $n\leftrightarrow p$ interconversion
cross sections, one then has to take into account the decays of the
strange baryons in the final states
\begin{align}
\Sigma^{+} & \to
\begin{cases}
p \ensuremath{\pi^{0}} & 51.6\% \nonumber \\
n \ensuremath{\pi^{+}} &48.3\%
\end{cases} , &
\Lambda & \to
\begin{cases}
p \ensuremath{\pi^{-}} & 63.9\%\\
n \ensuremath{\pi^{0}} &35.8\%
\end{cases} , \\
\Sigma^{-} & \to \,\,\,\,\, n \ensuremath{\pi^{0}} \quad \,\,99.8\% , &
\Sigma^{0} & \to \,\,\,\,\, \Lambda \gamma \quad \,\,\,100\% .
\label{eq:Hyperon-branchings}
\end{align}
The cross section for each of the processes (\ref{eq:kprct}) and
(\ref{eq:kprct2}) can be obtained in terms of an isospin ($I$)
decomposition of the wave-functions of the reactants. Together with
the knowledge of the relative phases $\phi$ of the individual isospin
amplitudes, the cross sections can then be inferred from the total
$\ensuremath{K^{-}}$-absorption cross sections of a given~$I$,
\begin{align}
\label{eq:isospin-abs-cs}
\sigma_I \simeq \frac{4\pi b_I}{k} \left| \frac{1}{1-i k A_I} \right|^{2} ,
\end{align}
where $A_I= a_I + i b_I$ are complex scattering lengths; $k$ is the
c.m.~momentum of the incoming state.
Taking into account the deviations from charge independence due to
$n$, $p$, and $\ensuremath{K^{\pm}}$ and $K^0$ mass differences
following~\cite{Dalitz:1960du,Martin:1970je} and using $A_{0} = (-1.74
+ i 0.70 )\ \ensuremath{\mathrm{fm}}$, $A_{1} = (-0.05 + i 0.63 )\ \ensuremath{\mathrm{fm}}$,
$\phi_{\mathrm{th}} = -52.9^\circ$ as well as a ratio $0.34$ of
$\Lambda\pi^{0}$ to total $I=1$ hyperon
production~\cite{Martin:1970je} one arrives at the values for
(\ref{eq:kprct}) and (\ref{eq:kprct2}).
In a final step, we use the branching
ratios~(\ref{eq:Hyperon-branchings}) to obtain the inclusive threshold
cross sections for proton/neutron interconversion,
\begin{align}
\label{KmN}
\ensuremath{K^{-}} + p & \to n + X : \quad (\sigma v)^{\ensuremath{K^{-}}}_{pn} \simeq 32\ \ensuremath{\mathrm{mb}} ,\\
\label{KmN2}
\ensuremath{K^{-}} + n & \to p + X : \quad (\sigma v)^{\ensuremath{K^{-}}}_{np} \simeq 13\ \ensuremath{\mathrm{mb}} ,
\end{align}
where in~(\ref{KmN}) the Coulomb correction has been factored
out. Whereas the first cross section is in good agreement with the
value previously used in the BBN context, the second one is a factor
of two smaller than in~\cite{Reno:1987qw}. In this regard, we remark
in passing that the latter reactions~(\ref{eq:kprct2}) are only due to
$I=1$ scatterings.
In principle, the nucleons for some of the final states of (\ref{KmN})
and (\ref{KmN2}) can have energies on the order of 30 MeV. While
protons will be stopped before inducing any nuclear changes, neutrons
with such energies are stopped primarily via interactions with $p$ and
\ensuremath{{}^4\mathrm{He}}, and could split \ensuremath{{}^4\mathrm{He}}\ nuclei in the collisions into some of its
constituents. However, since \ensuremath{{}^4\mathrm{He}}\ is one order of magnitude less
abundant than $p$, and since the maximum energy of $n$ is close to the
\ensuremath{{}^4\mathrm{He}}\ desintegration threshold, we neglect such secondary effects.
In addition, we further note that due to isospin invariance we do not
need to consider hyperon production processes in $\ensuremath{K^{+}} n$ scatterings.
As in the case of $\ensuremath{\pi^{-}} + \ensuremath{{}^4\mathrm{He}}$, reactions of $\ensuremath{K^{-}}$ on helium have not
been considered in the BBN context. Fortunately, the measurements of
Ref.~\cite{PhysRevD.1.1267} provide us with a detailed list of
branching ratios for $\ensuremath{K^{-}}$ absorption on \ensuremath{{}^4\mathrm{He}}\ at rest. By accounting
for the decay modes~(\ref{eq:Hyperon-branchings}) we find the
following particle multiplicities per $\ensuremath{K^{-}}$ absorption
\begin{alignat}{2}
\xi_{\ensuremath{{}^3\mathrm{He}}} &\simeq \xi_{\ensuremath{\mathrm{T}}} \simeq 0.13 , & \xi_{n} & = 1.57 -
0.34\lambda_{\ensuremath{\mathrm{D}}} , \nonumber\\
\xi_{\ensuremath{\mathrm{D}}} & = 0.17 + 0.34\lambda_{\ensuremath{\mathrm{D}}},\qquad & \xi_{p} &= 1.29 -
0.34\lambda_{\ensuremath{\mathrm{D}}} .
\label{eq:branchings-he4-Kminus}
\end{alignat}
When a final state is not resolved we have, for simplicity, assumed
that a fraction of $\lambda_{\ensuremath{\mathrm{D}}}$ is released in form of \ensuremath{\mathrm{D}}\ and
$(1-\lambda_{\ensuremath{\mathrm{D}}})$ in form of nucleons, and we adopt
$\lambda_{\ensuremath{\mathrm{D}}}=0.5$. Thus, for example, in the reaction $\ensuremath{K^{-}}+\ensuremath{{}^4\mathrm{He}}\to
\Lambda (\Sigma^0) (pnn)$, which occurs with branching fraction
$22.5\%$, we assume $50\%$ $\Lambda (\Sigma^0) (Dn)$ and $50\%$
$\Lambda (\Sigma^0) (pnn)$. Thereby, we are neglecting a certain
fraction of mass-3 nuclei. In this regard, note that---unlike in the
case of the threshold reaction $\ensuremath{{}^4\mathrm{He}}+\ensuremath{\pi^{-}}$---an accurate computation of the secondary
\ensuremath{{}^6\mathrm{Li}}-yield is more difficult as the \ensuremath{{}^3\mathrm{He}}/\ensuremath{\mathrm{T}}\ injection energy
is now continuous.
However, as we have already seen in the previous section, \ensuremath{{}^6\mathrm{Li}}\
production in excess of observationally constrained levels is an issue
only in the region where (\ref{eq:li-depletion-chain}) looses its
efficiency. Thereby, we shall make the simple assumption that on
average $\ensuremath{\mathrm{T}}$ and $\ensuremath{{}^3\mathrm{He}}$ carry one third of the liberated energy,
$\VEV{E_{\ensuremath{\mathrm{T}}}}\sim 30\ \ensuremath{\mathrm{MeV}}$ and $\VEV{E_{\ensuremath{{}^3\mathrm{He}}}}\sim 50\ \ensuremath{\mathrm{MeV}}$,
respectively. (With our assumptions) the latter value is higher as
\ensuremath{{}^3\mathrm{He}}\ stems predominantly from processes with $\Lambda$ and not
$\Sigma$ production.
Finally, we note that the contribution of hyperfragments of
$_{\Lambda}^{4}\mathrm{He}$ is $\sim 2\%$~\cite{PhysRevD.1.1267} which
thus does not pose any further complication.
With the above multiplicities~(\ref{eq:branchings-he4-Kminus}) one
then obtains effective cross sections for each isotope in the final
state,
\begin{align}
\ensuremath{K^{-}} +\ensuremath{{}^4\mathrm{He}} \to N + \Pi: \quad (\sigma v)_N^{\ensuremath{K^{-}}} = \xi_N
(\sigma_{\ensuremath{{}^4\mathrm{He}}}^{\ensuremath{K^{-}}} v) ,
\end{align}
with $\Pi$ symbolizing an arbitrary pionic final state and $N =
\ensuremath{{}^3\mathrm{He}},\,\ensuremath{\mathrm{T}},\,\ensuremath{\mathrm{D}},\,n$ or $p$. The extraction of $(\sigma
v)_N^{\ensuremath{K^{-}}}$ is similar to the pion case and is given
by~(\ref{eq:sigma-Km-hef}) in Appendix~\ref{sec:pion-kaon-capture}.
In order to take into account the effects from electromagnetic energy
injection in kaon decays we recall from the previous section that
$\ensuremath{\pi^{\pm}}$ as well as $\mu^{\pm}$ are unlikely to initiate an
electromagnetic cascade for $\tau_{X}\gtrsim 10^{4}\ \ensuremath{\mathrm{sec}}$. The
multiplicities in the $\ensuremath{K^{\pm}}$-decay to $\mu^{\pm}$ (including muons
from $\ensuremath{\pi^{\pm}}$ final states) and to $\pi^0$ are $\xi_{\mu}^{\ensuremath{K^{\pm}}}\simeq
1$ and $\xi_{\pi^0}^{\ensuremath{K^{\pm}}}\simeq 0.3$, respectively. For the
electromagnetic energy release in the muon decay we again take
$E_{\mathrm{inj}}^{\mu}\simeq m_{\mu}/3$ and for $\pi^0$ we assume
that, in addition to their rest mass, they carry on average
approximately one third of the energy released in the particular decay
channel, yielding $E_{\mathrm{inj}}^{\pi^0}\sim 250\ \ensuremath{\mathrm{MeV}}$. Taken
together, this amounts to a ``branching fraction''
$\mathrm{Br}_{K^{\pm}\to {\mathrm{vis}}}\sim 0.55$ and a total
electromagnetic energy injection of $E_{\mathrm{inj}} \sim 2\times
\mathrm{Br}_{K^{\pm}\to {\mathrm{vis}}} m_{\ensuremath{K^{\pm}}}$ per $X$-decay. As
for simplicity we assume that the kaon kinetic energy is small, we do
not have to worry about the increase of the electromagentic deposition
by decaying in-flight kaons.
However, we emphasize that even in the case of slow kaons, and because
of the exponential sensitivity to $\tau_X$, $\Orderof{10}$ \ensuremath{K^{\pm}}\ per
baryon will have strong effect on nuclear abundances once a
photodissociation threshold is met for $\tau_X > 10^4\ \ensuremath{\mathrm{sec}}$.
\FIGURE[t]{
\includegraphics[width=0.6\textwidth]{contour_ka+em+piflight.eps}
\caption{\small Same as in Fig.~\ref{figure1}, but for the case of
$\ensuremath{K^{\pm}}$ injection and assuming one charged kaon pair per $X$-decay,
$\xi^{(X)}_{\ensuremath{K^{\pm}}} = 1$, being injected close to the kinematic
threshold. Because of the larger cross sections of $K^-$ on
nucleons and helium, one requires a fewer number of initial $X$
decays. The trend of depleting $\ensuremath{{}^7\mathrm{Be}}+\ensuremath{{}^7\mathrm{Li}}$ for larger values of
$Y_X$ is eventually reverted because of the enhanced production of
lithium in form of \ensuremath{{}^7\mathrm{Li}}\ due to numerous extra neutrons.}
\label{figure2}
}
The kaon-induced solution to the cosmological lithium problem is
plotted in Figure~\ref{figure2}. In contrast to the case of primary
pions at rest (see Fig.~\ref{figure1}), a reduction of $\ensuremath{{}^7\mathrm{Li}}/\ensuremath{\mathrm{H}}$ to
observed values~(\ref{eq:LIobs}) is already possible for one injected
kaon per baryon. This is evidently due to the fact that $\ensuremath{K^{-}}$ cross
sections on nucleons and helium are significantly larger than for
thermal $\ensuremath{\pi^{-}}$. Moreover, pions from kaon decays only amplify any effect on
the light elements. Notice, however, that for too high of an
$n$-abundance (large $Y_X$) $\ensuremath{{}^7\mathrm{Li}} + \ensuremath{{}^7\mathrm{Be}}$ starts to increase
again. Though $\ensuremath{{}^7\mathrm{Be}}$ is still being depleted, net production of
lithium in form of $\ensuremath{{}^7\mathrm{Li}}$ eventually takes over. Similar to the case
of primary $\ensuremath{\pi^{\pm}}$ we find that solutions corresponding to
$\tau_{X}\gtrsim 10^4\ \ensuremath{\mathrm{sec}}$ are not viable because of \ensuremath{{}^6\mathrm{Li}}\
overproduction. This again excludes the potentially interesting region
with $\tau_{X}=\mathrm{few}\times 10^5\ \ensuremath{\mathrm{sec}}$ in which previously
formed deuterium is destroyed in photo-dissociation and brought back
to its SBBN prediction.
We now turn to the case of injected neutral kaons $K^0,\bar
K^0$---propagating as $K_L$ and $K_S$. Whereas $K_S$ definitely decay before
interacting with the nuclear background, the relatively large $K_L$
lifetime $\tau_{K_L} = 5.1\times 10^{-8}\ \ensuremath{\mathrm{sec}}$ may render its
impact on the BBN output even more drastic then in the $\ensuremath{K^{\pm}}$ case. In
contrast to $\ensuremath{K^{\pm}}$, however, $K_L$ are not stopped by electromagnetic
interactions so that one should take into account the energy dependence
of their cross sections with nucleons. Again, we avoid this complication by assuming the
injection of kaons close to the threshold.
For $K_L$-nucleon scattering we exploit charge indepence and use
isospin relations of the strong interaction processes. Indeed,
conservation of strangeness implies that the inelastic scattering of
$K_L$ on nucleons essentially resembles the one of $\bar K^0$, as
$K_L = 2^{-1/2}(K^0-\bar K^{0})$ up to small
$CP$-violating corrections. Thus, up to corrections due to $\ensuremath{K^{\pm}}$, $\bar K^0$,
and $n$, $p$ mass differences and other isospin-violating effects, one finds
\begin{align}
\label{eq:KLrct1}
\sigma(K_L p\to Y \pi) &\simeq \frac{1}{2} \sigma(\bar K^0 p\to Y \pi
) \simeq \frac{1}{2} \sigma(K^{-} n \to \tilde Y \tilde\pi)
\\
\label{eq:KLrct2}
\sigma(K_L n\to Y' \pi') &\simeq \frac{1}{2} \sigma(\bar K^0 n\to Y' \pi'
) \simeq \frac{1}{2} \sigma(K^{-} p \to \tilde Y' \tilde\pi')
\end{align}
where $Y,Y'(\pi,\pi')$ stands for a hyperon (pion) and $\tilde
Y,\tilde Y'(\tilde\pi,\tilde\pi')$ are the associated states with
flipped third isospin component---the final states
of~(\ref{eq:kprct2}) and (\ref{eq:kprct}), respectively. The energy
dependence of the cross sections not far from the threshold is
inferred from~(\ref{eq:isospin-abs-cs}). Indeed, since
(\ref{eq:KLrct1}) is a pure $I=1$ process, $ \sigma(K_L p\to Y \pi) =
\sigma_1/2$. With the branching fractions
(\ref{eq:Hyperon-branchings}) the inclusive cross sections for $p\to n
$ as well as for $n\to p$ conversion are then readily found from
(\ref{eq:KLrct1}) and (\ref{eq:KLrct2}), respectively.
In addition to the reactions listed above, the following processes are
possible: $K_{L} n \to \ensuremath{K^{-}} p$, $K_{L} p \to \ensuremath{K^{+}} n$ and $K_{L} p \to
K_{S} p$. The cross sections for the former two processes are
approximately equal, with $K_L$ scattering being mediated by its
$\bar{K}^0$ or $K^0$ component, so that they can be inferred from an
expression for $\ensuremath{K^{-}} p \to \bar{K}^0 n$ in~\cite{Martin:1970je} by
detailed balancing. Though the above processes potentially contribute
to the loss of $K_{L}$ and/or generation of $\ensuremath{K^{\pm}}$, the probability of
such processes within a kaon lifetime is very small so that they can
be safely neglected.
In addition to the processes discussed above, one should again account
for the induced charged pion population from to $K_L$ and $K_S$
decays. The respective branching fractions read
\begin{align}
K_L \to \ensuremath{\pi^{+}} X, ~ \ensuremath{\pi^{-}} X :\quad 46.3\% ,
\qquad
K_S \to \ensuremath{\pi^{+}} \ensuremath{\pi^{-}} :\quad 69.2\% ,
\end{align}
where $X$ is the inclusive particle final state without charged pions
(not to be confused with $X$-relics). As in the case of charged kaons,
the kinetic energy of outgoing pions can be as large as $m_K/2 - m_\pi
\simeq 110$ MeV, and already in the regime where $p+\ensuremath{\pi^{-}}$ cross
sections is enhanced. Finally, we would like to make a comment on
\ensuremath{{}^4\mathrm{He}}-$K_L$ reactions. As we have just seen, $K_L$ scattering on
nucleons can be related to $\ensuremath{K^{\pm}}$-reactions using charge independence
and isospin symmetry. On these grounds there is no obvious reason to
suspect that $K_L$-reactions on helium will be drastically different
in magnitude when compared to $\ensuremath{K^{-}}$.
Instead of attempting to infer the \ensuremath{{}^4\mathrm{He}}-$K_L$ cross sections we shall
rather outline some general features connected to the $K_L$ case. By
turning off the $\ensuremath{K^{-}}$ reactions on helium in the Boltzmann code we can
directly assess the difference between the charged and neutral kaon
case. Whenever the $K_L$ energy in the final state is very small,
\emph{i.e.} on the order of a few MeV, the results are essentially
identical to Fig.~\ref{figure2}.
To conclude the kaon section, we would like to comment that several
avenues for improvement still exist after our analysis. For example,
the treatment of pions from kaon decays is done via assigning them a
simplistic average energy, while of course a continuous energy
spectrum would be required. Furthermore, the effects of incomplete
stopping of $\ensuremath{K^{-}}$ and energetic $K_L$ would also need to be included
if one intends to explore the BBN sensitivity to a wider range of
$m_X$.
Nevertheless, we also remind the reader that the effect of in-flight
pions is not as drastic as naively expected so that the corrections
due to continuously distributed $\ensuremath{\pi^{\pm}}$ from kaon decays will be only
very moderate. Moreover, we close this section by remarking that, in
principle, no further conceptual complications arise when considering
more energetic injections of $\ensuremath{K^{\pm}}$ and $K_L$ and that such an
analysis can readily be performed by employing a treatment along the
lines of the previous section.
\subsection{Decays to muons and neutrinos}
Stopped muons in the final state of the $X$-decay are sourcing
energetic muon- and electron neutrinos.
In order to include these neutrinos in the set of Boltzmann equations
we shall use the fact that $\Gamma^{\nu}_{\rm stop}$ is slower than
the Hubble rate. We account for the continuous neutrino injection and
energy redshifting by calculating the neutrino phase space
distribution function $f(T,E_\nu)$. Previous analysis of BBN modified
in the presence of the energetic neutrinos \cite{Kanzaki:2006hm}
concentrated on the effects of energy deposition and neglected the
direct nuclear-chemical impact of
neutrinos~\cite{1984MNRAS.210..359S}, which turns out to be a more
important effect at early times.
The calculation of $f(T,E_\nu)$ depends on the injection rate
$\Gamma_{\mathrm{inj}}$, and on the primary spectrum of neutrinos at
the time of injection,
\begin{eqnarray}
\label{eq:primary-nu-spectrum}
F^0_{e,\, \mu}(E,E_0) = \left\{
\begin{array}{rll}
\nu_e,\,\overline\nu_e: &12E^2(E_0-E)E_0^{-4} &{\rm for}\quad 0<E<E_0,\\
\nu_{\mu},\,\overline\nu_{\mu}: &2E^2(3E_0-2E)E_0^{-4} &{\rm for}\quad 0<E<E_0,\\
&0 &{\rm for}\quad E>E_0.
\end{array}
\right.
\end{eqnarray}
where $E_0 \simeq m_{\mu}/2$ is the neutrino end-point energy in the
muon decay, and $F^0_{e,\, \mu}$ is normalized to unity, $\int F^0_{e,\,
\mu} dE = 1$. Once injected, neutrinos are subject to flavor
oscillations so that the primary energy spectrum will be
``distorted''.
Apart from vacuum oscillations, the neutrino propagation is affected by
the coherent neutral- and charged-current interactions with particles in primordial
plasma. To assess the importance of neutrino-refraction we can
compare the vacuum contributions in the neutrino Hamiltonian,
\begin{align}
\label{eq:vac-osc}
\frac{\Delta m_{\mathrm{sol}}^2}{4E} \gtrsim 10^{-13}\ \ensuremath{\mathrm{eV}} , \qquad \frac{|\Delta
m_{\mathrm{atm}}^2|}{4E} \gtrsim 10^{-11}\ \ensuremath{\mathrm{eV}},
\end{align}
with the matter-induced potential $V_{\mathrm{M}}$; $E < E_0$. Here,
$\Delta m_{\mathrm{sol}}^2 \simeq 7.7\times 10^{-5}\ \ensuremath{\mathrm{eV}}^2$ and
$|\Delta m_{\mathrm{atm}}^2| \simeq 2.4 \times 10^{-3}\ \ensuremath{\mathrm{eV}}^2$ are the
respective mass-squared differences responsible for solar and
atmospheric neutrino mixing~\cite{Amsler:2008zzb}. For $V_{\mathrm{M}}$
one can write (see \textit{e.g.} \cite{Raffelt:1996wa})
\begin{align}
\label{eq:mat-osc}
V_{\mathrm{M}}
& = \pm \left( 8\times 10^{-19} \ \ensuremath{\mathrm{eV}} \right) T_9^3 \times
\begin{cases}
( - 1 + 4 Y_{\nu_e} + 3 Y_e) & \mathrm{for}\ \nu_e (\overline\nu_e) \\
( - 1 + 2 Y_{\nu_e} + Y_e) & \mathrm{for}\ \nu_{\mu,\tau} (\overline\nu_{\mu,\tau})
\end{cases} ,
\end{align}
where the $Y_i$ denote the particle-antiparticle asymmetries
normalized to baryons. The numerical factor in front of (\ref{eq:mat-osc}) is $G_F n_b/\sqrt{2}$ with
the overall $+(-)$ sign for $\nu (\overline{\nu})$. Unlike the case of charged leptons
where $Y_e = \Orderof{1}$, the asymmetry in the neutrino sector could be large.
In this paper we do not consider such scenarios and limit the asymmetry
in the neutrino sector by requiring
$|Y_{\nu}| < 10^4$. The last condition corresponds to $|\xi_{\nu}|
\equiv |\mu_{\nu}|/T < 5\times 10^{-5}$ with $\mu_{\nu}$ being the
neutrino chemical potential. Comparing (\ref{eq:vac-osc}) with
(\ref{eq:mat-osc}) then implies that the flavor-evolution of injected
neutrinos is given by their vacuum oscillations. Note that the
restriction on the neutrino chemical potentials also renders
neutrino-neutrino self-interactions unimportant which usually have the
effect of locking the neutrino modes to each other, leading to a
coherent oscillatory behavior~\cite{Pastor:2001iu}.
A major simplification in calculation
of $f(T,E_\nu)$ occurs due to the large rates for vacuum
oscillations,
\begin{align}
\Gamma_{i,\,\mathrm{osc}} = 1.2\times 10^8 \left(\frac{1\
\ensuremath{\mathrm{MeV}}}{E}\right)\left(\frac{|\Delta m_i^2|}{1\ \ensuremath{\mathrm{eV}}^{2}}\right) \gtrsim
\begin{cases}
175\ \ensuremath{\mathrm{sec}}^{-1} & i = \mathrm{sol} \\5.5\times 10^3\ \ensuremath{\mathrm{sec}}^{-1}
&i = \mathrm{atm}
\end{cases}
\end{align}
compared to Hubble rate and neutrino injection rate,
$ \Gamma_{i,\,\mathrm{osc}} \gg H, \Gamma_X.$
This allows us to replace the primary injection spectrum $F^0_{e}$ in
(\ref{eq:primary-nu-spectrum}) by an effective one,
\begin{align}
\label{eq:primary-effective}
F_e &= \VEV{P_{ee}} F_e^0 + \VEV{P_{\mu e}}F_{\mu}^{0}
\end{align}
where the $\VEV{P_{ee}}$ and $\VEV{P_{\mu e}}$ are the
${\nu_e}(\overline\nu_e)$-survival and $\nu_{\mu}(\overline\nu_{\mu})
\to \nu_e (\overline\nu_e)$-appearance probabilities, averaged over
oscillations,
\begin{align}
\VEV{P_{ee}} & = 1 - \frac{1}{2} \sin^2{2\theta_{12}} \simeq 0.57 , \\
\VEV{P_{\mu e}} &= \VEV{P_{ e\mu}} =
\frac{1}{2}\sin^2{2\theta_{12}}\cos^2(\theta_{23}) \simeq 0.23 .
\end{align}
The vacuum mixing angles are given by $\sin^2\theta_{12} = 0.312$ and
$\sin^2\theta_{23} = 0.466$~\cite{Amsler:2008zzb} and we have assumed
$\theta_{13} = 0 $. Note that (\ref{eq:primary-effective}) already
accounts for the appropriate reduction of the electron neutrino flux
due to $\nu_e(\overline\nu_e)\to\nu_{\mu}(\overline\nu_{\mu})$ and
$\nu_e(\overline{\nu}_e)\to\nu_{\tau}(\overline{\nu}_{\tau})$
disappearances
\footnote{In a similar fashion, $F_{\mu} = \VEV{P_{\mu\mu}} F_\mu^0 +
\VEV{P_{e \mu}}F_{e}^{0}$ and $F_{\tau} = \VEV{P_{e\tau }} F_e^0 +
\VEV{P_{\mu \tau}}F_{\mu}^{0}$ with $\int dE (F_e + F_{\mu}
+F_{\tau}) = 2$.}
Defining $f(T,E_\nu)$ in such a way that $\int f(T,E_\nu)dE_\nu $ is
equal to the total number of energetic neutrinos per baryon, we arrive
at the neutrino distribution function,
\begin{align}
\label{eq:neut-distro}
f_{e}(T,E_\nu) = \int_{T}^{\infty} \fr{dT_1\Gamma_{\rm inj
}Y_X(T_1)}{H(T_1)T_1} F_{e}\left(E_{\nu},\fr{E_0 T}{T_1}\right) ,
\end{align}
where $\Gamma_{\mathrm{inj}} = \Gamma_X$, since we assume that the
neutrino-sourcing muons result from the decays of the $X$-relics.
Electron antineutrinos will have the largest effect on the BBN
network. Integrated with the weak cross-section over energy, the
distribution function gives the energetic neutrino-induced rate of
$p\to n$ conversion:
\begin{align}
\label{eq:avg-cs-nubar}
\overline{\nu}_{e} + p &\to n + e^{+} : \quad \Gamma^\nu_{pn} = n_b(T)
\int_{0}^{E_0} \sigma_{pn}^{\bar\nu}(E_{\nu}) f_e(T,E_\nu) dE_\nu
\end{align}
where $n_b(T)$ is the number density of baryons at temperature $T$.
The cross section for quasi-elastic neutrino nucleon scattering reads
(see e.g.~\cite{Bemporad:2001qy})
\begin{align}
\label{eq:cs-nubar}
\sigma_{pn}^{\bar\nu}(E_{\nu}) =
0.0952\times 10^{-42} \left(\frac{p_e E_e}{1\ \ensuremath{\mathrm{MeV}}^2}\right)
S(E_{\nu})\ \ensuremath{\mathrm{cm}}^{2} ,\quad Q \simeq -1.8\ \ensuremath{\mathrm{MeV}} ,
\end{align}
where $E_{\nu}$ and $E_e = E_{\nu} - \Delta m_{np}$ are the energies
of $\overline{\nu}_e$ and $e^{+}$ in the rest frame of the proton,
respectively; $\Delta m_{np} \simeq 1.293\ \ensuremath{\mathrm{MeV}}$ is the neutron-proton
mass difference and $p_e$ is the positron momentum. We introduce a
correction factor $S(E_{\nu}) = (1-0.0063 {E_{\nu}}/{\ensuremath{\mathrm{MeV}}})$ which
improves the agreement between the simple formula (\ref{eq:cs-nubar})
and a precise evaluation of $\sigma_{pn}^{\bar\nu}$
in~\cite{Strumia:2003zx} to better than $1\%$ in the $E_{\nu}$-regime
of interest.
In our code we also account for exoergic $ n\to p $ conversions via
$\nu_e + n \to p + e^{-}$. This process, however, is only important
for $\tau_X\lesssim 180\ \ensuremath{\mathrm{sec}}$, \textit{i.e.} prior to $n$
consumption by \ensuremath{{}^4\mathrm{He}}. The associated cross section $
\sigma_{np}^{\nu}(E_{\nu})$ is obtained from (\ref{eq:cs-nubar}) with
$E_e = E_{\nu} + \Delta m_{np}$ and $S(E_{\nu})= 1$. As in
(\ref{eq:avg-cs-nubar}) we average $ \sigma_{np}^{\nu}(E_{\nu})$ over
$f_e$ in order to obtain the rate $\Gamma^\nu_{np}$ for
$ n\to p $ conversion.
\FIGURE[t]{\includegraphics[width=0.6\textwidth]{contour_mu+em.eps}
\caption{\small Same as in Fig.~\ref{figure1}, but for $X$ decaying
into $\mu^{+}\mu^{-}$ pairs which eventually decay into neutrinos
(and $e^{\pm}$). Note the smaller values $\tau_{X}$ in comparison
to the previous figures. Energetic neutrinos are accumulating over
time so that $X$-decays as early in time as $t=10\ \ensuremath{\mathrm{sec}}$ can
affect the BBN network. Noteworthy is also the increased \ensuremath{{}^4\mathrm{He}}\
abundance for large values of $Y_X$ over the whole $\tau_{X}$-range
since the $X$-matter density is contributing to $H$. In this figure,
$m_{X} \simeq 2m_{\mu}$. A novel feature is also the appearance of a
second observationally favored region for $\tau_X\simeq 10^4\
\ensuremath{\mathrm{sec}}$. No additional \ensuremath{{}^6\mathrm{Li}}\ in excess of SBBN values is produced
and electromagnetic energy injection depletes previously formed
D/H. For the parameters marked by the star the temporal evolution of
the light elements is shown in Figure~\ref{evolution2}.}
\label{neutrinos}
}
Muon neutrions $\nu_{\mu}$ and $\overline{\nu}_{\mu}$ which are
sourced from muon decays (at rest) are not capable of interconverting
protons and neutrons as their maximum injection energy $m_{\mu}/2$
lies below the reaction threshold $\Orderof{100\ \ensuremath{\mathrm{MeV}}}$. However,
${\nu}_{\mu}({\overline\nu}_{\mu})$ along with
${\nu}_{e,\tau}({\overline\nu}_{e,\tau})$ are in principle capable of
dissocia\-ting \ensuremath{{}^4\mathrm{He}}\ via their neutral current interactions. Among the
possible final states, only the ones with mass-3 elements have an
appreciable cross section in the low energy regime
$E_{\nu}<m_{\mu}/2$~\cite{2008ApJ...686..448Y}. We include the
following neutral current (NC) and charged current (CC) reactions into
our Boltzmann network
\begin{align}
\label{eq:nu-he4-rct}
\mathrm{NC:\ }
\begin{cases}
\ensuremath{{}^4\mathrm{He}} + \stackrel{(-)}{\nu}_{\!\!\!e,\mu,\tau} \to \stackrel{(-)}{\nu}_{\!\!\!e,\mu,\tau} + p +\ensuremath{\mathrm{T}} \\
\ensuremath{{}^4\mathrm{He}} + \stackrel{(-)}{\nu}_{\!\!\!e,\mu,\tau} \to \stackrel{(-)}{\nu}_{\!\!\!e,\mu,\tau}+ n +\ensuremath{{}^3\mathrm{He}}
\end{cases},
\quad
\mathrm{CC:\ }
\begin{cases}
\vphantom{ \stackrel{(-)}{\nu}_{e,\mu,\tau}}
\ensuremath{{}^4\mathrm{He}} + \nu_e \to e^{-} + p +\ensuremath{{}^3\mathrm{He}} \\
\vphantom{ \stackrel{(-)}{\nu}_{e,\mu,\tau}}
\ensuremath{{}^4\mathrm{He}} + \overline\nu_{e} \to e^{+}+ n +\ensuremath{\mathrm{T}}
\end{cases}.
\end{align}
For the NC reactions of muon and tau neutrinos we infer the reaction
rates averaged over $f_{\mu,\tau}$ by following the same logic from
(\ref{eq:primary-effective}) through (\ref{eq:neut-distro}). For the
cross sections we use fitting formulas interpolating the tables
provided in~\cite{2008ApJ...686..448Y}.
\FIGURE[t]{\includegraphics[width=0.6\textwidth]{MuEvol.eps}
\caption{\small Temperature evolution of light nuclei, meta-stable
parent $X$ particles and daughter $\mu^{\pm}$ for $Y_X = 5\times
10^4$ prior to decay and $\tau_{X} = 8.5\times 10^3\ \ensuremath{\mathrm{sec}}$.
Starting off with the previously observed effects which are induced
by an elevated neutron abundance (suppressed \ensuremath{{}^7\mathrm{Be}}/H, elevated
\ensuremath{\mathrm{D}}/H), electromagnetic energy injection eventually dissolves
deuterium and beryllium for $T_{9}\lesssim 0.06$. This not only
reinstates the SBBN D/H prediction (corresponding dotted line) but
also implies that lithium is mainly produced in form of \ensuremath{{}^7\mathrm{Li}}.}
\label{evolution2}
}
Given that the efficiency for neutron-to-proton conversion per each
injected neutrino is so low [cf. (\ref{estimate2})], a high initial
$X$-abundance (in comparison to baryons) is needed in order to achieve
an appreciable reduction of $\ensuremath{{}^7\mathrm{Li}}+\ensuremath{{}^7\mathrm{Be}}$. In our code we take the
extra contribution of $\rho_X$ to the total energy density into
account. By comparing $\rho_X$ (prior to decay) with the radiation
energy density~$\rho_{\rad}$,
\begin{align}
\frac{\rho_X}{\rho_{\rad}} \simeq \frac{10^{-6} Y_{X}}{T_9} \left(
\frac{m_{X}}{1 \ensuremath{\mathrm{GeV}}} \right) ,
\end{align}
we see that a GeV-scale relic $X$ starts contributing to the Hubble
rate appreciably during BBN for $Y_X\gtrsim 10^5$.
In Figure~\ref{neutrinos} we show the results of our computation for
$X\to \mu^{+}\mu^{-}$ in the usual $(\tau_X,\xi^{(X)}_{\mu} Y_X)$ plane
for $\xi^{(X)}_{\mu} =1$. For simplicity, again, we assume the muon injection close to
the kinematic threshold, $m_{X} = 2m_{\mu}$, although this assumption
is less crucial compared to the hadronic case. Note that this time we have extended the
$\tau_X$ range to lifetime values as small as $10\ \ensuremath{\mathrm{sec}}$. This is
because the usual temporal correlation between $X$-decay and its
effect on the BBN yields breaks down. For example, neutrinos injected at
$t=10\ \ensuremath{\mathrm{sec}}$ may still induce $p\to n $ conversion at $t=100\
\ensuremath{\mathrm{sec}}$. As expected, the results show a similar pattern as in the
pion and kaon cases with $Y_{X}$ scaled to larger values. In the neutrino case,
however, one finds an increased \ensuremath{{}^4\mathrm{He}}\ abundance for $Y_X\gtrsim 10^5$
over the whole $\tau_{X}$-range. This is because the $X$ energy
density is contributing to the Hubble rate, leading to an earlier
$n/p$ freezeout and thereby to a higher \ensuremath{{}^4\mathrm{He}}\ abundance. Though this
makes results sensitive to $m_{X}$, it is nevertheless only a mild
dependence for $m_{X}\lesssim 1\ \ensuremath{\mathrm{GeV}}$ in the interesting region
$Y_{X} \sim 10^4$, in which lithium is reconciled with observations.
As it further turns out, the neutrino
reactions dissociating $\ensuremath{{}^4\mathrm{He}}$ are giving at most a marginal correction
to the BBN yields even for $X$-abundances as high as $Y_X = 10^6$. The
reasons are that the processes of (\ref{eq:nu-he4-rct}) possess
threshold energies $E_{\mathrm{th}}\sim 20\ \ensuremath{\mathrm{MeV}}$ and because the
associated cross sections are significantly smaller in comparison
to~(\ref{eq:cs-nubar}). This has another interesting consequence: in
contrast to the pion- and kaon-scenarios essentially no \ensuremath{{}^6\mathrm{Li}}\ is
produced for $\tau_X\gtrsim 10^4\ \ensuremath{\mathrm{sec}}$, allowing for a second
region of cosmologically favored \ensuremath{{}^7\mathrm{Li}}\ abundances in which also high
D/H is photo-dissociated. As discussed earlier, this region is
necessarily fine-tuned in $\tau_X$ because of the exponential
sensitivity in injected electromagnetic energy. Indeed, for even
larger lifetimes $\tau_X\gtrsim 10^6\ \ensuremath{\mathrm{sec}}$ one would observe what
could be called the ``photo-dissociation catastrophe'' with \ensuremath{{}^4\mathrm{He}}\
along with all other elements being dissociated (for large enough
$Y_X$). Nevertheless, we find this scenario rather distinct if not remarkable.
The injection of muons with relic $X$ decays with lifetimes of $10^4$ seconds
is capable of reducing \ensuremath{{}^7\mathrm{Li}}\ abundance while keeping other abundances close to
the their SBBN-predicted values. This occurs because of the
non-monotonic evolution of D/H: first the increase due to neutrino-induced neutron enrichment, followed by
the decrease due to energy injection.
The temporal evolution of elemental abundances for a single point in the parameter space marked by the star in
Figure~\ref{neutrinos} is illustrated in Figure~\ref{evolution2}.
The most prominent feature is the ``bump''
in the neutron abundance at $T_9\sim 0.06$ marking the point in time
in which deuterium is subjected to photo-destruction. The associated
decrease in $\ensuremath{\mathrm{D}}/\ensuremath{\mathrm{H}}$ from its elevated values down to its original
SBBN prediction (dotted lines) are clearly seen in this Figure. Also \ensuremath{{}^7\mathrm{Be}}\ is
dissociated around the same time so that the overall lithium abundance
comes essentially in the form of \ensuremath{{}^7\mathrm{Li}}.
\section{Metastable GeV-scale states and the lithium abundance}
\label{sec:metastable-gev-scale}
In this section we consider some specific models of GeV-scale relics
and their impact on the BBN predictions. All models can be subdivided
into broad categories of WIMPs and super-WIMPs, where "super-" refers
to the superweak interaction strength with SM-states. WIMPs are
thermally excited above their mass scale, then depleting their number
density via annihilation at $T<m_{\mathrm{WIMP}}$, and---in our
case---decaying after the start of BBN. Superweakly interacting
particles typically have very small production rates throughout the
whole history of the Universe. This justifies the assumption of
super-WIMPs being initially absent as the Universe enters its thermal
radiation-dominated stage {\em e.g.} after inflation and reheating,
so that only some small super-WIMP abundance develops due to the
thermal leakage from the SM states. Let us stress again that the
notion of a (super-)WIMP here is not to be confused with a dark matter
state (which has to be stable on cosmological timescales.) Of course,
some of metastable particles presented in this section are motivated
by their potential as mediators of the interactions between dark
matter particles and the SM sector~\cite{Pospelov:2007mp,Finkbeiner:2007kk}.
The simplest WIMP model without any need of UV completion
\cite{McDonald:1993ex,Burgess:2000yq} uses the so-called Higgs
portal. In this model, a singlet scalar field $S$ interacts with the
rest of the SM via its coupling to the Higgs doublet $H$,
\begin{equation} {\cal L}_{\rm H-portal} = \fr12(\partial_\mu S)^2 -
V(S) - (\lambda SS + A S )(H^\dagger H).
\label{S-model}
\end{equation}
Here we assume an approximate $Z_2$ symmetry, $S\to -S$, broken only
by the last trilinear term $ A S (H^\dagger H)$, and we further take
$A$ to be very small; $V(S)$ is the scalar potential in the secluded
sector. If the $\lambda$-coupling is above $O(10^{-10})$, the
$S$-particles are guaranteed to be in thermal equilibrium with the
SM-states as soon as the plasma temperature is around the electroweak
scale. Smaller values of $\lambda$ make $S$ to a
super-WIMP~\cite{McDonald:2001vt}. The physical mass of $S$ particles
comes from the potential term $m_0^2S^2/2 $ and the electroweak vacuum
expectation value (VEV) $v\simeq 246\ \ensuremath{\mathrm{GeV}}$, $m_S^2 = m_0^2 + \lambda
v^2$. Having
\begin{align}
\label{Smodel-param}
A,\ \lambda,\ \mathrm{and}\ m_S^2\qquad \text{(S-portal)}
\end{align}
at our disposal, we can always choose the region of parameter space
where the lithium abundance is reduced along the lines described in
the previous section. Of course, in this model the GeV scale is not
special, and metastable $S$-particles at the electroweak scale can
also be used for the same purposes, exploiting nucleons in the decay
products of $S$.
Going away from the simplest possibility, we introduce a \us\ vector
portal model~\cite{Holdom:1985ag},
\begin{align}
\label{Vportal} {\cal L}_{\rm V-portal} = -\frac{1}{4} V_{\mu\nu}^2
-\frac{\kappa}{2}\, F^Y_{\mu\nu}V^{\mu\nu} + |D_\mu \phi |^2 -V(\phi),
\end{align}
where the connection between the \us\ field strength $V_{\mu\nu}$ and
the hypercharge field strength $F^Y_{\mu\nu}$ is mediated via the
kinetic mixing parameter $\kappa$. The \us\ covariant
derivative is given by $D_{\mu}= \partial_{\mu}+i e' V_{\mu}$, with
$V_{\mu}$ being the new vector-state associated with $V_{\mu\nu}$ and
$e'$ the gauge coupling strength in the secluded sector.
Since we are going to consider GeV-scale phenomenology, one can substitute
the hypercharge with the photon field strength, $F^Y_{\mu\nu}\to
F_{\mu\nu}$ and absorb the cosine of the Weinberg angle into the new definition of $\kappa$.
After the spontaneous breaking of the \us\ gauge group
by a Higgs$'$ field $\phi$, the low-energy Lagrangian can be written
as
\begin{equation} {\cal L}=-\frac{1}{4} V_{\mu\nu}^2+\frac{1}{2}m_V^2
V_\mu^2 +\frac{1}{2}(\partial_\mu h')^2-\frac{1}{2}m_{h'}^2 h'^2
+{\cal L}_{\rm int},
\end{equation}
where $m_V = e'v'$ becomes the mass of $V_{\mu}$ and $m_{h'}$ is the
mass of the physical Higgs$'$ field $h'$, $\VEV{\phi} =
v'/\sqrt{2}$. Assuming a standard Higgs potential in the \us\ sector,
the interaction terms are given by
\begin{equation} {\cal
L}_{\rm int}=-\frac{\kappa}{2}\,V_{\mu\nu}F^{\mu\nu}+\frac{m_V^2}{v'}
h' V_\mu^2 +\frac{m_V^2}{v'^2}\,h'^2 V_\mu^2 - \frac{m_{h'}^2 }{2
v'} h'^3 - \frac{m_{h'}^2}{8 v'^2} h'^4.
\end{equation}
Thus, like in the previous example, the model is characterized by only
a handful of free parameters:
\begin{align}
\label{Vportal-param}
\alpha',\ \kappa,\ m_{h'},\ \mathrm{and}\ m_V\qquad \text{(V-portal)},
\end{align}
where, as usual, $\alpha'=e'^2/4\pi$.
The strength of the mixing angle $\kappa$ is undoubtedly a very
important parameter, as it allows to make the link between the SM- and
\us-sector arbitrarily weak. Indeed, for $\kappa< 10^{-12}$ any
production rate of $h'$ and $V$ particles is smaller than the Hubble
rate, and the model becomes a good candidate for the super-weak
regime. It is also important to notice that long lifetimes of
particles from the \us\ sector can be achieved {\em without} requiring
exceedingly small~$\kappa$. This happens rather naturally in the
regime $m_{h'} < m_V$, where the decay amplitude of the Higgs from the secluded
sector is suppressed by $\kappa^2$
\cite{Batell:2009di,Pospelov:2008zw}. As mentioned in the
introduction, the (sub)-GeV mass scales for \us\ states are motivated
by the possible enhancement of annihilation of weak-scale WIMPs into
leptons \cite{ArkaniHamed:2008qn,Pospelov:2008jd}.
\subsection{WIMP regime}
Here we would like to determine the values of parameters for both
models, (\ref{Smodel-param}) and (\ref{Vportal-param}), that lead to
the desirable depletion of the \ensuremath{{}^7\mathrm{Li}}\ abundance. Since in the WIMP
regime the coupling constants are not necessarily small, the results
of this subsection might be relevant for the direct searches of new
physics at GeV energy scales.
For the Higgs portal model in the WIMP regime there are several
generic choices of parameters that lead to the desirable range of the
lifetime--mass--abundance "islands" that suppress the overall lithium
abundance. These regions can be readily found using the results for
the lifetime \cite{Batell:2009jf} and freeze-out abundance of
$S$-particles \cite{Burgess:2000yq,Bird:2004ts,Bird:2006jd}. For
example, the following choice of parameters,
\be Y_S \simeq 3\times 10^4, ~~ \tau_X \sim 500\ {\rm sec},~~m_S = 250
~{\rm MeV}~~\Longrightarrow~~ \lambda^2 \simeq 2\times 10^{-2}, ~~ A =
4\times 10^{-8}~ {\rm GeV},
\label{Sest}
\ee
leads to the depletion of lithium via the decays of $S$-particles to
muon pairs, generating electron anti\-neutrinos. Higher masses of $S$
particles, $m_S > 2 m_\pi$ that generate hadrons in the decay product
would typically require $\lambda \sim O(1)$ in order to have $Y_S \la
10^2$. The estimate (\ref{Sest}) is performed for a Higgs mass value
of 100 GeV but can be easily rescaled for heavier values. Notice that
the $A$ parameter does not enter in the abundance calculation, and can
always be adjusted to obtain a desirable lifetime.
Interestingly enough, the mass range of $m_S \la 2$ GeV is being
probed through the search of missing energy decays of $B$-mesons, $B
\to K^{(*)}SS$, \cite{Bird:2004ts,Bird:2006jd,Badin:2010uh} as the
eventual decays of $S$ particles occur far away from the $B$-decay
vertex. When the phase space suppression can be neglected, the
prediction for the missing energy branching ratio at $m_h =100 $ GeV
is ${\rm Br}_{B\to K E\!\!\!/} \sim 4\times 10^{-6} + 3\times 10^{-4}
\lambda^2 $ \cite{Bird:2004ts}, where the first term stands for the SM
contribution via $B\to K\nu\bar\nu$. The current upper bound on the
branching ratio stands at $1.5 \times 10^{-5}$, excluding models with
$\lambda\sim O(1)$. The choice of parameters (\ref{Sest}) is still
(barely) allowed, both by the $B$ and $K$ decays with missing energy.
We conclude that only this low $m_S$ option (or alternatively $m_S \ge
2$ GeV) is capable of solving the lithium problem without enhancing
the missing energy branching ratios of $B$-mesons beyond what is
observed. Perhaps the most interesting consequence of large values for
$\lambda$ required by the solution to the lithium problem are the
implications for Higgs physics. For Higgs masses of 150 GeV and
lighter the missing energy decays $h\to SS$ will dominate over the
Standard Model width and lead to a significant enhancement of the
missing energy channel and suppression of "visible" decay modes
\cite{Burgess:2000yq}. In short, the viability of the proposed
scenario will be tested at the LHC, (super-)$B$-factories and new kaon
facilities.
Metastable GeV-scale particles from a \us\ sector is another
interesting possibility. In the WIMP-regime we are drawn to consider
$h'$ since the vectors $V$ decay well before BBN through their now
appreciable kinetic mixing $\kappa \gg 10^{-10}$ [see
Eq.~(\ref{GammaVlep}) below],
\begin{align}
\label{eq:lifetime-vector}
\tau_V \leq 0.05\ \ensuremath{\mathrm{sec}}\times \left( \frac{ 10^{-10}}{\kappa}
\right)^{2} \left(\frac{500\ \ensuremath{\mathrm{MeV}}}{m_V}\right) \quad
\mathrm{for}\quad m_V \gtrsim m_{e}.
\end{align}
Indeed, the longevity of $h'$ particles can be achieved rather
naturally in the regime $m_{h'} < m_V$. The lifetime of the $h'$
particles due to the four-body decays and one-loop induced amplitudes
was calculated in Ref.~\cite{Batell:2009yf}. For example, the decay
width to muons in the regime $2 m_\mu < m_{h'} \ll m_V$ is dominated
by loop effects and is given by
\begin{align}
\tau_{h'} \sim (10^{3}\div 10^4)\, \ensuremath{\mathrm{sec}} \times \left(
\frac{\alpha}{\alpha'} \right) \left( \frac{3.4\times
10^{-5}}{\kappa} \right)^{4} \left( \frac{{250~\rm MeV}}{m_{h'}}
\right) \left( \frac{m_V}{{500~\rm MeV}} \right)^2.
\end{align}
This formula remains approximately valid in the regime $m_{h'} \sim
m_V$, and breaks down at $m_V - m_{h'} \ll m_{h'}$. With $m_V\sim 500\
\ensuremath{\mathrm{MeV}}$ and $\alpha'\sim \alpha$ it is easy to see that for $m_{h'} =
250$ MeV and $\kappa \simeq (3\div 10)\times 10^{-5}$ the decay rate
of $h'$ is in the right ballpark, while for the same value of
couplings and masses the decay of vectors happens very fast, $\tau_V <
10^{-16}\ \ensuremath{\mathrm{sec}}$.
Since we take the $h'$ particle to be the lightest in the \us\ sector
and long-lived, what physical mechanisms could possibly deplete its
abundance to an acceptable level? It turns out that possible {\em
endothermic} excitations into the $V$ state, $h'\to V$ followed by
$V$ decays, although exponentially sensitive to the mass parameters in
the \us\ sector, are in principle sufficient for the $h'$
depletion\footnote{MP would like to thank Neal Weiner for a very
useful discussion of that point.}. In particular, the following
processes are capable of suppressing the $h'$ abundance:
\begin{eqnarray}
h' + h' \to V+V, &~~& \Gamma_1 \propto (\alpha')^2 \kappa^0 \exp(-m_{h'}/T-2\Delta m/T) \\
h' + V \to l^+l^-, &~~& \Gamma_2 \propto \alpha'\alpha \kappa^2 \exp(-m_{h'}/T-\Delta m/T)\\
\label{depletion}
h'+ l^\pm \to V + l^\pm, &~~& \Gamma_3 \propto \alpha'\alpha \kappa^2 \exp(-\Delta m/T),
\end{eqnarray}
where $\Delta m = m_V - m_{h'}$, and $l^\pm$ are the charged particles
of the SM. The last process in (\ref{depletion}) is especially
important because it comes with the least amount of exponential
suppression. We now estimate the freeze-out abundance of $h'$ due to
this process.
In the norelativistic regime, the equilbrium number density
$n_{h'}^{\mathrm{eq}}$ of $h'$ follows the well-known curve, and the
freeze-out abundance can be estimated by equating the depletion rate
$\Gamma_3$ with the Hubble rate, $H(T_f) = \Gamma_3(T_f)$,
\begin{align}
Y_{h'} &\simeq n^{eq}(T_f)/n_b(T_f),\\
n_{h'}^{\mathrm{eq}}(T) &= \left(\fr{ m_{h'}T}{2\pi}
\right)^{3/2}\exp(-m_{h'}/T).
\end{align}
We would like to determine the dependence of the freeze-out
temperature $T_f$ on $\Delta m$, and specialize it to two cases
considered in the previous section: decays to muons and decays to
pions/kaons. Considering the exponential dependence on the mass
splitting $\Delta m$ we are making a series of simplifying
approximations: $m_e\ll T_f \ll \Delta m \ll m_V,m_{h'}$. In that case
the cross section for the $h' + e^\pm \to V + e^\pm$ process is given
by
\be \sigma = \frac{16\pi\kappa^2\alpha\alpha'}{m_V^4} \frac{\Delta m
(E_e-\Delta m)^2}{E_e} ~~\Longrightarrow ~~ \Gamma_3(T) \simeq
\frac{64\kappa^2\alpha\alpha'T^3(\Delta m)^2}{\pi m_V^4} \exp(-\Delta
m/T), \ee
where $E_{e}$ is the electron energy. For $T_f$ larger than 100~MeV,
other charged SM states would have to be included in the
coannihilation process~(\ref{depletion}). However, since we are in
need of a suppression of the $h'$ abundance by a large factor, the
freeze out of a GeV-scale $m_h'$ has to happen below 100~MeV, so that
only $e^{\pm}$ will contribute to depletion process.
Introducing two dimensionless variables $x_f= m_{h'}/T_f$ and $\delta
= \Delta m /m_{h'} = m_V/m_{h'}-1$, we arrive at the following
analytic estimate of the freeze-out temperature and the resulting
abundance as a function of mass splitting in the \us\ sector:
\begin{align}
x_f &\simeq \fr{1}{\delta}\left\{13+
\ln\left[\frac{\alpha'}{\alpha} \fr{\delta^2}{(1+\delta)^4}
~\fr{15}{x_f}\left(\frac{\kappa}{3\times 10^{-5}}\right)^2
\fr{250~{\rm MeV}}{m_{h'}} \right] \right\},
\\
\label{eq:hWimpabd}
Y_{h'} &\simeq 4.2\times 10^8 \times x_f^{3/2}\exp(-x_f).
\end{align}
It is easy to see that for mass splittings $\delta \simeq 0.3\div 0.5$ the
abundance varies in the interval $10^{-1}\div 10^2$, which will be
suitable for the solution of the lithium problem using the decays of
$h'$ to pions and kaons. Mass splittings of order $\delta \simeq
0.8\div 1.4$ correspond to abundances on the order $10^4\div 10^6$, which are
suitable for suppressing lithium via the decays of $h'$ to muons. We
remark in passing that~(\ref{eq:hWimpabd}) represents the fraction of
$h'$ relative baryons after $e^{\pm}$-annihilation but prior to their
decay.
Since the realization that a new attractive \us\ gauge force can be
used for the explanation of the PAMELA anomaly
\cite{ArkaniHamed:2008qn,Pospelov:2008jd}, a lot of dedicated work has
been done in order to understand the prospects of searching/detecting
particles from a putative \us\ sector. The most promising avenues for
the discovery of new GeV-scale particles are high-luminosity
medium-to-low energy experiments
\cite{Batell:2009yf,Reece:2009un,Essig:2009nc}, although some
prospects of discovering such particles in the fragmentation of heavy
exotic particles produced in the high-energy collisions have also been
investigated~\cite{Baumgart:2009tn}. The mixing angles deduced in
this section, $\kappa \sim 3\times 10^{-5}$, and vector masses in the
range of $400$~MeV and larger, represents one of the most challenging
corners in the $m_V$-$\kappa$ parameter space. These angles are small
enough so that the search of $V$ in this range at even the highest
luminosity $e^+e^-$ machines is not possible. Therefore, the search
for $V$ in this regime with fixed target experiments is perhaps the
only realistic option. Vector particles are relatively short lived,
$c\tau \le 1$~cm, so that in the set-up with a detector at some
macroscopic distance behind the target all the decays will happen
before reaching the detector. With proton beams, where the detector
is typically tens or hundreds of meters behind the target, such a
search might prove to be very challenging. Therefore, the best
discovery potential for $V$ in this parameter range would probably be
a high-intensity electron beam on a thin
target~\cite{Bjorken:2009mm}. Lastly, the lifetimes of mediators in
excess of milliseconds lead to interesting effects in the annihilation
of dark matter, when the decays of mediators occur away from the point
where the annihilation occur
\cite{Batell:2009zp,Schuster:2009au,Rothstein:2009pm}. This leads to
novel signatures in the indirect detection of dark matter in models
with light mediators. Finally, for overdensity constraints on some
variants of these models see, \emph{e.g}~\cite{Chen:2009ab}.
\subsection{GeV-scale super-WIMPs}
In the previous examples we considered metastable particles that
initially had thermal abundances. This is not the only possibility,
and in the following we will look at the abundances of vectors $V$ and
scalars $S$ in the super-WIMP regime. Our assumption is that the link
between the SM and the secluded sectors is extremely weak, and given
by a single parameter, $\kappa$ for (\ref{Vportal}), and $A$ for
(\ref{S-model}). The states $X=S$ or $V$ are then produced with
sub-Hubble rates from the thermal scattering of SM particles. In this
case, both, the production and the decay rates are proportional to the
square of the small parameters $\kappa$ or $A$. Consequently, the
product of abundance and lifetime, $Y_X\tau_X$ is independent on $A$
or $\kappa$ and is only a function of the mass $m_X$ and the SM
couplings/masses. This scaling holds for both cases despite the fact
that the physical production mechanisms of $S$ and $V$ are vastly
different. An extra vector particle that appears just as another
massive photon is mostly produced at temperatures comparable to its
mass $T \ge m_V$ whereas a scalar particle with mixing to the Higgs
boson is most efficiently produced at $T\sim m_W$, where the
$T/m_h$-dependence of the production rate relative to the Hubble rate
is maximized.
We begin by analyzing the \us\ model~(\ref{Vportal}) in the super-WIMP
regime. The decay widths of the vector particles to leptons and
hadrons are given by~\cite{Batell:2009yf}
\begin{align}
\label{GammaVlep}
\Gamma_{ V \rightarrow \overline{l}l } &=\frac{1}{3} \alpha \kappa^2 m_V \sqrt{1-\frac{4 m_l^2}{m_V^2}}
\left(1+\frac{2 m_l^2}{m_V^2}\right),\\
\label{GammaVhad}
\Gamma_{ V \rightarrow {\rm hadrons} } &=\frac{1}{3} \alpha \kappa^2
m_V \sqrt{1-\frac{4 m_\mu^2}{m_V^2}} \left(1+\frac{2
m_\mu^2}{m_V^2}\right) R(s= m_V^2),
\end{align}
where $R$ is the experimentally measured ratio of the inclusive
hadroproduction in $e^+e^-$ collisions to the direct muon production,
$ R= \sigma_{e^+ e^- \rightarrow {\rm hadrons}}/ \sigma_{e^+ e^-
\rightarrow \mu^+ \mu^-}$. It follows immediately [see also
Eq.~(\ref{eq:lifetime-vector})] that a desirable lifetime range
$\tau_{V}\gtrsim \Orderof{100\ \ensuremath{\mathrm{sec}}}$ requires $\kappa$ in the
ballpark of $10^{-12}$.
The production of GeV-scale $V$-bosons in the early Universe cannot be
calculated "exactly" because of the unsurmountable difficulty in
treating hadrons in the intermediate regime $T\sim \Lambda_{\rm QCD}$.
In particular, all production modes involving free quark and gluons,
$q\bar q \to V$, $q + g \to q +V$ etc are not calculable because of
the strong QCD coupling. Cases that can be reliably calculated are
$m_V \ll 100$~MeV, and $m_V \gg 1$~GeV, which are, unfortunately,
exactly opposite to the regime we are interested in. Nevertheless, we
can obtain an order-of-magnitude estimate that captures the main
behavior of the production mechanism as a function of $m_V$. The main
feature of the production mechanism is that it receives an exponential
cutoff $\exp(-m_V/T)$ in regime $T<m_V$, and this property is not
modified by strong dynamics. Therefore, the production effectively
stops at $T \sim m_V$ with the residual abundance of $V$ particles
parametrically dependent on the ratio of the production rate over the
Hubble rate,
\begin{align}
\label{YV}
Y_V = \fr{s}{n_b}\left.\fr{n_V}{s}\right|_{f} \simeq (1.2 \div 4.9)
\times\fr{s}{n_b} \fr{1}{h_{\mathrm{eff}}(m_V)} \fr{\Gamma_{V\to
e^{+}e^{-}}}{H(m_V)} \sim 0.3\times
\left(\fr{10^3\ \ensuremath{\mathrm{sec}}}{\tau_V}\right)\left(
\fr{\ensuremath{\mathrm{GeV}}}{m_V}\right)^{2} \left( \fr{40}{g_{\mathrm{eff}}}
\right)^{3/2} .
\end{align}
The estimated range follows from a computation of the collision
integral for the inverse decays of Fermi-Dirac distributed pairs
$e^{\pm}$ (lower value) and massless
$e^{\pm},\,\mu^{\pm},\,u\bar{u},\,d\bar{d},\,s\bar{s}$ (upper value);
for a related calculation see also~\cite{Redondo:2008ec}. The upper
value is saturated for ``freeze-out'' temperatures above the QCD
hadronization scale, $\Lambda_{\mathrm{QCD}}<T_f\lesssim\Orderof{1\
\ensuremath{\mathrm{GeV}}}$, where the perturbative treatment of the light quarks becomes
accessible. On the other hand, lower mass vectors whose production
ceases later are guaranteed to receive a contribution to their
abundance from electron-positron pairs. This limits $Y_V$ from below.
In the last relation we have normalized the effective number of
entropy degrees of freedom $h_{\mathrm{eff}}\simeq g_{\mathrm{eff}}$
on a typical value during the QCD hadronization epoch and further made
the assumption that $\tau_V$ is dominated by the channels which are
also responsible for the inverse decay.
The results suggest that a sub-GeV state with a lifetime of the order
of a thousand seconds is abundant enough to produce $\Orderof{1}$
pions/kaons per baryon, and thus capable of reducing the lithium
abundance by a factor of a few. In the most optimal range with $m_V
\sim m_\rho$, the pion branching dominates over lepton branching,
while the energy of injected pions is close to the
delta-resonance. This enhances the the efficiency of $p\to n$
conversion, Fig. \ref{figure1b}, so that $Y_V\sim 2$ is required.
Since $Y_V$ is not calculable exactly, it is unfortunately not
possible to give a final judgement whether the required abundance can
be achieved or falls marginally short.
We also notice that $m_V$ below the di-pion threshold, $2m_\mu < m_V <
2m_\pi$ can only be created in abundances less than $\Orderof{10^2}$
which renders this parameter range incapable of reducing lithium via
$V$-decays into muons. On the other hand, a heavier mass range for a
super-WIMP $V$ is also interesting. For example, $V$-bosons in the
mass range of 10 GeV or more are less abundant than protons, but will
contain nucleons among their decay products. In this case one can
easily suppress the lithium abundance via direct injection of extra
nucleons along the lines of Refs.~\cite{Reno:1987qw,Jedamzik:2004er}.
The model with the singlet $S$ mixed to the Higgs boson via the
$ASH^\dagger H$ coupling can be treated very similarly.
Interestingly, it turns out that in this model the abundance of $S$
particles can be calculated without significant QCD uncertainties
because it dominatly occurs at electroweak scale temperatures, whereas
the lifetime of the $S$ boson with a mass in the GeV range is
notoriously difficult to handle.
In principle many SM scattering processes may contribute to the
emission of the $S$ boson. These include $f_1\bar f_2 \to VS$, $f_1V
\to f_2 S$, $hh\to SV$, $VV\to SV$, $tg\to tS$, $t\bar t \to gS$ etc,
where $f_1,f_2$ stand for the SM fermions, $V=W,Z$ for massive gauge
bosons, and $g$, $t$ are gluons and top quark. An exact treatment of
the production mechanisms in this model falls outside the scope of
this paper. However, despite this rather large number of production
channels, the asymptotic behavior of the production mechanisms are
quite obvious. In the regime $T \gg v$ the production rate has to
scale as $\Gamma \propto A^2/T$, and in the low-energy regime it is
$\propto A^2 T^3 m_h^{-4}$. This implies that $\Gamma$ is naturally
peaked at temperatures around the electroweak scale.
In order to avoid straightforward but tedious calculations of the $S$
abundance in the most minimal model (\ref{S-model}), we go to the
two-Higgs doublet model analogue of (\ref{S-model}), and couple the
$S$ scalar to a "mixed" Higgs portal:
\be ASH^\dagger H \to \fr{1}{2}AS(H_1H_2 + h.c.). \ee
Assuming a mild hierarchy of scales in the Higgs sector together with
a ratio of the Higgs VEVs $ \tan \beta = v_2/v_1 \gg 1$, $H_2$
represents the SM Higgs doublet while $H_1$ contains heavi(er)
physical scalars $H$, $A$, $H^\pm$ with common mass scale $m_H$. The
production of scalars $S$ is peaked around $m_H$. In what follows we
calculate the abundance of $S$ resulting from $H_1 \to H_2 S$ decays
using $m_H \gg m_h, m_{W(Z)}$. In exact analogy to
(\ref{eq:meson-decay-rate}) the decay rate can be written as
$\Gamma_H\langle m_H/E \rangle$, where \be \Gamma_H = \fr{A^2}{16\pi
m_H} \ee is the decay rate of $H_1$ particles at rest. Accounting
for that $H_1$ carries four degrees of freedom we arrive at the
following integral formula for the freeze-out (or rather "freeze-in")
abundance of $S$ bosons realtive to baryons.
\begin{align}
\label{YS}
Y_S = \fr{s}{n_b} \int \fr{m_H dT}{H(T) T s(T)} \int
\fr{4d^3p}{E(2\pi)^3} \fr{\Gamma_{H}}{\exp(E/T) -1}
\end{align}
To good approximation we take $h_{\mathrm{eff}}(T)$ in $s = 2\pi^2
h_{\mathrm{eff}} T^3/45$ to be that of the SM, $h_{\mathrm{eff}} =
427/4 $. The integrals in (\ref{YS}) can be taken exactly, resulting
in \be Y_S = \fr{s}{n_b} \fr{135\zeta(5)}{2\pi^3}
\frac{\Gamma_H}{NH(m_H)}\simeq 3.8\times 10^5 \times
\fr{A^2M_{P}}{m_H^3} \ee where $H(m_H)$ is the value of the Hubble
constant at $T=m_H$ and $\zeta(x)$ is the Riemann Zeta-function.
The decay of the $S$-bosons at late times proceed to muons and hadrons
depending on what channels are kinematically allowed. At $m_S \ll 1$
GeV the decay rates can be obtained within the framework of chiral
perturbation theory and by low-energy
theorems~\cite{Voloshin:1985tc,Truong:1989my}. For $m_{S}\gtrsim
1.5$~GeV or so, perturbative QCD starts getting applicable. Around
these energies the decay rate is dominated by $S$ decaying into
$s$-quarks with a $\sim 25\%$ contribution to muons:
\be \Gamma_S \simeq \fr{3m_S}{ 8 \pi} \left( \fr{A m_s
\tan\beta}{m_H^2} \right)^2 \,\left(1 + \fr{m_\mu^2}{3m_s^2}
\right)
\label{GS}
\ee
Notice that heavy-quark-mediated $S\to gg$ decays are suppressed
relative to the SM by a factor of 9 because in the large $\tan\beta$
regime only $b$-quarks contribute, and $t,c$ contributions are
subdominant. Since $m_s \ge m_\mu$, the decays to strange-antistrange
pairs happen in at least $75\%$ of the cases. When $m_S$ is below the
di-hyperon threshold, each of these decays lead to either $K^-$ or
$K_L$ in the final state. The last step in checking the suitability of
this model involves trading $A^2$ in the formula for the abundance
(\ref{GS}) for the lifetime of $S$. Taking $m_s \sim 100$ MeV, we get
\be Y_S = 3.4 \times \left(\fr{10^3\ \ensuremath{\mathrm{sec}}}{\tau_X}\right)
\left(\fr{2~{\rm GeV}}{m_S}\right) \left( \fr{m_H
\tan^{-2}\beta}{5~{\rm GeV}}\right), \ee
where all parameters are normalized on their "natural" values. It is
in a way remarkable that in these models many numbers of very
different orders of magnitude conspire to yield abundances in the
right ballpark necessary for a suppresion of the \ensuremath{{}^7\mathrm{Li}}\ abundance via
the emission of kaons (and pions).
\section{Conclusions}
\label{Sec:Conclusions}
The current discrepancy between the prediction of the lithium
abundance in SBBN and its observation at the Spite-plateau value
prompted a number of particle-physics related explanations. Previous
works have mainly concentrated on the effects of heavy, electroweak
scale particles on the \ensuremath{{}^7\mathrm{Li}}\ abundance. Such models typically involve
decays of electroweak scale WIMPs with nucleons among the decay
products, or the catalytic suppression of \ensuremath{{}^7\mathrm{Li}}\ by metastable charged
particles, that have their masses again in the electroweak range. In
this paper we have considered a related but nevertheless distinct
possibility: decays of (sub-)GeV-scale neutral relic
particles. Because of their kinematic constraints, neither of these
particles can be charged nor can their decay products contain nucleons
in the final state. Nevertheless, the reduction of the lithium
abundance may occur simply because pions and kaons in the final state
have a significant probability of interacting with protons and
creating a neutron-excess. We have shown that
$\Orderof{1}-\Orderof{100}$ abundances of metastable particles with
respect to baryons and with lifetimes around a few hundred to ten
thousand seconds can easily reduce the lithium abundance by a factor
of a few and still be consistent with measurements of primordial
deuterium. We have also shown that if the decays to the charged
hadrons are kinematically not possible, the muons in the final state
can also reduce the overall lithium abundance. In this case the main
effect comes from electron antineutrinos that induce $p \to n$
interconversions. It turns out that $\Orderof{10^5}$ muon decays per
proton around a few hundred seconds are needed in order to reduce the
lithium abundance to observationally favored levels. Moreover,
injection of pions and muons not far from thresholds also leave less
"damage" in form of \ensuremath{{}^6\mathrm{Li}}, compared to scenarios with electroweak-mass
decaying relics. This is because the \ensuremath{{}^6\mathrm{Li}}\ production requires
significant amounts of energy to split \ensuremath{{}^4\mathrm{He}}\ and create energetic
$A=3$ nuclei, which in case of the muons/pion injection is far less
efficient. This broadens the acceptable lifetime range to up to
$O(10^4)$ seconds.
We have done a careful numerical investigation of this problem by
including charge exchange and absorption reactions of pions, kaons and
(anti)neutrinos on nucleons and \ensuremath{{}^4\mathrm{He}}. In doing so we have updated the
values for the most relevant reactions, and integrated them into a
Kawano-based nuclesynthesis code. We have also accounted for the
change in the neutrino spectrum due to the Hubble expansion and due to
neutrino oscillations. The results are presented in the form of
abundance-lifetime plots, Figures~\ref{figure1}-\ref{neutrinos}, where
we have assumed a maximal branching into each species. We can see
that the elevated deuterium abundance is an inevitable consequence of
all such scenarios, resulting from the "removal" of extra neutrons by
the $np\to d\gamma$ reaction. The only exception is the case of relic particles
decaying to muons at $\sim 10^4$ seconds. In this case, the late energy injection,
combined with early neutrino-induced $p\to n$ interconversion can lead to the
SBBN value of deuterium, and achieve the suppression of \ensuremath{{}^7\mathrm{Li}}\ by a factor of a few.
In order to relate this proposal to some concrete model realizations,
we have considered two minimalistic extensions of the SM. We chose to
extend the SM by a singlet scalar $S$ coupled via the so-called Higgs
portal, or a new \us\ group that has kinetic mixing with the photon
field strength. Both models, in principle, possess new long-lived
states, and in the case of the \us\ model this is either the vector
$V$ or the Higgs$'$ particle $h'$ of the \us-breaking sector. The
cosmological history in these models then depends on the strength of
the couplings that connect the secluded sectors to the SM
particles.
If the couplings to the Standard Model are large enough, the new
GeV-scale states $X$ are thermally excited (WIMP regime). An
acceptable abundance of $X=S,\ h'$ prior to decay is then either
achieved via annihilation, {\em e.g.} $SS\to f\bar f$, or via the
excitation to the unstable state, $ h' e \to V e \to e\bar e e$. We
have shown that in the WIMP regime both models provide viable
solutions to the lithium problem, and that both, the $\pi,K$-mediated
or the antineutrino-mediated suppression of \ensuremath{{}^7\mathrm{Li}}\ indeed work. We
have identified the plausible choice for the parameters where such
reduction will occur. The same parameters may be chosen to fit the
PAMELA and FGST anomalies, when the mediator models are complemented
with electroweak-scale WIMPs. In addition to cosmological probes,
there is a certain chance of detecting these new GeV-scale states in
laboratory experiments, such as in rare decays of mesons, in Higgs
decays, and in fixed target experiments. Last but not least, we remark
that a $m_{h'} < m_V$ mass pattern naturally arises in straightforward
supersymmetric generalizations of the \us\ model.
We have further shown that the super-WIMP regime of these models also
have a potential to provide a viable resolution to the lithium
problem. In this case, the metastable states are never in thermal
equilibrium and are only produced via an extremely small thermal
rate. Such models are relatively predictive because the mass and the
coupling of $X$ can be translated directly into lifetime- and
abundance-predictions. Unfortunately, the extreme smallness of the
coupling constants prevents direct experimental searches of such
particles in the laboratory. On the other hand, the sensitivity that
the early Universe exhibits to new physics is once again emphasized by
the fact the lithium and deuterium abundances are capable of probing
effective coupling constants as small as $\alpha \kappa^2 \sim
10^{-26}$. It remains to be seen whether the \ensuremath{{}^7\mathrm{Li}}\ reduction can be
achieved in another well-motivated model of super-WIMPs in the form of
heavy sterile neutrinos.
To close up, the problem of the \ensuremath{{}^7\mathrm{Li}}\ abundance definitely deserves
special attention, as it hints at one of the very few inconsistencies
in the standard cosmological picture. Of course, at this point it is
difficult to insist that particle physics must necessarily be key for
solving the current lithium problem. Yet, as shown in this paper,
several particle physics scenarios that reduce the \ensuremath{{}^7\mathrm{Li}}\ abundance are
quite natural and perhaps deserve further detailed considerations. As
to the lithium problem, only future progress in cosmology,
astrophysics, particle, and nuclear physics may help to resolve this
intriguing problem.
{\bf Acknowledgments} Research at the Perimeter Institute is supported
in part by the Government of Canada through NSERC and by the Province
of Ontario through MEDT. JP wants to thank the Galileo Galilei
Institute for Theoretical Physics for the hospitality and the INFN for
partial support during the final stages of this work.
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'State of the Union': Nick Hornby on Covering an Entire Relationship in Ten Conversations
He also explains why Rosamund Pike and Chris O'Dowd's performances prove this story "found its best form" on TV.
Rosamund Pike and Chris O'Dowd in "State of the Union"
Parisatag Hizadeh/Confession Films/SundanceTV
When Nick Hornby started writing the SundanceTV series "State of the Union," he knew where he wanted the season to end. At first, the endpoint he decided on may not seem all that revolutionary – but like most elements of the experimental, 10-minute an episode, 10-episode season, context adds so much more than that.
"I wanted the last words to be 'I love you,' and for it not to be exactly as that seems," Hornby said.
But aside from how those last words are framed, it's the format of the series that lends itself to something with additional insight than the usual "couple in strife" story. In the show, Rosamund Pike and Chris O'Dowd play Louise and Tom, an on-the-outs, estranged married couple who meet once a week before their scheduled therapy session at a pub right across the street.
Hornby wrote the entire series, which was in turn directed by Stephen Frears. (The latter's 2000 film "High Fidelity" adapted Hornby's novel of the same name.) Having this story play out in smaller increments made sense, given the time that these characters would have together. The trick then became to make that weekly format take full advantage of the premise.
'State of the Union' Trailer: Patricia Clarkson and Brendan Gleeson Take Over as Season 2 Couple in a Crisis
'State of the Union' Season 2 Ordered at SundanceTV, Short-Form Emmy Winner Sets New Cast
45 Directors Pick Favorite Horror Movies: Bong Joon Ho, Tarantino, del Toro and More
"When I thought about setting it after their therapy, all the energy went out of it, because they would only be able to talk about what they just talked about. That would feel like you just missed a dramatic moment," Hornby said. "Whereas before, you've got a week of life, a previous therapy session and all the things that are going on with them from the previous however many years as well. There's a wealth of material for them to pick through."
More than a formal curiosity, "State of the Union" presents a chance for these conversations to really dig into language. Aside from the usual idea that there's plenty of subtext lurking underneath each of Louise and Tom's conversations, the two parse out metaphors and crossword clues as an alternate way into the marital problems the two are trying to reconcile.
"I really like writing dialogue. The difficult thing is to get to the end of ten pages and realize you haven't gotten anywhere that you wanted to go. Then you back to the beginning and start compressing and giving it a shape and making sure that each episode has a story arc, as well as the entire series. Those were the trickier parts of it, but mostly it was a blast to write," Hornby said.
Parisatag Hizadeh/Confession Fil
As with most conversations these days, those conversations also turn to the overall state of the world. With the U.K. navigating its removal from the European Union, Louise and Tom are forced to look at the ways Brexit mirrors their own predicament. Hornby wasn't afraid to bring the topic into the show — if anything, he said that in a weird way, it was one form of keeping the show timely.
"This series could come out in three years and it would not seem dated. Brexit is a very, very imprecise point on our historical timeline. It really does look like it's going to go on forever. So of all the things that I didn't worry about, that was one of them because I knew it would still be relevant by the time we got back to filming," Hornby said.
Louise and Tom's story will live on in other formats. A version of their conversations is now available in book form, separated out in the same ten segments. But even though this is an idea that could easily lend itself to a podcast or a play, there's one all-important element of "State of the Union" that Hornby said would be lost in any other version.
"One of the real joys of the series, for me, is the faces of the actors. I think their faces are so animated and they're so good at reacting and speaking," Hornby said. "A couple people have asked me if I've thought of turning it into a play, but the idea of not having access to those faces while you're sat there, watching the dialogue, I can't quite get that. I think maybe it's found its best form."
So instead of taking this one couple's story and translating it to another medium, he's working on adapting this TV format to handle the story of two new people. While there are endless possibilities for how to build on this foundation — Hornby said that the next iteration could forego married couple and focused on a different kind of parental or familial relationship — he wants to follow people who are older than Louise and Tom.
""I would like to write a second season of this, but it would be two different people at a different stage. Different age, different life experience. We're trying to think about that now," Hornby said. "We all know couples that split up in their 40s with kids and everyone recovers and ends up with new partners. But with older couples, there's a whole different set of problems and a whole different set of regrets, maybe. The stakes get higher as you get older."
"State of the Union" is now available to stream via SundanceTV.
This Article is related to: Television and tagged Nick Hornby, State Of The Union, SundanceTV
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\section{Introduction}
Recently, deep neural networks is used to model implicit representations of 3D shapes has been widely applied for reconstruction \cite{duggal2021secrets}, generation \cite{chen2019learning}, compression \cite{tang2020deep} and rendering \cite{remelli2020meshsdf, takikawa2021neural}.
As one of the most popular methods, DeepSDF~\cite{park2019deepsdf} represents the zero-level surface of the whole 3D shape by regressing its continuous signed distance function (SDF).
However, the effectiveness of such models depends on the complexity of 3D shapes and the capacity of neural networks.
In Fig~\ref{fig:compare2degeneration}~(d) compared to (c) we demonstrate an example, where the capacity of the model is insufficient. Consequently, the reconstruction of one complex 3D shape is of poor quality.
To alleviate this problem, we propose to learn a set of local SDFs to represent the whole surface. In this case, each local SDF is responsible for a part of the reconstructed shape.
Learning such local SDFs is much easier. Here we use the assumption that the complexity of one local part of the 3D shape is much simpler than the whole and usually similar to other local parts.
However, learning a set of local SDFs instead of one global SDF makes the training process of the model more difficult, since the distribution of training data is not uniform, especially when more SDFs are considered in a local region.
One of the possible solutions is to make these local SDFs learnable using both the database and the local latent code. Here we utilize the reasonable assumption that similarity of the neighbor parts means similarity in the local code of the shape.
Recently, graph neural networks (GNNs) have demonstrated high effectiveness in geometric learning field, especially in 3D reconstruction problem~\cite{ranjan2018generating,bouritsas2019neural,zhou2020fully,hanocka2019meshcnn}.
The discussed works show that it is possible to decode latent representation of the shape geometry with highly accurate reconstruction, even for finer details.
The GNNs provide geometric constraints to facilitate the information smoothed over the graph as the original purpose for clustering vertices of a graph~\cite{kipf2016semi, defferrard2016convolutional}.
This property leads to a message exchange mechanism among the vertices and results in their local similarity.
Consequently, we introduce Global-to-Local~(G2L) network based on GNNs to learn local latent codes that came from one global code. Thus, our G2L combines the advantages of the GNNs and SDF approaches.
We assume that geometrical locality is learnable in the space of latent code as in the original geometric space of the shape.
In addition, we also include one geometric similarity loss function based on the geometric structure to enhance the effectiveness of the GNN in local latent code learning.
Remind our assumption is that the neighboring local regions should have similar latent codes in general. Our experimental results confirm the mentioned assumption with the local latent codes learning.
Several methods~\cite{chabra2020deep, jiang2020local, genova2020local, tretschk2020patchnets, hao2020dualsdf} learn the SDF of 3D shape locally and have shown promising results.
However, all of them need either the voxel representation of 3D shape to align the local latent codes or explicit parametric models (for example, a sphere) to fit.
Both approaches face problems of the volumetric representation that in many cases highly difficult to solve or even is insolvable.
In contrast, our work leverages geometric learning techniques to model local SDFs of 3D shapes directly in the latent code space.
Contributions of our work are:
\begin{itemize}
\item We propose the Local Geometry Code Learning method, where we learn the shape as zero-surfaces with local latent codes.
\item We use graph neural networks to generate local latent codes and distribute them on the 3D shape, which does not request voxelization of the 3D shape as it is in locally modeling methods.
\item We introduce a geometric similarity in the loss function that helps to learn and reduce the fluctuation of the reconstructed surface.
\item Our experimental validation shows that the proposed approach could keep more details of the reconstructed shape in comparison with the original SDF decoder.
\end{itemize}
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\linewidth]{figs/pipeline.pdf}
\end{center}
\caption{(a) Overview of the LGCL model; (b) Local region separation, where one 3D shape sample is represented with mesh data and projected to a 2D plane.}
\label{fig:geo_sdf_decoder}
\end{figure*}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\linewidth]{figs/gcn.pdf}
\end{center}
\caption{Architecture of the G2L network.}
\label{fig:g2l}
\end{figure}
\section{Related work}
\subsection{SDF learning methods}
Learning a signed distance function to represent the 3D shape as a set of iso-surfaces recently receives extensive attention in the field.
Chen and Zhang~\cite{chen2019learning} proposed to assign a value to each point in 3D space and use a binary classifier to extract an iso-surface.
Mescheder~\etal~\cite{mescheder2019occupancy} utilize a truncated SDF to decide the continuous boundary of 3D shapes. In contrast with~\cite{chen2019learning}, they predict the probability of occupancy in voxels, which could be used in a progressive multi-resolution procedure to get refined output.
Park \etal~\cite{park2019deepsdf} learn a continuous field of 3D shape instead of the discrete representation of SDF in the grid and is understandable as a learned shape-conditioned classifier for the decision boundary of 3D shape surface.
Atzmon and Lipman~\cite{atzmon2020sal} leveraged sign agnostic learning (SAL) to learn from unsigned distance data, such as the distance directly from points to triangle soups.
Gropp \etal~\cite{gropp2020implicit} suggested using a geometric regularization paradigm to approximate the signed distance function, which can be achieved without 3D supervision and/or a direct loss on the surface of the shape.
Inspired by incorporating derivatives in a regression loss leads to a lower sample complexity, Atzmon and Lipman~\cite{atzmon2020sald} generalized SAL to include derivatives and show a significant improvement in the quality of 3D shape reconstruction.
Note that it is extremely hard to obtain the ground-truth data about the signed distance from sampling points to the surface of 3D shape for training.
This is the main motivation for us to propose a method that can avoid such a voxalization used in algorithms that we discuss in the above paragraphs.
\subsection{Modelling local SDFs}
Instead of learning a single SDF for representing a whole shape, Jiang \etal~\cite{jiang2020local} designed an embedding of local crops of 3D shapes during training, and optimize a set of latent codes on a regular grid of overlapping crops with one single shared decoder when run on inference.
Inspired by DeepSDF~\cite{park2019deepsdf}, Chabra \etal~\cite{chabra2020deep} replaced the dense volumetric SDF representation used in traditional surface reconstruction with a set of locally learned continuous SDFs defined by a single parameterized neural network.
In contrast with the voxels(grid)-based representation of SDFs, Genova \etal~\cite{genova2019learning} proposed a network to encode shapes into structured implicit functions (SIF) as a composition of local shape elements.
Tretschk \etal~\cite{tretschk2020patchnets} designed an explicit parametric surface model, which fits an implicit SDF in each local patch separated from one sphere.
Hao \etal~\cite{hao2020dualsdf} represent 3D shapes as two levels of granularity with SDF, which provides interpretability for latent space of SDF in local parts of the shape.
They introduced a novel shape manipulation method by editing the primitives of local SDFs.
In the most recent works, Genova \etal~\cite{genova2020local} developed the SIF to learn a set of local SDFs that are arranged and blended according to a SIF template.
The method associates a latent vector with each local region that can be decoded with the SDF to produce finer geometric detail.
\subsection{Geometric learning on 3D shapes}
Generalize neural networks to data with the non-Euclidean structure are known as Graph Neural Networks (GNNs) in the domain of geometric learning.
Ranjan \etal~\cite{ranjan2018generating} proposed to learn a non-linear representation of human faces by spectral convolutions with Chebychev biasis~\cite{kipf2016semi, defferrard2016convolutional} as filters.
Bouritsas \etal~\cite{bouritsas2019neural} replaced the convolution kernel with operators applied along a spiral path around the graph vertices.
Hanocka \etal~\cite{hanocka2019meshcnn} leveraged the intrinsic geodesic connections of edges to define convolution operators, which inherited the direction invariant property as in 3D points convolution methods~\cite{qi2017pointnet, qi2017pointnet++}.
Zhou \etal~\cite{zhou2020fully} further improved the reconstruction precision by using locally adaptive convolution operators for registered mesh data.
We incorporate several ideas of this and the previous subsection in our method, but do it in different ways. More details are in the next section below.
\section{Local geometry code learning method}
\subsection{Modeling SDF locally}
A shape $\mathcal{S}$ can be represented as the zero level-set of $f_{\theta}(\mathbf{x}, \mathbf{z})$ as:
\begin{equation}
\mathcal{S} = \{ \mathbf{x} \in \mathbb{R}^3 \mid f_{\theta}(\mathbf{x,z})=0 \},
\end{equation}
where $f_{\theta}(\mathbf{x},\mathbf{z}):\mathbb{R}^3 \times \mathbb{R}^m \rightarrow \mathbb{R}$ is a signed surface distance function implemented as a neural network (usually as multilayer perceptron network) with learnable parameters $\theta$. The latent code $\mathbf{z}$ decides output shape of $f_{\theta}$ along with the sampling coordinates $\mathbf{x}$.
Similar with DeepLS~\cite{chabra2020deep}, we want to make $f_{\theta}$ model the whole shape as a composition of its local parts:
\begin{equation}
\mathcal{S} = \{ \mathbf{x} \in \mathbb{R}^3 \mid \bigcup\nolimits_{i} \mathbbm{1}_{\mathbf{x} \in \mathcal{R}_i} f_{\theta}(T_i(\mathbf{x}),\mathbf{z}_{i}^{L})=0 \},
\end{equation}
where $T_i(\cdot)$ supposes to transfer global location $\mathbf{x}$ to the local coordinate system $\mathbf{x}_{i}^{L}$ of a local region $\mathcal{R}_i$, and $\mathbf{z}_i^{L}$ indicates its related local latent code, as illustrated in Fig~\ref{fig:geo_sdf_decoder}~(a).
Different from splitting the 3D space into volumes~\cite{jiang2020local,chabra2020deep} or explicitly parametric surface model~\cite{tretschk2020patchnets, hao2020dualsdf}. we define the local region $\mathcal{R}_i$ with a key point $\mathrm{p}_i$ as:
\begin{equation}
\mathcal{R}_i = \{ \mathbf{x} \in \mathbb{R}^3 \mid \argmin\nolimits_{\mathbf{p} \in \mathcal{P}} d(\mathbf{x}, \mathbf{p}) = \mathbf{p}_i\},
\end{equation}
where $\mathcal{P}$ is a set of key points and $d(\cdot)$ is a distance function, \eg Euclidean distance.
Note that key points $\mathcal{P}$ are only used for aligning the sampling points to their corresponding local latent code $\mathbf{z}_i^L$. Thus,these key points do not necessary in the training or inference.
One simple illustration for our region division method is shown Fig~\ref{fig:geo_sdf_decoder}~(b). One patch with different color from their neighbours indicates a local region, which owns a corresponding latent code.
\subsection{Geometric leaning on local latent codes}
Different from the volume representation in DeepLS~\cite{chabra2020deep} and LDIF~\cite{genova2020local}, we do not know which local region includes the 3D shape, and it will lead to defining some unnecessary local latent codes.
Another problem is that these local latent codes are learned independently, which will lead to inconsistent surface estimates at the local region boundaries as mentioned in~\cite{chabra2020deep}.
Inspired by the geometric learning of COMA~\cite{ranjan2018generating}, Neural3DMM~\cite{bouritsas2019neural}, FCM~\cite{zhou2020fully}, we introduce the mesh structure of 3D shape as a "scaffold" to put key points and propagate information between local latent codes.
Then we can get two benefits that it allows to keep each local region has an intersection with the 3D shape and construct communications among them by representing the mesh as a graph.
Let us assume that a 3D surface mesh can be defined as a set of vertices $\mathcal{V}$ and edges $\mathcal{E}$, where $\mathcal{V}$ is replaced by $\mathcal{P}$ to define the local regions as shown in Fig \ref{fig:geo_sdf_decoder} (b).
Cooperate with several graph convolution layers to construct a graph network $\mathrm{G2L}(\mathcal{E}, \mathbf{z}^G): \mathbf{z}^G \rightarrow \{ \mathbf{z}^L_i \}$, we could predict the local SDF with the local latent codes aligned to these local regions, as shown in Fig \ref{fig:geo_sdf_decoder} (a).
Such graph neural network provides geometric deformations on each local latent code.
Consequently, each local latent code can contribute to the shape representation since each key point is on the shape.
Since our method gets benefits from modeling SDF in \textbf{Local} with latent codes and learning these latent codes through \textbf{Geometric Learning} with graph neural networks, thus we name it Local Geometry Code Learning (LGCL).
\subsection{Loss function}
\noindent {\bf Sign agnostic learning.}
Due to the advanced works of SAL~\cite{atzmon2020sal} and IGR~\cite{gropp2020implicit}, we do not request the true distance of sampling points to the surface of shape during training.
Instead of getting the true distance in a rendering way, directly calculating the distance from a point to a triangle soup is more convenient and efficient, and also without the requirement of watertight structures.
Thus, we construct basic loss function as:
\begin{equation}
\mathcal{L}_{\mathrm{basic}} = \mathcal{L}_{\mathrm{sal}} + \mathcal{L}_{\mathrm{igr}} ,
\end{equation}
where $\mathcal{L}_{\mathrm{sal}}$ just needs the unsigned distance $d_{\mathrm{u}}$ from point $\mathbf{x}$ that sampled from whole space $\Omega$ to the shape $\mathcal{S}$, and it is defined as:
\begin{equation}
\mathcal{L}_{\mathrm{sal}} = \mathbb{E}_{\mathbf{x}\in\Omega} \big| \left|f_{\theta}(T_i(\mathbf{x}), \mathbf{z}_i^L) \right| - d_{\mathrm{u}}(\mathbf{x}, \mathcal{S}) \big|.
\end{equation}
For the $\mathcal{L}_{\mathrm{igr}}$, we use its variant type from Siren~\cite{sitzmann2020implicit} as:
\begin{equation}
\begin{aligned}
\mathcal{L}_{\mathrm{igr}} &= \lambda_{\mathrm{grad}} \mathbb{E}_{\mathbf{x}\in\Omega}(\| \nabla_{\mathbf{x}}f_{\theta} \|_2 -1)^2 +\\
&\quad \quad
\mathbb{E}_{\mathbf{x}\in\Omega_0} \| \nabla_{\mathbf{x}}f_{\theta} - n(\mathbf{x}) \|_2,
\end{aligned}
\end{equation}
where $\Omega_0$ means the domain of zero-iso surface of the shape, $\|\cdot\|_2$ is the euclidean 2-norm.
\noindent {\bf Geometric similarity loss.}
There is a contradiction between the distributions of key points and sampling points.
Consider a complex part of the shape, it needs more key points to get more local latent codes for better modeling.
However, the more key points are allocated, the harder the optimization of local latent codes is.
Since each local region would be smaller and get fewer sampling points for training.
Even if increasing the number of sampling points, it is still difficult to ensure assigning enough sampling points to each local latent code.
To alleviate this problem, we propose a loss $\mathcal{L}_{\mathrm{sim}}$ to make these local latent codes not only learn from the sampling points but also learn from each other.
The assumption here is the difference between the adjacent local latent codes is smaller than the ones that are far away from each other.
On the other hand, it provides a kind of regularization effect on the local latent codes that cannot get sufficient training, which forces them to be similar to their neighbors.
Again, we use the geometric structure as a graph to calculate $\mathcal{L}_{\mathrm{sim}}$ as:
\begin{equation}
\mathcal{L}_{\mathrm{sim}} = \frac{1}{N_{\mathrm{v}}} \sum^{N_{\mathrm{v}}}_{i} \left| \mathbf{z}_i^L - \sum^{K}_{k} G_{\mathrm{l}}(\mathbf{z}_i^L, \mathcal{N}_k(i)) \right|
\end{equation}
where $G(x_i, \mathcal{N}(i)): x = \frac{1}{\left| \mathcal{N}(i) \right|} \sum_{j\in \mathcal{N}(i)}(x_j)$ means to update the value of $x_i$ by the average value of its neighbours.
Here $\mathcal{N}_k(i)$ means the neighbours of vertex $i$ in the $k$ layer.
And for better learning, we increase the neighbor region of one local latent code by $K$ layers.
We use $K=3$ layers for our experiments.
\noindent {\bf Total loss}
Our final loss function consists of above losses and a regular term $\| \mathrm{z}^G\|$ as:
\begin{equation}
\mathcal{L}_{\mathrm{total}} = \lambda_{\mathrm{sim}} \mathcal{L}_{\mathrm{sim}} + \mathcal{L}_{\mathrm{basic}} + \lambda_{reg} \|\mathrm{z}^G\|_2
\end{equation}
In our experiments, we use the setting of $\lambda_{\mathrm{grad}}=0.1$, $\lambda_{\mathrm{sim}}=1.0$ and $\lambda_{\mathrm{reg}}=0.001$ if without extra explanation.
\section{Experiments}
In our experiments, all of the used models are trained and evaluated mainly on a subset of the D-Faust dataset~\cite{bogo2017dynamic}, which is the No.50002 subset of mesh registrations for 14 different actions about the human body, such as leg and arm raises, jumps, etc.
Due to the low variation between adjacent scans, we sample the used dataset at a ratio of 1:10 and then split them randomly with 90\% for training and 10\% for the test.
Our data pre-processing method inherits from both IGR~\cite{gropp2020implicit} and SAL~\cite{atzmon2020sal}, which will generate 600K sampling points from each object, 300K are sampled on the object surface with their normals and the other 300K are sampled from two Gaussian distributions centered at points on the object surface.
\subsection{Reconstruction}
As one of the main baselines, we train the DeepSDF with the same setting in ~\cite{park2019deepsdf} on the completed No.50002 sub dataset of the D-FAUST dataset.
In addition, we also train other two different sizes of DeepSDFs with the loss function $\mathcal{L}_{\mathrm{basic}}$: SDF-8, which is similar to the original DeepSDF, but with 8 fully connected layers and 512 neurons in each hidden layer. The dimension of its latent code is 256. One skip connection is also used at the fourth layer with the latent code. SDF-4, has 4 fully connected layers, 128 neurons in each one and none skip connection, and the length of latent code is 64. Each fully connected layer in both SDF-8 and SDF-4 except the last one is followed by Softplus activation and initialized as in ~\cite{atzmon2020sal}.
In our LGCL-based method, we use a 4-layers G2L (as shown in Fig.~\ref{fig:g2l}) network followed by an SDF-4 network. The graph convolution kernels in the G2L are from~\cite{ranjan2018generating} as chebConv or~\cite{zhou2020fully} as vcConv, we gave the difference of their performance in the following results.
we re-implemented the kernels in a more compact form, and the parameters of one graph convolution layer can be represented as $(\mathrm{n},\mathrm{m},\mathrm{k})$, where $\mathrm{n}$ means the size of input channel and $\mathrm{m}$ is the size of the output channel. $\mathrm{k}$ stands for the size of the Chebyshev filter when using chebConv and the number of weight basis when using vcConv.
More details about the architecture of our models can be viewed in Table~\ref{tab:arch}.
The local latent code for each vertex of the G2L is set to an 8 length vector.
\begin{table}
\begin{center}
\begin{tabular}{l|c|c}
\hline
& LGCL-VC & LGCL-Cheb \\
\hline
\multirow{4}{*}{G2L} & vcConv(8,8,8) & chebConv(8,8,6) \\
& vcConv(8,16,16) & chebConv(8,16,6) \\
& vcConv(16,32,32) & chebConv(16,32,6) \\
& vcConv(32,64,64) & chebConv(32,64,6) \\
\hline
\multirow{4}{*}{SDF-4}
& \multicolumn{2}{c}{Linear(67,128)} \\
& \multicolumn{2}{c}{Linear(128,128)} \\
& \multicolumn{2}{c}{Linear(128,128)} \\
& \multicolumn{2}{c}{Linear(128,1)} \\
\hline
\end{tabular}
\end{center}
\caption{Setting of the architecture in the LGCL method. LGCL-VC means using the graph convolution kernels from~\cite{zhou2020fully} and LGCL-Cheb is from~\cite{ranjan2018generating}.}
\label{tab:arch}
\end{table}
We train all the models with 300 epochs at a learning rate of 5e-4 for the parameters of neural networks and 1e-3 for latent codes optimization.
Both learning rates are decayed to half after 200 epochs.
We evaluate all the methods on the split test dataset.
As same as in DeepSDF~\cite{park2019deepsdf}, the latent code will be estimated with the frozen neural network before the inference.
\begin{table}
\begin{center}
\setlength{\tabcolsep}{2pt}{
\begin{tabular}{l|c|c|c|c|c|c}
\hline
\multirow{2}{*}{Model} & Net & Latent & \multirow{2}{*}{CD} & \multirow{2}{*}{HD} & \multicolumn{2}{c}{ED} \\
& Params & Params & & & Mean & Std \\
\hline
DeepSDF & 1.84 M & 0.26 K & 0.28 & 68.11 & 12.51 & 16.37 \\
SDF-8 & 1.58 M & 0.26 K & 0.22 & 59.86 & 6.94 & 10.30 \\
SDF-4 & 41.86 K & 0.06 K & 2.20 & 119.56 & 24.45 & 31.19 \\
LGCL-Cheb & 58.42 K & 55.12 K & 2.56 & 108.58 & 2.51 & 1.87 \\
LGCL-VC & 0.19 M & 55.12 K & 1.55 & 71.67 & 3.35 & 2.44 \\
\hline
\end{tabular}}
\end{center}
\caption{Quantitative evaluation. CD means Chamfer Distance, HD means Hausdorff Distance and ED stands for Euclidean Distance of the point to surface.
All the distances are represented in millimeters.
We directly run the DeepSDF code as the baseline.}
\label{tab:qe}
\end{table}
\begin{table}
\begin{center}
\setlength{\tabcolsep}{1pt}{
\begin{tabular}{l|c|c|c|c|c}
\hline
\multirow{2}{*}{Model} & \multicolumn{2}{|c}{Error(mm)} & \multicolumn{3}{|c}{Percentage($\%$)}\\
& $<50\%$ & $<90\%$ & $>5$ mm & $>10$ mm & $>20$ mm\\
\hline
DeepSDF & 6.19 & 37.79 & 57.42 & 33.27 & 17.44 \\
SDF-8 & 3.55 & 18.59 & 34.79 & 14.14 & 6.39 \\
SDF-4 & 11.91 & 73.42 & 34.80 & 43.07 & 25.37 \\
LGCL-Cheb & 2.13 & 5.21 & 11.44 & 0.03 & 0.00 \\
LGCL-VC & 2.92 & 6.85 & 24.47 & 0.98 & 0.00 \\
\hline
\end{tabular}}
\end{center}
\caption{Statistics of reconstruction errors.}
\label{tab:statis}
\end{table}
To evaluate the performance of reconstruction, we measure the Euclidean Distance (ED) from the vertices of ground truth to the surface of reconstruction generated from different methods.
We also report our results under the metrics of Chamfer Distance (CD) and Hausdorff Distance (HD).
Due to CD and HD are applied on the point cloud, then we sample 30000 points on both surfaces of ground truths and reconstruction for it.
For a more fair comparison, we also list the size of network parameters and latent codes of different methods, while both are necessary to represent 3D shapes.
All of these quantitative results are shown in Table~\ref{tab:qe}.
As one can see that SDF-only-based methods need much more parameters of network to achieve comparable performance with our method.
We can see that there is a positive correlation between the size of the DeepSDF network and its performance on reconstruction, as all the quantitative results of SDF-4 are worse than SDF-8's.
By introducing local latent codes, our LGCL-based model outperforms SDF-4 by approximately one order of magnitude under the metric of Euclidean distance.
Even compared to the DeepSDF-8 which has a huge size of parameters, our results still has competitive advantages and obtain the smallest Euclidean error as $2.51 \pm 1.87$ mm with chebConv kernels.
More details about the Euclidean errors of different methods can be found in Table~\ref{tab:statis}.
Consequently, LGCL-Vc decreases the errors of CD and HD of SDF-4 by about 30\% and 40\% respectively.
We visualize two examples about the Euclidean error of each vertex shown in Fig~\ref{fig:compare2degeneration}.
It obviously shows that the small size of the DeepSDF network struggles to reconstruct the details, note that it almost loses the whole hands of the human.
In contrast, our LGCL model could keep more information in local regions though it causes more fluctuation.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\linewidth]{figs/compareerror-00115-00170.pdf}
\end{center}
\caption{Visualization of the per-vertex Euclidean error of the reconstructions.
GT means the ground truth shape, the model of Ours here used the LGCL-VC.}
\label{fig:compare2degeneration}
\end{figure}
\subsection{Ablation study}
\label{sec:ablation}
We perform ablative analysis experiments to evaluate the influence of our proposed geometric similarity loss $\mathcal{L}_{\mathrm{sim}}$.
It is controlled by adjusting its coefficient $\lambda_{\mathrm{sim}}$ to constrain the similarity between local latent codes with their neighbours.
As shown in Table~\ref{tab:lambda_influence},
the geometric similarity loss takes different influences on LGCL-VC and LGCL-Cheb.
Specifically, it tends to get better ED results with less constraint on similarities of local latent codes of LGCL-VC.
In contrast, one needs more similar local latent codes to decrease the errors of CD and HD since the large freedom of graph convolution kernels that used in LGCL-VC.
And for LGCL-Cheb, it implicitly has a stronger geometric constraint set by its ChebConv kernels.
Thus the extra geometric similar loss takes a little impact on the errors of CD and HD, but it should be patient to pick the adopted when you consider the ED errors.
\begin{table}
\begin{center}
\setlength{\tabcolsep}{3pt}{
\begin{tabular}{l|c|c|c|c|c|c}
\hline
\multirow{2}{*}{Model} &
\multirow{2}{*}{$\lambda_{\mathrm{sim}}$} &
\multirow{2}{*}{CD} &
\multirow{2}{*}{HD} & \multicolumn{3}{c}{ED} \\
& & & & Mean & Std & Median \\
\hline
\multirow{3}{*}{LGCL-VC}
& 0.1 & 2.66 & 105.59 & 2.45 & 1.72 & 2.16 \\
& 1.0 & 1.55 & 71.67 & 3.35 & 2.44 & 2.92 \\
& 10.0 & 1.45 & 66.05 & 4.07 & 2.97 & 3.56 \\
\hline
\multirow{3}{*}{LGCL-Cheb}
& 0.1 & 2.43 & 108.65 & 2.72 & 1.91 & 2.39 \\
& 1.0 & 2.56 & 108.58 & 2.51 & 1.87 & 2.13 \\
& 10.0 & 2.74 & 108.69 & 4.44 & 3.11 & 3.94 \\
\hline
\end{tabular}}
\end{center}
\caption{Influence of geometric similarity loss, all results are shown in millimetres.}
\label{tab:lambda_influence}
\end{table}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\linewidth]{figs/failed_results.pdf}
\end{center}
\caption{Failed results of different methods for getting local latent codes.
(a) is the ground truth~(GT);
(b) uses G2L but with the same initial input of global latent code for each graph vertex;
(c) uses a pyramid G2L to get local latent codes;
(d) does not include $\mathcal{L}_{\mathrm{sim}}$.
Please check more details in Sec~\ref{sec:ablation}.}
\label{fig:failed_results}
\end{figure}
We found some interesting explorations on the influence of different methods for getting local latent codes, as shown in Fig~\ref{fig:failed_results}.
In our LGCL-based methods, we split the global code $\mathrm{z}^G$ evenly into parts, which is equal to the number of the graph vertices, and then align these parts to different vertices.
However, for the result in Fig~\ref{fig:failed_results}~(b), we directly align the same initial input, which is the global latent code, to each vertex of the G2L.
Since each local latent code has the same initial value, it provides a similarity constraint implicitly between them.
Then we do not introduce the $\mathcal{L}_{\mathrm{sim}}$.
In this case, each local region of reconstruction tends to shrink to the same type of mini polyhedron.
We attribute this degeneration of modelling local SDFs to the over-constraint on the similarity among the local latent codes, which introduces a limitation to geometric deformation of them.
We also show the results without the usage of $\mathcal{L}_{\mathrm{sim}}$ in our LGCL-based methods in Fig~\ref{fig:failed_results}~(d).
It is obvious to see the dramatic vibrations in some local regions.
We argue that it is caused by insufficient training of some local latent codes in corresponding local regions.
Furthermore, as shown in Fig~\ref{fig:failed_results}~(c), the result looks like fall in between (b) and (d), which has both similar polyhedron and vibrations that exist in local regions.
We got this result by modifying the G2L network as a pyramid structure as in the COMA~\cite{ranjan2018generating} decoder.
The modification changes the graph structure of each graph convolution layer with pooling layer.
The pyramid structure provides a cluster mapping from top to bottom in this G2L variant.
In other words, the mentioned modification adds an extra similarity constraint on local latent codes within a more large geometric range.
However, the approach is also limited to deform the local latent codes and leads to a compromise result compared to (b), (d) and our major results in Fig~\ref{fig:compare2degeneration}.
\section{Conclusions}
In this work, we propose the LGCL method which is based on a new architecture for the local geometry learning handling.
The idea is to perform the learning regression process directly in the latent code space. Consequently, our approach makes general GNNs more flexible, compact, and simple in realization.
The experimental results show that our method considerably outperforms baselines DeepSDFs both in accuracy and model size.
We think that our architecture is novel, promising, and can be further improved in future work.
\section{Acknowledgements}
This work has been supported by the Spanish project TIN2015-65464-R, TIN2016-79717-R (MINECO/FEDER) and the COST Action IC1307 iV\&L Net (European Network on Integrating Vision and Language), supported by COST (European Cooperation in Science and Technology).
We acknowledge the CERCA Programme of Generalitat de Catalunya.
Shun Yao acknowledges the Chinese Scholarship Council (CSC) grant No.201906290108.
We also acknowledge the generous GPU support from Nvidia.
{\small
\bibliographystyle{ieee_fullname}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction.}
Recent developments in nanophysics have made possible the use of the charge and spin dynamics
of electrons to develop new technologies like spintronics and quantronics. This has moreover led to the possibility of
using the electron spin, or other more complex entities like the phase in Josephon junctions
to fabricate qubits for quantum computing. However,
the utility of these nanosystems as qubits is strongly limited by
their coupling to the omnipresent dissipative environment. The environment destroys the coherence of the qubit over a certain time
scale and a lot of recent theoretical and experimental activity has
focused on ways and means to increase this time scale.
This clearly emphasizes the importance of understanding the effects
engendered by the coupling of a two level system to a dissipative bath.
The fact that the environment plays a crucial role in the
physics of small quantum systems has been well known since the pioneering
work of Ref.{\onlinecite{Caldeira-Leggett}}, where it was shown that the coupling of a
two level system to an Ohmic boson bath could
effectively suppress the tunneling of the two level system. In the context of decoherence,
the most commonly studied problem is the spin-boson model \cite{QDS} which describes the effect
of a dissipative bosonic bath on a central spin, where the spin can be an effective description
of a system whose discrete lowest energy levels dominate the physics at low enough temperatures
and the bosons are often the phonons present in the system. A physical manifestation of the spin boson
problem is a nanomagnet (described by a giant spin) coupled to phonons \cite{stamp-review}.
However, for many practical realisations of a
central spin or a qubit (spin $1/2$), a spin bath comprising other spins
might be the principal source of decoherence. This is indeed the case in semiconducting
quantum dots, where the nuclei with non zero spins constitute the spin bath and interact with
the central electronic spin in the dot via the hyperfine interaction \cite{Coish-Loss,Marcus}.
Another manifestation of a spinbath occurs in Si:P \cite{Hu}.
The abundance of spin baths in real systems, necessitates an understanding of their effect on decoherence.
Unlike the case of bosonic baths often
modeled as a collection of harmonic oscillators, spin baths can exhibit a wide range of phenomena
depending on the interactions between the spins, residual anisotropies etc. Clearly one expects any resulting
decoherence of the central spin to depend rather crucially on the
underlying nature of the spin bath and its coupling to the central spin.
Earlier studies which considered independent spins in the bath seemed to indicate that spin baths
were not qualitatively different from bosonic baths \cite{Hanggi,Hanggi1}.
More interestingly, recent studies of decoherence induced by spin baths
described by mean field Hamiltonians have demonstrated that
interactions between the bath spins can be used as a lever to augment
the time scales over which the system decoheres
\cite{italian-meanfield, sib,Yuan}. These results were however
obtained either numerically for a bath with a small number of spins or
for the special case where the bath Hamiltonian commutes with the bath-central spin
coupling Hamiltonian leading to an effective classical decoherence.
A more robust treatment of intrabath interactions was presented in
Ref.\onlinecite{cb} where the authors studied numerically the zero temperature {\it quantum }decoherence of two coupled spins engendered by a bath described by the random transverse Ising model.
The authors used this model to argue that the central spin decoheres differently, depending
on whether the spinbath has a regular or chaotic spectra.
Despite their various drawbacks, these works collectively highlight the
importance of interactions and disorder in the bath.
Moreover, disordered spin baths warrant further attention because
both interactions (often dipolar) and disorder are present in real spin bath systems like
quantum dots in semiconducting heterojunctions and in Si:P.
In this paper, we re-examine the decoherence
induced by disordered interacting spinbaths at finite
temperatures. More precisely, we study the effect of an Ising
bath with random spin-spin interactions on the coherence of a
single spin $1/2$. The random interactions are
characterized by their variance $\Delta^2$ where $\Delta$ is analogous
to the cut off frequency for a bosonic bath as well as a mean value $J_0$
which has no bosonic counterpart. Our choice of an Ising bath is primarily to
facilitate an analytical study of the problem at finite temperatures in the
thermodynamic limit. To ensure a quantum decoherence of the central
spin in our model, the central spin is coupled to the transverse spin
components of the bath. To better comprehend the effect of the bath, we consider
a model in which the time evolution of the central spin is exclusively governed
by its coupling to the bath. For such a model, the problem is exactly soluble
for a bath comprising independent spins/bosons. However, when intra-bath interactions are
present, no exact solution can be obtained and one has to take recourse to approximate
methods. In this paper, we only study the limit of a
weak coupling of the central spin to the bath, where robust analytical methods are
available to study the problem in an unbiased manner.
We use the resolvent operator method, which takes us beyond
the oft used Markovian master equation approaches to study the decoherence of the central
spin induced by a perturbative coupling to the Ising spin bath.
An interesting aspect of our work is that in the absence of any dynamics intrinsic to the central spin,
for weak coupling to the bath, the decoherence is primarily dictated by the local-field distribution
or equivalently, the dynamic structure factor of the bath spins. For a bath described by an Ising spin chain,
we obtain the Markovian decoherence time scale and the non Markovian corrections as a function of
the temperature and the parameters of the bath.
We also discuss the case of Ising spins on various lattices in the
high temperature regime and the case of the infinite-ranged Sherrington-Kirkpatrick
spin glass model.
The paper is organized as follows: we present the model and
derive a general expression for the decoherence of the central spin in \sref{sec:model} followed by
a discussion of the weak coupling regime in \sref{sec:wcr}.
We then present our results for different disordered Ising spin baths in \sref{sec:DIsb}.
\section{Model}\label{sec:model}
We present the model used to study the decoherence of the central spin
${{\mbox{\boldmath$\sigma$}}}_c$, weakly coupled
to a bond-disordered bath of $N$ Ising spins ${{\mbox{\boldmath$\sigma$}}}_i$ in the thermodynamic limit $N\to\infty$.
The total Hamiltonian describing the combined system of the central spin and the spin bath is
given by
\begin{eqnarray}
H&=&H_B+\sigma^x_cV \\
&\equiv&-\sum_{(ij)} J_{ij} \sigma_i^z \sigma_j^z -\sigma^x_c \sum_{i} \lambda_{i} \sigma_i^x
\label{H}
\end{eqnarray}
where,
$\sigma^x_c$ is the $x$-component Pauli operator of the central spin and $\sigma_i^x$ and $\sigma_i^z$
denote the Pauli operators of the bath spins and $J_{ij}$ are the
interaction strengths between the bath spins.
Depending on the details of the model studied, $(ij)$ could represent
interactions between nearest neighbour spins or interactions of infinite range.
In contrast to the models studied in Ref.~\onlinecite{italian-meanfield}, where the bath hamiltonian $H_B$ and the
bath operator
$V$ that couples to the central spin commute, here our choice of $V$ is such that $[H_B, V] \neq 0$.
The central spin and spin bath coupling is characterized by the parameters $\lambda_i$.
Since, we are interested in the influence of disorder as well as the tendency of the
system to order, we consider random interaction energies $J_{ij}$ which are
quenched random variables drawn from a distribution $p(J)$ with mean
$J_0$ and variance $\Delta^2$.
Though the central spin does not have any intrinsic dynamics, its coupling
to the bath generates a non trivial dynamical behaviour. We note that $H_B$ is the usual
Ising spin glass Hamiltonian which has been well studied in the past\cite{spinglassbook}. Depending on the
distribution of the spin-spin interactions and the dimensionality, this model can exhibit ferromagnetic, antiferromagnetic or even spin glass order in some temperature range. Since these phenomena have
ramificiations for the collective behaviour of the bath, it is reasonable to expect the
resulting decoherence to depend crucially on the underlying order in the bath.
It is important to note that the formalism developed in this section and \sref{sec:wcr} is {\it a priori}
applicable to any
bath hamiltonian $H_B$ (bosonic baths, Heisenberg spin baths, baths with both spins and bosons etc).
For a Hamiltonian of the form \eqref{H}, since $\sigma^x_c$ is a constant of motion, it is
convenient to directly study the time evolution of the reduced
density matrix of the
central spin
\begin{equation}
\rho(t)=\mathrm{Tr}_B \left( e^{-iHt} \Omega e^{iHt} \right)
\end{equation}
where $\Omega$ is the initial density matrix of the composite system consisting of the central
spin and the bath and $\mathrm{Tr}_B$ denotes the partial trace over the bath degrees of freedom. We use the units $\hbar=k_B=1$ in this paper.
Denoting the eigenstates of $\sigma^x_c$ by $|\! \gets \rangle$ and $|\! \to \rangle$, we see that due to the property of the density matrix $\mathrm{Tr} [\rho(t)] = 1$ and the
stationarity of $\mathrm{Tr} [\rho(t)
\sigma^x_c]$, the diagonal elements of the density matrix given by the
populations $\langle \gets \! | \rho(t) | \! \gets \rangle$
and $\langle \to \! | \rho(t) | \! \to \rangle$ remain
constant. Only
the coherences $\langle \to \!| \rho(t) | \! \gets \rangle=\langle \gets \!| \rho(t) | \! \to \rangle^*$ representing the off-diagonal elements of
the density matrix $\rho(t)$ change with time. Often these coherences vanish at long times resulting in a decoherence of the central
spin i.e.,
the asymptotic state of the central spin is a mixture of the states $|\! \gets \rangle$ and $|\! \to \rangle$
irrespective of the nature of the initial state $\Omega$.
To simplify the calculation, we assume that at time $t=0$, the central spin and the bath are
disentangled resulting in a factorizable initial density matrix: $\Omega=\rho(0) \otimes \rho_B$.
We further suppose that initially the central spin is in a pure state
$ |\psi \rangle = \alpha|\! \gets \rangle + \beta|\! \to \rangle$ ($\rho(0) = |\psi \rangle \langle \psi |$)
and that the bath is at thermal equilibrium with temperature $T\equiv 1/\beta$\,:
\begin{equation}
\rho_B = \frac{e^{-\beta H_B}}{Z} \label{rhoB}
\end{equation}
where $Z=\mathrm{Tr} \, \exp(-\beta H_B)$ is the bath partition function.
With these initial conditions we obtain the time evolved reduced density matrix
\begin{widetext}
\begin{equation}
\rho(t)= |\alpha|^2 \vert \!\gets \rangle \langle \gets \! \vert +
|\beta|^2 \vert \! \to \rangle \langle \to \! \vert +
M(t) \alpha^* \beta \vert \! \to \rangle \langle \gets \! \vert
+ M(t)^* \alpha \beta^* \vert \! \gets \rangle \langle \to \! \vert
\end{equation}
\end{widetext}
where the factor
\begin{equation}
M(t)= \mathrm{Tr} \left( e^{-i(H_B+V)t} \;\rho_B\; e^{i(H_B-V)t} \right) \label{Mt}
\end{equation}
is a measure of the decoherence induced by the bath at time $t$.
Here $\mathrm{Tr}$ denotes the usual trace as $H_B$ and $V$ are operators in the bath Hilbert space.
Under a rotation of $\pi$ around the $z$-axis,
while $H_B$ and $\rho_B$ remain unchanged, $V \to -V$. Consequently the coherence $M(t)$ is a real number.
As mentioned in the introduction, though the coherence $M$ is easy to evaluate for baths consisting of
non-interacting spins/bosons (see \aref{Isb}), it is rather difficult to estimate for an interacting spin bath
for arbitrary values of the coupling to
the central spin.
We remark that $M(t)$ is related to Loschmidt echoes of the bath which characterize
its sensitivity to perturbations in its equations of motion \cite{LE,LE1,fazio}.
\section{Weak coupling regime}\label{sec:wcr}
In this section, we present a perturbative formalism to calculate $M(t)$ valid for weak
coupling to the bath i.e., the energy scale of the operator $V$ is much smaller than all the energy scales of the bath.
We use the resolvent operator method which goes beyond the Born-Markov approach and, though
perturbative, is intrinsically capable of handling non-Markovian time evolutions. We obtain a tractable expression
for the decoherence in the weak coupling regime.
\subsection{Resolvent operator method}
To determine the complete time evolution of the coherence $M$ in the weak coupling limit $V \to 0$, we first express it
in terms of a {\it self energy} $\Sigma$\cite{ROM,ROM1}. To obtain the self energy, it is
convenient to work with the Laplace transform of $M(t)$
\begin{equation}
{\tilde M}(z)= -i \int_0^\infty dt \, e^{izt} M(t) \label{LTdef}
\end{equation}
where $z$ is a complex variable with $\mathrm{Im} z >0$.
As shown in \aref{Dse}, this Laplace transform can be written as
\begin{equation}
{\tilde M}(z)= \left[ z-\Sigma(z) \right]^{-1} \label{LT}
\end{equation}
\noindent
where the self-energy $\Sigma$ is given by
\begin{equation}
\Sigma(z)= \mathrm{Tr} \Big[ {\cal L}_V \rho_B + {\cal L}_V {\cal Q} (z-{\cal Q}\, {\cal L} \,{\cal Q})^{-1}
{\cal Q} {\cal L}_V \rho_B \Big] \label{Sigma}
\end{equation}
In the above expression, the superoperators ${\cal Q}$, ${\cal L}_B$, ${\cal L}_V$ and ${\cal L}$ are defined by their actions on any operator $A$ acting on the bath Hilbert space:
${\cal Q} A = A - \mathrm{Tr} (A) \rho_B$, ${\cal L}_B A = [H_B, A]$, ${\cal L}_V A = V A + A V$ and ${\cal L}={\cal L}_B+{\cal L}_V$.
As $\rho_B$ is a density matrix, ${\cal Q}$ is a projection operator, i.e. ${\cal Q}^2={\cal Q}$.
Note that ${\cal L}_V$ is not a Liouville operator whereas ${\cal L}_B$ is the Liouville operator
corresponding to the bath Hamiltonian $H_B$. We reiterate that the above derivation for \eqref{LT} and \eqref{Sigma} is independent of the specific
Hamiltonian $H_B$ and coupling operator $V$ considered in this paper.
The self-energy $\Sigma$ can now be expanded perturbatively in the interaction operator $V$.
The second order result is
given by the expression \eqref{Sigma} with ${\cal L}$ replaced
by the bath Liouvillian ${\cal L}_B$.
The first-order term $ \mathrm{Tr} ( {\cal L}_V \rho_B )=2\mathrm{Tr} (V \rho_B )$ vanishes for the Ising
spin bath Hamiltonian $H_B$ and the interaction operator $V$ defined by \eqref{H}.
Therefore, the first non-zero contribution to the self-energy
is given by the second-order term $\Sigma_2$ which can be rewritten in terms of the time-dependent
symmetrised correlation function of $V$ (see \aref{Sose}) :
\begin{equation}
\Sigma_2(z) = -2i \int_0^\infty dt\, e^{izt} \left[ \langle V(t)V\rangle + \langle VV(t)\rangle \right] \label{soSigma}
\end{equation}
where $\langle...\rangle$ refers to the thermal average over bath spin configurations. Neglecting higher order contributions to $\Sigma$ in \eqref{LT} is equivalent to the Born approximation \cite{QDS}.
We will see in the following that this approximation can describe the decoherence at all timescales whereas
a direct expansion of the coherence $M$ gives only the short-time evolution.
We remark that in the case of Heisenberg spins, an underlying magnetic order could result in a first order
contribution to the self energy which would then lead to an oscillatory behaviour of $M(t)$.
The advantage of the resolvent operator formalism is that we can use the analyticity properties of the self
energy $\Sigma$ to obtain a
tractable expression for the coherence $M$.
As shown in \aref{Dse}, since $H_B$ and $V$ are Hermitian operators,
the spectrum of the operator ${\cal L}$
is real and hence the self-energy $\Sigma$ is analytic in the upper (lower) half plane.
Furthermore, since the spectrum of $\cal L$ in the thermodynamic limit is expected to be a continuum
for the models considered in this paper, the self-energy $\Sigma$ manifests a branch cut on the real axis.
The coherence $M$ can thus be written
in terms of the real functions $\Lambda$ and $\Gamma$ defined by
\begin{equation}
\Lambda (E)-i\Gamma(E)= \lim_{\eta {\to} 0^+} \Sigma(E+i\eta) \label{DeltaGamma}
\end{equation}
where $E$ is real. Since $M(t)$ is real, the functions $\Lambda$
and $\Gamma$ satisfy $\Lambda(-E)=-\Lambda(E)$ and $\Gamma(-E)=\Gamma(E)$.
Performing the inverse Laplace transform of \eqref{LT} and
taking the limit $\eta {\to} 0^+$, we obtain
\begin{equation}
\Theta(t) M(t) = \frac{i}{2\pi} \int dE \frac{e^{-itE}}{E-\Lambda(E)+i\Gamma(E)} \label{ThetaM}
\end{equation}
where $\Theta(t)$ is the Heaviside step function. Moreover,
integrating $\Sigma(z)/(z-E)$ along an appropriate contour in the upper half plane, one obtains the Kramers-Kronig
like dispersion relation
\begin{equation}
\Lambda(E)-i\Gamma(E) = -\frac{i}{\pi} {\cal P} \int \frac{dE'}{E'-E} \left[ \Lambda(E')-i\Gamma(E') \right]
\label{Ht}
\end{equation}
where ${\cal P}$ denotes the Cauchy principal value. This shows that $\Theta(t) M(t)$
is exclusively determined by $\Gamma$ (or $\Lambda$).
\subsection{Non-Markovian evolution }\label{sec:NME}
In this section, we analyze \eqref{ThetaM} to determine $M(t)$ at any time $t$ in the weak coupling regime.
We first note that differentiating \eqref{ThetaM} with respect to the time $t$ yields the conditions
$M(0)=1$ and $\partial_t M(0)=0$.
Differentiating \eqref{ThetaM} further
and taking
the limit $V\to 0$, we find, to second order in $V$,
\begin{eqnarray}
\Theta(t) \partial^2_t M & \simeq &- \frac{1}{2\pi} \int dE e^{-iEt} [ \Gamma_2(E) + i\Lambda_2(E) ] \nonumber \\
& \simeq &- \frac{1}{\pi} \Theta(t) \int dE \, e^{-iEt} \, \Gamma_2(E)
\end{eqnarray}
where $\Gamma_2$ and $\Lambda_2$ denote the second-order terms of the functions $\Gamma$ and $\Lambda$. Note that to obtain the second equality we have used the relation \eqref{Ht} and the Fourier
transform of $2\Theta(t)-1$.
A solution to the above differential equation, for $t > 0$, is
\begin{equation}
M(t) \simeq 1 - \frac{2}{\pi} \int dE\, \frac{\sin (tE/2)^2}{E^2} \Gamma_2(E) \label{Mapproxst}
\end{equation}
where we have taken into account the symmetry of the function $\Gamma_2$. Though this Fermi golden rule like
approximation yields the correct short time behaviour, it leads to the false result
$M(t) \simeq -\Gamma_2(0)t$ for $t \to \infty$
and hence is invalid for an arbitrary time $t$.
To obtain the long-time decoherence, we evaluate the integral
\eqref{ThetaM} using the analytic continuation of $\Sigma$ from the upper half plane
to the lower half plane (second Riemann sheet). If the functions $\Gamma$ and $\Lambda$
are analytic in the vicinity of $E=0$, the analytic continuation of $\Sigma$ in
the second Riemann sheet is given by
\begin{equation}
{\tilde \Sigma} (z) = - i\Gamma(0) + \Lambda'(0) z + O(|z|^2)
\end{equation}
for small $z$ where $\Lambda'$ denotes the derivative of the function $\Lambda$
with respect to $E$. Note that the symmetry properties of the functions $\Lambda$ and $\Gamma$
ensure $\Lambda(0)=0$ and $\Gamma'(0)=0$.
The coherence \eqref{ThetaM} is principally determined by the singularities of
$[z-{\tilde \Sigma} (z)]^{-1}$, one of which is a pole at $z_0 = - i\Gamma_2(0) + O(V^4)$
close to the real axis. Every other singularity lies beyond a finite distance $\epsilon^{-1}$
from the real axis determined by the temperature and
the bath parameters. Consequently, for times
$t \gg \epsilon$, the residue $\exp(-iz_0t)[1-\partial_z {\tilde \Sigma} (z_0)]^{-1}$
dominates and thus in the weak coupling limit $V \to 0$,
\begin{equation}
\ln M(t) \simeq - \Gamma_2(0)t + \Lambda_2'(0) \quad . \label{Mapproxlt}
\end{equation}
Combining the approximations for short times and long times given by \eqref{Mapproxst} and \eqref{Mapproxlt}
respectively, we see that
in the weak coupling regime the decoherence at any time $t$ is described by
\begin{equation}
\ln M(t) \simeq - \frac{2}{\pi} \int dE\, \frac{\sin (tE/2)^2}{E^2} \Gamma_2(E) \label{Mapprox} \quad .
\end{equation}
This equation shows that $M(t)$ in the weak coupling regime is determined by the entire function $\Gamma_2$.
For asymptotic times, we see from \eqref{Mapprox} that the coherence of the central spin is essentially given by $M(t) \simeq \exp[-\Gamma_2(0)t]$
which
is simply the solution of the Markovian master equation obtained
within the Born-Markov approximation \cite{QDS}.
However, for the short and intermediate time evolution of $M$, the full energy dependence of $\Gamma_2$ comes into play
which can then lead to a {\it non-exponential decay } i.e., non-Markovian behaviour of the coherence. Depending on the temperature and
bath parameters, the asymptotic
Markovian regime could even disappear, provided that $\Lambda_2'(0)$ goes to infinity. We remark that Eq.\eqref{Mapprox}
is valid to all orders in $V$ for a bath of independent bosons.
\section{Disordered Ising spin baths}\label{sec:DIsb}
In this section, we use the formalism of the previous section to determine the decoherence induced by
various disordered Ising spin baths. Though the spinbath models of real systems are expected to be more
complicated than the Ising Hamiltonians considered here, we nonetheless study these systems in various
dimensions to understand the effect of these simpler systems on the decoherence.
More precisely, we study the effect of the disordered Ising spin chain on the coherence
$M$ of the central spin as a function of the temperature and the bath parameters $J_0$ and $\Delta$.
We also make predictions for the infinite-ranged Ising model also known as the Sherrington-Kirkpatrick
model and for Ising models in higher dimensions in the high temperature regime. Before we embark on
in-depth calculations of $M$, we show that for any Ising bath, the decoherence of the central
spin in the weak coupling regime is intimately linked to the local-field distribution of the bath.
\subsection{Local-field distribution}
As shown in \sref{sec:wcr}, the decoherence in the weak coupling regime is determined by the time-dependent
correlation function $\mathrm{Re} \langle V(t) V \rangle$, through \eqref{Mapprox} and \eqref{soSigma}.
For the case of the Ising bath Hamiltonian defined in (\ref{H}), this correlation
can be obtained from the local-field distribution of the bath. To see this,
it is useful to work in the eigenbasis $|\{\sigma_i\} \rangle$ of $H_B$ ~: $H_B|\{\sigma_i\} \rangle=-
\sum_{(ij)} J_{ij} \sigma_i \sigma_j|\{\sigma_i\} \rangle$.
As the matrix element $\langle \{\sigma_i\} | \sigma_k^x |\{\sigma'_i\} \rangle$ is nonvanishing only for spin configurations
$\{\sigma_i\}$ and $\{\sigma'_i\}$ where $\sigma'_i=\sigma_i(1-2\delta_{ik})$, the time-dependent correlation of $V$ is a sum
of local correlations : $\langle V(t) V \rangle=\sum_k \lambda_k^2 \langle \sigma_k^x(t) \sigma_k^x \rangle$. We find
\begin{equation}
\mathrm{Re} \langle V(t)V\rangle = \frac{1}{Z} \sum_{k,\{\sigma_i\}} \lambda_k^2 e^{\beta \sum_{(i j)} J_{ij} \sigma_i \sigma_j}
\cos \left( 2t \sum_{i}\nolimits^{(k)}J_{ki}\sigma_i\right) \label{VV1}
\end{equation}
where $\sum_{i}^{(k)}$ denotes a sum over the spins ${{\mbox{\boldmath$\sigma$}}}_i$ interacting with the spin ${{\mbox{\boldmath$\sigma$}}}_k$.
Note that the term $\sum_{i}^{(k)} J_{ki} \sigma_i$ in \eqref{VV1} is the effective local field acting on the spin at site $k$
generated by the configuration $\{\sigma_i\}$. Since $\mathrm{Re} \langle \sigma^x(t) \sigma^x \rangle=\cos(2th)$ for an isolated spin {\boldmath $\sigma$}
in a field $h$ parallel to the $z$-axis, we rewrite \eqref{VV1} as
\begin{equation}
\mathrm{Re} \langle V(t)V\rangle = \sum_k \lambda_k^2 \int dh\, P_k(h) \cos(2th)
\end{equation}
where $P_k(h)= \left\langle \delta \left( h- \sum_{i}^{(k)} J_{ki} \sigma^z_i \right) \right\rangle$ can be interpreted as the distribution of
the local field $h$ at site $k$ at temperature $T$. This interpretation
is also valid for the bath thermodynamic quantities such as magnetisation or specific heat \cite{lfd}.
It is now rather straightforward to obtain the function $\Gamma_2$ which is the
crucial ingredient to determine the coherence $M$ in the weak coupling regime. To avoid computational
complexity, in the rest of the paper,
we consider equal couplings $\lambda_k=\lambda N^{-1/2}$. With this choice, the function $\Gamma_2$ reads
\begin{equation}
\Gamma_2(E) = 2\pi \lambda^2 P (E/2) \label{GammaP}
\end{equation}
where the local-field distribution $P$ is the spatial average
\begin{equation}
P(h)=\frac{1}{N} \sum_k P_k(h) = \frac{1}{N} \sum_k \left\langle \delta \left( h- \sum_{i}\nolimits^{(k)} J_{ki} \sigma^z_i \right) \right\rangle \quad .
\label{lfd}
\end{equation}
\noindent
The thermodynamic limit of this expression is unambiguously defined and $P$ is self-averaging \cite{salfd}
i.e., in the limit $N \to \infty$, $P$ is given by
bond disorder average of any distribution $P_k$. For an Ising spin system, the local-field
distribution $P$ is temperature dependent and determines the thermodynamic quantities
and the dynamic linear response of the system \cite{lfd,Plefka}. Eq. (\ref{GammaP}) shows that the local-field
distribution is also an important characteristic of an Ising system considered as a bath.
We remark that for a bath of independent spins \cite{Hanggi}, i.e. $H_B=-\sum_i h_i \sigma_i^z$ in \eqref{H},
the function $\Gamma_2$ is given by \eqref{GammaP} with the field distribution
$P(h)=\sum_k (\lambda_k/\lambda)^2 \delta(h-h_k)$.
In this case, since the spins are non-interacting, the distribution $P$ is arbitrary and temperature independent
and the ensuing
decoherence of the central spin is temperature independent.
This feature of a temperature independent decoherence induced by a bath of independent spins
seen in the weak coupling limit
is also seen in the exact non perturbative result for the coherence $M(t)$ obtained in \aref{Isb}.
\subsection{Ising spin chain}
In this section we consider a 1D Ising bath described by the Hamiltonian
\begin{equation}
H_B =-\sum_{i=1}^{N-1} J_{i} \sigma_i^z \sigma_{i+1}^z \quad . \label{1DH}
\end{equation}
The spin at site $i$ interacts with its nearest neighbors with interaction strengths $J_i$ and $J_{i-1}$.
The $J_i$ are a quenched set of random bonds drawn independently from a distribution $p$ with mean
$J_0$ and variance $\Delta^2$.
\subsubsection{Born self-energy}
To obtain the coherence $M$ we need the time-dependent correlation of
the coupling operator $V=-\sum_i \lambda_i \sigma_i^x$. As shown earlier, this correlation
is given by $\mathrm{Re} \langle V(t) V \rangle=\sum_k \lambda_k^2 \mathrm{Re} \langle \sigma_k^x(t) \sigma_k^x \rangle$.
Here the time dependent spin-spin correlations can be written in terms of the static correlation
function as
\begin{eqnarray}
\mathrm{Re} \langle \sigma_k^x (t) \sigma_{k}^x \rangle &= &\cos(2tJ_{k-1}) \cos(2tJ_{k}) \\ &&
- \langle \sigma_{k+1}^z \sigma_{k-1}^z \rangle \sin(2tJ_{k-1}) \sin(2tJ_{k}) \nonumber
\end{eqnarray}
where $\langle \sigma_{k+1}^z \sigma_{k-1}^z \rangle=\tanh(\beta J_{k-1}) \tanh(\beta J_{k})$ is related
to the derivative of the partition function
\begin{equation}
Z=2^N \prod_{i=1}^{N-1} \cosh (\beta J_{i})
\end{equation}
with respect to the interaction strengths $J_{k}$ and $J_{k-1}$.
The choice $\lambda_k=\lambda N^{-1/2}$ yields a variance of
$\mathrm{Re} \langle V(t)V\rangle$ of order $N^{-1}$ and a mean of order $N^0$.
Consequently, $\mathrm{Re} \langle V(t)V\rangle$ and hence $\Sigma_2$ are self-averaging
in the thermodynamic limit. The second-order self-energy is thus given by the average of \eqref{soSigma}
over bond disorder. We obtain
\begin{widetext}
\begin{equation}\label{Gamma1D}
\Gamma_2(E)=2\pi \lambda^2 \int dJ \, p_e (J) p_e (J+E/2) \left[ 1 - \tanh(\beta J)\tanh(\beta J+\beta E/2) \right]
\end{equation}
\end{widetext}
where $p_e (J)=[p(J)+p(-J)]/2$ is the symmetrized bond distribution.
Note that the integral on the right side is the local-field distribution \eqref{lfd}
for the spin chain\cite{salfd,1Dlfd}. Equation \eqref{Gamma1D} is valid for all bond distributions $p$.
Moreover, for distributions $p$ symmetric around their mean values,
as $\Gamma_2$ depends only on the symmetrized distribution $p_e$ the self-energy $\Sigma_2$ is the same for opposite
means $\pm J_0$. This shows that, in this case, though the decoherence is influenced by the interactions in the bath,
it is insensitive to the
underlying ferromagnetic or antiferromagnetic nature of the interactions. This feature can be understood as follows.
The spin chain Hamiltonian \eqref{1DH} and the local field operator
$J_{k-1} \sigma_{k-1}^z + J_{k} \sigma_{k+1}^z$ are invariant under
the transformation $[J_{i},\sigma^z_i] \to [-J_{i},(-1)^{i-k} \sigma^z_i]$.
Consequently, the local-field distribution $P$ and hence
the function $\Gamma_2$ are invariant
under the transformation $J_0 \to -J_0$. To complete the determination of the second-order self-energy, its real part
$\Lambda_2$ can be obtained from the function $\Gamma_2$ using \eqref{Ht}.
In \fref{fig:gamma}, we plot
$\Gamma_2$ for different values of $J_0$ and the temperature $T$.
\begin{figure}
\centering \includegraphics[width=0.45\textwidth]{Gamma.eps}
\caption{\label{fig:gamma} The real part of the dimensionless second-order self-energy
$\Gamma_2 \Delta / \lambda^2$
as a function of $E/\Delta $ for a Gaussian bond distribution
with mean values $J_0=0$, $2\Delta$ and temperatures $T=0.02\Delta$, $\Delta$, $10\Delta$.}
\end{figure}
\subsubsection{Markovian evolution}
As shown in \sref{sec:wcr}, in the weak coupling limit $\lambda \to 0$, the evolution of
the coherence $M$ is essentially Markovian : $M(t) \simeq \exp(-\gamma t)$. For the
disordered Ising chain, the decoherence
rate $\gamma$ is given, to lowest order in $\lambda$, by
\begin{equation}
\gamma=\Gamma_2(0)=2\pi \lambda^2 \int dJ \, p_e (J)^2 \left[ 1 - \tanh(\beta J)^2 \right] \quad . \label{rate}
\end{equation}
The rate $\gamma$ increases monotonically from $\gamma=0$ at $T=0$ and saturates to the value
\begin{equation}
\gamma_\infty = 2\pi \lambda^2 \int dJ \, p_e (J)^2 \label{ratehT}
\end{equation}
as $T \to \infty$. At low temperatures, $\gamma \simeq 4\pi\lambda^2 p (0)^2\,T$ for disorder distributions with $p(0) \neq 0$.
The vanishing of $\gamma$
at zero temperature can be understood as follows \cite{salfd}. At $T=0$, the local
field distribution is completely dictated by the spin structure of the ground state
$| \{ \sigma_i \} \rangle$ of the bath Hamiltonian \eqref{1DH}. Since $J_i \sigma_i \sigma_{i+1} > 0$ for any pair
of neighboring spins in the ground state, this inevitably leads to nonvanishing local fields
$J_{i-1} \sigma_{i-1} + J_i \sigma_{i+1} = (|J_{i-1}| + |J_i|) \sigma_i$. This implies $P(h=0)=0$ and hence $\gamma=2\pi\lambda^2 p(0)=0$.
We now study the influence of the bond distribution on the rate $\gamma$.
In the weak disorder regime $\Delta \ll |J_0|$,
the distribution $p_e(J)^2$ practically vanishes for $J\neq \pm J_0$ thus the rate $\gamma$ is given by
\begin{equation}
\gamma \simeq \pi \lambda^2 \int \! dJ \, p(J)^2 \;\; \left[ 1 - \tanh(\beta J_0)^2 \right] \quad . \label{wdapprox}
\end{equation}
In this regime, the temperature variations of $\gamma$ are independent of the form of the bond distribution
$p$ and are exclusively determined by the mean value $J_0$.
In \fref{fig:rate}, we plot the temperature dependence of $\gamma$
for various values of the mean
interaction strength $J_0$ in the case of a Gaussian bond distribution
\begin{equation} \label{Gbd}
p(J)=\frac{e^{-(J-J_0)^2/2\Delta^2}}{\sqrt{2\pi} \, \Delta } \quad .
\end{equation}
The linear regime at low temperature given by $\gamma \simeq 2(\lambda/\Delta)^2 \exp(-J_0^2/\Delta^2)\,T$ is visible
only for $|J_0| < 2 \Delta $. For higher interaction strengths $J_0$, $\gamma$ remains
practically zero in the low-temperature regime. The maximal rate obtained as $T \to \infty$
is given by \eqref{ratehT} : $\gamma_\infty = \sqrt{\pi} \lambda^2 [1+\exp(-J_0^2/\Delta^2)]/2\Delta $.
For $|J_0| > 2 \Delta$, the high-temperature rate $\gamma_\infty$ is essentially independent
of $J_0$ and the temperature dependence of $\gamma$ is well described by the weak disorder
approximation $\gamma \simeq (\sqrt{\pi} \lambda^2 /2\Delta ) [ 1 - \tanh(\beta J_0)^2 ]$. Note that the agreement with
this approximation improves with increasing $J_0$.
We now consider a uniform distribution for the intra-bath interaction strength:
\begin{eqnarray}
p(J) &=& (2\sqrt{3}\Delta )^{-1} \quad \mathrm{for} \quad \vert J-J_0\vert < \sqrt{3}\Delta \nonumber \\
&=& 0 \quad ~~~~~~~~~~~~\mathrm{otherwise} \quad .
\end{eqnarray}
In this case, the integral \eqref{rate} can be evaluated exactly for any temperature $T$ :
\begin{widetext}\begin{eqnarray}
\gamma &=& \frac{\pi \lambda^2 T}{12\Delta^2} \left\{ \tanh \left[ \frac{|J_0|+\sqrt{3}\Delta}{T} \right] -
3 \tanh \left[\frac{ |J_0|-\sqrt{3}\Delta}{T} \right]\right\} \quad \mathrm{for} \quad |J_0|< \sqrt{3} \Delta \;\;\; \\
&=& \frac{\pi \lambda^2 T}{12 \Delta^2} \left\{ \tanh \left[ \frac{|J_0|+\sqrt{3}\Delta}{T} \right] -
\tanh \left[ \frac{|J_0|-\sqrt{3} \Delta}{T}\right]\right\} \quad \mathrm{for} \quad |J_0|>\sqrt{3} \Delta \nonumber \quad .
\end{eqnarray} \end{widetext}
For $|J_0|<\sqrt{3} \Delta $, $\gamma \propto T$ at low temperatures with
a slope $\pi \lambda^2/3\Delta^2$ independent of $J_0$ whereas,
for $|J_0|>\sqrt{3} \Delta $ the low temperature behavior is not linear.
For $|J_0| > 4 \Delta $, the agreement
between the exact result and the weak disorder approximation
$\gamma \simeq (\pi \lambda^2 /2 \sqrt{3} \Delta ) [ 1 - \tanh(\beta J_0)^2 ]$ is excellent.
Moreover,
in this regime, the Gaussian and uniform bond distributions cannot be distinguished
($\sqrt{\pi/3} \simeq 1.02$). An interesting feature of our results is that a non-zero $J_0$
favours the coherence of the central spin via a robust short range ordering of the bath spins.
\begin{figure}
\centering \includegraphics[width=0.45\textwidth]{rategamma.eps}
\caption{Dimensionless rate $\gamma \Delta/\lambda^2$ as a function of the temperature $T/ \Delta$
for a Gaussian bond distribution with mean $J_0 = 0$, $\Delta$ and $2\Delta$. The weak disorder
approximation \eqref{wdapprox} is shown for $J_0 = 2\Delta$.}
\label{fig:rate}
\end{figure}
Comparing the above results with those obtained for a bath comprising free spins,
we see that the timescale of the decoherence generated by the spin chain bath $\gamma_\infty^{-1} \sim \Delta /\lambda^2$
(for weak coupling $\lambda \ll \Delta $) is much longer
than the decoherence time $\lambda^{-1}$ (for $\lambda \ll \sqrt{N}$) obtained in the free case (see \aref{Isb}).
As we will demonstrate later, $\gamma_\infty^{-1} \gg \lambda^{-1}$ for any dimensionality
of the spin bath lattice.
This clearly illustrates the fact that interactions
in the bath significantly slow down the decoherence of the central spin.
Another interesting comparison is to an Ohmic boson bath.
Contrary to the Ising bath, the decoherence rate
$\gamma_{bos}$ of an Ohmic boson bath is proportional to $T$ in the whole temperature range
and thus does not saturate at high temperatures. This forces the question as to whether
interactions between the
bosons also lead to a saturation of the rate $\gamma_{bos}$. This warrants further work which is
beyond the scope of the present paper. If the central spin is coupled both to a spin
bath and a boson bath, the resulting coherence $M(t)$ is given by the product of \eqref{Mt} and
a similar factor, with $H_B$ replaced by the boson bath Hamiltonian $H_b$ and
the interaction operator $V$ by an analogous boson operator $V_b$. Then in
the Markovian regime at weak coupling, the total decoherence rate is simply the sum $\gamma_{bos}+\gamma$.
Consequently, at high temperatures the Ohmic boson bath dominates but for temperature
$T \,\hbox{\bbfnt\char'056}\, \Delta$ the Ising bath has to be taken into account.
The question of their relative dominance depends on the various bath coupling constants and
can vary from system to system.
\subsubsection{Non-Markovian regime}
Here, we discuss the non-Markovian aspects of the decoherence of the central spin
essentially seen at low temperatures and at short and intermediate time scales.
We first consider the case of a bond distribution with zero mean.
The time evolution of the coherence $M(t)$ within the Born approximation,
shown in \fref{fig:nonMarkov}, is obtained by a numerical evaluation of $\Gamma_2(E)$,
$\Lambda_2(E)$ and $M(t)$ using the expressions \eqref{Gamma1D}, \eqref{Ht} and \eqref{ThetaM} for a Gaussian
bond distribution \eqref{Gbd} with $J_0=0$.
Note that the agreement with the weak coupling approximation \eqref{Mapprox} is remarkably good even for
reasonably large values of $\lambda$ i.e., of the order of $0.1 \Delta$.
Though, in \fref{fig:nonMarkov}, we illustrated the equivalence of the Born and weak coupling approximations
for $M(t)$ in the weak coupling regime, we nonetheless expect, based on the analyticity arguments presented
in Sec. \ref{sec:wcr}, the weak coupling approximation \eqref{Mapprox} to be an exact description of the full
coherence $M$ for low enough $\lambda$.
We now discuss the influence of
temperature on the behavior of the coherence $M(t)$.
In \fref{fig:nonMarkov}, we see that at high temperatures, $M$ remains
practically constant for times $t \,\hbox{\bbfnt\char'056}\, \Delta^{-1}$
and then decays exponentially
as described by \eqref{Mapproxlt}. For temperatures $T \,\hbox{\bbfnt\char'056}\, \Delta $,
the Markovian regime is preceded by a novel intermediate time regime $\Delta^{-1} \,\hbox{\bbfnt\char'056}\, t \,\hbox{\bbfnt\char'056}\, \beta$.
The difference between the high-temperature
and the low-temperature decoherence can be traced back to the temperature dependence of
the function $\Gamma_2$. At high enough temperatures,
since $\Gamma_2(E)$ is essentially a peak of width $\Delta$, one crosses over from
the short time regime to the Markovian regime for $t \simeq \Delta^{-1}$.
As the temperature is lowered, $\Gamma_2(0)$ steadily decreases but the curvature $\Gamma''_2 (0)$
remains negative. However, below a certain
temperature $\Gamma''_2 (0)$ becomes positive, see \fref{fig:gamma} and
the function $\Gamma_2$ can be effectively characterized by two energy scales:
$T$ and $\Delta $. At low temperatures,
$\Gamma_2$ remains practically constant for $|E| \,\hbox{\bbfnt\char'056}\, T$, increases
linearly for larger energies with a slope $(\lambda/\Delta )^2$
and finally vanishes for $|E| \,\hbox{\bbfnt\char'046}\, \Delta $.
These three energy regimes result in three different time regimes for the coherence
\eqref{Mapprox}. The short-time behaviour ($t \,\hbox{\bbfnt\char'056}\, \Delta^{-1}$) is determined
by the long-energy tails of $\Gamma_2$. For intermediate times $\Delta^{-1} \,\hbox{\bbfnt\char'056}\, t \,\hbox{\bbfnt\char'056}\, \beta$,
the linear regime of $\Gamma_2$ yields a power law decay of the coherence
$M(t) \propto t^{-\epsilon}$ where the exponent is given by
\begin{equation}
\epsilon=\frac{2}{\pi} \left( \frac{\lambda}{\Delta } \right)^2 \quad .
\end{equation}
For long times ($t \,\hbox{\bbfnt\char'046}\, \beta$) the integral \eqref{Mapprox} is dominated by the energies
$|E| \,\hbox{\bbfnt\char'056}\, T$ for which $\Gamma_2(E) \simeq \gamma$ and hence
the decoherence is essentially exponential with the rate $\gamma$.
However, one should be cautious about extending the above results to ultra-low temperatures
because the contribution of the energies $|E| \,\hbox{\bbfnt\char'046}\, T$, given by
$\Lambda_2^\prime(0)$, diverges in the limit $T \to 0$.
This divergence is logarithmic with a prefactor $\epsilon$.
At zero temperature, the Markovian regime disappears and the coherence vanishes
in the limit $t \to \infty$ according to the power law $t^{-\epsilon}$.
The low temperature behavior of $M(t)$ obtained here is very similar to the
decoherence
induced by a boson bath in the strict Ohmic case \cite{QDS} i.e. for an Ohmic
spectral
function with a cut-off frequency $\omega_c$ far larger than $T$ and
$1/t$.
In this case, the coherence of the spin coupled to the boson bath is given by
$\ln M(t)=-K \ln[(\omega_c / T \pi) \sinh (\pi T t)]$ where $K$ is the
spin-bath
coupling strength. We recover a Markovian behavior,
$\ln M(t) \simeq K \ln(2\pi T / \omega_c )-K\pi T t$, for times $t \gg
\beta$ and a power law, $M(t)=(\omega_c t)^{-K}$, at zero temperature.
\begin{figure}
\centering \includegraphics[width=0.45\textwidth]{wcahT.eps}
\vskip0.9cm
\includegraphics[width=0.45\textwidth]{wca.eps}
\caption{$(\Delta/\lambda)^2 \ln M$ as a function of time $t\Delta$
within the Born approximation for a Gaussian bond distribution with zero
mean for coupling strengths $\lambda=0.1 \Delta$, $0.2 \Delta$
and temperatures (a) $T =10 \Delta $ and (b) $T=0.02 \Delta$.
The full line is the weak coupling approximation and the dotted line corresponds
to a free spin bath.}
\label{fig:nonMarkov}
\end{figure}
We now present a more detailed comparison of our numerical results with the weak coupling
approximation \eqref{Mapprox}, shown in \fref{fig:nonMarkov}.
For given $\lambda$ and $T$, the Born approximation deviates from the expression \eqref{Mapprox}
as time increases. This difference, which is more pronounced at low temperatures,
can be easily quantified in terms of higher order corrections
stemming from the expansion of the analytically continued self-energy discussed in Sec. \ref{sec:NME}.
At long times, the first correction to $\ln M(t)$ arises at $O(\lambda^4)$ and takes the form
\begin{eqnarray}\label{correction}
\ln M(t) - \left[ \Lambda_2^\prime(0) - \Gamma_2(0) t \right]
&\simeq& - \Gamma_2(0)\Gamma_2^{\prime\prime}(0) +
{\textstyle \frac{1}{2}} [\Lambda_2^\prime(0)]^2 \nonumber \\
&&
- \Gamma_2(0) \Lambda_2^\prime(0) \, t \quad .
\end{eqnarray}
In the high temperature regime, since $\Lambda_2^\prime(0)$ is positive, the weak coupling approximation
given by the second-order terms in \eqref{correction}
slightly overestimates $\ln M$. As the temperature is lowered, the evolution of the function $\Gamma_2$ discussed earlier (see \fref{fig:gamma}) results in a sign change of $\Lambda_2^\prime(0)$. Thus for low enough
temperatures, $\Lambda_2^\prime(0)$ is negative and the weak coupling approximation underestimates $\ln M$.
This analysis is indeed very consistent with our results shown in \fref{fig:nonMarkov}.
This discussion clearly highlights the efficiency of our approach based on the analyticity properties
of the self-energy, obtained within any approximation, to describe the long-time decoherence.
We remark that other fourth order corrections to $\ln M(t)$ exist beyond the Born approximation.
We now consider the case $J_0 \neq 0$. For weak disorder
$|J_0| \gg \Delta$, the function $\Gamma_2$ at low temperature consists
of two Gaussian peaks of width $2\sqrt{2}\Delta$ centered around $\pm 4J_0$.
The low-temperature self-energy in this regime is thus very different
from that for $J_0=0$, see \fref{fig:gamma}. Nonetheless, the qualitative
behavior of the coherence $M$ at intermediate and long times, determined
by the low-energy $\Gamma_2$, is very similar
to the one discussed above for $J_0=0$. For short times $t \,\hbox{\bbfnt\char'056}\, \Delta^{-1}$,
contrary to the $\ln M(t) \propto -t^2$ behavior seen for $J_0=0$, here
the coherence $M(t)$ oscillates at the frequency $2J_0/\pi$.
At zero temperature,
the long-time decoherence decays as a power law, $M(t) \propto t^{-\eta}$ where the exponent
$\eta=\epsilon \exp(-J_0^2/\Delta^2)$. As seen earlier for the Markovian decay
rate $\gamma$, the low-temperature behavior seen here also signals a slowing down
of the decoherence by a nonzero $J_0$.
\subsection{Higher-dimensional spin lattices}
Here we consider other geometries for an Ising spin bath described by
the Hamiltonian $H_B=-\sum_{\langle i j \rangle} J_{ij} \sigma_i^z \sigma_j^z$ where the
spins
occupy the sites $i$ of a regular lattice of arbitrary dimensionality and are coupled
by nearest-neighbor interactions. The bonds $J_{ij}$
are drawn independently from the Gaussian distribution $p$ with mean $J_0$ and variance
$\Delta^2$ given by \eqref{Gbd}. An important difference between the spin chain model and
the generic Ising model on higher dimensional lattices is the presence of frustration
arising from geometric constraints and/or randomness.
Though frustration can give rise to novel ground states and related dynamical behaviour,
it renders any analytic study of these models very difficult.
In this section, we focus on the high temperature regime where one can use a controlled
analytical method like the high temperature series expansion
to study the effect of higher dimensions on the decoherence.
To obtain the coherence $M$ at high temperatures, we expand the second-order self-energy
\eqref{soSigma} in terms of the inverse temperature $\beta$. To do so, we first rewrite the time-dependent
correlation \eqref{VV1} as
\begin{widetext}
\begin{equation}
\mathrm{Re}\langle V(t)V\rangle = \frac{1}{Z'} \sum_k \lambda_k^2 \sum_{\{\sigma_i\}}\prod_{\langle i j \rangle } (1+\sigma_i \sigma_j \kappa_{ij})\mathrm{Re}
\prod\nolimits_i^{(k)} \big[ \cos(2tJ_{ik})+i\sigma_i \sigma_k \sin(2tJ_{ik}) \big] \label{VtVhT}
\end{equation}\end{widetext}
where $\prod\nolimits_i^{(k)}$ denotes a product over the nearest neighbors of site $k$,
$Z'=\sum_{\{\sigma_i\}} \prod_{\langle ij \rangle} (1+\sigma_i \sigma_j \kappa_{ij})$ and $\kappa_{ij}=\tanh(\beta J_{ij})$.
Since
$\kappa_{ij} \to 0$ as $\beta \to 0$, we consider the above expression as a power series
in $\kappa_{ij}$. Multiplying out the two products in Eq.\eqref{VtVhT} generates a series
of products of nearest-neigbour spin pairs\,: $(\sigma_i \sigma_j \sigma_k \sigma_l \ldots)$.
Since Eq.\eqref{VtVhT}
involves sums over configurations $\{\sigma_i\}$, each of these spin pair products contributes
only if it simplifies to $1$. This implies, for instance, that the $n^\mathrm{th}$ order
terms in the expansion of $Z'$ in $\kappa_{ij}$ correspond to closed loops comprising
$n$ bonds on the lattice. An immediate consequence is that $Z'=1$ up to third order.
However, in the expression \eqref{VtVhT} there exists another type of relevant spin pair products
which involve repeated bonds. We discuss these terms in the following.
As $T \to \infty$, for equal couplings to the central spin $\lambda_k=\lambda N^{-1/2}$,
the time-dependent correlation \eqref{VtVhT} becomes
$\mathrm{Re}\langle V(t)V\rangle = \lambda^2 N^{-1} \sum_k \prod\nolimits_i^{(k)} \cos(2tJ_{ik}) $.
Note that this correlation remains unchanged under $J_{ik} \to -J_{ik}$. This invariance
stems from the equiprobability of every spin configuration at infinite temperature. This infinite-temperature correlation is self-averaging in the thermodynamic limit
and hence the corresponding second order self-energy is given by the average of \eqref{soSigma}
over the bond distribution. This leads to
\begin{equation}
\Gamma_2(E) = 2 \lambda^2 \int_{-\infty}^{\infty} \!\! dt \; e^{iEt} e^{-2\Delta^2 t^2 s} \cos(2J_0
t)^s
\label{GammahT}
\end{equation}
where $s$ is the coordination number of the lattice. As remarked earlier for the 1D case,
here also $\Gamma_2$ is the same for $\pm J_0$ for bond distributions symmetric around
their mean values. Eq.\eqref{GammahT} shows that in the infinite temperature limit
the only characteristic of the bath lattice which intervenes in the decoherence is $s$.
In particular, there is no explicit dependence on the dimensionality. However, we shall
show later that the detailed geometric characteristics of the lattice manifest themselves in
the higher order corrections. Though static properties
of the spin system are independent of $J_0$ and $\Delta$ in the infinite temperature limit,
these parameters strongly influence
the correlation $\mathrm{Re}\langle V(t)V\rangle$ and hence the decoherence.
Using \eqref{GammahT}, for the case $J_0=0$, we obtain from \eqref{Mapprox}
\begin{equation}
\ln M(t) = - 2 \sqrt{\pi} \lambda^2 \tau t \, \mathrm{erf} (t/\tau)
+ 2 \lambda^2 \tau^2 \left( 1-e^{-t^2/\tau^2} \right)
\end{equation}
where $\mathrm{erf}$ is the error function and the characteristic time $\tau$ is defined by
\begin{equation}
\tau= \Delta^{-1} (2s)^{-1/2} \quad .
\end{equation}
For times $t \gg \tau$, we recover the Markovian regime where
$\ln M(t) \simeq - 2 \sqrt{\pi} \lambda^2 \tau t + 2 \lambda^2 \tau^2$.
For $t \ll \tau$, one finds the usual short time
evolution\,:
$\ln M(t) \simeq - 2 \lambda^2 t^2$. The same time regimes
exist for $J_0 \neq 0$. For short times $t \ll \tau$ we find, for an even $s$,
\begin{equation}
\ln M(t) = - \frac{\lambda^2}{2^{s-2} J_0^2} \sum_{n=0}^{s/2-1}
\frac{s! \sin[J_0 t(2n-s)]^2 }
{n!(s-n)! (2n-s)^2} -
\frac{s! \lambda^2 t^2}{2^{s-1}[(s/2)!]^2} \quad . \label{sthT}
\end{equation}
If $s$ is an odd number, the short-time decoherence is described by
\eqref{sthT} without the quadratic term in $t$ and with a sum running from $0$ to $(s-1)/2$.
Clearly the effect of a nonzero $J_0$ is to induce oscillations in the coherence $M(t)$.
The decoherence rate in the Markovian regime is given by
\begin{equation}
\gamma_\infty = \frac{\lambda^2}{2^{s-1} \Delta } \sqrt{\frac{\pi}{2s}}
\sum_{n=0}^{s} \frac{s!}{n!(s-n)!} e^{-J_0^2(2n-s)^2/2s \Delta^2} \quad .
\label{ratehTs}
\end{equation}
For $s=2$ this expression simplifies to the result \eqref{ratehT} obtained for the spin chain.
As anticipated, the decoherence time $\gamma_\infty^{-1}$ is of the order of $\Delta/\lambda^2$
and hence far longer than that for free spins which is of the order of $\lambda^{-1}$.
The infinite-temperature rate manifests the influence of the lattice geometry:
for lattices with an odd $s$ like the honeycomb lattice ($s=3$), $\gamma_\infty \to 0$ in
the weak disorder limit $\Delta \ll |J_0|$ whereas for an even $s$, triangular and square
lattices for example, $\gamma_\infty \neq 0$ in this limit.
We now evaluate the leading order corrections in $\beta$ to the decoherence rate $\gamma$.
To illustrate the significance of the lattice geometry we consider various bidimensional lattices.
For a triangular lattice, the lowest order terms in the sum over the configurations $\{\sigma_i\}$
in \eqref{VtVhT} take the form $-\sin(2tJ_{ik})\kappa_{ij}\sin(2tJ_{jk})$ where
the site $i$ is a neighbor of the site $k$ and the site $j$ is a neighbor of
both sites $k$ and $i$. Consequently, the rate up to first order reads
\begin{widetext}\begin{equation}
\gamma^T=\gamma^T_\infty - {\frac{\sqrt{3\pi}}{16}} \frac{\lambda^2 \beta
J_0}{\Delta } \left[ 2-2e^{-4J_0^2/3\Delta^2}+e^{-J_0^2/3\Delta^2}
-e^{-3J_0^2/\Delta^2} \right]+O(\beta^2)
\end{equation}\end{widetext}
where $\gamma^T_\infty$ is given by \eqref{ratehTs}
with $s=6$. The explicit appearence of $J_0$ in this expansion is
linked to the fact that the first nonzero correction for this lattice
occurs at first order in $\beta$. Moreover, since this correction is positive (negative) for baths
with a majority of antiferromagnetic (ferromagnetic) bonds, the ensuing decoherence time
is longer for a predominantly ferromagnetic bath. This clearly highlights the importance of frustration
in determining decoherence. Thus at high temperatures, the decoherence induced by the triangular lattice
is very different than that by the linear chain\,: the convergence of $\gamma^T$ to $\gamma^T_\infty$
is slower and depends on the sign of $J_0$.
We also remark that though the cubic and triangular lattices have the same coordination $s=6$
and hence the same infinite-temperature rate, the corrections are different since there is no
three-bond closed loop on the cubic lattice.
For the honeycomb lattice, the first corrections arise at second order. These corrections
correspond to loops with two repeated bonds\,: $-\kappa_{ki}\sin(2tJ_{ik})\kappa_{kj}\sin(2tJ_{jk})$.
Here the contribution of closed loops becomes relevant at higher orders. The resulting
decoherence rate is
\begin{widetext}\begin{equation}
\gamma^H=\gamma^H_\infty - \sqrt{\frac{\pi}{24}}
\frac{\lambda^2 \beta^2 }{\Delta } \left[
\left( 3{\Delta^2} + 4 J_0^2 \right) e^{-J_0^2/6\Delta^2}
+{\Delta^2} e^{-3J_0^2/2\Delta^2} \right]+O(\beta^4)
\end{equation}\end{widetext}
where $\gamma^H_\infty$ is given by \eqref{ratehTs} with $s=3$.
Finally for the square lattice we obtain the decoherence rate
\begin{widetext}\begin{equation}
\gamma^S=\gamma^S_\infty - \frac{1}{4} \sqrt{\frac{\pi}{2}} \frac{\lambda^2 \beta^2
}{\Delta}
\left[\left( \frac{3}{4} \Delta^2-2 J_0^2 \right) e^{-2J_0^2/\Delta^2}
+ 3(\Delta^2+J_0^2) e^{-J_0^2/2\Delta^2}+
\frac{9}{4}\Delta^2+5 J_0^2 \right]+O(\beta^4)
\end{equation}\end{widetext}
where $\gamma^S_\infty$ is given by \eqref{ratehTs} with $s=4$.
We mention that for the square lattice both closed loops and loops with repeated bonds
contribute to the lowest order correction. Interestingly, the convergence
of the rates $\gamma^H$ and $\gamma^S$ to their respective infinite-temperature limits
is reminiscent of that of the chain. Moreover, as seen for the spin chain,
our results for $\gamma^H$ and $\gamma^S$ are independent of the sign of $J_0$.
This feature can be attributed to the bipartite nature of these lattices.
As explained for the chain, the respective local field operators and Hamiltonians
are invariant under a sign change of the bonds $J_{ij}$ coupled with
a suitable unitary transformation for the spins. Since this argument is valid in the entire
paramagnetic phase, the higher order corrections to $\gamma^H$ and $\gamma^S$
are expected to be independent of the sign of $J_0$ in this phase. On the other hand, due to its
non-bipartite nature no analoguous transformation exists for the triangular lattice.
It would be interesting to study the evolution of these rates behave as one lowers temperature and enters
a non--paramagnetic phase.
\subsection{Infinite-ranged Ising bath}
We now consider a bath described by the mean field Sherrington-Kirkpatrick model
where all the Ising spins interact with each other. We restrict ourselves to the well studied
case of spin-spin interaction strengths distributed around a zero mean with a variance $\Delta^2$.
This model exhibits a spin glass phase transition at a
finite temperature $T_{sg}=\Delta$ below which the spins freeze \cite{sk}. The spin glass phase is
characterized by the large number of metastable states present which then lead to
anomalous dynamic behaviour.
An interesting question is whether the
decoherence manifests novel features as goes from the high temperature paramagnetic
phase to the low temperature spin glass phase.
As shown in \sref{sec:DIsb}, the coherence $M(t)$ of the central spin in the weak coupling regime
can be obtained through a knowledge of the local-field distribution $P(h)$.
Though the infinite-range model has been extensively studied in the past using the replica approach
and numerical simulations, it is not easy to obtain the local-field distribution for all fields and temperatures.
Here, we do not delve into the problem of calculating $P(h)$ but use existing results \cite{salfd}
to make predictions for the coherence of the central spin coupled to this mean field bath.
In the high temperature paramagnetic phase, the local-field distribution is
\begin{equation}
P(h)= \frac 1{2\Delta \sqrt{2\pi}} \left[e^{-(h-\beta \Delta^2)^2/2\Delta^2}
+e^{-(h+\beta \Delta^2)^2/2\Delta^2} \label{phhT} \right] \quad .
\end{equation}
In the spin glass phase, calculations based on the replica formalism
yield the following result for $P(0)$ provided the temperature
$T \simeq T_{sg} $:
\begin{equation}
P(0)= \frac 1{\Delta \sqrt{2\pi e}} \left[ 1- \left(1- \frac{T}{T_{sg}} \right) -\frac 1 3 \left( 1- \frac{T}{T_{sg}} \right)^4 \right] + ... \quad .
\end{equation}
These analytic results are sufficient to determine the decoherence in the Markovian regime\,:
the decoherence rate is given by
$\gamma = \sqrt{2\pi} \lambda^2 \exp[-\frac 1 2 (T_{sg}/T)^2]/\Delta$ for $T > T_{sg}$ and
$\gamma = \sqrt{2\pi/e} \lambda^2 [ 1- (1- T/T_{sg}) -\frac 1 3 (1- T/T_{sg})^4 ]/\Delta$ for temperatures in
the vicinity of the spin glass transition temperature i.e., $ T \simeq T_{sg}$ .
Clearly, in the Markovian regime, one does not see any sign of the spin glass transition.
Note that $\gamma$ saturates at infinite temperature to a value comparable to that obtained earlier
for finite-ranged lattice models. This implies that even in the case of a maximally frustrated bath,
the central spin decoheres at timescales longer than those for free spins.
For lower temperatures, only numerical solutions exist for the local-field distribution \cite{salfd,Plefka}.
These results suggest a continuous variation of $P(h)$ with temperature. The only significant signature
of the transition is a flattening of $P$ at $T=T_{sg}$ which has no manifest effect on the decoherence.
Moreover, numerical extrapolations of $P(0)$ to low temperatures indicate a rate $\gamma \sim (\lambda/\Delta)^2 \, T$
which is very similar to the linear $T$ behaviour seen in the Ising spin chain system at low enough
temperatures. Again, since $\gamma=0$ at zero temperature, the decoherence is no longer Markovian.
The form of $M(t)$ is then dictated by the low-energy behavior of $\Gamma_2$ through \eqref{Mapprox}.
Based on the numerical inference $P(h) \propto h$ for $T=0$, the central spin is expected to decohere
as a power law at $T=0$ \cite{salfd,Plefka}. To conclude, we see that both the thermal transition and
the spin glass order of the Sherrington-Kirkpatrick model do not have any palpable effect
on the asymptotic decoherence in the weak coupling regime.
\section{Conclusions}
In this paper, we have studied the decoherence induced by a Ising spin bath with
random intra-bath interactions. The resolvent operator formalism was used to determine
the coherence $M$ of the central spin for weak coupling to the bath.
We then obtained detailed analytical results for the disordered Ising spin chain bath for arbitrary
temperature.
The decoherence was found to be independent of the sign of the mean
intra-bath interaction strength $J_0$ for symmetric bond distributions. Three regimes were identified
in the time evolution of $M(t)$\,: a short time Gaussian decay, an intermediate time power law behavior
and the usual asymptotic Markovian regime. The relative sizes of these regimes are fixed by temperature.
At zero temperature, the Markovian regime was found to vanish and the decoherence is essentially described by
a power law decay. We also studied the decoherence induced by an infinite-ranged Ising spin glass bath
and Ising models on lattices in dimensions greater than one. For all these baths, the Markovian rate
was found to saturate to a finite value at infinite temperature, which is much smaller than the corresponding rate
for a free spin bath. Our results clearly indicate that intra-bath interactions significantly increase the timescales
over which the central spin decoheres.
For the infinite-ranged Ising spin glass, our conclusions based on existing results
suggest that the thermal spin glass transition has no visible effect on the decoherence. Plausibly this is an artefact of
the infinite-ranged interactions and/or the Ising nature of the spins. This raises the general issue of
the influence of thermal and quantum phase transitions and the resulting orders in finite-ranged spin baths
on the decoherence of the central spin. In most realistic cases, the spin environment consists of
Heisenberg spins. In this case, one expects the dynamics of the bath to be richer and this may
have interesting consequences for the decoherence. This however, is beyond the scope of the analytic
work presented in this paper.
An interesting question is the effect of a strong coupling
between the central spin and the bath. For a bath of independent spins/bosons the results obtained
in the weak coupling regime
are qualitatively valid even for strong coupling. However, in the presence of interactions, this is not necessarily
the case and one can expect novel dynamical behaviors.
A natural extension of our work would be to include the intrinsic dynamics
of the central spin and study the relaxation
induced by the spin bath.
These and other questions are left for future work.
\begin{acknowledgments}
R.C. acknowledges support from ACI Grant No. JC-2026.
\end{acknowledgments}
\begin{appendix}
\section{Independent spin bath}\label{Isb}
Here we derive an exact expression for the coherence $M(t)$ given by \eqref{Mt}
in the case of a bath comprising non-interacting spins described by the Hamiltonian $H_B=-\sum_i h_i \sigma_i^z$.
Since the spins are independent, the trace \eqref{Mt} can be factorized as
\begin{equation}
M(t)=\prod_{k=1}^N \frac{ \mathrm{Tr} \left[
e^{\beta h_k \sigma_k^z}e^{-it( h_k \sigma_k^z - \lambda_k \sigma_k^x )}
e^{it( h_k \sigma_k^z + \lambda_k \sigma_k^x )} \right]}{2\cosh (\beta h_k)} \quad .
\end{equation}
To evaluate the factors in this expression we require the diagonal elements
of $2\times 2$ matrices of the form $\exp[-ibA(a)]\exp[ibA(a)]$ where
$A(a)=a\sigma^z+\sigma^x$. Diagonalizing $A(a)$, we find that these diagonal elements
are equal and given by
\begin{equation}
\langle \sigma | e^{-ibA(a)} e^{ibA(a)} | \sigma \rangle =
\frac{1}{1+a^2} \left[ a^2+ \cos\left( 2b \sqrt{1+a^2}\right) \right] \quad .
\end{equation}
The resulting coherence $M(t)$ is independent of the temperature and
can be written as
\begin{equation}
\ln M(t) = \sum_{k=1}^N \ln \left[
1- \frac{2\lambda_k^2}{\lambda_k^2+h_k^2} \sin \left( t \sqrt{\lambda_k^2+h_k^2}\right)^2 \right] \quad .
\end{equation}
\noindent
For $h_k=0$, the above expression leads to the coherence, $M(t)
=\prod_k \cos (2 \lambda_k t)$ which then culminates in a Gaussian decay $M(t)=\exp(-2t^2 \sum_k \lambda_k^2)$
for weak coupling to the bath.
For nonzero local fields $h_k$ the coherence in the weak coupling regime reads:
\begin{equation}
\ln M(t) = -2 \lambda^2 \int dh \; P(h) \frac{\sin(ht)^2}{h^2}
\end{equation}
where the coupling strength $\lambda$ and the field distribution $P$ are defined by
$\lambda^2= \sum_k \lambda_k^2$ and $P(h)=\sum_k (\lambda_k/\lambda)^2 \delta(h-h_k)$.
This result is the same as that obtained earlier in \eqref{Mapprox} for the specific
bath considered here.
\section{Derivation of the self-energy}\label{Dse}
In this Appendix, we derive the expressions \eqref{LT} and \eqref{Sigma} for the Laplace transform ${\tilde M}(z)$
of the coherence \eqref{Mt} and we show that ${\tilde M}(z)$ is analytic in the upper and lower half
planes.
We first rewrite the expression \eqref{Mt} as
\begin{equation}
M(t)=\mathrm{Tr} \left( e^{-i{\cal L} t} \rho_B \right)
\end{equation}
where ${\cal L}$ is a superoperator in the bath Liouville space defined by
${\cal L} A = (H_B+V) A - A (H_B-V)$ where $A$ is any operator in the bath Hilbert space.
The Laplace transform \eqref{LTdef} can then be written as ${\tilde M}(z)=\mathrm{Tr} [G(z) \rho_B]$
where $G(z)=(z-{\cal L})^{-1}$ is the resolvent of the operator ${\cal L}$.
We now introduce the superoperators ${\cal P}$ and ${\cal Q}$
defined by ${\cal Q}=1-{\cal P}$ and ${\cal P} A=\mathrm{Tr} (A) \rho_B $ where $A$ is any
operator in the bath Hilbert space. Since $\mathrm{Tr} ( \rho_B) =1$, ${\cal P}$ and ${\cal Q}$
are projection operators.
Using ${\cal P}$, ${\cal Q}$ and $(z-{\cal L})G(z)=1$, we obtain the following coupled equations
\begin{eqnarray}
{\cal P}(z-{\cal L}){\cal P}\,{\cal P}G(z) \rho_B + {\cal P}(z-{\cal L}){\cal Q} \,{\cal Q}G(z) \rho_B\!\!&=&\!\!\rho_B \nonumber \\
{\cal Q}(z-{\cal L}){\cal P}\,{\cal P}G(z)\rho_B + {\cal Q}(z-{\cal L}){\cal Q}\,{\cal Q}G(z)\rho_B\!\!&=&\!\!0
\end{eqnarray}
for the operators ${\cal P}G(z) \rho_B$ and ${\cal Q}G(z) \rho_B$. Solving the latter
for ${\cal Q}G(z) \rho_B$ in terms of ${\cal P}G(z) \rho_B$ and then substituting in the former yields
\begin{equation}
{\cal P} \left[ z-{\cal L} - {\cal L} {\cal Q} (z-{\cal Q}{\cal L}{\cal Q})^{-1}
{\cal Q} {\cal L} \right] {\cal P}G(z) \rho_B = \rho_B \quad .
\end{equation}
Finally tracing both sides gives $[z-\Sigma(z)]{\tilde M}(z)=1$ where $\Sigma(z)$ is given by \eqref{Sigma}.
We now discuss the analyticity properties of the Laplace transform ${\tilde M}(z)$.
Consider an eigenoperator $A$ of ${\cal L}$ with the eigenvalue $\lambda$ : ${\cal L} A = \lambda A$.
The scalar product $\mathrm{Tr}(A^{\dag} {\cal L} A)=\lambda \mathrm{Tr}(A^{\dag} A)$ can also be written
\begin{eqnarray}
\mathrm{Tr}\left( A^{\dag} {\cal L} A \right)&=&\mathrm{Tr}
\left[ A^{\dag}(H_B+V)A - (H_B-V)A^{\dag} A \right]^*\nonumber \\
&=& \mathrm{Tr}\left(A^{\dag} {\cal L} A \right)^*=\lambda^* \mathrm{Tr}(A^{\dag} A)
\end{eqnarray}
where the first equality is obtained using the Hermiticity of $H_B$ and $V$ and the second one
using the cyclic property of the trace. Consequently, the eigenvalues of ${\cal L}$ are real
and thus the resolvent $G(z)$ and the Laplace transform ${\tilde M}(z)$ are analytic in the upper and lower half planes.
\section{Second-order self-energy}\label{Sose}
Here, we show that the second-order self-energy $\Sigma_2$ can
be rewritten in terms of the time-dependent correlation of the interaction
operator $V$ as given by \eqref{soSigma}. Let $| \alpha \rangle$ denote the eigenstates of the Hamiltonian $H_B$ and
$|\alpha, \beta ) = | \alpha \rangle \langle \beta |$ denote the eigenstates of the corresponding Liouville operator
${\cal L}_B$. Any superoperator
${\cal F}$ in the Liouville space can be expanded in this eigenbasis as
\begin{equation}
{\cal F}= \sum_{\alpha, \beta, \gamma, \delta} ( \alpha, \beta | {\cal F}|\gamma, \delta ) \; |\alpha, \beta )( \gamma, \delta |
\end{equation}
where the scalar product in the Liouville space is defined by $(A|B)=\mathrm{Tr} (A^{\dag} B)$.
The following decompositions are useful for our purpose :
\begin{eqnarray}
&&{\cal P} = \rho_B \sum_{\beta} ( \beta , \beta | = \sum_{\alpha, \beta} \langle \alpha | \rho_B | \alpha \rangle |\alpha, \alpha ) ( \beta , \beta |
\\
&&(z-{\cal Q}{\cal L}_B {\cal Q})^{-1} = \sum_{\alpha, \beta} \frac{1}{z-E_\alpha+E_\beta} |\alpha, \beta ) ( \alpha , \beta | \nonumber \\
&&{\cal L}_V = \sum_{\alpha, \beta,\gamma} \Big[ \langle \alpha | V | \gamma \rangle |\alpha, \beta ) ( \gamma , \beta | +
\langle \gamma | V | \beta \rangle |\alpha, \beta ) ( \alpha , \gamma | \Big] \nonumber
\end{eqnarray}
where ${\cal P}=1-{\cal Q}$ and $E_\alpha$ is the eigenergy corresponding to the eigenstate
$| \alpha \rangle$.
Using these results we find
\begin{eqnarray}
\Sigma_2 (z) =&& 2 \sum_{\alpha, \beta} \langle \alpha | \rho_B | \alpha \rangle |\langle \alpha
| V | \beta \rangle|^2 \\
&&~~ \times \left[ \frac{1}{z-E_\alpha+E_\beta}
+ \frac{1}{z-E_\beta+E_\alpha} \right]
\quad .\nonumber
\end{eqnarray}
Comparing this expression with the time-dependent correlation
\begin{eqnarray}
\langle V(t)V \rangle& =& \mathrm{Tr} \left( \rho_B e^{itH} V e^{-itH} V \right) \nonumber \\
&=& \sum_{\alpha, \beta} \langle \alpha | \rho_B | \alpha \rangle |\langle \alpha | V | \beta \rangle|^2 e^{it(E_\alpha-E_\beta)}
\end{eqnarray}
we infer that the Laplace transform of $4\mathrm{Re}\langle V(t)V \rangle$ is $\Sigma_2 (z)$ as given by \eqref{soSigma}.
\end{appendix}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,760
|
Q: How to validate inputs and prevent save actions using databinding in eclipse? I want to create input forms which validate user input and prevent the model from being saved with invalid data. I have been using databinding which works up to a point but my implementation is not as intuitive as I would like.
Imagine an input which contains '123' and the value must not be empty. The user deletes the characters one by one until empty. The databinding validator shows an error decoration.
However, if the user saves the form and reloads it, then a '1' is displayed in the field - i.e. the last valid input. The databinding does not transmit the invalid value into the model.
I have a ChangeListener but this is called before the databinding so at that point the invalid state has not been detected.
I would like the error to be displayed in the UI but the model remains valid (this is already so). Also, for as long as the UI contains errors, it should not be possible to save the model.
/**
* Bind a text control to a property in the view model
**/
protected Binding bindText(DataBindingContext ctx, Control control,
Object viewModel, String property, IValidator validator)
{
IObservableValue value = WidgetProperties.text(SWT.Modify).observe(
control);
IObservableValue modelValue = BeanProperties.value(
viewModel.getClass(), property).observe(viewModel);
Binding binding = ctx.bindValue(value, modelValue, getStrategy(validator), null);
binding.getTarget().addChangeListener(listener);
ControlDecorationSupport.create(binding, SWT.TOP | SWT.LEFT);
return binding;
}
private UpdateValueStrategy getStrategy(IValidator validator)
{
if (validator == null)
return null;
UpdateValueStrategy strategy = new UpdateValueStrategy();
strategy.setBeforeSetValidator(validator);
return strategy;
}
private IChangeListener listener = new IChangeListener()
{
@Override
public void handleChange(ChangeEvent event)
{
// notify all form listeners that something has changed
}
};
/**
* Called by form owner to check if the form contains valid data e.g. before saving
**/
public boolean isValid()
{
System.out.println("isValid");
for (Object o : getDataContext().getValidationStatusProviders())
{
ValidationStatusProvider vsp = (ValidationStatusProvider) o;
IStatus status = (IStatus)vsp.getValidationStatus()
.getValue();
if (status.matches(IStatus.ERROR))
return false;
}
return true;
}
A: Your best bet is to steer clear of ChangeListeners - as you've discovered, their order of execution is either undefined or just not helpful in this case.
Instead, you want to stick with the 'observable' as opposed to 'listener' model for as long as possible. As already mentioned, create an AggregateValidationStatus to listen to the overall state of the DataBindingContext, which has a similar effect to your existing code.
Then you can either listen directly to that (as below) to affect the save ability, or you could even bind it to another bean.
IObservableValue statusValue = new AggregateValidationStatus(dbc, AggregateValidationStatus. MAX_SEVERITY);
statusValue.addListener(new IValueChangeListener() {
handleValueChange(ValueChangeEvent event) {
// change ability to save here...
}
});
A: You can use AggregateValidationStatus to observe the aggregate validation status:
IObservableValue value = new AggregateValidationStatus(bindContext.getBindings(),
AggregateValidationStatus.MAX_SEVERITY);
You can bind this to something which accepts an IStatus parameter and it will be called each time the validation status changes.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,364
|
\subsection*{Abstract}
\begin{abstract}
Content providers build serving stacks to deliver content to users.
An important goal of a content provider is to ensure \emph{good user
experience}, since user experience has an impact on revenue. In this
paper, we describe a system at Yahoo{} called YTrace{} that diagnoses
bad user experience in near real time. We present the different components
of YTrace{} for end-to-end multi-layer diagnosis (instrumentation, methods
and backend system), and the system architecture for delivering
diagnosis in near real time across all user sessions at Yahoo{}.
YTrace{} diagnoses problems across service and network layers in the
end-to-end path spanning user host, Internet, CDN and the datacenters,
and has three diagnosis goals: detection, localization and root cause
analysis (including cascading problems) of performance problems
in user sessions with the cloud. The key component of the methods
in YTrace{} is capturing and discovering
causality, which we design based on a mix of instrumentation API,
domain knowledge and blackbox methods.
We show three case studies from production that span a large-scale
distributed storage system, a datacenter-wide network, and an end-to-end
video serving stack at Yahoo{}. We end by listing a number of open
directions for performance diagnosis in cloud and content providers.
\end{abstract}
\section{Introduction}
Large content providers \hide{such as Yahoo, Google, Netflix and Facebook}{}
serve users from large-scale serving stacks in geographically distributed
datacenters on the Internet. They can be modeled as cloud infrastructure
that consists of multiple datacenters and a Content Distribution Network
(CDN) (Figure \ref{fig:Model-CP}). Users interact with the content provider
by making RPCs (also called \emph{user sessions}) to the CDN and the
datacenters. The \emph{user experience} of
a user session with the provider depends on several factors from the
serving stack, to the datacenter network and the Internet, to the content. \emph{Bad} user experiences
result in loss of users and revenue \cite{websiteperf}.
Content providers build for good user experience by building high-performance
serving stacks and network infrastructure. Serving stacks are compositions
of services, and services are usually
large distributed systems comprising of hundreds to thousands of
hosts -- on top of the datacenter network and inter-datacenter wide area
paths. Serving stacks include latency-tolerant distributed execution techniques
such as parallelism and redundancy \cite{dean2013tail}. For example,
a user request for a personalized web page could be served by ``assembling''
parts of the page, each generated by a service\footnote{Such designs are also called
service-oriented and microservices architectures.}. In order to do this,
services (specifically, hosts in a service) make RPCs to each other
over the underlying network paths.
\begin{figure}
\begin{centering}
\includegraphics[clip,width=0.9\columnwidth]{model}
\par\end{centering}
\protect\caption{Model of a large content provider showing the end-to-end path for
a user session.\label{fig:Model-CP} Lower figure shows canonical
execution graph to determine instrumentation; ``S'' and ``NW''
represent services and network respectively.\label{fig:Canonical-execution-tree}}
\end{figure}
Due to the composition scale and heterogeneity of a serving stack, it is prone
to performance problems that span multiple layers -- from the infrastructure
layer such as network and servers, to the higher layers such as the
OS, containers and service processes within a server, to the distributed
systems layer -- and localized among nodes in the end-to-end
path (Figure \ref{fig:Model-CP}). Detecting and troubleshooting bad
user experience is a complex and tedious problem at scale, since it often involves
multiple services and layers, and hence, coordination between multiple teams across service tiers
and underlying layers. It is hence equally important to build systems
that continuously monitor and diagnose bad user experiences. Such
systems help troubleshoot to quickly fix performance problems, and
know where to allocate resources in the medium-term. Further, near real time
\emph{diagnosis as a service} is a useful primitive to optimize
existing systems against performance problems. Content providers
have designed and deployed several systems in practice \cite{chow2014mystery,kim2013root,krishnan2009moving,ren2010google,sigelman2010dapper,sun2011identifying,yu2011profiling,zhu2012latlong};
however, these systems do not diagnose performance problems end-to-end
and across layers.
We present YTrace{}, a system that we are building at Yahoo{} to
diagnose end-to-end performance problems that impact user experience.
YTrace{} has three components: instrumentation to collect data, diagnosis
methods that run on the data, and an efficient backend to index the
data and execute diagnosis queries (Figure \ref{fig:YPerf-architecture}).
In this paper, we focus on the first two components and touch upon the
third. We consider dynamic
web content that is tailored for users -- perhaps the most common
on the Internet. Our definition of user experience depends on the content type:
for web content, we estimate user experience as the
page load time -- the latency between the user's content request
and the Javascript OnLoad event; and for video streams, we consider
duration of rebuffering events. Our work can easily be extended to
diagnose performance problems with other content types and definitions
of user experience.
\begin{figure*}
\begin{centering}
\includegraphics[width=0.9\textwidth]{ytrace-backend-overview-mdc}
\par\end{centering}
\protect\caption{YTrace{} architecture and components.\label{fig:YPerf-architecture}}
\end{figure*}
When building an end-to-end diagnosis system, there are key requirements
for large content providers:
\begin{itemize}
\item Tie to user experience: Instrumentation and diagnoses should directly
relate to user experience of real users.
\item The diagnosis output should be general enough to help troubleshoot
\emph{almost all} performance problems, including casding failures.
\item Multi-layer: The diagnosis should span as many layers in the serving
stack as possible. At a minimum, it should include all services, the
host machines and the underlying network layer.
\item Instrumentation should have low overhead, so it does not affect the
user experience.
\item Accuracy: Diagnosis should have low false positive and false negative
rates for the use cases. It should be able to diagnose tail latency.
\end{itemize}
The key ideas behind YTrace{} rely on identifying concurrent event execution,
both at the service level and in the network. Knowing the context of concurrency
enables YTrace{} to compute the most important information for diagnoses -- the
critical path in the execution. In order to find concurrency, YTrace{}
records and mines causal relationships between events in a user session at
the service and network layers. It aggregates diagnoses across user sessions
and renders an interactive dashboard geared towards troubleshooting.
\textcolor{red}{}%
\begin{comment}
\textcolor{red}{bz analysis?}
challenges?
builds on prev work
move rel work to top?
add summary para
\end{comment}
\section{Problem Statement}\label{sec:Formulation}
There are three broad classes of use cases of YTrace{}: troubleshooting,
resource provisioning and service adaptation. Troubleshooting aims
to fix performance problems that users face after the problems occur.
It requires the system to deliver near real time, actionable, insights
into performance problems. Resource provisioning is a relatively longer-term
task that involves querying the system for aggregate views of diagnosis
to find where to add resources\footnote{Resource provisioning also requires answers to ``what-if'' questions.
This is outside the scope of our current work.}. Service adaptation uses YTrace{} as
a near real time diagnosis-as-a-service to build serving stacks that optimize for
user experience. For example, the traffic engineering service
at a CDN may route users to CDN nodes based on diagnoses of Internet
paths; the rate adaption module in a video player may make strategic rate
choices if it had diagnoses. Since this involves pre-defined queries, the system may materialize
such queries to minimize query times\footnote{Large query delays can be detrimental
to performance, e.g., in load balancing \cite{mitzenmacher2000useful}.}.
\begin{comment}
\begin{figure*}[t]
\includegraphics[width=1\textwidth]{cet.eps}
\protect\caption{Canonical execution tree for a user session for determining necessary
and sufficient instrumentation. Nodes ``S'' and ``NW'' represent
services and datacenter network respectively. The three colors denote
entities.}
\end{figure*}
\end{comment}
Based on discussions with teams across Yahoo, we formulate a problem
statement whose solution provides actionable
input for the three use cases. YTrace{} has three goals for every
user session:
\begin{description}
\item [{Detection:}] Is a user session seeing a performance problem?
\item [{Localization:}] Where are the performance problems in the end-to-end
path (and across all layers)?
\item [{Root~cause~analysis:}] Why are the performance problems occurring?
\end{description}
In addition to per-session diagnoses, the YTrace{} backend
supports (and materializes views of) aggregate queries over multiple user sessions, such as
clusters of users (e.g., ISP and geography), and over a service
in the datacenter in a time window. Aggregate queries with such predicates
enable statistically significant analyses while conditioning on confounding
variables.
\section{Instrumentation}\label{sec:Instrumentation-Space}
The first step towards performance diagnosis of a user session
is instrumentation of components that participate in the session. The
instrumentation should not add significant latency to the session.
The key is to determine \emph{necessary and sufficient} instrumentation for diagnosis.
We implement optimized libraries for
instrumentation so that the instrumentation overhead
is very low relative to end-to-end latency.
One way to determine instrumentation is by considering the canonical
end-to-end user session graph, whose nodes are
components (which impact user experience) that participate in user sessions and whose edges represent
point-to-point communication between nodes; it should cover all components and layers
that are necessary for diagnosis. Figure \ref{fig:Canonical-execution-tree}
shows the graph for large content providers that spans: the user
end-host, the CDN, the serving stack
spanning one or more datacenters, and the underlying network infrastructure.
In order to diagnose performance problems with each
node in the canonical graph, the necessary and sufficient instrumentation
will include performance data from every node in the
graph (necessary condition) and will not include redundant instrumentation
between edges (sufficient). The necessary and sufficient instrumentation
for root cause analysis at a node depends on the attributes YTrace{}
needs to be able to fingerprint and match problem signatures (\S\ref{sec:Diagnosis}).
YTrace{} includes two forms of instrumentation: (1) synchronous instrumentation
that is in-band with the user session, and (2) asynchronous instrumentation
from components that cannot be modified for instrumentation (such as network
devices). We implement synchronous instrumentation in the form of
\emph{distributed tracing}, which allows YTrace{} to tie performance of any
component into the user experience. YTrace{} uses causal relationships in
instrumentation data to diagnose performance problems.
\subsection{Synchronous Instrumentation}
\noindent\textbf{User-side instrumentation.} User end-host
instrumentation enables YTrace{} to diagnose performance problems
with events at the end-host (includes browser, any containers, and the content itself).
In general, content providers cannot alter the browser (e.g., by introducing
plugins), which leaves them with a limited set of user-side performance
measurements.
The work of content in a browser can be modeled as a sequence of events
spanning fetching resources (either via local cache or network), execution
and rendering. The W3C Navigation Timing (NavTiming) recommendation
\cite{navtiming} describes such an event model for origin content
(the page HTML) and exposes it via a Javascript API to the web page.\footnote{A similar event API for the other resources that the page requires
is supported by the W3C Resource Timing recommendation \cite{restiming}.}
We use the NavTiming event model for user-side instrumentation for web content in
YTrace{}. This enables us to break down the user experience (page
load time) into timing of events for origin content. There is a causal
relationship between all NavTiming events: for example, DNS lookup (if any)
causes TCP connect to the CDN. In particular, all events measured by
NavTiming and Resource Timing have well-defined causal relationships.
Having causal relationships helps us understand the events that resulted
in bad user experience, since they are necessary to construct the
latency critical path of the session.
In the case of video streaming (which is a linearizable sequence
of RPCs per-segment to the CDN), YTrace{} uses an event model that includes timing
of per-segment events at the video player. These events span
segment RPCs, and decoding and rendering of those segments to the screen.
\textcolor{red}{}
\begin{figure}
\begin{centering}
\includegraphics[bb=0bp 70bp 683bp 312bp,clip,width=1\columnwidth]{timeseq}
\par\end{centering}
\textcolor{red}{\protect\caption{Inferring causality in RPC patterns.\label{fig:Inferring-causality}}
}
\end{figure}
\vspace{0.1in}\noindent\textbf{Distributed tracing.}
YTrace{} synchronously traces user sessions (i.e., RPCs from user agent) through all
execution nodes in the serving stack, including the CDN services and the user host
(see user-side instrumentation above). Distributed tracing is a
common monitoring primitive in large-scale serving stacks~\cite{chow2014mystery,sigelman2010dapper},
and involves two steps: assigning a globally unique ID to a user session, and
propagating this ID through all nodes in the serving stack. The ID propagation
is typically implemented by adding the ID to all RPCs during session execution
-- for example, in the form of a serialized header. YTrace{} records the timing
of events related to each RPC during session execution using node-local clocks.
When implementing distributed tracing, there are a few design constraints
that arise from large-scale environments. Such environments are highly
heterogeneous, not only in the platforms used, but also in runtime complexity
such as RPC execution patterns (see Figure \ref{fig:Inferring-causality}),
serialization formats and protocols. We find two forms of RPC-level concurrency
in distributed execution: parallelism and redundancy -- both in the context of
RPC ``fanout'' implementations. Parallelism includes parallel RPCs; the opposite of
which is serialized RPC execution. Redundancy is a case where a service doing
the fanout only waits for a few responses before sending back a response to the caller;
this is typically used in search engine stacks.
Perhaps the most important requirement in distributed tracing implementations is to
\emph{capture causality} among \emph{all} events in the end-to-end execution. Ideally,
causality among events should be described by the services themselves (during tracing),
and not inferred offline using tracing data (an approach adopted by
prior work \cite{aguilera2003performance,chow2014mystery}).
The reason for this is dynamic
behavior in web services: for example, the concurrent/serialized and redundant
RPC execution patterns shown in Figure~\ref{fig:Inferring-causality} can be triggered
as functions of session attributes, performance history and runtime environment. Such
dynamicity makes offline inference of causal relationships between RPC edges hard
(without additional data that may not be within the scope of a tracing system).
YTrace{} captures causality between RPC ``edges'' (requests and responses),
since causality in RPC execution patterns such as concurrency and redundancy
exists between RPC edges. This RPC model is a key difference between YTrace{}
and prior tracing systems such as Dapper (and Zipkin), which capture causality
at the granularity of RPCs. YTrace{} captures causality in two forms: during
tracing using service-level APIs designed to capture causality, and offline using
(well-defined) happens-before relationships~\cite{lamport1978time}.
Services call the YTrace{} tracing API at each RPC edge. The API returns an immutable
session context (passed as a handle) for each incoming RPC, until that RPC is fully served. The
API times each RPC edge, and any annotations\footnote{The annotations are optional service-specific
timestamped key-value pairs, such as lock events. They are used by developers to
understand service-specific performance, such as impact of lock contention on
end-to-end performance.} across the session. It consumes
and returns all headers that are/should be serialized in RPCs.
\begin{lstlisting}
void *handle = create(/*String*/ in_header);
String out_header = sendtonext(handle);
recvfromnext(handle, /*String*/ in_header);
String out_header = sendtoprev(handle);
annotate(handle, /*String*/key, /*String*/val);
close(handle);
\end{lstlisting}
The API captures causality in two ways. First, the session context (handle)
allows YTrace{} to capture parent-child relationships between RPC edges.
Second, the headers that the API generates include an RPC ID and parent RPC ID
(that are unique to the session), and these IDs record
causality between the request and response(s) (if any) of an RPC. The parent-child
RPC IDs also record global ordering of RPCs in the session.
We implement the YTrace{} tracing API as a userspace library
that services call during execution. The library provides a 99th percentile runtime SLA of
3$\mu s$ per RPC. The low runtime overhead of the library is due to two reasons.
First, the APIs are stateless, since the session state (handle) in a
service is immutable. The state, instead, is passed over the network
in the serialized headers; an example of such state is the child-parent
RPC IDs in the session. This design choice trades-off expensive state
maintenance in the API with a few additional bytes on the network; and
also avoids any need for synchronization at session-level in a service.
Second, all logging in YTrace{} is asynchronous.
The API captures a significant set of causal relationships in sessions,
but it does not capture causality (or lack of it) in RPC execution patterns such as parallelism and redundancy in
the execution graph \cite{chow2014mystery,dean2013tail}; see Figure
\ref{fig:Inferring-causality} -- we found that doing so makes the API
complex (which slows down adoption). YTrace{} uses happens-before
relationships to infer this causality (see $\S$\ref{sec:Diagnosis}).
\begin{figure}
\begin{centering}
\includegraphics[width=1\columnwidth]{flowchart}
\par\end{centering}
\protect\caption{YTrace{} diagnosis flow.\label{fig:flowchart}}
\end{figure}
\vspace{0.1in}\noindent\textbf{CDN instrumentation.}
The wide area Internet path can be a significant source of
performance problems that impact user experience.
In order to diagnose these problems and their
impact on user experience, we need to instrument the Internet path in isolation.
Typical approaches explored in literature include active probing
(which is asynchronous) or model-based methods that infer path
performance user-side and/or CDN-side measurements of the trace.
Both approaches have limitations: the former adds network traffic
and may not be causally related to the session (since it is asynchronous),
while the latter methods are prone to user host problems (which are
not uncommon).
In order to diagnose and isolate Internet performance problems,
we take periodic snapshots of measurements that the TCP stack in
the CDN kernel maintains, for the TCP connection used by user host
RPCs. Specifically, we snapshot \texttt{tcp\_info} structures
from the Linux kernel. The structure contains end-to-end
statistics of the TCP connection that affect serving performance,
such as packet retransmissions, reordering, RTT, sender and receiver
windows, etc.; it hence measures the Internet path as the flow sampled
it. YTrace{} uses the TCP connection statistics to localize throughput
bottlenecks to sender (content generation), receiver and path-based
limitations; and to diagnose download bottlenecks with user hosts.
Note that we do not include diagnosis of CDN traffic
engineering-related performance problems; in other words, YTrace{}'s
diagnosis is conditioned on the traffic engineering decision for a
user session (see discussion, \S\ref{sec:Discussion-and-Conclusion}).
\vspace{0.1in}\noindent\textbf{Process profiling.} We are adopting
Continuous Profiling \cite{anderson1997continuous,ren2010google}
in Yahoo{} services for a small fraction of user sessions. At a
high level, Continuous Profiling collects performance counters exposed
by modern CPUs. Performance counters allow us to understand host-level
bottlenecks and localize bad user experience down to code using the
associated program counters (in conjuction with the process binaries).
At the end of each session, YTrace{} records a directed trace graph that
includes: \emph{(i)} the end-to-end execution graph with compute,
serialization and RPC timings at each node and event causality, \emph{(ii)} user-side event
timings and event causality, and \emph{(iii)} TCP-layer measurements of
user-side Internet at the CDN.
\subsection{Asynchronous Instrumentation}
Despite service stack-level concurrency, the underlying network is a shared resource
and typically underprovisioned (e.g., fat-tree datacenter topologies).
The network can introduce performance problems in RPCs during session
execution, which can impact user experience.
YTrace{} collects asnchronous instrumentation from components in the
end-to-end serving stack that cannot be modified to do tracing. Such
components are typically in the underlying network layer, such as the
datacenter network devices.
YTrace{} collects \emph{syslogs} from all datacenter network devices.
Syslogs include detailed and fine-grained state information of each
device and the root cause. In conjunction with syslogs, YTrace{} uses
network topology to localize user session performance
problems to network devices. It collects network topology snapshots
of: \emph{(i)} the wide area Internet paths from the CDN to \emph{client
clusters} and to the datacenters using traceroutes, and \emph{(ii)}
each datacenter network using device configurations.
\section{Diagnosis}\label{sec:Diagnosis}
YTrace{} uses synchronous and asynchronous instrumentation to diagnose performance problems
that impact user experience. In this section, we sketch the diagnosis
methods. See Figure \ref{fig:flowchart} for a flow overview.
\vspace{0.1in}\noindent\textbf{Detection.} The first step towards
performance diagnosis is to detect performance problems. Since our focus is on user experience, we frame
the detection problem around it: Is the page load time\footnote{We can ask similar detection questions about user experience metrics
for other content types, e.g., rebuffering in video.} \emph{large} for the user session? YTrace{} answers this question
by \emph{estimating a baseline }(normal behavior) for the page load
time based on history, and finding if the session has a statistically
significant deviation from the baseline. The page load time is measured
at the user-side. Note that detection algorithms have to be aware
of confounding variables such as the web page (session) attributes, the user attributes and
time of day; YTrace{} conditions the detection based on domain knowledge
of pre-defined confounding variables.
YTrace{} also supports detection based on other definitions of user
experience, or not based on user experience. For example, video quality of experience,
abnormal service latencies or unusual execution graphs for a session.
YTrace{} currently estimates a simple baseline as the historic inter-quartile range,
since we are interested in understanding performance behind both low and high
user experience metrics.
\subsection{Content Diagnosis}
For sessions that were detected as having performance problems,
YTrace{} uses the user-side instrumentation to determine whether there were performance
problems that were localized to the user agent (browser). To do this,
it checks whether the latency of user-side events from the Navigation
Timing API \cite{navtiming} (e.g., DOM processing and rendering)
are significant relative to the OnLoad time for the page. Note that
NavTiming events are causally related and the critical path of user-side
events includes all events.
In general, resources on the page are fetched (and executed) concurrently
with the origin page -- and the concurrency depends on the origin
content, ordering of resource arrivals and execution latency (parsing,
DOM construction, etc.). This dependency between resources leads to
blocking periods; however, such analysis requires browser modifications
\cite{wang2013demystifying}. YTrace{} currently treats all content
(origin and resources) performance as independent (note that
origin content is typically the dynamic, non-cacheable
content in personalized pages). We are investigating dependency and
blocking time measurement that avoids browser changes.
\subsection{Service Diagnosis}
Service localization refers to the question of which services in the session execution
graph \emph{caused} a performance problem for sessions that were
detected as having performance problems. YTrace{} localizes service-level
problems by estimating the critical path(s) -- in terms of service
latencies -- in the session execution graph. In order to compute the
critical path, YTrace{} needs context of concurrency in the
execution graph, which is determined by causality between RPC edges.
YTrace{} tracks causality as follows. The synchronous instrumentation
system tracks two forms of RPC causality during tracing:
parent-child RPC relationships, and request-response causality.
In order to keep the tracing API simple to use and reduce usage errors, we do not track
(at the API-level) causality \emph{between} sibling RPC edges at a single node. Causality
between sibling RPC edges may (or may not) exist, depending on the
RPC execution patterns used (see Figure~\ref{fig:Inferring-causality}).
Since such patterns are typically dynamic, based on session and environment
attributes, we cannot use blackbox methods that mine causal relationships
from offline session trace data \cite{aguilera2003performance,chow2014mystery}.
The parallelism and redundancy RPC patterns lend well to happens-before
relationships (directly follows from their definition).
Consider outgoing RPC edges $e_{1}\dots e_{n}$ at a service node $A$ (Figure \ref{fig:Inferring-causality});
and let the responses to the outgoing edges be the edges $r_{1}\dots r_{n}$
(incident on $A$). Denote the first byte and last byte timestamps ($A$'s
wall clock) of an edge $e$ by $C_{f}(e)$ and $C_{l}(e)$. Timestamps
are taken from user space. Note that $e_{i}$ causes $r_{i}$ for all $i$,
denoted as $e_{i}\rightarrow r_{i}\forall i$.\\
\noindent Causality in \emph{non-parallelism}: $r_{i}\rightarrow e_{j}$:
$C_{f}(r_{i})<C_{f}(e_{j})$; $r_{i}\nrightarrow e_{j}$ otherwise.\\
\noindent\emph{Redundancy:} If edge $r_{0}$ is $A$'s reply to calling
node, $r_{i}\rightarrow r_{0}$ $\forall$ $r_{i}$: $C_{f}(r_{i})<C_{l}(r_{0})$;
$r_{i}\nrightarrow r_{0}$ otherwise.
YTrace{} uses edge causality to estimate
the critical path in the execution graph, defined as the \emph{causal} round trip path in
the graph with the largest total (service and network) latency.
Latency at a service with an incoming edge $e_{0}$ and a causal outgoing
edge $e_{1}$ is the computation time: $l_{01}^{s}=C_{f}(e_{1})-C_{l}(e_{0})$.
The network latency is the RPC edge (de)serialization time at the caller node. Note that the
critical path in the same execution graph can change based on RPC
causality: for example, if service $A$ makes two RPCs $e_{1}$ and
$e_{2}$ to $B$, the critical path may include one or both of $e_{1}$
and $e_{2}$ depending on $e_{1}-e_{2}$ causality:\\
\includegraphics[clip,width=0.8\columnwidth]{critical-path}
YTrace{} estimates the contribution of a service as the sum of computation
latencies for all incoming-causal outgoing edge pairs $l_{ij}$. It
reports the service-level localization output as the top service contributors,
and their fraction of end-to-end (user-side) latency, amongst services in
the critical path.
\subsection{Network Diagnosis}
\label{sec:netdiag}
YTrace{} uses syslogs from network devices to diagnose datacenter network
problems. It uses TCP stack measurements at the CDN to isolate problems
on the user-to-CDN Internet path (note that we cannot instrument the Internet path).
We first look at datacenter network problem diagnosis.
\noindent\textbf{Datacenter network diagnosis.}
The critical path found during service localization represents the subset of execution
that contributed to user latency, and it includes time spent by
RPCs in the network. The network can degrade the performance of RPCs
by inducing latency, packet losses and reordering, which increase
the RPC time and reduce throughput (especially for RPCs with large
payloads). Our goal in network diagnosis is to localize and find
root causes of datacenter network problems. We are interested in
localizing cascading problems and finding root causes that propagate
across the network stack; for example, a hardware problem in a switch
that cascades into problems in the connected router as both L2 and routing
plane problems.
YTrace{} uses syslogs and the datacenter network topology to diagnose
cascading problems. Each datacenter network device emits a stream of
syslog messages, which are semi-structured text that include a timestamp,
severity level and semantics of the problem (network interface, problem
type and attributes, etc.). Our goal is to represent a problem as a
structured graph that describes the causal activity (the cascade) in the problem.
It uses domain knowledge to preprocess syslogs: mapping them to structured
``templates'' (including equivalence classes of problem types), and
extracting device attributes (if any). We leverage some of the prior
work on template extraction \cite{Qiu:2010:HMN:1879141.1879202}.
The domain knowledge is a one-time
input to YTrace{} and does not need changes unless the syslog templates
change (e.g., due to vendor or major OS changes, which are infrequent).
The first step towards diagnosing session performance due to network problems
is to find RPCs in the session critical path that impacted user experience.
YTrace{} computes a candidate list of RPCs per-session as follows.
Consider two services $A$ and $B$, and an RPC $e$ from $A$. The local clock
at node $A$ is $C_{f}^{A}$ (we consider timestamps at the start
of each RPC request/response to avoid self-loading effects). For diagnosis,
we are interested in the variable component of the one-way delay
-- typically queueing delays. YTrace{} estimates queueing delay during
the RPC as $d_{AB}(e)=C_{f}^{B}(e)-C_{f}^{A}(e)-\min_{\Delta}(d_{AB})$,
where the $\min$ is taken over the recent $\Delta$ time window\footnote{The time window should be large to include queue dissipation ($\mu s$
in datacenter networks) but smaller than clock drift timescales (mins.).}; the $\min$ is an estimate of the constant components of the one-way
delay. YTrace{} detects an RPC as having a performance problem if the
queueing delay is significant relative to the end-to-end latency.
It then computes a set of network devices that the RPC could have traversed.
The typical way to localize network-level performance problems is boolean
network tomography \cite{Duffield:2003:SNP:948205.948232} on end-to-end
observations of RPC queueing delays. Tomography takes as input observations
of paths -- good and bad -- that \emph{overlap} with a problem path.
It aims to isolate the part of the network that led to the bad observations.
Tomography is not directly applicable in datacenter networks, since such
networks use multi-path routing -- hence, the path an RPC takes is not
deterministic. This makes the problem combinatorial and expensive to solve.
Syslogs provide a single network-wide solution that addresses both localization
and root cause analysis.
YTrace{}'s network diagnosis module has two components: real time
problem graph mining that ingests all syslogs from a datacenter, and
asynchronous low-volume learning that periodically generates \emph{causal rules}
as input for the real time component.
\begin{figure}
\begin{center}
\includegraphics[width=.8\columnwidth]{sample_graphs-png}
\caption{Example network problem graphs from our datacenter.}
\label{f:causal_graph_example}
\end{center}
\end{figure}
More formally, a problem graph is a directed graph of syslog templates,
where an edge $T_i \rightarrow T_j$ implies that template $T_i$ caused
template $T_j$. A problem graph could exist within a single device or
span multiple devices. A causal rule connects two templates by a causal
relationship: $T_i \rightarrow T_j$; depending on whether $T_i$ and $T_j$
happen within a single device or different devices, a causal rule could be
either intra-device or inter-device. For example, Figure~\ref{f:causal_graph_example}
shows two instances of problems from a Yahoo datacenter
(colors encode layer in protocol stack). The left graph shows
a multi-layer problem that spans aggregate and top-of-rack tiers in
the fat-tree network, and multiple layers in the protocol stack.
It encodes a cascading problem: a module failure causes a link down event,
which triggers a spanning tree protocol status change, and causing an
interface status change on a peering device. The right graph shows a
problem within a top-of-rack devices that is an Ethernet (L2) flapping issue.
YTrace{}'s diagnosis module mines problem graphs as follows.
It divides the syslog timeline across the datacenter into small time windows.
Within each time window, it maps syslog lines into templates and uses
the corpus of causal rules to iteratively construct problem graphs,
starting from intra-device edges and then adding inter-device edges.
At any point of time, we typically have 100-200 causal rules. Hence,
the runtime overhead of mining problem graphs in a small time window
of syslog messages across the datacenter is relatively low (it can run
on a single machine).
The problem of mining causal relationships between syslog templates is
relatively harder, since it is the problem of finding needles in a haystack
of syslogs. In such cases, happens-before relationships result in
significant false positive rates. We adopt statistical causality mining
techniques to discover causal rules -- in particular, we use Quasi
Experimental Design (QED). First, we find (in a larger time window) template
pairs that have a statistically significant correlation in their timeseries.
For each template pair ${T_i,T_j}$ that is correlated, QED finds causality
by testing the hypothesis that an element of the \emph{treated} set is much more
likely than an element of the \emph{untreated} set. The treated set consists
of instances when $T_i$ and $T_j$ exists together at any time; while
the untreated set has instances when $T_i$ exists but not $T_j$.
If the treated set is more likely, QED assigns a causal relationship
$T_i \rightarrow T_j$.
For each RPC in a session that is detected as having a performance
problem, YTrace{} summarizes the set of problem graphs on devices
that the RPC could have traversed. At this point, the network diagnosis
in YTrace{} is meant to show \emph{possible} problems in the network that
impact an RPC, since these problems may not manifest as performance
problems in all RPCs. We are working on methods to establish causal
relationship between a network problem and RPC performance. A limitation
of syslog-based diagnosis is that it will only mine problems that syslogs
can describe. We believe that our syslog-mining methodology can be
applied on logs from any multi-layer distributed service.
We refer the reader to our prior work \cite{netarxiv} for details
of the network diagnosis methods.
\noindent\textbf{Internet path diagnosis.}
YTrace{} synchronously instruments the user-to-CDN (and user-to-datacenter)
path. In the context of Internet path performance, it captures userspace
RPC timing at the user host, and RPC timing at the CDN and TCP stack measurements
of the RPC at the CDN node. In practice, we observed that a common source
of performance problems is the user host. Hence, the measurements taken from
the browser (or any container on top) include a mix of problems in the user host
and the Internet path (even after we measure and account for CDN-side latencies).
We use measurements from the TCP stack in the CDN host kernel as estimators
of the Internet path performance (as sampled by TCP). The TCP measurements
include path RTT, RTT variation, segment retransmissions, congestion windows and
reordering. YTrace{} uses the TCP measurements to estimate the impact of
the Internet path on RPCs from user host, and isolate Internet problems from
the user host performance. We are looking into using tomography on the TCP
measurements to localize bottlenecks on the Internet (in conjunction with
topology measurements).
\subsection{Ongoing Work}
\vspace{0.1in}\noindent\textbf{Process localization.} A part of our
localization goal is to simplify performance debugging by localizing
user session performance problems to source code. One approach requires
YTrace{} to track performance counters for processes within each
service and associate the counters with code; and it has to be low-overhead.
The performance counters provide a context for fingerprinting runtime
behavior of code (for node-local root cause analysis), and include
program counters that help associate with code. This is early-stage work.
\vspace{0.1in}\noindent\textbf{Root cause analysis.} Root cause analysis
in operational practice typically relies on fingerprinting performance problems
based on domain knowledge and experience. While YTrace{} includes root cause
analysis of network problems using syslogs, an open question is how to
incorporate service and network operator inputs (domain knowledge) to
do service-level root cause analysis. The key to this is to provide a
suitable model of performance problems that operators can input,
using the following grammar:
\begin{lstlisting}
SYMPTOM symptom
PATHOLOGY pathology DEF ( symptom | NOT symptom )
symptom := symptom1 AND symptom2
symptom := symptom1 OR symptom2
symptom := ( symptom )
PROCEDURE symptom funcname
\end{lstlisting}
We model a performance problem as a boolean-valued expression on one or more boolean-valued
\emph{symptoms}. A symptom is a function of instrumentation.
For each detected problem, we evaluates matching problem expressions
to identify the root cause(s). The root cause analysis is based
on prior work on network root cause analysis \cite{DBLP:conf/usenix/KanuparthyD14}.
\begin{comment}
As an example, YTrace{} records a number of TCP-layer attribute timeseries
at each CDN node for user-CDN and CDN-datacenter connections. A root
cause in this case could be a congested link. Symptoms for identifying
congestion could be based on TCP round trip time and retransmission
measurements.
\end{comment}
\section{YTrace Backend}
A key aspect of YTrace{} is a high-performance
analytics backend that enables near real time and accurate diagnoses.
Figure~\ref{fig:YPerf-architecture} shows an overview of the backend.
The backend ingests YTrace instrumentation data (a timeseries of events)
and runs statistical analyses and diagnosis on the event stream.
The events and analyses are written to a persistent store that drives
an interactive visualization system.
\vspace{0.1in}\noindent\textbf{Data transport.}
The first step after instrumentation is to transport the data to the
indexing and analysis systems. YTrace{} uses a publish-subscribe
messaging system to transport instrumentation events.
Since the YTrace{}
libraries and the transport system implement asynchronous write semantics,
instrumentation events can incur delivery delays
or be delivered out of order, be lost, or sometimes be duplicated.
This is particularly the case for all tracing events in a session, where
it is not always feasible to determine if all data for a session has arrived for analyses.
Moreover, due to event asynchrony, there may be statistical biases in
certain analytics leading to false diagnosis. In our implementation, we trigger analysis
of an event after a delay $\delta$; $\delta$ is pre-computed as the
minimum duration after which any event is delivered with a high likelihood.
In order to find biases, the YTrace{} backend measures event volume
as a function of service and datacenter; and uses it to estimate the expected
volume at the current time. If there is a bias, it does not trigger analysis
for that statistic. Inferring and avoiding bias is a part of our ongoing work.
\vspace{0.1in}\noindent\textbf{Indexing.}
The indexing system provides a high-throughput write, low-latency read
interface for structured data. Data in YTrace{} is a timeseries of graphs
from the network (topology) and application layers (session traces). Since
events for a session are transferred asynchronously, it is important that
the writes are idempotent and session updates do not require any reads. The
ETL process materializes a number of indices for sessions for common queries.
We currently use an Apache HBase cluster as our persistent store. Domain-specific
queries such as the network paths connecting two server hosts or Internet path
to a client host are processed by an API tier. Such queries are useful for
diagnosis such as tomography.
\vspace{0.1in}\noindent\textbf{Making it real time.}
In order to make the diagnosis near-real time, we would need to:
(1) minimize the latency between end of a session and when the events in
the session are analyzed, and
(2) build a high-throughput analytics backend.
A significant factor that contributes to the latency above is data
movement from multiple datacenters into a central indexing system
in a single datacenter. Note that on the contrary, a central
indexing system improves the analytics throughput; however, the
latency induced by wide area data movement degrades performance
more significantly since wide area links have limited bandwidth
and are shared resources.
To reduce wide area data movement, we are working on a federated
database that is partitioned across all datacenters. Each datacenter
includes a local indexing system, and the data partitioning
is based on the datacenter-locality of events in user sessions.
Events inside a datacenter are transported within the datacenter; hence,
the events for a session that is served by two datacenters will reside
in two indexing systems. In the ideal case, the processing for a query
would be done at the relevant indexing systems, and the aggregated output(s)
returned to the federation layer -- the aggregations are relatively
low-volume. This is, however, not true of some queries such as joins,
which may require inter-datacenter data movement.
We are also adding support for approximate queries to speed up
query processing and reduce wide area data movement. The database
has to be aware of data biases, both from transport and partitioning,
in order to minimize statistical bias in query output. Our work builds
on prior work in wide area and approximate query processing (e.g., the
recent work on WANalytics \cite{vulimiri2015wanalytics} and BlinkDB \cite{agarwal2014knowing}).
\section{Case Studies}
In this section, we show some experiences and results using YTrace{}
in production.
\begin{figure}
\begin{centering}
\includegraphics[angle=270,width=1\columnwidth]{sherpa-plot}
\par\end{centering}
\protect\caption{Effect of network delays on about 2m user operations with the Sherpa{}
data store from two Yahoo{} datacenters.\label{fig:Results-sherpa}}
\end{figure}
\subsection{Distributed Storage}
We consider a hosted large-scale, low-latency, distributed key-value
storage system, Sherpa{}~\hide{\cite{cooper2008pnuts}}{}, that
is used as a common storage backend in serving stacks at Yahoo in Figure
\ref{fig:Results-sherpa}. Sherpa{} aims for an SLO of 2ms for key reads.
We look at a multi-layer analysis of latencies in Sherpa{}.
Operations with Sherpa{} traverse router nodes and are served by Storage Unit (SU) nodes --
all connected by the datacenter network. We use YTrace{} data to
look at the impact of round-trip network latencies\footnote{Latency computations
in this study use a single clock. Network latency is: (router-SU
RPC exchange at router) - (processing delay at SU).} on latency of
two million Sherpa{} operations in two datacenters. The figure shows that the network
contributes to a significant fraction of operation latency. The tail
of the distribution (top-10\%) includes operations that saw variable
delays in the network (e.g., due to congestion or non-shortest path
routing).
Using a simple model of network delay for a key read payload, we
can show that the minimum delay for a read RPC to traverse the router
and SU nodes and back is 0.5 to 0.9ms (depending on the number of round-trips
TCP takes). Under per-hop queueing or non-shortest path routing conditions
(a router-SU path normally traverses 1-2 ToR and/or one AGG device), the
delay can be 0.7-1.3ms. Hence, in order to optimize for operation latency
and maintain SLOs, the storage system could be designed to minimize the number of
network hops traversed by RPCs.
\subsection{Datacenter Network}
We look at datacenter-wide problems from a single Yahoo datacenter
using YTrace{}'s network diagnosis output. The datacenter consists of
a large fat-tree network topology with thousands of network devices.
The topology is made of multiple ``tiers'': traversing bottom-up,
Top-of-Rack (ToR) devices that connect servers (running services),
multiple aggregation (AGG) tiers and a core tier that connects the datacenter
to the Internet. RPCs between services within the datacenter typically
traverse the ToR and AGG tiers; hence, any problems in the two tiers
will impact a significant fraction of RPCs in the datacenter.
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{dcnt_layer}
\caption{Number of problem graphs for each network tier.}
\label{f:gsig_cnts_layer}
\end{center}
\end{figure}
Figure~\ref{f:causal_graph_example} shows two examples of problem
graphs from ToR and AGG tiers in the datacenter network; see \S\ref{sec:netdiag}
for details. Figure~\ref{f:gsig_cnts_layer} shows the distribution of
different problem classes across the three network tiers. Over 93\% of
the problems occur in the ToR switches (which dominate in number and are
relatively low-cost devices). A large fraction of ToR and AGG problems occur
in the lower layers (PHY and L2), and sometimes in higher layers such as
the routing plane. On the other hand, middleboxes (that can be topologically placed anywhere
in the network) show problems mostly in the higher layers (L3 and L4). The
duration of the problem graphs can last between a few seconds to hundreds
of seconds -- which makes the likelihood of RPCs being affected high.
We refer the reader to our prior work \cite{netarxiv} for more results and
details of network diagnosis methods.
\begin{figure}[t]
\begin{centering}
\includegraphics[angle=270,width=1\columnwidth]{cdf_fb_server_srtt}
\end{centering}
\caption{User-side RPC first-byte latency broken down by the dominant bottleneck (Internet or CDN).\label{fig:cdn-fb}}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[angle=270,width=1\columnwidth]{netrtt-plot}
\par\end{centering}
\protect\caption{Network RTT variation of user and datacenter paths in 350m user sessions
with the Yahoo{} CDN.\label{fig:Results-cdn}}
\end{figure}
\subsection{Video Stack}
Video serving stacks can be modeled as three tier architectures, spanning
the video player (user-side), the CDN and a backend store. A video
playback is a sequence of RPCs by the player to the CDN; the CDN makes an
RPC to the backend store if it does not have the response cached.
We trace RPCs from the video player through the CDN, while synchronously
instrumenting the TCP stack in the CDN kernel periodically over the
course of the RPC. We use the TCP measurements as the source of truth
for the Internet path performance, since the delays induced by the kernel
space are relatively very low. We collect data for all user sessions
over a course of two weeks for this case study.
We first look at the impact of backend and Internet performance on the
user experience. We quantify the user experience as the first-byte delay
for each RPC. Figure~\ref{fig:cdn-fb} shows the distribution of user-side
latencies after dividing the set of RPCs into two parts: RPCs that are
bottlenecked by the Internet path and RPCs that are bottlenecked by the
CDN or backend. About 95\% of the RPCs are bottlenecked by the Internet
path as we would expect. In the remaining 5\% RPCs that are bottlenecked
by the CDN/backend, the cache miss rate is 40\% as compared to 2\% overall.
Further, the user experience degradation due to RPCs bottlenecked by
the CDN/backend is tens of milliseconds higher than RPCs bottlenecked
by the Internet. This shows that in order to troubleshoot or fix tail
latencies, we should focus on the CDN/backend.
\begin{figure}
\begin{centering}
\includegraphics[angle=270,width=1\columnwidth]{cdf_d_ds}
\par\end{centering}
\protect\caption{User-side stack latency estimates (lower bound).\label{fig:dds}}
\end{figure}
When there is a cache miss, the CDN makes an RPC to the backend.
We look at RTT of TCP connections at the Yahoo{} CDN -- for RPCs from the user
host and to the backend in Figure \ref{fig:Results-cdn}. The
RTT is the delay between a TCP data segment and the corresponding
TCP acknowledgement at the CDN node's TCP stack. For 350m user sessions,
we analyze the \emph{variation} in RTT of each TCP connection, defined
as the ratio of RTTvar and smoothed RTT in the kernel.
We see that the RTT variation is significantly
higher for user connections than datacenter connections; however,
the tail RTT variation is dominated by the datacenter connections.
This also makes the case for troubleshooting tail latency problems
by looking at the CDN/backend.
Finally, we show that it is possible to estimate content download
latencies in the user host by tracing instrumentation alone (i.e.,
timing at user and CDN hosts, and TCP kernel variables). The first-byte
delay at the user host for video segment RPCs to the CDN ($\delta_{fb}$)
includes an RTT ($\delta_{rtt}$) on the Internet, CDN and backend
(if any) latencies ($\delta_{cdn}$ and $\delta_{be}$), and the download
stack latency at the user host ($\delta_{ds}$). Considering the TCP Retransmission Timeout ($\delta_{rto}$)
as a conservative estimate for $\delta_{rtt}$, we can estimate a lower
bound for $\delta_{ds}$: $\delta_{ds} \ge \delta_{fb} - \delta_{cdn} - \delta_{be}- \delta_{rto}$.
Figure~\ref{fig:dds} shows the distribution of the lower bound of
download stack latencies across video sessions (truncated to
positive real numbers). We see that the user host contributes tens to
hundreds of milliseconds of latency when delivering data to the
video player application running on the browser. Although user-side
download stack delays are not under the control of a content provider,
providers can avoid the effect of such problems by ordering
RPCs to mask the problem. We refer the reader to our prior work \cite{2016arXiv160504966G}
for more results on the video stack.
\section{Related Work}
Diagnosis systems are typically designed for diagnosing a subset
of the end-to-end path or a specific layer of the stack.
YTrace{} is an attempt to build an end-to-end
multi-layer diagnosis system at web-scale, since performance troubleshooting
activities typically rely on such insights. In doing so, it
builds on some prior systems. We capture representative work in this section.
\noindent\textbf{Distributed tracing systems.}
Distributed tracing is a common instrumentation primitive in content
providers. Capturing, recording or mining causality between events in a distributed
trace is necessary to make sense of session performance. Systems in
prior work differ in the amount of instrumentation and trace analysis
complexity -- in fact, there is a tradeoff between instrumentation overhead
and analysis complexity to do the same amount of diagnosis.
Systems implement causality synchronous with execution \cite{fonseca2007x,sigelman2010dapper}
or mine using historic traces \cite{chow2014mystery}. For example,
Dapper (and its derivative, Zipkin) capture causality between \emph{spans},
which are combinations of requests and associated responses. While
span-level causality is useful, it is not expressive enough to model
RPC executions such as parallelism and redundancy. History-based
causality mining helps minimize instrumentation overhead in production;
it relies, however, on resources for offline mining of causal relationships.
It works well in homogeneous environments, where there is a common RPC
library and RPC execution patterns are predictable, but
may not be feasible in heterogeneous and dynamic runtime
environments due to runtime transitions to non-causality within a
session. Magpie \cite{barham2004using} lies on the instrumentation side of
the spectrum -- it captures detailed instrumentation, such as OS events
and packet traces, to infer causality without needing offline analysis.
While this yields detailed diagnoses, it may not be feasible in production.
Project 5 \cite{aguilera2003performance}, Mystery Machine \cite{chow2014mystery}
and Pinpoint \cite{} lie on the analysis side of the spectrum -- they require
offline resources to mine causality.
X-Trace is an experimental system that requires session
tracing support from network devices; having such support helps do
multi-layer causal discovery synchronously with the session (a limitation
of YTrace{}), but network support may not be feasible in practice.
\noindent\textbf{Network diagnosis.}
There has been significant research on network diagnosis methods. Sun et al.
capture TCP variables at the CDN to localize performance bottlenecks
\cite{sun2011identifying}; they require OS kernel changes in the
critical path, since they require TCP instrumentation outside of the
\texttt{tcp\_info} structure. WhyHigh \cite{krishnan2009moving} and LatLong \cite{zhu2012latlong}
further discover client clusters with performance problems and diagnose
user-to-CDN path problems at an aggregate level (instead of per-session).
Yu et al. \cite{yu2011profiling} and Ghasemi
et al. \cite{Ghasemi:2013:RDT:2537148.2537156} diagnose datacenter
network performance using detailed instrumentation (e.g.,
socket logs and packet traces). At large serving rates, such logging
may be infeasible. Network tomography techniques
\cite{Duffield:2003:SNP:948205.948232} localize bad performance to
network interfaces; they assume that the path between two hosts is
known -- uncommon in datacenter networks.
MonitorRank \cite{kim2013root} uses similar tomography-based localization.
\noindent\textbf{Log mining.}
Service and network log mining are common diagnosis methods.
Distalyzer compares anomalous logs with known baseline
logs \cite{nagaraj2012structured} for diagnosis.
Spectroscope compares two trace logs to understand
differences between them \cite{sambasivan2011diagnosing}.
Xu et al. mine log features \cite{xu2009detecting}.
Syslogs have been used to study network-specific failures in datacenters~\cite{Gill:2011:UNF:2018436.2018477,Potharaju:2013:NCE:2523616.2523638,Potharaju:2013:DDS:2504730.2504737},
but not for root cause analysis. Prior work has not
explored causal discovery for log analysis -- this becomes
particularly necessary when looking for a small number of
cascading problems in large log volumes.
\noindent\textbf{Code and content localization.}
Binary profiling \cite{anderson1997continuous,attariyan2012x,ren2010google}
and code mining methods \cite{186221} have been used to diagnose
performance problems in single hosts down to code. These systems
do not track code-level problems with user experience.
More recently, Pivot Tracing \cite{mace2015pivot} allows users
to insert breakpoints in running code and log them while tracing
(synchronously) -- mainly tailored towards debugging within a distributed
system (as opposed to a content provider). Content diagnosis methods used browser
modifications \cite{wang2013demystifying} and middleboxes \cite{Kiciman:2007:APR:1294261.1294264}.
We are exploring the feasibility of these methods in YTrace{}.
\section{Discussion and Conclusion}\label{sec:Discussion-and-Conclusion}
In this paper, we presented the design of YTrace{}, a system for
end-to-end multi-layer performance diagnosis in large content providers.
We formulated a problem statement that covers diagnosis use cases, and
presented instrumentation and methods for diagnosis.
Our discussion opens several research questions that we cover next.
If an RPC is observed to have high latency in the datacenter,
YTrace{} currently lists correlated network problems
from devices that the RPC potentially traversed (based on syslogs).
The longer the problem, the more likely that RPCs that traversed the
devices will be impacted. Such correlations are useful towards
troubleshooting (esp. when looking at aggregated data).
Going from correlation to causation -- in other
words, whether a network problem \emph{caused} performance problems with
RPCs -- is an open question. It requires apriori knowledge of
devices the RPC traversed, e.g., using Netflow (since datacenter networks use multi-path routing),
and inferring causal relationships between problems in those devices
and RPC performance. One approach is to look for symptoms in RPCs
instrumentation that are caused by each network pathology.
YTrace{} diagnoses distributed root causes such as cascades in the datacenter
network, but does not diagnose distributed root causes across services.
Such problems create runtime dependencies between services that impact
performance (despite RPC parallelism and redundancy).
A common service-level cascading problem is backlog that builds up across
services (typically calling services). Distinguishing these from backlogs
that arise due to problems within the host requires appropriate
instrumentation and diagnosis methods. We are looking into adopting
causality-based joint mining of service logs and network syslogs to
diagnose distributed root causes.
We assumed that Traffic Engineering (TE) at the CDN is a given: YTrace{}'s
diagnosis is conditioned on the TE for a user session. Diagnosing
performance\emph{-sub-optimal} TE for a session (i.e., whether a user
was directed to a CDN node that caused bad user experience) requires
knowledge of Internet path performance from the user to all CDN nodes
at that time; accurately doing it is an open research direction. A
related system, LatLong, diagnoses \emph{average} latency \cite{zhu2012latlong}.
Content providers may not have a complete view of the user end-host
stack performance (hardware, OS environment, browser, etc.) as the
content is parsed, executed and rendered. Analysis similar to WProf
\cite{wang2013demystifying} that does not require browser modifications
would help diagnose bottlenecks that reside in the user end-host,
and could be exposed to the content provider similar to Navigation
Timing \cite{navtiming}.
In order to reduce overhead due to instrumentation, YTrace{}
supports sampling a fraction of user sessions. Sampling leads to challenges
in analyzing tail latency -- it requires inversion of the
sampled distribution of execution graphs to estimate high quantiles. For example,
prior work on latency looked at estimating confidence intervals for
latency quantiles \cite{sommers2010multiobjective}.
In order to localize performance problems to networks and inter-domain
links on the Internet, we are looking into adopting tomography methods that
work on TCP measurement data. Tomography methods
assume that the Internet path for a user IP address is known.
In practice, provider networks may use multipath routing. Without additional active probing
or data (e.g., Netflow) at the time of the RPC flow \cite{pan2014end},
it is challenging to find the sequence of Internet hops that a given RPC took.
Finally, Internet and transit providers can deploy traffic management
mechanisms that do not follow conventional wisdom and can impact
performance. For example, traffic shaping leads to changes in link
capacity, which can impact long-running
flows. YTrace{} currently diagnoses such mechanisms as a part of
the user-CDN Internet path; diagnosing such root causes, however,
is an open problem. Recent work on tomography shows that the methods
can be used (under sufficient sample size) to find content discrimination
\cite{Zhang:2014:NNI:2740070.2626308}, under assumptions of static routing.
Web-scale performance diagnosis requires re-thinking from ground up:
the instrumentation design, algorithms and systems design to enable
near real time diagnoses. There is an inherent tradeoff between complexity
of and how detailed diagnosis could be, versus the amount of per-session
instrumentation volume we can collect in production at scale. Traditional
methods such as tomography and blackbox RPC causality learning are hard to
apply in large-scale heterogeneous cloud environments. YTrace{} is an attempt to accomplish
performance diagnosis at scale.
\begin{small}
\section*{Acknowledgements}
Ahmed Mansy led the TCP instrumentation effort.
We would like to thank many Yahoos who we had insightful
discussions with, and who contributed to YTrace instrumentation, deployment
and backend: Jayanth Vijayaraghavan, Vahid Fatourehchi, Joshua Blatt, Christophe Doritis,
Vidya Srinivasan, Srikanth Sampath Kumar, Arvind
Murthy, Nishant Mishra, Jacob Cherackal, Powell Molleti, Antonia Kwok,
Bhupendra Singh, Seema Datar, Evan Torrie, Rupesh Chhatrapati, Maurice
Barnum, Amit Jain, Ramachandran Subramaniam,
Amar Kamat, Tague Griffith, Archie Russell, Tim Miller, Scott Beardsley,
Amotz Maimon, Ian Flint, Rick Hawes and Benoit Schillings.
We thank Chen Liang (Duke) for syslog analysis and Jennifer Rexford
(Princeton) for helpful discussions on the video stack.
\end{small}
\balance
{\footnotesize{}\bibliographystyle{acm}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,003
|
\section{Introduction}
\label{sec:intro}
Conversational agents such as Alexa, Siri, and Google Assistant should help users discover, learn, and retain novel factual information.
More generally, systems for conversational information-seeking should help users develop their information need, be mixed-initiative, incorporate user memory, and reason about the utility of retrieved information as a combined set~\citep{Radlinski2017ATF}.
We focus on a curiosity-driven, information-seeking scenario where a user starts a conversation with an assistant by asking an open-ended question and then drills down into interest areas (Figure~\ref{fig:example}).
\begin{figure}[t]
\centering
\begingroup
\addtolength\leftmargini{-5mm}
\begin{itemize}
\setlength\itemsep{-1.25mm}
\it
\fontsize{10}{12}\selectfont
\item[] U: <assistant wake-word>, tell me about Tahiti.
\item[] A: It's the largest island in French Polynesia, near the center of the Pacific
\item[] U: What is its history with France?
\end{itemize}
\endgroup
\caption{
An example of information-seeking dialog that the Curiosity{} dataset aims to support.
Assistants should answer user questions \emph{and} convey information that inspires meaningful followup questions.
}
\label{fig:example}
\end{figure}
In this setting, what policies should assistants pursue to maintain the user's interest in the topic?
Theories of human learning, such as Vygotsky's zone of proximal development, propose that learning novel information should be rooted in pre-existing knowledge and skills of the learner~\citep{chaiklin-03}.
Considering this, a good policy may give general information about Tahiti; a better policy would select information related to the user's knowledge (e.g., familiarity with France).
We hypothesize that engagement is correlated with policies that integrate a user's pre-existing knowledge, and test this through a large-scale, Wizard-of-Oz~(\abr{w}{\small o}\abr{z}{}) style collection~\citep{Kelley1984AnID,Wen2016ANE} that captures assistant policies, user reactions, and topically relevant entities that the user knows about.
The Curiosity{} dataset has 0{} English dialogs annotated with sentence-level knowledge grounding, the user's prior knowledge, dialog acts per message, and binary ratings per message.\footnote{
Dataset and code at \href{http://curiosity.pedro.ai}{curiosity.pedro.ai}{}.
}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{2020_emnlp_curiosity/figures/rover-dialog}
\vspace{-16pt}
\caption{
We sample pre-existing knowledge by asking users to indicate which \topic{topically} related entities they already \emph{know}.
The assistant paraphrases facts related to either known entities (rooted facts), an aspect (aspect facts), or the topic generally (general facts).
The user expresses engagement through a like button.
Dialog acts are annotated in a separate crowd-source task.
}
\vspace{-16pt}
\label{fig:ex-dia}
\end{figure}
In our dialog task (Figure~\ref{fig:ex-dia}), one worker takes the role of a curious user learning about a geographic entity and the other of a digital assistant with access to Wikipedia facts (Section~\ref{sec:data}).
At the start of each dialog, the user is assigned an entity as their \topic{topic} (e.g., \topic{Puerto Rico}) along with two \aspect{aspects} (e.g., \aspect{history} and \aspect{demographics}) to investigate.
Beforehand, we show the user a list of \entity{entities} related to the \topic{topic}, and they mark which they know; these entities are a sample of their pre-existing knowledge.
The user engages in open-ended discovery while the assistant simultaneously answers the user's questions and proactively introducing facts likely to prompt followup questions.
Section~\ref{sec:analysis} uses dialog act annotations combined with explicit and implicit user feedback to compare assistants' content selection and presentation policies.
For example, in interactions where the user asks a question and the assistant paraphrases a fact, how often does the user ask a followup question versus trail off in disinterest?
Most datasets (Section~\ref{sec:rel}) do not have enough annotations to answer these questions: it requires message-level dialog act annotations and feedback signals.
We compare three assistant policies: using a fact with a rooted entity, a fact from the user's aspect, or a generic fact about the topic.
The policies are compared through user ``likes'' of assistant messages and by the dialog act of their subsequent message~(e.g., did they ask a specific followup or change topic).
In Section~\ref{sec:method}, we design models that predict the policies used by the assistant: what type of message to send and which fact to use~(if any).
All models are trained jointly with a multi-task objective function.
We compare an end-to-end \textsc{bert}{}~\citep{Devlin2018BERTPO} model to our task-specific Hierarchical Recurrent Encoder model~\citep{Serban2015BuildingED} and show that our model improves over the baseline.
In summary, we make three main contributions: (1) we design an experiment to test the efficacy of personalizing conversational information systems through a user's prior knowledge, (2) introduce the Curiosity{} dataset---the first dialog dataset combining sentence-level knowledge groundings, per message ratings, \emph{and} per message dialog act annotations, allowing for robust and fine-grained structural learning of dialog policies for similar applications, and (3) design a multi-task model that incorporates the user's prior knowledge and improves over a natural \textsc{bert}{} baseline.
\section{Building the Curiosity{} Dataset}
\label{sec:data}
This section describes the construction of the Curiosity{} dataset.
Dialog topics consist of prominent world geographic entities.
The \emph{worldwide} spread of entities makes each novel to most users, the consistent topic type makes starting dialogs easier, and their rich histories, demographics, and economics add topical diversity.
For example, most people are only vaguely familiar with the history of \topic{Puerto Rico}, but most know about related concepts such as the \entity{United States} or \entity{Hurricane Maria}.
Section~\ref{sec:geo} describes how we select geographic topics, aspects, and derive a set of facts to ground against.
We collected the dataset in two steps: (1) collecting dialogs with a custom interface (Section~\ref{sec:ints}) and (2) after-the-fact dialog act annotation (Section~\ref{sec:da}).
Sample dialogs from Curiosity{} are in Appendix~\ref{apx:samples}.
\subsection{Geographic Topics, Aspects, and Facts}
\label{sec:geo}
We select 0{} geographic pages from Wikipedia that have separate geography and history pages (e.g., \topic{Puerto Rico}, \entity{Geography of Puerto Rico}, and \entity{History of Puerto Rico}).\footnote{
The existence of these pages implies that the topic has ample historical and geographical knowledge to draw from.
}
We use sentences from each page to build a set of 0{} facts.
We run an entity linker over the content~\citep{gupta-etal-2017-entity} and index each fact by its source page (\topic{topic}), source section (\aspect{aspect}), and mentioned entities.
Finally, we fit a \textsc{tf-idf} text matcher~\citep{rajaraman_ullman_2011} with Scikit-Learn~\citep{scikit-learn}.
While conversing, assistants are shown facts filtered by topic, aspect, or mentioned entities, that are ranked by textual similarity to the dialog.
\subsection{User and Assistant Dialog Interfaces}
\label{sec:ints}
To collect dialogs, we build user and assistant interfaces for annotators.
The user's interface samples their prior knowledge of a topic, captures which assistant messages interest them, and manages the dialog context.
The assistant's interface provides contextually relevant facts.
Appendix~\ref{apx:int-photos} has screenshots and details of each interface component.
\paragraph{Sampling User's Prior Knowledge}
When deployed, digital assistants can draw from prior interactions~\citep{Ram2018ConversationalAT} to estimate what a user knows.
However, since we do not have these prior interactions, we collect information about what users know.
Instead of exhaustively asking about every entity related to the topic, we sample this knowledge.
Before the dialog begins, we show the user fifteen related entities that range from commonplace to obscure (\entity{United States} versus \entity{Ta\'ino}).
Users mark the entities they could (1) locate on a map or (2) summarize succinctly in one sentence.
\paragraph{Like Button for User Interest}
As part of our collection, we aimed to determine what fact-grounded utterances users found interesting.
Users ``liked'' the assistant's message if they found it ``interesting, informative, and relevant to their topic.''
\paragraph{Assistant's Topic Summary and Fact Bank}
The worldwide spread of Curiosity{}'s entities makes them unfamiliar to most crowd-workers, including the assistants.
So that the assistant can still engage the user, the assistant interface provides contextually relevant information.
First, the interface shows a topic summary from Wikipedia.
Second, the assistant paraphrases facts from a contextually updated fact bank (box 2 in Figure~\ref{fig:ex-dia}).
To reduce information overload, we use simplified topic descriptions from SimpleWikipedia and show a maximum of nine facts at a time.\footnote{
If a description exists in \href{https://simple.wikipedia.org/}{simple.wikipedia.org}, we use that; otherwise, we use the description from \href{https://en.wikipedia.org/}{en.wikipedia.org}.}
We encourage assistants to ``stimulate user interest and relate information to things they already know or have expressed interest in.''
Assistants are instructed to select relevant facts, click the ``use'' button, and paraphrase the content into their next utterance.
Like~\citet{dinan2019wizard}, the fact bank shows facts to the assistant using \abr{tf-idf}{} textual similarity to recent dialog turns but differs by incorporating the user's prior knowledge.
We show the assistant nine facts: three facts that mention an entity familiar to the user (rooted facts), three facts from their assigned aspects (aspect facts), and three from anywhere on the page (general facts).
By construction, rooted facts overlap with the exclusive categories of aspect and general facts.
For each category, we find the nine highest \abr{tf-idf} scoring facts and then randomize their order.
To avoid biasing the assistant, we do not inform them about the user's known entities or distinguish between types of facts.
\subsection{Dialog Act Annotation}
\label{sec:da}
Inducing structure on conversations through dialog acts is helpful for analysis and downstream models~\citep{tanaka-etal-2019-dialogue}.
We introduce structure---beyond knowledge groundings---into Curiosity{} by annotating dialog acts for each message.
In a separate collection, we annotate all utterances with dialogs acts using a custom interface~(Appendix~\ref{apx:acts}).
The annotation schema is based on \abr{iso} $24617$-2~\citep{Bunt2010TowardsAI,Bunt2012ISO2A} with customized sub-categories for our scenario.
Table~\ref{tbl:acts} shows our schema, descriptions, and examples.
\begin{table*}[ht]
\centering
\small
\begin{tabular}{l r l l}
\toprule
\textbf{Dialog Act} & \textbf{Count} & \textbf{Description} & \textbf{Example} \\
\midrule
request topic & $10,789$ & A request primarily about the topic. & I'd like to know about \topic{Puerto Rico}. \\
request aspect & $41,701$ & A request primarily about an aspect. & Could you tell me about its \aspect{history}? \\
request followup & $4,463$ & A request about mentioned concept. & Do you know more about the \entity{Ta\'inos}? \\
request other & $10,077$ & Requests on unmentioned concepts. & What is there to know about cuisine? \\
\midrule
inform response & $59,269$ & Directly answer an info request. & \entity{Ta\'inos} were caribbean indigenous. \\
inform related & $6,981$ & Not a direct answer, but related info. & I do not know, but\ldots \\
inform unrelated & $557$ & Does not answer question, not related. & Politics is tiring! \\
\midrule
feedback positive & $26,946$ & Provide positive feedback & Thats quite interesting! \\
feedback negative & $176$ & Provide negative feedback & Thats pretty boring. \\
feedback ask & $36$ & Ask for feedback & Do you find \textless~info~\textgreater~interesting? \\
\midrule
offer topic & $91$ & Offer to discuss topic & Want to learn about \topic{Puerto Rico}? \\
offer aspect & $1,440$ & Offer to discuss aspect & How about more on its \aspect{demographics}? \\
offer followup & $63$ & Offer to discuss mentioned concept. & I could say more about the \entity{Spanish}. \\
offer other & $1,619$ & Offer to discuss unmentioned concept. & How about I tell you about its exports. \\
offer accept & $1,727$ & Accept offer of information. & I'd love to learn about its \topic{history}. \\
offer decline & $405$ & Decline offer of information & Sorry, I'm not interested in that. \\
\bottomrule
\end{tabular}
\caption{
Counts, abbreviated descriptions and examples of the dataset's dialog acts.
}
\label{tbl:acts}
\end{table*}
\subsection{Data Quality}
\label{sec:collection}
We crowd-sourced conversations in two phases using \abr{p}{\small arl}\abr{ai}{}~\citep{miller2017parlai}.
In the first, pilot studies collect feedback from individual workers.
Based on feedback, we create task guidelines, sample dialogs, a \abr{faq}, tutorial videos, and qualification tests.
These materials were used to train and qualify crowd-workers for the second phase.
During the second, we monitor the interface usage and removed workers that ignored instructions.
\begin{table}
\small
\centering
\begin{tabular}{ c c c }
& Annotator 1 & Annotator 2 \\
\toprule
Utterance 1, Label A & Yes & No \\
Utterance 1, Label B & Yes & No \\
Utterance 2, Label A & Yes & Yes \\
Utterance 2, Label B & Yes & Yes \\
\bottomrule
\end{tabular}
\caption{
Consider a task where each utterance has labels A and B.
In the single-label version, each utterance is labeled as either A or B.
The table shows the outcome of converting the multi-label version to single-label by creating a row for each example--label combination.
Cell values are binary indicators.
}
\label{table:krip-multi}
\end{table}
Using Krippendorff's $\alpha$~\citep{kripp2004}, we validate the quality of dialog act annotations.
Dialog acts are multi-class and multi-label: a message can have none, one, or multiple dialog acts (e.g., positive feedback and followup).
However, Krippendorff's $\alpha$ is computed for single-label tasks from a table where rows represent examples, columns represent annotators, and cells indicate the singular class label.
We convert our multi-label problem to a single label problem by making each combination of example and label class a row in the table (Table~\ref{table:krip-multi}).
Since there are few dialog acts per utterance, most annotations agree; however, since Krippendorff's $\alpha$ focuses on disagreement, it is appropriate for this scenario.
Using a separate annotation interface (Appendix~\ref{apx:acts}), we doubly annotate 4,408 dialogs and the agreement score 0.00{} is higher than the 0.8 threshold recommended by \citet{kripp2004}.
Next, we analyze the annotated dialogs and introduce our model.
\section{Dataset Analysis}
\label{sec:analysis}
This section shows statistics of the Curiosity{} dataset and that users prefer aspect-specific, rooted facts.
\subsection{Dataset Statistics}
\label{sec:stats}
Table~\ref{tbl:stats} shows the basic statistics of the Curiosity{} dataset.
In total, our dataset contains 0{} dialogs with 0{} utterances.
The fact database contains 0{} facts; of those, 0{} ($81\%$) were shown to the assistants and 0{} ($29\%$) were used in at least one message.
We randomly split dialogs into training, validation, and testing folds.
\begin{table}[t]
\centering
\scalebox{0.75}{
\begin{tabular}{l r r r r r}
\toprule
\textbf{Metric (\# of)} & \textbf{Total} & \textbf{Train} & \textbf{Val} & \textbf{Test} & \textbf{Zero} \\
\midrule
Dialogues & 0{} & 0{} & 0{} & 0{} & 0{} \\
Utterances & 0{} & 0{} & 0{} & 0{} & 0{} \\
Likes & 0{} & 0{} & 0{} & 0{} & 0{} \\
Topics & 0{} & 0{} & 0{} & 0{} & 0{} \\
Facts Total & 0{} & NA & NA & NA & NA \\
Facts Shown & 0{} & 0{} & 0{} & 0{} & 0{} \\
Facts Used & 0{} & 0{} & 0{} & 0{} & 0{} \\
\bottomrule
\end{tabular}
}
\caption{
Curiosity{} has 0{} dialogs. On average, dialogs have $12.9$ utterances.
$60\%$ of the assistants' 90,534 utterances were liked.
}
\label{tbl:stats}
\end{table}
\subsection{What Facts do User Prefer?}
In Section~\ref{sec:intro}, we hypothesized that when assistants use facts that mention previously known entities (rooted facts), users will be more likely to engage.
In our data collection, we incorporate two mechanisms to test this hypothesis.
The first mechanism is explicit: we directly ask users---through a like button---to indicate what messages they preferred.
The second mechanism is implicit and derived by mining dialogs for specific sequences of dialog acts that suggest engagement with the content.
For each of these mechanisms, we compute the likelihood $P(\text{Prefer}\, | \,\text{Fact Source})$ of a user preferring utterances grounded to each fact source (Rooted, Aspect, or General).
Figure~\ref{fig:prefs} shows this likelihood and indicates that users prefer: (1) facts relevant to aspects versus general ones and (2) rooted facts in three of four scenarios.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{student_prefs}
\caption{
User engagement is measured by dialog act followups (left) and like button usage (right).
We compare reactions to messages that use a fact mentioning an entity the user knew about (rooted) and whether the fact is general or aspect-specific.
Pairwise differences are statistically significant ($99\%+$) with a two proportion z-test \emph{except} for dialog act followups between rooted and non-rooted general facts.
Overall, users prefer on-aspect, rooted facts.
}
\vspace{-12pt}
\label{fig:prefs}
\end{figure}
\subsubsection{Likes for Explicit Preference Elicitation}
Explicit preference is computed directly from like button usage and shown on the right panel of Figure~\ref{fig:prefs}.
Overall, users liked $60\%$ of messages, and they prefer on-aspect, rooted facts.
\subsubsection{Mining Acts for Implicit Preferences}
When users ask specific followup questions---as opposed to generic ones---about an assistant's fact, it shows that the user implicitly prefers these kinds of messages.
For example, asking about an entity like the Ta\'inos is more specific than asking about history and therefore indicates engagement.
We identify these interactions by mining for pairs of assistant-user messages where the assistant uses a fact and their message is labeled with an ``inform'' dialog act.
With these, we compute the likelihood
$$P(\text{Outcome}=\text{request\ followup}\, | \,\text{Fact Source})$$
that the user's message has the ``request followup'' dialog act given the source.
Similarly to likes, users engage more with aspect-oriented and rooted facts.
\subsubsection{Paraphrase Analysis}
\label{sec:para-analysis}
Although our work does not include a paraphrase model, we manually analyze a random sample of two hundred and fifty assistant messages where facts were used.
Of these messages, $51\%$ were acceptable paraphrases, $27\%$ were verbatim copies, $12\%$ were contextualizations of near copies, and the remainder were errors such as incorrect paraphrases or did not incorporate the fact.
Appendix~\ref{apx:para} shows descriptions, counts, and random examples of each category.
This analysis estimates that about half of grounded messages have non-trivial signal for future paraphrase models to use.
\section{Models}
\label{sec:method}
We design a machine learning model that predicts assistant and user actions.
We introduce a multi-task architecture for \textbf{C}uriosity that \textbf{H}ier\textbf{ar}chically \textbf{M}odels (\abr{charm}{}, Figure~\ref{fig:model}) dialogs to:
(1) predict the dialog acts of the user message~(utterance act prediction),
(2) select the best fact~(fact prediction),
(3) choose the best set of dialog acts for the next message~(policy act prediction),
and (4) predict if the assistant message will be liked~(like prediction).
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{2020_emnlp_curiosity/figures/charm-model}
\caption{
\textbf{Architecture}: \abr{charm} builds a dialog context up to $t=i-1$ to predict the current message's dialog acts~(policy prediction) and the best facts to use.
The model uses this combined with the current utterance to classify it's dialog acts and if it will be liked.
}
\label{fig:model}
\end{figure*}
\subsection{Text Representation}
\label{subsec:method:text}
\abr{charm}{} jointly encodes the text of utterances and facts with one encoder.
$E$ is a bi-directional \abr{lstm}~\citep{Sutskever2014SequenceTS} over \abr{gl}{\small o}\abr{ve}{}~\citep{pennington2014glove} word embeddings and Wikipedia2Vec~\citep{yamada2018wikipedia2vec} entity embeddings.\footnote{
In \abr{charm}{}, \textsc{bert}{} was not as effective an encoder.
}
The text $t_i^u$ of utterance $u_i$ in dialog $D$ is represented as $E(t_i^u$).
Similarly, fact~$f_j$ on turn~$i$ is represented as $E(t_{i,j}^f)$ where~$j$ indexes facts shown on that turn.
\subsection{Dialog Representation}
\label{subsec:method:dialog}
In our models, we use a hierarchical recurrent encoder (\abr{hre}{}) architecture~\citep{Sordoni2015AHR,Serban2015BuildingED} where a forward \abr{lstm} contextualizes each utterance to the full dialog.
We modify the \abr{hre} model by adding additional inputs beyond the utterance's textual representation.
First, we represent user's known entities
\begin{equation}
\bm{k}=\text{avg}(E_{\text{entity}}(e_1),\ldots,E_{\text{entity}}(e_k)))
\end{equation}
as the average of entity embeddings.
An entity embedding also represents the topic
\begin{equation}
\bm{t}=E_{\text{entity}}(\text{topic)}
\end{equation}
of the dialog.
Next, we create trained speaker embedding $\bm{v}_s$ for the user and $\bm{v}_t$ for the assistant.
Given the set of all dialog acts $\mathcal{A}$, each utterance has a set of dialog acts $\mathcal{A}_u\in\mathcal{P}(\mathcal{A})$ where $\mathcal{P(X)}$ denotes the set of all subsets of $\mathcal{X}$.
Finally, we use an act embedder $A$ to compute an act representation
\begin{equation}
\bm{a}^i=\frac{1}{|\mathcal{A}_u|}\sum_{a_k\in\mathcal{A}_u} A(a_k)
\end{equation}
by averaging embeddings at each turn.
The input at each step is the concatenation
\begin{equation}
\bm{c}^i=[E(t_i^u);\bm{a}^i;\bm{t};\bm{k};\bm{v}]
\end{equation}
of the representations for text, speaker, topic, known entities, and utterance dialog acts.\footnote{
The speaker embedding $\bm{v}$ alternates between $\bm{v}_s$ and $\bm{v}_t$.
}
With this joint representation, the contextualized dialog representation
\begin{equation}
\bm{h}^{i-1}=\text{LSTM}(\bm{c}^1,\ldots,\bm{c}^{i-1})
\end{equation}
is the final \abr{lstm} state and includes time step~$t=i-1$.
The dialog up to and including time $i$ is
\begin{equation}
\bm{d}^i=[\bm{h^}{i-1};\bm{c}^{i}]
\end{equation}
which emphasizes the current utterance and makes multi-task training straightforward to implement.
\subsection{Tasks and Loss Functions}
\label{subsec:method:task}
In our model, we jointly learn to predict fact usage, user likes, utterance acts, and policy acts.
\paragraph{Fact Prediction}
For every assistant turn, the model predicts which fact(s) from
$$\{f_1,\ldots,f_k\}\in \mathcal{F}^{(i)},\mathcal{F}^{(i)}\in\mathcal{P}(\mathcal{F})$$
the assistant marked as ``used'' where $\mathcal{F}$ is the set of all facts.
We frame this task as pointwise learning to rank~\citep{Li2008LearningTR}.
A fact prediction network
\begin{equation}
\bm{s}_{j}^{f,(i)}=\text{GELU}\left(\left[\bm{W}^f\cdot \bm{h}^{(i-1)} +\bm{b}^f;E(t_j^f)\right]\right)
\end{equation}
with parameters $\bm{W}^f$ and $\bm{b}^f$ using a Gaussian Error Linear Unit~\citep{Hendrycks2017BridgingNA} outputs salience scores for each fact.
The network does not use utterance $u_i$ since it contains signal from the choice of fact.
The predictions
\begin{equation}
\bm{\hat{y}}^{f,(i)}_j=\text{softmax}(\bm{s}_{j}^{f,(i)})
\end{equation}
are converted to probabilities by the softmax
\begin{equation}
\text{softmax}(\bm{q})=\frac{exp(\bm{q})}{\sum_{j=1}^k exp({\bm{q}_j})}
\end{equation}
over $k$ labels. Using this, we compute the fact loss
\begin{equation}
\mathcal{L}_f=\frac{1}{|\mathcal{F}^{(i)}|}\sum_{i,j} \ell_{ce}(\bm{\hat{y}}^f_{i,j},\bm{y}_{i,j})
\end{equation}
where labels $\bm{y}^{f,(i)}_{j}$ indicate if fact from utterance $i$ in position $j$ was used and
\begin{equation}
\ell_{ce}(\bm{\hat{y}},\bm{y})=\sum_{p=1}^k\bm{y}_p\log(\bm{\hat{y}}_p).
\end{equation}
is the cross entropy loss.
To mitigate class imbalance, we also scale positive classes by nine~\citep{Japkowicz2002TheCI}.
\paragraph{Policy Act and Utterance Act Prediction}
Each utterance may have multiple dialog acts so we treat policy and utterance act prediction as a multi-label task.
The goal of policy prediction is to choose the best act for the next utterance; the utterance act classifies the last message's acts.
To predict these acts, we create a policy act network
\begin{equation}
\bm{s}^{p,(i)} = \text{GELU}(\bm{W}^p\cdot \bm{h}^{i-1} + \bm{b}^p)
\end{equation}
and an utterance act network
\begin{equation}
\bm{s}^{u,(i)} = \text{GELU}(\bm{W}^u\cdot \bm{d}^i + \bm{b}^u)
\end{equation}
where the probability of act $a_k$ is $p^{*,i}_k=exp(\bm{s}^{*,(i)}_k)$.
From these, we derive the policy act loss
\begin{equation}
\mathcal{L}_p=\sum_k^{\left|\mathcal{A}\right|}y^a_{i,k}\log p^{p,i}_k + (1-y^a_{i,k})\log (1-p^{p,i}_k)
\end{equation}
and utterance act loss
\begin{equation}
\mathcal{L}_u=\sum_k^{\left|\mathcal{A}\right|}y^a_{i,k}\log p^{u,i}_k + (1-y^a_{i,k})\log (1-p^{u,i}_k)
\end{equation}
for an utterance at $t=i$ with act labels $y^a_{i,k}$.
\paragraph{Like Prediction}
For every assistant message, the model predicts the likelihood of the user ``liking'' the message.
We treat this as binary classification, predict the ``like'' likelihood
\begin{equation}
\hat{y}^l_i=\text{softmax}(\text{GELU}(\bm{W}^l\cdot \bm{h}^i + \bm{b}^l)),
\end{equation}
and use it to compute the like loss
\begin{equation}
\mathcal{L}_l=\ell_{ce}(\hat{y}^l_i,y^l_i)
\end{equation}
where $y^l_i$ indicates if the message was liked.
We train the model jointly and optimize the loss
\begin{equation}
\mathcal{L}=\mathcal{L}_f+\mathcal{L}_l+\mathcal{L}_p+\mathcal{L}_u.
\end{equation}
See Appendix~\ref{apx:method:train} for training details.
\section{Modeling Experiments}
\label{sec:exp}
\abr{charm}{} improves over a \textsc{bert}{} model in most tasks.
\begin{table*}[t]
\small
\centering
\IfFileExists{2020_emnlp_curiosity/commit_auto_fig/experiment-table.tex}{\input{2020_emnlp_curiosity/commit_auto_fig/experiment-table}}{\input{2020_emnlp_curiosity/auto_fig/experiment-table}}
\caption{
The \abr{charm}{} model outperforms end-to-end \textsc{bert}{} on most tasks.
We compare fact selection with \abr{mrr}{}, dialog act prediction with micro-averaged \abr{f}$_1${}, and like prediction with accuracy.
Ablating dialog history degrades context-dependent tasks (fact selection and policy act prediction), but not tasks more dependent on one message.
}
\label{tab:experiments}
\end{table*}
\subsection{Evaluation}
We evaluate each sub-task with separate metrics.
Fact selection is evaluated with mean reciprocal rank (\abr{mrr}{}).
For utterances with at least one selected fact, we compute the \abr{mrr}{} using the selected facts as relevant documents.
We compare like prediction with binary classification accuracy.
For utterance and policy act prediction, we compare models with micro-averaged \abr{f}$_1${} scores so that frequent classes are weighted more heavily.
For each metric, we report validation and test set scores.
\subsection{Baselines}
\textsc{bert}{}~\citep{Devlin2018BERTPO} is a standard baseline for many \abr{nlp} tasks.
We use a multi-task extension of an uncased \textsc{bert}{} model as our primary baseline and fine-tune it for our unique set of tasks~(\abr{e2e} \bert{}).
Specifically, we use the \abr{cls} representation of each utterance to replace the \abr{hre}{} representation as a time-distributed input to the same multi-task decoders~(Section \ref{subsec:method:task}).
The context-less \abr{charm}{} ablation replaces the dialog contextualizer \abr{lstm} with a per-timestep projection layer.
Lastly, we report majority class accuracy for classification tasks.
\subsection{Discussion}
The proposed \abr{charm}{} model for conversational curiosity is more effective than \abr{e2e} \bert{}{} for most of the tasks in Curiosity{} (Table~\ref{tab:experiments}).
Specifically, \abr{charm}{} improves significantly in fact prediction ($13$ \abr{mrr}{} points) and both dialog act prediction tasks ($5$ \abr{f}$_1${} points), demonstrating the efficacy of the structural encoding of the various input modalities.
Generally, models accurately predict utterance acts and likes, but their \abr{mrr}{} and \abr{f}$_1${} scores on fact selection and policy act prediction is comparatively worse.
To a degree, this is expected since there is not always one best fact or one best action to take as the assistant; there may be various reasonable choices, which is common in information retrieval tasks.
Nonetheless, models that specifically reason about the relationship between prior knowledge and entities would likely yield improvement.
For example, \citet{Liu2018KnowledgeDF} predict the most relevant unmentioned entity while~\citet{Lian2019LearningTS} model a posterior distribution over knowledge.
We leave these improvements to future work.
\section{Related Work}
\label{sec:rel}
Our work builds on knowledge-grounded conversational datasets and modeling.
\paragraph{Datasets} Although there are numerous grounded datasets, we did not find one for conversational information seeking that contained fine-grained knowledge groundings, message-level feedback from the user, and dialog acts.
Table~\ref{tab:datasets} compares the Curiosity{} dataset to several others according to six factors: (1) is the goal of the task information seeking, (2) is the dataset collected from natural dialog with one participant always taking the role of an assistant, (3) are dialog responses constrained, (4) are document groundings annotated---as opposed to distantly supervised---and fine-grained, (5) is there message level feedback for the assistant, and (6) is the dataset annotated with dialog acts.
\begin{table*}[t]
\centering
\small
\begin{tabular}{p{5.3cm}cccccc}
\toprule
\multicolumn{1}{c}{Dataset} &
\multicolumn{1}{p{1.15cm}}{\centering Info Seeking} &
\multicolumn{1}{p{1.5cm}}{\centering Dialog w/Assistant} &
\multicolumn{1}{p{1.3cm}}{\centering Free Response} &
\multicolumn{1}{p{1.45cm}}{\centering Annotated Grounding} &
\multicolumn{1}{p{1.3cm}}{\centering Message Feedback} &
\multicolumn{1}{p{1cm}}{\centering Dialog Acts} \\
\midrule
Curiosity{} (ours) & \cmark & \cmark & \cmark & \cmark & \cmark & \cmark \\
\midrule
{Topical Chat}~\citep{Gopal2019topical} & \cmark & \dmark & \cmark & \cmark & \cmark & \dmark \\
{Search as a Conversation}~\citep{ren2020search} & \cmark & \cmark & \cmark & \cmark & \xmark & \xmark \\
{Wizard of Wikipedia}~\citep{dinan2019wizard} & \cmark & \cmark & \cmark & \cmark & \xmark & \xmark \\
\textsc{q}{\small u}\textsc{ac}{}~\citep{ChoiQuAC2018} & \cmark & \cmark & \xmark & \cmark & \xmark & \dmark \\
\abr{cmu dog}~\citep{Zhou2018ADF} & \cmark & \cmark & \cmark & \dmark & \xmark & \xmark \\
\abr{ms} {Marco Conv.}~\citep{Nguyen2016MSMA} & \cmark & \xmark & \abr{n/a} & \abr{n/a} & \abr{n/a} & \abr{n/a} \\
\abr{o}{\small pen}\abr{d}{\small ial}\abr{kg}{}~\citep{moon-etal-2019-opendialkg} & \xmark & \cmark & \cmark & \cmark & \xmark & \xmark \\
\textsc{c}{\small o}\textsc{q}{\small a}{}~\citep{Reddy2018CoQAAC} & \xmark & \cmark & \dmark & \cmark & \xmark & \xmark \\
{Holl-E}~\citep{Moghe2018TowardsEB} & \xmark & \dmark & \cmark & \cmark & \xmark & \xmark \\
{Commonsense}~\citep{Zhou2018CommonsenseKA} & \xmark & \xmark & \cmark & \xmark & \xmark & \xmark \\
{Reddit+Wiki}~\citep{Qin2019ConversingBR} & \xmark & \xmark & \cmark & \xmark & \xmark & \xmark \\
\bottomrule
\end{tabular}
\caption{
\cmark~indicates a dataset has the feature,~\dmark~that it does with a caveat, and~\xmark~that it does not.
Conversational \abr{ms marco} is a search dataset but has inquiry chains we want assistants to induce (exemplar in Appendix~\ref{apx:marco}).
Topical Chat and Search as a Conversation are motivationally similar.
While our dataset's combination of (human) annotation is unique, all three datasets are steps forward in resources for conversational information-seeking.
}
\label{tab:datasets}
\end{table*}
Our dataset is most similar to those for information-seeking such as \textsc{q}{\small u}\textsc{ac}{}~\citep{ChoiQuAC2018}, Wizard of Wikipedia~\citep[\abr{w}{\small o}\abr{w}{}]{dinan2019wizard}, \abr{cmu dog}~\citep{Zhou2018ADF}, \abr{ms marco}~\citep{Nguyen2016MSMA}, Topical Chat~\citep{Gopal2019topical}, the \abr{trec} Conversational Assistance track~\citep[\abr{ca}{\small s}\abr{t}{}]{Dalton2020TRECC2}, and Search as a Conversation~\citep[\abr{s}{\small aa}\abr{c}{}]{ren2020search}.
\textsc{q}{\small u}\textsc{ac}{} constrains assistant responses to spans from Wikipedia, which makes it better for conversational \emph{question answering}, but prevents more sophisticated assistant policies.
\textsc{q}{\small u}\textsc{ac}{} also provides dialog acts, but they exist so that the assistant can inform the user of valid actions; we annotate dialog acts after-the-fact so that we can compare \emph{freely chosen} user responses.
Like \textsc{q}{\small u}\textsc{ac}{}, Topical Chat, \abr{s}{\small aa}\abr{c}{}, and \abr{w}{\small o}\abr{w}{} have annotated knowledge-groundings for each message, but responses are free-form.
\abr{s}{\small aa}\abr{c}{} is a contemporaneous, \abr{ca}{\small s}\abr{t}{}-based dataset that shares our motivation to make conversation a medium for information-seeking.
Topical Chat includes user feedback, but instead of explicitly defined roles, workers implicitly take dual and alternating roles as the user and assistant through knowledge asymmetry; followup work added automatically annotated dialog acts to Topical Chat~\citep{hedayatnia2020policy}.
Many tasks instruct annotators to take on a specific role in the dialog.
For example, in Wizard of Wikipedia, annotators assume an assigned persona~\citep{zhang-etal-2018-personalizing} in addition to being the user or assistant.
Consequently, many dialogs revolve around personal discussions instead of teaching about a topic.
Additionally, annotators may not have the background to play their role.
In contrast, we ask annotators to take roles that---as humans---they already know how to do: read about and convey interesting information on a topic (assistant) and engage in inquiry about a novel topic (user).
Our work is one of many in knowledge-grounded conversational datasets.
For example, \citet{Moghe2018TowardsEB} have workers discuss movies and ground messages to plot descriptions, reviews, comments, and factoids; however, one worker plays both roles.
In \abr{o}{\small pen}\abr{d}{\small ial}\abr{kg}{}~\cite{moon-etal-2019-opendialkg}, annotators ground messages by path-finding through Freebase~\citep{Bast2014EasyAT} while discussing and recommending movies, books, sports, and music.
\citet{Qin2019ConversingBR} use Reddit discussion threads as conversations and ground to web pages.
Similarly, \citet{ghazvininejad2018a} collect Twitter three-turn threads and ground to Foursquare restaurant reviews.
Our work adds to this dataset compendium.
\paragraph{External Knowledge in Models} Our model is related to those that incorporate external information like facts in question answering~\citep{Weston2015MemoryN,Sukhbaatar2015EndToEndMN,kvnets2016}, knowledge base triples in dialog models~\citep{Han2015ExploitingKB,He2017LearningSC,Parthasarathi2018ExtendingNG}, common sense~\citep{Young2017AugmentingED,Zhou2018CommonsenseKA}, or task-specific knowledge~\citep{Eric2017KeyValueRN}.
Similarly to~\citet{kalchbrenner-blunsom-2013-recurrent,khanpour-etal-2016-dialogue}, \abr{charm}{} predicts the act of the current message, but also next message's act like~\citet{tanaka-etal-2019-dialogue} do.
\section{Future Work and Conclusion}
\label{sec:fw}
We see two immediate directions for future work.
The first is to augment our \abr{charm}{} model with a text generation module to make a digital version of our human assistants.
This involves contextualizing and paraphrasing facts which our dataset supports.
Second, dialog act sequences could identify additional data-driven policies that could be used to define rewards or losses.
By conditioning on dialog acts or sequences of dialog acts, textual outputs could be better-controlled~\citep{Sankar2019DeepRL,See2019WhatMA} and combined with knowledge grounding~\citep{hedayatnia2020policy}.
However, text is not the native modality of digital assistants.
We envision digital assistants participating in information-seeking, which means handling speech input.
Consequently, automatic speech recognition (\abr{asr}) introduces transcription errors which are especially prevalent in knowledge-oriented text like question answering~\citep{peskov2019noisy}.
\citet{gopalakrishnan2020asr} show this is also problematic in information-seeking dialog by comparing models on textual and \abr{asr} versions of Topical Chat.
To close the loop in conversational information-seeking, models need to account for the speech-based environment of digital assistants.
In summary, this work introduces Curiosity{}: a large-scale conversational information seeking dataset.
With Curiosity{}'s unique set of annotations, we design \abr{charm}{} which jointly learns to choose facts, predict a policy for the next message, classify dialog acts of messages, and predict if a message will be liked.
We hope that our dataset will encourage further interest in curiosity-driven dialog.
\section{Components of Dialog Interfaces}
\label{apx:int-photos}
In this section, we provide short descriptions and screenshots of every component of the user and assistant dialog interfaces.
\subsection{User's Interface}
Figure~\ref{fig:entity-quiz} shows the interface that we use to sample the user's prior knowledge of entities related to the topic.
To derive a diverse sample, we use Wikipedia page views as a proxy for how well known the entity is.
All experiments use the English Wikipedia dump generated on July 23, 2019.
We divide entity mentions into ten buckets based on the frequency of page views, and round-robin sample fifteen entities from those buckets.
The interface is shown before the user starts chatting with the assistant.
\begin{figure}[ht]
\centering
\nicebox{
\includegraphics[width=\linewidth]{2020_emnlp_curiosity/figures/entity-quiz.pdf}
}
\caption{
In this example, the user is assigned to learn about \topic{Lesotho}, specifically its \aspect{culture} and \aspect{history}.
In addition to their training with guidelines and videos, we repeat the instructions here.
The related entities span relatively common ones like the \entity{United States} or \entity{Africa} to less known ones such as \entity{Basutoland}.
}
\label{fig:entity-quiz}
\end{figure}
We elicit how ``interesting'' a user finds each of the assistant's messages through the like button in Figure~\ref{fig:like-button}.
Only users can ``like'' a message; the assistant cannot ``like'' user messages.
Users are instructed to ``like'' messages if they are ``interesting, informative and/or entertaining'' and ``relevant to their topic and/or aspects.''
They are specifically instructed not to ``like'' messages that are devoid of factual content, only express feelings, or only contain greetings or farewells.
\begin{figure*}[ht]
\centering
\nicebox{
\begin{minipage}{.35\textwidth}
\includegraphics[width=\linewidth]{2020_emnlp_curiosity/figures/like-instructions}
\end{minipage}
\begin{minipage}{.60\textwidth}
\includegraphics[width=\linewidth]{2020_emnlp_curiosity/figures/like-button}
\end{minipage}
}
\caption{
The user expresses the ``interestingness'' of the assistant's messages through a ``like'' button (right of message).
The instructions are shown prominently in the full interface and repeated in training material.
}
\label{fig:like-button}
\end{figure*}
\paragraph{Switching Aspect}
Users are randomly assigned two aspects for each dialog and told to spend time discussing each.
The guidelines instruct them to spend at least two turns per topic, but we do not specify any further time requirements.
When the user changes aspects, we instruct them to click a button (Figure~\ref{fig:switch-aspect}) to indicate when and which aspect they are switching to.
Additionally, this event triggers a reset in the context we use to rank the assistant's facts.
\begin{figure}[ht]
\centering
\nicebox{\includegraphics[width=.98\linewidth]{2020_emnlp_curiosity/figures/switch-aspect}}
\caption{
The user is assigned two aspects about their topic.
After they are satisfied with what they have learned about the first aspect, they click a button and switch to the next aspect.
While the button click is not communicated to the assistant (the user must send a corresponding message), it resets the fact contextualizer; we observe that without this, too many facts were related to the previous aspect.
}
\label{fig:switch-aspect}
\end{figure}
\subsection{Assistant Interface}
By design, we intend for most workers to not be familiar in depth with most of the geographic topics.
Thus, the most important responsibility of the assistant interface is to provide enough information---without overwhelming them---to be engaging conversational partners.
The first interface shown is a short description of the topic from either Simple Wikipedia or the English Wikipedia.
This component helps the assistant reach a general understanding of the topic so that they can choose better facts.
\begin{figure}[ht]
\centering
\nicebox{\includegraphics[width=.98\linewidth]{2020_emnlp_curiosity/figures/summary}}
\caption{
A short topic description is always visible to the assistant.
The goal is to ensure the assistant always has a general understanding of the dialog topic.
}
\label{fig:summary}
\end{figure}
The most important component of the assistant interface is their list of available facts.
These facts have high textual similarity with the most recent three turns and are broken into three categories: facts related to entities the user knows about (rooted facts), facts related to an aspect (aspect facts), and facts from anywhere on the page (general facts).
Feedback from pilot collections showed that six facts was too few which caused a lack of relevant facts, but twelve facts overwhelmed annotators.
Thus, we use nine facts so that we can also balance equally across each type of fact.
When composing their reply, the assistant can use any number of facts as in Figure~\ref{fig:grounded-msg}.
To discourage verbatim copying, we disable the paste feature in javascript.
We also drop repeatedly unused facts.
\begin{figure*}[ht]
\centering
\begin{minipage}{\textwidth}
\includegraphics[width=\linewidth]{2020_emnlp_curiosity/figures/fact-bank}
\end{minipage}
\begin{minipage}{\textwidth}
\nicebox{
\includegraphics[width=.98\linewidth]{2020_emnlp_curiosity/figures/grounded-message}
}
\end{minipage}
\caption{
The assistant could incorporate any number of facts into their reply to the user.
Their goal was to answer the user's immediate questions, and anticipate what information they would be most interested in.
}
\label{fig:grounded-msg}
\end{figure*}
\section{Dialog Act Annotation}
\label{apx:acts}
To annotate dialog acts, we create a separate annotation interface (Figure~\ref{fig:da-iface}).
The interface shows one dialog at a time, and the same annotator annotates all the utterances.
In addition to the utterances, the interface shows the topic, aspects, and sender of each message.
Lastly, we incorporate a ``Report Dialog'' feature to help identify and remove inappropriate dialogs.
\begin{figure*}[ht]
\centering
\includegraphics[width=\linewidth]{2020_emnlp_curiosity/figures/da-interface}
\caption{
To annotate dialog acts, we develop an interface that showed each utterance on a separate line.
Annotators assign zero or more dialog acts to each utterance using grouped dropdowns.
}
\label{fig:da-iface}
\end{figure*}
\section{Sample Dialogs}
\label{apx:samples}
Tables~\ref{tab:dialog_example1} and~\ref{tab:dialog_example2} show Curiosity{} dialogs and highlight the dataset's features.
Typos and grammatical errors made by annotators are left unaltered.
\begin{table*}[t]
\small
\begin{center}
\setlength\tabcolsep{3.5pt}
\begin{tabular}{p{0.2cm}p{3.8cm}p{11cm}}
\toprule[\heavyrulewidth]
& \multicolumn{1}{r}{\textbf{Topic}:} & \underline{Djibouti} \\
& \multicolumn{1}{r}{\textbf{Aspects}:} & \textit{Economy}, \textit{Politics} \\
& \multicolumn{1}{r}{\textbf{Known Entities}:} & France, Africa \\
\midrule
& Dialog Acts & Messages \\
\midrule
U & \texttt{request topic} & Hi can you tell me about Djibouti? \\
A & \texttt{inform response} & The country of Djibouti is officially called the Republic of Djibouti is located in the horn of Africa. It was created out of French Somaliland. It is also recorded the country was colonized by France in1800s and later gained its independence on June 27, 1977. \\
U & \texttt{request aspect} & Can you tell me about their Politics \\
A & \texttt{inform response} & After its independence, The country had two regiments commanded by the French officers. The country also maintains its close ties with Somalia, Ethiopia, France and the United States. \\
U & \texttt{request aspect} & Any other information about the French and their impact of the politics? \\
A & \texttt{inform response} & There are no direct information on its effect on the country's politics but even though the country is already independent. The 13th Demi-brigade of the French Foreign Legion is still stationed in this country. It is also the largest french military presence abroad. commanded by a 3-star general. \\
U & \texttt{request aspect} & Heavy French infeleunce. what other interesting inofrmation on politics do you have? \\
A & \texttt{inform response} & In recent years, the country has improved its training techniques, military command, and information structure. They also became more self-reliant in supplying their military and assist in peacekeeping missions and provide military assistance to countries that officially ask them for help. \\
U & \texttt{request aspect, feedback positive} & Wow Interesting, Can you now tell me about the economy they have? \\
A & \texttt{offer aspect} & Sorry, I do not have that information. But I can tell you more about Politics. \\
U & \texttt{offer accept} & Sure \\
A & \texttt{inform response} & Camp Lemonnier is rented by the United State as a Naval Expeditionary Based for \$63 Million a year. While Japan and France each pays \$30 million a year and China pays \$20 Million a year. \\
U & \texttt{request other} & Lastly, any any fun facts? \\
A & \texttt{inform response} & I am not sure if this is a fun fact, but the country's gross domestic product expanded by more than 6\%. From \$341 Million to 1.5 Billion \\
U & \texttt{} & That's a huge increase. thank you for all your help \\
A & \texttt{} & You are welcome \\
\bottomrule[\heavyrulewidth]
\end{tabular}
\end{center}
\caption{Example dialog \#1 from Curiosity{}. (U: User, A: Assistant)}
\label{tab:dialog_example1}
\end{table*}
\begin{table*}[t]
\small
\begin{center}
\setlength\tabcolsep{3.5pt}
\begin{tabular}{p{0.2cm}p{3.8cm}p{11cm}}
\toprule[\heavyrulewidth]
& \multicolumn{1}{r}{\textbf{Topic}:} & \underline{British Columbia} \\
& \multicolumn{1}{r}{\textbf{Aspects}:} & \textit{Government and politics}, \textit{Culture} \\
& \multicolumn{1}{r}{\textbf{Known Entities}:} & Canada, Seattle \\
\midrule
& Dialog Acts & Messages \\
\midrule
U & \texttt{request topic} & Hi! Can you help me learn some basic information about British Columbia? I don't know much except that it's located in Canada. \\
A & \texttt{inform response} & Yes, British Columbia is the westernmost province of Canada and is located between the Rocky Mountains and the Pacific Ocean. \\
U & \texttt{request aspect, feedback positive} & I didn't know it was on the coast! What can you tell me about government and politics there? \\
A & \texttt{inform response} & One interesting fact about the government is that the Green Part plays a larger role in this province than it does in other provinces of Canada. \\
U & \texttt{request followup, feedback positive} & Interesting. What can else you tell me about the Green Party? \\
A & \texttt{inform response} & The New Democratic Party and the Green Party caucuses together control 44 seats. Which seems like a lot but the British Columbia Green Party only takes up 3 of those 44 seats. \\
U & \texttt{request aspect} & That's a pretty small influence. Can you tell me some fun culture facts about British Columbia? \\
A & \texttt{} & I am sorry I do not have any information on their culture right now. \\
U & \texttt{request topic} & That's okay. What other fun facts can you share? \\
A & \texttt{inform response} & Interestingly, Queen Victoria chose British Columbia to distinguish what was the British sector of the Columbia District from the United States which became the Oregon Territory on August 8, 1848. \\
U & \texttt{request aspect} & So that's why it has "British" specifically as part of it's name! Makes sense. Are there any sports or outdoor activities that are popular in British Columbia? \\
A & \texttt{inform response} & Horseback riding is enjoyed by many British Columbians. \\
U & \texttt{} & Thanks for your help today. Now I know more than I did before. \\
A & \texttt{} & No problem, it was a pleasure. \\
\bottomrule[\heavyrulewidth]
\end{tabular}
\end{center}
\caption{
Example dialog \#2 from Curiosity{}. (U: User, A: Assistant).
After mentioning the Green Party, the user asks a specific followup question; we use these interactions to estimate implicit preference.
}
\label{tab:dialog_example2}
\end{table*}
\section{Paraphrase Analysis and Samples}
\label{apx:para}
In Section~\ref{sec:para-analysis}, we describe the results of a manual analysis on two hundred and fifty assistant paraphrases.
Annotations were completed by the authors and shown in Table~\ref{tab:para}.
We break messages into four categories: paraphrases, copies, errors, and unrelated.
Paraphrases include messages that incorporate the selected fact and possibly additional information.
Copies include verbatim copying, cherry-picked phrases, and trivial contextualizations like replacing an entity with a pronoun.
Table~\ref{tab:par-ex} shows ten randomly selected paraphrases from the two hundred and fifty manual annotations.
\begin{table*}
\small
\centering
\IfFileExists{2020_emnlp_curiosity/commit_auto_fig/paraphrase-table.tex}{\input{2020_emnlp_curiosity/commit_auto_fig/paraphrase-table}}{\input{2020_emnlp_curiosity/auto_fig/paraphrase-table}}
\caption{
We analyze the paraphrases annotators use through manual categorization.
The ``Copy'' category includes cherry-picked verbatim phrases, verbatim copies, and contextualized copies (e.g., changing a named entity to ``it'').
The majority of paraphrases are correct and only incorporate the provided fact, but a few weave in other information.
$7.2\%$ of paraphrases are either unrelated to the selected facts or paraphrase the fact incorrectly.
Overall, $51.2\%$ of messages have valid paraphrases.
}
\label{tab:para}
\end{table*}
\begin{table*}
\small
\centering
\begin{tabular}{p{15cm}}
\toprule
\textbf{Label:} paraphrase-correct \\
\textbf{Message:} I do not have the information available but I can tell you 10 million tones were exported annually for, Cardiff dock in the years leading up to the First World War. \\
\textbf{Fact:} In the years leading up to the First World War , more than 10 million tonnes of coal was exported annually from Cardiff Docks . \\
\midrule
\textbf{Label:} copy \\
\textbf{Message:} Media and journalists in Namibia are represented by the Namibia chapter of the Media Institute of 'southern Africa and the Editors Forum of Namibia. \\
\textbf{Fact:} Media and journalists in Namibia are represented by the Namibian chapter of the Media Institute of Southern Africa and the Editors ' Forum of Namibia . \\
\midrule
\textbf{Label:} paraphrase-multiple \\
\textbf{Message:} The film industry has a rich history in Indonesian culture and the Indonesian Film Festival has been held regularly since 1955. \\
\textbf{Fact:} Indonesia has held annual film festivals and awards , including the Indonesian Film Festival ( Festival Film Indonesia ) that has been held intermittently since 1955 . \\
\midrule
\textbf{Label:} context \\
\textbf{Message:} EU means European Union. GDP per capital ranges from 188\% of EU average in Bratislava to 54\% in eastern Slovakia \\
\textbf{Fact:} GDP per capita ranges from 188 \% of EU average in Bratislava to 54 \% in Eastern Slovakia . \\
\midrule
\textbf{Label:} paraphrase-correct \\
\textbf{Message:} Sudan was considered the 17th growing economy in the world. Its faster development of the country from oil profits. \\
\textbf{Fact:} Sudan , officially the Republic of the Sudan , is a country in Northeast Africa . \\
\midrule
\textbf{Label:} paraphrase-correct \\
\textbf{Message:} There is a mountain range called the Annamite Range that runs along eastern border to Vietnam and another range, named Luang Prabang Range, near the Thailand border. \\
\textbf{Fact:} The Mekong River forms a large part of the western boundary with Thailand , where the mountains of the Annamite Range form most of the eastern border with Vietnam and the Luang Prabang Range the northwestern border with the Thai highlands . \\
\midrule
\textbf{Label:} copy \\
\textbf{Message:} In 1603, the kingdoms of England, Scotland and Ireland were united in a personal union when James VI, king of Scots, inherited the crowns of England and Ireland and moved his court from Edinburgh to London; each country nevertheless remained a separate political entity and retained its separate political, legal and religious institutions. Would you like to know anything else? \\
\textbf{Fact:} In 1603 , the kingdoms of England , Scotland and Ireland were united in a personal union when James VI , King of Scots , inherited the crowns of England and Ireland and moved his court from Edinburgh to London ; each country nevertheless remained a separate political entity and retained its separate political , legal , and religious institutions . \\
\midrule
\textbf{Label:} copy \\
\textbf{Message:} The A31 connects to the A35 at Bere Regis , and passes east through Wimborne and Ferndown to Hampshire where it later becomes the M27 \\
\textbf{Fact:} The A31 connects to the A35 at Bere Regis , and passes east through Wimborne and Ferndown to Hampshire , where it later becomes the M27 . \\
\midrule
\textbf{Label:} context \\
\textbf{Message:} I'm not sure of that. One fact I do have is that by 1857, Chicago was Illinois' largest city. \\
\textbf{Fact:} By 1857 , Chicago was Illinois 's largest city . \\
\midrule
\textbf{Label:} paraphrase-correct \\
\textbf{Message:} On March 2, 2008, Dmitry Medvedev was elected President of Russia and Vladimir Putin became Prime Minister. \\
\textbf{Fact:} On 2 March 2008 , Dmitry Medvedev was elected President of Russia while Putin became Prime Minister . \\
\bottomrule
\end{tabular}
\caption{
A random sample of ten manually labeled paraphrases from the assistant.
The top row indicates the label we (the authors) annotated, the middle row the message, and the bottom row the original fact from Wikipedia.
The original fact is shown as displayed to crowd-workers including punctuation tokenization.
}
\label{tab:par-ex}
\end{table*}
\section{Like Prediction Comparison}
\label{apx:like-compare}
Like prediction is the one task where \abr{charm}{} was not the best model.
To better understand the differences between the \abr{charm}{} and \textsc{bert}{} model, we randomly sample thirty dialogs in the test set and find assistant messages where the model predictions disagree.
Of the 202 assistant messages in the thirty dialogs, the \textsc{bert}{} like prediction is correct 174 times ($86.1\%$) and \abr{charm}{} 170 times ($84.2\%$).
The models disagree on eight predictions ($3.96\%$); \textsc{bert}{} is correct on six of these and \abr{charm}{} on the other two.
Table~\ref{tab:like-compare} shows the messages, predictions, and labels.
Although we might guess at why the predictions differ (e.g., perhaps something in the Florida dialog confused \textsc{bert}{}), without further analysis we cannot establish causality.
\begin{table*}[ht]
\small
\centering
\begin{tabular}{l l p{12cm}}
\toprule
Liked & Correct Model & Message \\
\midrule
No & \textsc{bert}{} & You are welcome! \\
\midrule
Yes & \textsc{bert}{} & I'm sorry I don't have anymore information about the etymology of Tunisia, but what I can tell you is that Tunisia Sports City is a whole sports city being constructed in Tunis \\
\midrule
Yes & \textsc{bert}{} & Yes Buddhism is a dominant influence in Lao culture. It has been great helping you. \\
\midrule
Yes & \abr{charm}{} & Florida is a state in the southeast United States. What would you like to know? \\
Yes & \textsc{bert}{} & They have an average daily temperature of 70.7, it's the warmest state in the U. S. \\
Yes & \abr{charm}{} & Yes, I can. Florida is nicknamed the ``Sunshine State'', but severe weather is a common occurrence. \\
\midrule
Yes & \textsc{bert}{} & Hello, Indonesia is part of the Malay Islands and is in Southeast Asia. Would you like to know more about the history? \\
Yes & \textsc{bert}{} & I do not have etymologic information, would you like to know more about the economy? I can tell you thank Indonesia develops military and commuter aircraft. \\
\bottomrule
\end{tabular}
\caption{
To compare like prediction between models, we randomly sample thirty dialogs and obtain predictions from \abr{charm}{} and \textsc{bert}{}.
The table only shows messages where the model predictions disagree and indicates which model was correct.
Dialogs are delineated by horizontal lines.
Unfortunately, from only these examples we cannot determine why the \abr{charm}{} model errors in most of these predictions.
}
\label{tab:like-compare}
\end{table*}
\section{Model Training, Implementation, and Computation}
\label{apx:method:train}
We implement all models with PyTorch~\citep{paszke2017automatic} and \abr{A}{\small llen}\abr{nlp}{}~\citep{Gardner2018AllenNLPAD}.
The learning rates for models is set using the built-in learning rate finder in \abr{A}{\small llen}\abr{nlp}{}.
Model losses were optimized with Adam~\citep{Kingma2014AdamAM}; the \textsc{bert}{} model uses a learning rate of $.0001$ and \abr{charm}{} a learning rate of $.001$ with otherwise default parameters.
We train for a maximum of forty epochs and early stop based on the sum of validation losses.
The \abr{charm}{} model uses batch size $64$ and the \textsc{bert}{} model batch size $4$.
Our best model (\abr{charm}{}), has $26,970,475$ parameters, takes two hours and eighteen minutes to train, and early stops on epoch fifteen.
In our models, text encoders for utterances and facts share parameters.
Models were developed on a single machine with eighty Intel $2.0$GHz \abr{cpu}s, $256$\abr{gb} \abr{ram}, and eight Tesla V100 graphics cards.
Each model was trained and evaluated on a single graphics cards with hyper-parameter sweeps parallelized across the eight cards.
\abr{A}{\small llen}\abr{nlp}{} configuration files and software dependencies (including version) are included in our code at \href{https://github.com/facebookresearch/curiosity}{github.com/facebookresearch/curiosity}.
\section{MS Marco Conversational Sample Queries}
\label{apx:marco}
Conversational \abr{ms marco} is a search dataset that partially inspired this work.
Assistant messages should prompt followup queries like in Table~\ref{tab:marco}.
\begin{table*}[t]
\small
\begin{center}
\begin{tabular}{l}
\toprule
Query \\
\midrule
What is a physician's assistant? \\
What are the educational requirements required to become a physician's assistant? \\
What does the education to become a physician's assistant cost? \\
What's the average starting salary of a physician's assistant in the UK? \\
What's the average starting salary of a physician's assistant in the US? \\
What school subjects are needed to become a registered nurse? \\
What is the physician's assistant average salary vs a registered nurse? \\
What the difference between a physician's assistant and a nurse practitioner? \\
Do nurse practitioners or physician's assistant's make more? \\
Is a physician's assistant above a nurse practitioner? \\
What is the fastest way to become a nurse practioner? \\
How much longer does it take to become a doctor after being a nurse practitioner? \\
What are the main breeds of goat? \\
Tell me about boer goats. \\
What goat breed is good for meat? \\
Are angora goats good for meat? \\
Are boer goats good for meat? \\
What are pygmy goats used for? \\
What goat breed is the best for fiber production? \\
How long do Angora goats live? \\
Can you milk Angora goats? \\
How many Angora goats can you have per acre? \\
Are Angora goats profitable? \\
\bottomrule
\end{tabular}
\end{center}
\caption{
An exemplar query chain from the conversational variant of \abr{ms marco}.
An ideal assistant should answer these questions \emph{and} inspire these types of followup questions.
}
\label{tab:marco}
\end{table*}
\section*{Acknowledgments}
We thank Rajen Subba, Stephen Roller, Alborz Geramifard, and Scott Yih for insightful discussions.
Thanks to Becka Silvert for improving the task guidelines and Victor Ling for building the dialog act annotation tool's backend.
Thanks to Sara LaRocca and Hal Daum\'e III for advice on adapting Krippendorff's $\alpha$.
We thank our anonymous \abr{acl} and \abr{emnlp} reviewers as well as Shi Feng, Jordan Boyd-Graber, Joe Barrow, Karthik Gopalakrishnan, and \abr{umd} \abr{clip} members for feedback on the paper.
Rodriguez's contributions to this work while at \abr{umd} were supported by \abr{nsf} Grant \abr{iis}-1822494.
Any opinions, findings, conclusions, or recommendations expressed here are those of the authors and do not necessarily reflect the view of the sponsor.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 7,148
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{"url":"https:\/\/caicedoteaching.wordpress.com\/tag\/cofinality\/","text":"## 580 -II. Cardinal\u00a0arithmetic\n\nFebruary 5, 2009\n\nHomework problem 5. (${\\sf ZF}$). Show that if $\\omega\\preceq{\\mathcal P}(X)$ then in fact ${\\mathcal P}(\\omega)\\preceq{\\mathcal P}(X)$.\n\nQuestion. If $X$ is Dedekind-finite but ${\\mathcal P}(X)$ is Dedekind-infinite, does it follow that there is an infinite Dedekind-finite set $Y$ such that ${\\mathcal P}(Y)\\preceq X$?\n\nFrom now on, until we cover determinacy, we assume the axiom of choice unless stated otherwise.\n\nAdvertisements\n\n## Set theory seminar -Stefan Geschke: Cofinalities of algebraic\u00a0structures\n\nJanuary 6, 2009\n\nThis is a short overview of a talk given by Stefan Geschke on November 21, 2008. Stefan\u2019s topic, Cofinalities of algebraic structures and coinitialities of topological spaces, very quickly connects set theory with other areas, and leads to well-known open problems. In what follows, compact always includes Hausdorff. Most of the arguments I show below are really only quick sketches rather than complete proofs. Any mistakes or inaccuracies are of course my doing rather than Stefan\u2019s, and I would be grateful for comments, corrections, etc.\n\nDefinition. Let $A$ be a (first order) structure in a countable language. Write ${\\rm cf}(A)$ for the smallest $\\delta$ such that $A=\\bigcup_{\\alpha<\\delta}A_\\alpha$ for a strictly increasing union of proper substructures.\n\nSince the structures $A_\\alpha$ need to be proper, ${\\rm cf}(A)$ is not defined if $A$ is finite. It may also fail to exist if $A$ is countable, but it is defined if $A$ is uncountable. Moreover, if ${\\rm cf}(A)$ exists, then\n\n1. ${\\rm cf}(A)\\le|A|$, and\n2. ${\\rm cf}(A)$ is a regular cardinal.\n\nExample 1.\u00a0Groups can have arbitrarily large cofinality. This is not entirely trivial, as the sets $A_\\alpha$ may have size $|A|$\n\nQuestion 1. Is every regular cardinal realized this way?","date":"2017-10-22 04:28:25","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\": 21, \"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\": 0.8566957712173462, \"perplexity\": 443.5198687762508}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-43\/segments\/1508187825141.95\/warc\/CC-MAIN-20171022041437-20171022061437-00538.warc.gz\"}"}
| null | null |
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