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Битва за Моротай () — боевые действия в Тихом океане, продолжавшиеся с 15 сентября 1944 года до конца войны в августе 1945 года. Бои начались, когда объединённые силы США и Австралии высадились на юго-западном побережье острова Моротай, который союзники планировали использовать в качестве опорной базы при освобождении Филиппин в том же году. Силы вторжения значительно превосходили небольшой гарнизон защитников острова и в течение двух недель смогли захватить необходимые объекты. Подкрепление к японцам прибывало с сентября по ноябрь, однако они были не в состоянии эффективно атаковать оборонительный периметр союзников. Бои с перерывами продолжались до конца войны, японские войска несли значительные потери из-за болезней и голода.
Превращение острова в базу союзников началось вскоре после высадки, а уже в октябре два крупных аэродрома были готовы к использованию. Эти и другие сооружения сыграли важную роль в освобождении Филиппин в 1944—1945 годах. Торпедные катера и самолёты, базирующиеся на Моротае, также значительно изматывали японские силы в Голландской Ост-Индии. Позже военные объекты острова были расширены для поддержки Борнейской операции 1945 года. Моротай оставался важным транспортным узлом и командным центром.
Литература
Конфликты 1944 года
Сражения Японии во Второй мировой войне
Сражения войны на Тихом океане
Сражения США во Второй мировой войне
Сражения Великобритании
Сражения Австралии
Сражения Нидерландов
Сентябрь 1944 года
Октябрь 1944 года
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Q: PHP Functions declaration increasing memory usage I am trying to optimize memory usage in my PHP application, so I started placing memory_get_usage function at specific places like after includes, arrays declarations, etc. So I can see where the memory peak starts.
By doing this I found out that the memory used tripled after including a functions.php file which only has functions declarations (no executions or classes there), just the classic:
function myFunction(){
//some code
}
Searching the web didn't really help much, I could not find much information about memory usage on function declarations.
Any thoughts?
Thanks!
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Q: opening view in new tab in mvc3 I am using mvc3 and telerik mvc grid. In grid I gave custom command. When we click the command a new view should be opened in new tab. Say, view opened is "Home/Index" in which telerik grid is there and custome button inside the grid will open new view say "Posts/Index" in new tab 'post' is controller name and 'index' is action name.
How can I achieve the same?
A: @{
Html.Telerik().Grid<Model>()
.Name("Runs")
.DataKeys(keys => keys.Add(r => r.ID))
.DataBinding(dataBinding => dataBinding
//Ajax binding
.Ajax()
.Select("Run", "Management")
)
.Columns(columns =>
{
columns.Bound(r => r.Definition.Name).Width(100);
columns.Bound(r => r.CreateTime);
columns.Bound(r => r.FileName);
columns.Bound(r => r.Status);
columns.Bound(r => r.CreatedBy);
columns.Bound(r => r.ID).ClientTemplate(Html.ActionLink("View", "RunDetails", new { id = "<#= ID #>" }, new { @class = "t-grid-action t-button t-state-default t-grid-edit t-button-hover" }).ToString()).Width(200).Title("Commands").Filterable(false);
})
.Pageable(pager => pager.PageSize(20))
.Sortable( sorting => sorting.SortMode(GridSortMode.MultipleColumn))
.Filterable()
.Render();
}
Try this...
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{
"redpajama_set_name": "RedPajamaStackExchange"
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Tore Elias Hoel (born 14 December 1953 in Harstad, Norway) is a Norwegian poet, author and children's author.
Bibliography
1979 – Å fange en hval (poems)
1981 – Presidentens ro (poems)
1988 – Verdsmeisteren (novel)
1991 – Eg heiter Pawel (novel)
1999 – Fire dagar i Nairobi (novel)
2003 – Glaskula (novel)
2005 – Vinterfilm (novel)
2005 – Jomfru Rosenving på Santavajasø (libretto for Children's opera)
2007 – To gutar til Paris (novel)
External links
Tore Elias Hoel at NRK Authors
Living people
1953 births
Norwegian writers
People from Harstad
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Showing results for tags 'justin timberlake'.
Justin Timberlake announces that he is close to finishing his new album!
Anyone excited for this?
Joe Biden inauguration thread: Demi Lovato, Justin Timberlake, JLo performing
This will be Biden's official inauguration topic on Exhale. Updates will reside in this post JLo, Demi Lovato and Justin Timberlake, among others, are performing at Joe Biden's inauguration ceremony on January 20th. Related:
Pharrell Williams says nearly all the songs on Justin Timberlake's 'Justified' album were originally written for Michael Jackson
PickleSpears posted a topic in Music News
Pharrell said all but 1 song from Justified were written for Michael Jackson. Pharrell said: All but one song was written for Michael. Probably "Cry Me A River." Here's a link to the Billboard article and the video of the original interview: Related:
Watch the trailer for Justin Timberlake's new movie "Palmer"
Justin Timberlake is returning to the big screen! Sort of. His new movie "Palmer" hits Apple TV on January 29th. Exhale, how we feeling about JT's new role? Related:
Justin Timberlake calls Beyoncé "the greatest singer ever"
OnlyBeyonce posted a topic in Music Discussions
Justin Timberlake called Beyoncé the greatest singer ever in a new interview with Interview Mag. Do you agree with him?
This Day in Pop: Britney Spears performs 'Human Nature' with Madonna at her Sticky & Sweet Tour in 2008 (November 6)
PokemonSpears posted a topic in Britney Spears
On November 6, 2008, Britney Spears joined Madonna on her Sticky & Sweet Tour in Los Angeles at the Dodger Stadium to perform Madge's 1995 single Human Nature. This marked Britney's first performance since her appearance at the VMA's 2007, and the first one after being put under a conservatorship in early 2008. Britney wasn't the only ex-Mouseketeer to join Madonna onstage that night. Justin Timberlake appeared right after Britney left the stage to perform with Madonna their duet 4 Minutes. It's unknown whether Britney and Justin had some backstage time together or not. The Human Nature performance featured a backdrop that showed footage of Britney trapped in an elevator, and the version of the tour included Britney's iconic phrase "It's Britney, *****" from her 2007 song Gimme More. We were able to see a little bit of the chat that Madonna and Britney had backstage on MTV's For the Record which aired that year. Madonna reflects about the times that Britney was going through at the moment, and the reason to choose her for that particular song.
stickyandsweet
I was this old when I found out "Rock Your Body" was intended for Michael Jackson
So Justin's 2002 disco-fied, very Michael Jackson Off the Wall-esque track, "Rock Your Body," was intended For The Emperor of Pop himself Michael Jackson (you guy call the Biebs the king, I'm upgrading MJ's status in this bish then). Honestly, it makes sense. The guy who basically ripped the dance moves, almost erased his sisters legacy, I'm not surprised in his ultimate falsetto voice he would do the ultimate Michael Jackson track, and keep it for himself. https://www.facebook.com/watch/?v=929451290912073 And as this video host says anything in that Justified album, would probably be better vocally, musically and visually done better by Michael Jackson. If we're being honest, Justin Timberlake's entire career would be more epic if Michael Jackson did it since that's the entire sound and image and dance moves Timberlake ripped off.
Justin Timberlake: I'm voting for Joe Biden and Kamala Harris
Justin Timberlake announced he's voting for Joe Biden and Kamala Harris. We love to see it king. Related:
Justin Timberlake disses "Can't Stop The Feeling"
In a new interview with Apple Music, Justin Timberlake disses his last big global hit song, 2016's "Can't Stop The Feeling." He says the song never felt like his because he wrote it for the movie Smurfs, but RCA and Dreamworks said you need to sing it. He says he wrote it with someone else in mind. Someone who had kids. "This is a song I don't feel is mine. We wrote it for this movie." Funny thing is all the blogs are saying it's very ironic. JT is in a long list of artists who diss their last global hit. Ironically he has yet to have a hit as big as this song. Funny, the Trolls World Tour soundtrack also flopped (though the movie was a groundbreaking success as it was the first film released in an on-demand home rental release thx to Covid 19). It's a move that was a success, but at the same time was controversial as Universal faced severe backlash saying all theatrical films will also get the on-demand home release as well, which caused several chains to block future Universal studios films unless they reverse this decision. Check it out below:
Melanie C calls "can't stop the feeling" by Justin Timberlake a Spice girls song, none the less
In an interview with Philippines radio station for promotion of Her 8th studio album "Melanie C" She was asked what song would she give away to another artist, What song by another artist is actually a Spice Girls song in disguise, and out of her iconic girl groups catalog which one is the most Melanie C track the most Lizzo would get a deluxe track From her new album. The deluxe edition is only available in physical form. It took 4 years for the deluxe of version of me to go streaming and digital global. Justin Timberlake's Can't stop the feeling is basically A spice girls track in terms of the production and the overall message that's positive Melanie C would take Let love lead the way from the Spice girls catalog and basically call it a Melanie C track.
Vote: Which is Justin Timberlake's best single?
Iconic 4 U posted a topic in Music Discussions
JT has released quite a few memorable singles but which do you think is his best? My vote is going to Like I Love You, the bridge is so euphoric And the music video is a serve
Justin Bieber, Justin Timberlake & Justine Skye in the studio together with Timbaland
Are you here for this?
justine skye
Britney Spears Tribute Video
ryro_85 posted a topic in Britney Spears
Hey Exhale! I just came across this video I made 12 years ago (Circa 2008)...I know it's not the best quality video, but I was super proud of myself at the time considering the apps I had at that time to work with. I have always loved the song "LoveStoned / I Think She Knows" and made this video based on Britney's life at the time with the Paparazzi craziness she was constantly dealing with..."Those flashing lights come from everywhere. The way they hit her, I have to stop and stare. She's got me LoveStoned. Man, I swear she's bad and she knows. I think that she knows." I just love how I was able to cut and edit clips from Britney's music videos to marry them perfectly with the lyrics of this song, and especially the dance breaks and flashing lights that seemed to fit flawlessly with the song. I spent so much time working on this, and hope you can enjoy my creation even though the video quality isn't up to par with the amazing videos other fans have posted here in Exhale. Please take a look and let me know what you think! I hope you can find some enjoyment from this video. #FreeBritney
LoveStoned
I Think That She Knows
Britney Tribute
YouTube Video Compilation
Leave Britney Alone
Paparazzi Harassment
Gimme Gimme More
Britney LoveStoned
What is your favourite Justin Timberlake album?
Hey Exhale, idk how many JT fans there are on here But I was wondering what is your favourite album by him? Or songs, if you don't have a fave album
Timbaland teases a sequel to Justin Timberlake's "FutureSëx/LoveSounds"
Fans of Justin Timberlake's innovative sophomore solo album, FutureSëx/Love Sounds, could be in for a treat. Producer Timbaland, who produced that album, posted a photo of him and JT together and captioned it: "FUTURE **** LOVESOUNDS pt ✌🏾🤐🤐" It's unclear if it's actually something in the works, or a project he hopes to have happen. My initial thoughts... big yes! But at the same time, I fear the GP will think it's lame. Exhale, are you here for a FSLS 2?
Justin Timberlake and Jessica Biel welcome a new baby boy!
Dirk posted a topic in Entertainment
They had a secret pregnancy, but is it a secret if no-one was asking anyway? And the real question is... will this child be vaccinated?
I want Justin Timberlake to murder my *******
GODNEY IS A QUEEN posted a topic in Britney Spears
ladies...this is what you call a MAN whewww and lordT...that 3:11 mark his arms not even gonna lie, I blame prime JT for so making me the ***** **** I am today And yes this is britney related because he was hottest when he was with ha homegirl was so lucky...like imagine being ****ed down by that
Demo - Battle of the sexxxes / Duet song allegedly meant to be for Justin & Britney
Melina posted a topic in Britney Spears
Hello guys, I hope this has not been posted yet. I was on Instagram and I saw a post with a link towards a demo. It's called "Battle of the sexxxes". You can hear Justin and Timbaland on the demo. This song was apparently meant to be the duet song of Justin & Britney back in 2008. Here is a direct link to the Instagram post : Source : Instagram post That's not my instagram, all credits go to this Instagram account : @britneyjustinn Justin is singing Britney's part... Lyrics : (Timbaland) Don't hurt her Justin! That was the past Put it all behind you, homeboy (Justin ) Let me talk to you Let me, let me talk to you I just, I just wanna talk to you Let me, let me, let me talk to you I just wanna talk to you Is it alright if I talk to you? Just listen Baby girl, don't be afraid I know you remember me, I'm your boy from back in the day I heard you singin' you're back, and your mind seems right And I can't even lie to you, your body still drives me wild So don't be afraid to get nasty Don't be afraid to get wild Just point to your spot and ask me I'll make it worth your while Don't be afraid to get nasty Don't be afraid to get wild Just point to your spot and ask me I'll make it worth your while Let's explore explicit Let's express reflexes Exploit your pleasure It's the battle of the sexxxes Let's explore explicit Let's express reflexes Exploit your pleasure It's the battle of the sexxxes Talk to me Make it talk to me Talk to me Make it talk to me Talk to me Make it talk to me Talk to me Make (Meant to be sung by Britney ) Baby boy, I know you want my love Already gave you a hit so you know that it's like a drug You remember my flavor, you still know my taste I still got it, baby, it'd be a shame if it went to waste So don't be afraid to get nasty Don't be afraid to get wild Just point to your spot and ask me I'll make it worth your while Don't be afraid to get nasty Don't be afraid to get wild Just point to your spot and ask me I'll make it worth your while Let's explore explicit Let's express reflexes Exploit your pleasure It's the battle of the sexxxes Let's explore explicit Let's express reflexes Exploit your pleasure It's the battle of the sexxxes Talk to me Make it talk to me Talk to me Make it talk to me Talk to me Make it talk to me Talk to me For those who were not here in 2008, this was a huge deal. Britney was supposed to collab with them, and at the last minute, she denied them. I remember they got mad. Exhale was a mess. The good ol' days
britney spears Britney posts video dancing to Justin Timberlake's "Filthy," JT responds
Jordan Miller posted a topic in Britney Spears
Britney broke the Internet when she posted a video of her dancing to Justin Timberlake's "Filthy." The self-awareness in the caption I- Tell me this isn't her posting
Britney, JT and Christina reunion?
YvonneSpears posted a topic in Britney Spears
I love this montage! Would you be excited for a reunion with all three of them?
On a separate note, Justin Timberlake talked about Britney a few weeks ago on Lance Bass' podcast...
jennyj posted a topic in Britney Spears
Everyone's still going crazy about Britney's IG post today. But just a few weeks ago, on Lance Bass' podcast The Daily Popcast during his "The *Nterviews" segment, Justin was a guest where they talked about the iconic denim duo outfits they wore. Justin said: Article: https://www.today.com/style/justin-timberlake-looks-back-his-matching-denim-moment-britney-spears-t176671 Podcast: https://podcasts.google.com/?feed=aHR0cHM6Ly9mZWVkcy5zaW1wbGVjYXN0LmNvbS9IcHJPNGxRZQ&episode=NWFjMTE2Y2ItM2Q4Ni00YjljLThjNjMtNWFmODAyNDZkOTBj&hl=en&ved=2ahUKEwi-uLaUiqroAhVUCjQIHQMIBD8QieUEegQIAhAE&ep=6
FEATURED! Justin Timberlake recently liked a post about Britney on IG
ILikeChillinWithYou posted a topic in Britney Spears
Alexa, play Obsessed by Mariah Carey
justin timberlake Marsha Ambrosius reveals she's singing the ad-libs at the end of Justin Timberlake's "Cry Me A River"
Singer Marsha Ambrosius says she's the one actually singing the ad-libs in the last 20 seconds of Justin Timberlake's "Cry Me A River," NOT JT. Ambrosius is known for being one-half of the English R&B duo known as Floetry (along with Natalie Stewart). She's got credits on "Cry Me A River," as well as on Michael Jackson's "Butterflies" and has made guest appearances on projects for Solange, Busta Rhymes, Q-Tip, Nas, Jamie Foxx, Queen Latifah, Talib Kweli, Dr. Dre, Kendrick Lamar, Common, Snoop Dogg and Nipsey Hussle and more. I'm legit annoyed / bummed cause that's one of the best parts of the song and now I'll forever know. Ignorance is bliss.
justin timberlake Justin Timberlake accused of cultural appropriation for calling SZA "sis"
Justin Timberlake is in ho water for talking over SZA and calling her "sis" several times during their interview on Ellen. I am a JT stan, so maybe I'm a little biased / blind. Also, the white privilege in my life is real so perhaps I'm going to sound ignorant... but for the sake of transparency here in Exhale... are people are blowing this out of proportion? Did he intentionally talk over her? I'm not convinced. Did he call her "sis" because he's trying to represent something he's not? Also not convinced. Thoughts?
justin timberlake Justin Timberlake & Anderson Paak - "Don't Slack" (music video)
Justin Timberlake and Anderson Paak teamed up for a new song titled "Don't Slack," the latest release from the Trolls soundtrack. The song is a little messy, but I'm living for it. It caught my attention because it doesn't sound like anything out there right now. Drag me I don't care, but this slaps. Now get to work *****... don't slack. The video is out:
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Kiakola (persiska: کیاکلا), eller Kia Kola (کیا کلا), tidigare Jadid ol-Eslam (جدید الاسلام), är en ort i Iran. Den ligger i provinsen Mazandaran, i den norra delen av landet.
Kiakola är administrativt centrum för delprovinsen (shahrestan) Simorgh.
Källor
Orter i Mazandaran
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{"url":"https:\/\/blender.stackexchange.com\/questions\/135042\/alternatives-for-operators-report-function?noredirect=1","text":"# alternatives for operator's report() function?\n\nit seems that the report() function of Operator type lets all other routines finish before it actually reports. this makes it useless in many instances: for example, when reporting progress on baking (I have abandoned the idea of a progress bar and merely want to show a message: 'baking ith map out of n maps'). However, this message would always come at the end. Are there ways to supress this behavior and output a message instantaneously? In Excel VBA, there's DoEvents() built-in function. It lets all processes freeze until the ongoing processes finish\n\n\u2022 \u2013\u00a0batFINGER Mar 23 '19 at 4:40\n\u2022 @batFINGER thanks looks useful. I'll check it out and update \u2013\u00a0SerhiiPoklonskyi Mar 23 '19 at 16:34\n\nprobably, such thing as reporting to user interface must be the easiest thing to do in any API but... this is life. anyways, here's what I have found for myself.\n\nIdea in short: so, when blender script runs, Blender does not redraw screen any longer. So I start drawing text into a particular window with blf. Because Blender is not redrawn while the script runs, you actually don't see it. So I force redrawing. And then you can see the text. Don't know how robust that solution is but anyways.\n\nThis is the function which starts drawing:\n\ndef UI_report_start(message, space_type):\n\n\"\"\"\ndraws specified message, in the specified space_type. space_type is a value in:\n\n['Image Editor',\n'Node Editor',\n'UV Editor',\n'3D View']\n\nreturns the draw handler and the space type\nfor later deletion\n\"\"\"\n\n# the drawer function\ntry:\nblf\nexcept:\nimport blf\n\ndef draw_message(self, context):\nblf.position(0, 15, 30, 0)\nblf.size(0, 20, 72)\nblf.draw(0, message)\n\nif space_type == 'Image Editor':\nhandler = bpy.types.SpaceImageEditor.draw_handler_add(draw_message, (None, None), 'WINDOW', 'POST_PIXEL')\nreturn handler, space_type\nelif space_type == 'Node Editor':\nhandler = bpy.types.SpaceNodeEditor.draw_handler_add(draw_message, (None, None), 'WINDOW', 'POST_PIXEL')\nreturn handler, space_type\nelif space_type == '3D View':\nhandler = bpy.types.SpaceView3D.draw_handler_add(draw_message, (None, None), 'WINDOW', 'POST_PIXEL')\nreturn handler, space_type\nelif space_type == 'UV Editor':\nhandler = bpy.types.SpaceUVEditor.draw_handler_add(draw_message, (None, None), 'WINDOW', 'POST_PIXEL')\nreturn handler, space_type\nelse:\nraise Exception('UI_report_start', 'Incorrect Window type specified')\n\n\nthis is the function that stops drawing:\n\ndef UI_report_stop(reporter, space_type):\n\"\"\"\nstops drawing the report\nthe drawing handler and the space type must be input\n\"\"\"\n\nif space_type == 'Image Editor':\nbpy.types.SpaceImageEditor.draw_handler_remove(reporter, 'WINDOW')\nelif space_type == 'Node Editor':\nbpy.types.SpaceNodeEditor.draw_handler_remove(reporter, 'WINDOW')\nelif space_type == '3D View':\nbpy.types.SpaceView3D.draw_handler_remove(reporter, 'WINDOW')\nelif space_type == 'UV Editor':\nbpy.types.SpaceUVEditor.draw_handler_remove(reporter, 'WINDOW')\nelse:\nraise Exception('UI_report_stop', 'Incorrect Window type specified')\n\n\nthis is a hypothetical function taking much time to complete and reporting is embedded into it:\n\ndef execute(self, context):\nhandler, space = UI_report_start(\"does it have to be THAT difficult?!?!#\\$\", '3D View') # <--- this starts drawing blf but you don't see it yet!\nbpy.ops.wm.redraw_timer(type='DRAW_WIN_SWAP', iterations=1) # <--- this is redrawing the screen and you see the message!\n# from now on redrawing stops but the message is still 'frozen' in place and you can see it!\nfor i in range(0, len(bpy.data.images['heavy'].pixels)):\nbpy.data.images['heavy'].pixels[i] = random()\nUI_report_stop(handler, space) # <-- this removes the message!\nreturn {'FINISHED'}\n\n\nas you can see, it does not draw the progress bar, only a message but I can live with it.\n\nYou can also update the screen more frequently but redrawing and drawing blf takes resources, so do it only once in a while","date":"2020-04-03 22:29:06","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\": 0.3384644389152527, \"perplexity\": 14022.891110559493}, \"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-2020-16\/segments\/1585370518767.60\/warc\/CC-MAIN-20200403220847-20200404010847-00199.warc.gz\"}"}
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\section{Introduction}
Classical Cepheids (hereafter Cepheids) are among the most important tools of stellar astrophysics. Thanks to the period-luminosity relation, tight especially in the near-infrared bands \citep[see eg.,][]{BreuvalPL,RipepiPL,OwensPL}, they are among the most important distance indicators, essential for measuring the Universe and determining its expansion rate \citep[eg.][]{Freedman2012,Riess2019}. Classical Cepheids pulsate either in radial fundamental (F) mode or in the radial first overtone (1O) mode or in the radial second overtone (2O) mode. Double-mode pulsators, involving different pulsation modes (F+1O, 1O+2O, 1O+3O) or even triple-mode pulsators are also known and proved to be useful tools to test the stellar pulsation and evolution theories, or to constrain physical parameters of these important variables \citep[eg.][]{md05,Beaulieu2006,Pilecki2018,DeSomma2020}.
While single-mode large-amplitude radial pulsation of Cepheids is well understood, the mechanisms behind double-mode and triple-mode radial pulsation remain unclear \citep[see][for opposing views]{kollath2002,sm08b}. In the recent years, classical Cepheids revealed even more complex nature through the presence of plethora of additional low-amplitude periodicities that cannot be interpreted as due to radial pulsation. These low-amplitude periodicities appear to be common in first overtone pulsators and scarce in the fundamental mode Cepheids.
The discoveries of low-amplitude periodicities in classical Cepheids were possible thanks to long-term continuous monitoring of these stars by the Optical Gravitational Lensing Experiment \citep[OGLE;][]{o3,o4}. The OGLE Collection of variable stars (OGLE-CVS) includes virtually all classical Cepheids in the Magellanic Clouds \citep{o4_clouds_henrietta} and for majority of these stars high quality, homogeneous data, covering several observing seasons, is available, allowing us to detect periodic signals at the milli-magnitude level.
\cite{mkm04} and \cite{mk09} analysed OGLE LMC data and reported that in 8\,per cent of 1O Cepheids additional low-amplitude signals with frequencies close to the radial mode frequency are detected and concluded that these signals are most likely due to excitation of non-radial modes. Additional signals are coherent. Sometimes two peaks are detected in the frequency spectrum on the same side of the radial mode frequency.
The most interesting new form of double-periodic pulsation are classical Cepheids with dominant 1O variability and with additional low-amplitude variability of shorter period, with period ratios $\px/\pov$ in the $(0.60,\,0.65)$ range (conversely, with frequencies in the $(1.54,\,1.67)\fov$ range). The group was first identified through analysis of LMC data by \cite{mk08,mk09} and \cite{o3_lmc_cep}, with two sequences of slightly different period ratios reported in the latter study. In the SMC, \cite{o3_smc_cep} reported 138 double-periodic stars forming three distinct sequences in the Petersen diagram. This sample was analysed in detail by \cite{ss16}, who found additional signals in 35\,per cent of these stars, centered at sub-harmonic frequencies, $1/2\fx$. Interestingly, double-periodic pulsation of the same characteristics exists among first overtone RR~Lyr (RRc) stars, see eg., \cite{ns19} and was recently detected in an anomalous Cepheid with {\it TESS} \citep{PlachyTESS}. A common explanation was proposed by \cite{wd16} who associated the signals at $\fx$ with harmonics of non-radial modes of moderate degrees. The three sequences in the Petersen diagram for Cepheids were associated with excitation of non-radial modes of $\ell=7$ (top sequence), $\ell=8$ (middle sequence) and $\ell=9$ (bottom sequence). In this model, direct detection of non-radial modes, at $1/2\fx$ is less likely, as expected amplitude is low due to geometric cancellation. The expected amplitude is the largest for even $\ell$ and indeed majority of the detections at $1/2\fx$ correspond to the middle sequence in the Petersen diagram \citep{ss16}. We also note that beyond Magellanic Clouds, this form of pulsation was also detected in OGLE Galactic disk and Galactic bulge data \citep{Pietruk2013,Rathour2021} and recently in {\it TESS} data \citep{PlachyTESS}.
As the just described form of double-periodic pulsation is present both for RRc stars and 1O Cepheids, one may speculate that other forms of double-periodic pulsation discovered for RRc stars may also be present among 1O Cepheids. \cite{nsd15} reported the discovery of a puzzling group of double-periodic RRc stars, with additional periodicity of longer period; in the Petersen diagram, these double-periodic stars cluster tightly at $\pov/\px=0.686$. Additional periodicity is always coherent in these stars. The nature of this group remains puzzling \citep[see][for discussion]{wd16}. Double-periodic 1O Cepheids that share similar characteristic were reported by \cite{sa18b} who analysed the whole OGLE LMC and SMC sample in search of additional periodicities.
Other form of variability detected in classical Cepheids is periodic modulation of pulsation. \cite{mkm04} discovered long-period modulation of double-mode 1O+2O Cepheids. As analysed by \cite{mk09}, both modes are modulated with the same period and their amplitudes are anti-correlated. Recently modulation was detected in a double-mode, F+1O Cepheid \citep{Rathour2021}. Modulation was also detected in single-mode Cepheids: in the 2O Cepheid V473~Lyr \citep[see][]{MolnarSzabados}, in a few 1O Cepheids \citep{o4_clouds_multimode, Kotysz, Rathour2021} and in several single-mode F-mode Cepheids in the LMC \citep{s17} and one Cepheid in the Galactic disk \citep{Rathour2021}. Interestingly, for single-mode F-mode Cepheids, except these modulations, additional periodicities are hardly present in the frequency spectra, in sharp contrast to 1O pulsators \citep{s17,sa18b}.
In this paper, we conduct in-depth frequency analysis of 1O Cepheids from the OGLE-IV LMC and SMC sample using classical consecutive prewhitening technique, supplemented with the time-dependent prewhitening for non-stationary variations. For each star with additional variability beyond radial mode, manual analysis is conducted to extract all low-amplitude periodicities present in the data. Similar analysis was done by \cite{sa18b}, who used kernel-regression method for prewhitening \citep{sa18a} but searched only for one single additional periodicity.
Some preliminary results of this study were presented in \cite{MSniegProc} and \cite{OZProc}.
\section{Data analysis}
We analyse OGLE-IV \citep{o4} $I$-band data for first overtone classical Cepheids in the SMC and LMC included in the OGLE Collection of Variable Stars \citep[OGLE-CVS,][]{o4_clouds}. The sample consists of 1784 stars in the SMC and 1766 stars in the LMC\footnote{The collection is regularly updated with new stars. Current numbers may be slightly higher.}. We use a standard consecutive prewhitening technique \citep[see eg.,][]{mk09} to search for additional periodicities beyond the dominant radial first overtone mode. Significant periodicities are identified with the help of the discrete Fourier transform (DFT) and are included in a sine series of the following form
\begin{equation}
m(t)=A_0+\sum_{k} A_k\sin(2\pi \nu_k t+\phi_k)\,,\label{eq:sseries}
\end{equation}
fitted to the photometric data, $m(t)$. Above, $A_0$ is mean magnitude, index $k$ enumerates the sine terms for which $A_k$ and $\phi_k$ are amplitude and phase corresponding to variability with frequency $\nu_k$. The dominant terms correspond to Fourier series describing radial first overtone variability ($\fov$ and its harmonics). Residuals from the fit are then inspected with DFT to identify further periodicities of lower amplitude. In principle, signals with ${\mathrm{S/N}}>4$ were considered significant and included in the solution. Weaker signals, with ${\mathrm{S/N}}>3.5$, were also included in the solution, provided these were linear combinations of the already identified frequencies or were located at the expected frequency (eg. close to the sub-harmonic frequencies). 4-$\sigma$ outliers were successively removed from the data. Slow trends were modelled either using low-order polynomials or spline functions. In some cases time-dependent prewhitening (see below) was employed to remove slow trends. Two frequencies were considered resolved, provided their difference exceeds $2/\Delta T$, where $\Delta T$ is data span.
First, we applied an automatic procedure to identify stars which only show 1O pulsation and no signature of additional periodicity. To this aim, Fourier series was fitted to the data to the order satisfying $A_k/\sigma(A_k)>4$. Possible slow trends were modelled with second order polynomial function and severe (6-$\sigma$) outliers were removed from the data. DFT of the residual data was searched for significant signals in the $(0,\,6\fov)$ range. A star without significant signals was not analysed further. All other stars were subject to detailed manual analysis.
When new significant signal is detected in the data, we first check whether it can be represented as a linear combination of the previously detected frequencies. If so, no new independent frequency is introduced, just respective combination term is included in the solution. For low-amplitude double-periodic pulsation the most common combination is $\fx+\fov$. A special case is the periodic modulation of pulsation, which, in general, manifests as equidistant multiplets centred on radial mode frequencies, and on the harmonics, as well as through signal at the modulation frequency (and its harmonics), see eg., \cite{bsp11}. Separation within multiplet components corresponds to modulation frequency, $\fmodul=1/\pmodul$. In the ground-based data, typically triplets or doublets are observed. In the latter case, the modulation peaks are detected on one side of the radial mode frequency (and of its harmonics). To claim the modulation we require at least two modulation side peaks corresponding to the same modulation frequency are detected in the frequency spectrum (see Sect.~\ref{ssec:modulation} for a more thorough discussion).
After prewhitening the first overtone and its harmonics, quite often the dominant signal in the frequency spectrum is unresolved with the just prewhitened 1O signal. It indicates that 1O is non-stationary; its amplitude and/or phase vary on a long timescale (of order of data length or higher). Such signals are often strong, dominate the frequency spectrum and increase the noise level, thereby hindering the detection of additional low-amplitude periodicities. To get rid of such non-stationary variation we employ time-dependent prewhitening as applied to {\it Kepler} space data by \cite{pam15} (see their Appendix A) and later applied to OGLE data eg. in \cite{ss16} or \cite{ns19}. In a nutshell, the data are divided into groups of specified length, $\delta t$, typically corresponding to single observing season or its half, and sine series is fitted independently to each group, keeping the frequencies fixed to that determined from all data fit. Only zero point, amplitudes, and phases of sine terms are adjusted. Such determined fits are then used to compute the residuals on a group-to-group basis. As a result amplitude or phase changes on the time scale longer than $\delta t$ are removed from the data, while possible faster changes, that can be well resolved, are still preserved. The technique can also be used to remove slow trends in the data when only group-dependent zero-points are subtracted from the data. After the time-dependent pre-whitening, the non-stationary signal(s) are removed, which reduces their amplitudes and the overall noise level and new low-amplitude signals are often detected.
Non-stationary variability may also manifest through more complex structures in the frequency spectrum, power excesses or wide bands of power. For Cepheids with period ratios $\px/\pov$ in the $(0.60,\,0.65)$ range, these take various forms, eg., bell-shaped power excess, U-shaped power excess, nearly rectangular bands of power excess, as already noted by \cite{ss16} for 1O Cepheids or by \cite{ns19} for RRc stars showing similar variability. In these stars a power excess centred at sub-harmonic frequency, $1/2\fx$, is quite often detected, as illustrated in Fig.~\ref{fig:fspexample} for a few stars ($1/2\fx$ is marked with an arrow). To characterise such signals, we follow the simplest approach adopted also in the just quoted studies: we include a single sine term in the solution with frequency corresponding to maximum amplitude within the power excess (filled diamond in Fig.~\ref{fig:fspexample}). Remnant and quite often resolved power is then present in the frequency spectrum after prewhitening. One would have to include several sine terms of close frequencies to remove such power excess.
With OGLE data we face a common issue of ground-based data analysis: a possible confusion due to daily aliases. Aliases arise due to regularities in data sampling and their expected structure can be revealed by calculating the spectral window. OGLE data for Magellanic Clouds are seasonal, thus both daily and 1-year aliases are present and typically strong. For signals with ${\mathrm{S/N}}\lesssim 6$ a few daily aliases may be of comparable height: sometimes a `comb' of four to six aliases with minute differences of the ${\mathrm{S/N}}$ are present. Which alias corresponds to intrinsic stellar variability? The problem was faced eg., by \cite{om09} who studied RRc stars in $\omega$~Cen. For ambiguous cases, they provided two, or even three solutions with different interpretations (see their tab.~1). Two solutions were common: either a new double-periodic form of pulsation with period ratios $\px/\pov\approx 0.61$ or double-mode radial 1O+2O pulsation. With analysis of more data from various stellar systems, it became clear that the former explanation is correct \citep[see eg.][]{2009AcA....59....1S,2012MNRAS.424.2528S,2012A&A...540A..68C,2014A&A...570A.100S,pam15}. In this study, when selecting the intrinsic signal, whenever applicable, we make use of the prior knowledge on various forms of double-periodic pulsation among 1O Cepheids. This is illustrated in Fig.~\ref{fig:fspexample}, with the frequency spectra for \idlmc{0421} and \idlmc{0549}\footnote{In the following full OGLE IDs, eg., OGLE-LMC-CEP-0421, are abbreviated as \idlmc{0421}}.
For \idlmc{0421}, after prewhitening with first overtone frequency and its harmonic, the highest ${\mathrm{S/N}}=7.7$ signal is detected at $\nu=0.380\cd$ (marked with filled diamond). After prewhitening with this signal two daily aliases come to attention. The higher ${\mathrm{S/N}}=5.6$ signal is located at low-frequencies (open circle at $\nu=0.276\cd$) and the lower ${\mathrm{S/N}}=4.9$ signal at higher frequency (filled circle at $\nu=0.727\cd$). Since the latter signal corresponds to $P/\pov=0.621$ and we already detected a power excess centred at its sub-harmonic (with the highest signal at $\nu=0.380\cd$), we consider this alias a true periodicity. We include it in our solution and consider the star as a classical example of double-periodic star with period ratio $\px/\pov$ in the $(0.60,\,0.65)$ range. It nicely fits the middle sequence in the Petersen diagram, and has a signal centred at sub-harmonic frequency present.
For \idlmc{0549}, after prewhitening with first overtone frequency and its harmonics, three daily aliases come to attention. The highest ${\mathrm{S/N}}=7.7$ signal is located at $\nu=1.462\cd$ (not visible in Fig.~\ref{fig:fspexample}), then there is a ${\mathrm{S/N}}=7.4$ signal at $\nu=0.541\cd$ (open circle) and a ${\mathrm{S/N}}=7.0$ signal at $\nu=0.459\cd$ (filled circle). Since for the last signal we find $P/\pov=0.616$ and we can clearly see a signal centred at its sub-harmonic (arrow) we consider it a true periodicity, even thou it is the third highest signal in a 1-day alias family.
The above outlined procedure is subjective and in some cases may lead to identification of false periodicities that are, in fact, aliases of the correct ones. This is mitigated by conservative choices and selecting the lower alias only when ${\mathrm{S/N}}$ differences are small and there are good arguments behind (like given in the above examples). Error is still possible, but rather for a few Cepheids out of hundreds with additional signals considered. Consequently, conclusions about statistical properties of the studied groups of double-periodic pulsation are robust.
\begin{figure}
\noindent\includegraphics[width=\columnwidth]{OGLE-LMC-CEP-1005_fsp.eps}\\
\noindent\includegraphics[width=\columnwidth]{OGLE-LMC-CEP-1305_fsp.eps}\\
\noindent\includegraphics[width=\columnwidth]{OGLE-LMC-CEP-0421_fsp.eps}\\
\noindent\includegraphics[width=\columnwidth]{OGLE-LMC-CEP-0549_fsp.eps}
\caption{Exemplary frequency spectra for analysed stars, after prewhitening with the first overtone frequency and its harmonics (dashed lines). Filled circles and diamonds mark the highest signals detected in the $\px/\pov=(0.60,\,0.65)$ range and corresponding sub-harmonic range, respectively. Open symbols mark locations of the corresponding daily aliases. Arrows indicate the exact location of the sub-harmonic of the highest signal detected in the $\px/\pov=(0.60,\,0.65)$ range.}
\label{fig:fspexample}
\end{figure}
\section{Results}
\label{sec:results}
\subsection{Overview}
OGLE-IV photometric data on 1766 1O Cepheids from the LMC and 1784 1O Cepheids from the SMC were analysed. Of these, in 45\,per cent of stars in the SMC and in 25\,per cent of stars in the LMC, no significant additional signal beyond radial mode and its harmonics was detected in automatic procedure outlined in the previous section.
Additional signals were detected in over thousand 1O Cepheids in both Clouds. Corresponding period ratios (shorter to longer) are plotted in the Petersen diagrams in Fig.~\ref{fig:petall}, separately for SMC (left) and LMC (right). Triangles and circles correspond, respectively, to additional periodicities with period longer than 1O period and shorter than 1O period. Thus, either $\pov/\px$ vs. $\px$ (triangles), or $\px/\pov$ vs. $\pov$ (circles) is plotted.
\begin{figure*}
\includegraphics[width=\columnwidth]{pap_petallsmc.eps}
\includegraphics[width=\columnwidth]{pap_petalllmc.eps}
\caption{The Petersen diagrams (ratio of shorter to longer period, $P_{\rm S}/P_{\rm L}$, vs. the longer period, $P_{\rm L}$) for the SMC (left) and the LMC (right) with period ratios for all additional periodicities detected in the analysis, except those that can be interpreted as due to periodic modulation. If $\px>\pov$, then $\pov/\px$ vs. $\px$ is plotted with triangles, while for $\px<\pov$, $\px/\pov$ vs. $\pov$ is plotted with circles. Filled, dark gray symbols correspond to coherent additional variability, while open black symbols are used when unresolved power or power excess is detected in the frequency spectrum after prewhitening the data with additional periodicity. Data on radial double-mode and triple-mode Cepheids from OGLE-CVS are plotted with small colored squares (different colors for different stellar systems: LMC, SMC, Galactic bulge, BLG, and Galactic disk, GD, to provide context.}
\label{fig:petall}
\end{figure*}
While the number of 1O Cepheids in both Magellanic Clouds is similar, additional periodicities are much more frequent in the LMC. We discuss it in detail in Sect.~\ref{ssec:unclassified}. Here we just note that since SMC is further away than LMC, its Cepheids are dimmer, and the detection limit is higher. It seems, however, that higher incidence rate of stars with additional periodicities is an intrinsic property of the LMC sample, at least for some range of first overtone periods.
Based on the location in the Petersen diagram, we first classify the stars into a few already known classes of double-periodic Cepheids. These include three classes of double-mode radial pulsation, discussed in Sect.~\ref{ssec:dm}, double-periodic pulsation with period ratios, $\px/\pov$, in the $(0.60,\,0.65)$ range (Sect.~\ref{ssec:61}) and double-periodic pulsation with period ratios $\pov/\px$ clustered around $0.684$ (Sect.~\ref{ssec:686}). Then we discuss the remaining double-periodic variables in Sect.~\ref{ssec:unclassified}. Stars in which modulation side peaks were detected are discussed in Sect.~\ref{ssec:modulation}. In several stars multiple forms of double-periodic pulsation are present simultaneously.
\subsection{Double-Mode Radial Pulsation}\label{ssec:dm}
To identify the candidates for double-mode radial pulsation, we compare the location of double-periodic stars in the Petersen diagram, with the location of known double-mode radial pulsators from the OGLE-CVS in the LMC, SMC, Galactic bulge and Galactic disc \citep[][]{o4_clouds,o4_gal2}. In Fig.~\ref{fig:petall}, these stars are plotted with small filled squares, colour-coded according to the stellar system. 1O+2O stars from the OGLE-CVS form a rather tight progression with $\poov/\pov$ centred at $\approx 0.8$. The more significant scatter may be noted at the short and long-period ends of this progression. Metallicity dependence is weak, as may be inferred from comparison of data for the four stellar systems. Progression for F+1O stars is slanted, with $\pov/\pf$ decreasing from $\approx0.76$ at the short-period end, to $\approx0.69$ at the long-period end. As in the case of 1O+2O stars, higher dispersion of period ratios is visible at short and long-period ends of the progression. Metallicity dependence is clear as best visible comparing the progressions for the most numerous LMC and SMC samples. 1O+3O radial double-mode pulsation is scarce; interestingly, this combination of radial modes is more frequent in the triple-mode pulsators than in the double-mode pulsators. Data for both groups are used in Fig.~\ref{fig:petall}. The 1O+3O stars are clustered in two groups with a rather large spread of $\pooov/\pov$, in between $0.66$ and $0.68$.
In Fig.~\ref{fig:petrad}, we show new candidates for double-mode radial pulsators, discussed in more detail in the following sections.
\begin{figure}
\includegraphics[width=\columnwidth]{pap_petrad.eps}
\caption{The Petersen diagram with new candidates for double-mode radial pulsation marked with large open symbols. Small filled squares represent data on the double-mode and triple-mode pulsators from the OGLE-CVS in different stellar systems (indicated with different colors as described inside the plot).}
\label{fig:petrad}
\end{figure}
\subsubsection{Candidates for F+1O double-mode radial pulsation}\label{sssec:f1o}
In Fig.~\ref{fig:petall}, several new candidates for double-mode radial F+1O pulsation may be identified. These are stars for which $\px>\pov$, that fall along the progression of F+1O double-mode radial pulsators from the OGLE-CVS. 38 candidates were identified in the LMC and 7 candidates were identified in the SMC. Data for these new candidates are given in Tab.~\ref{tab:f1o}. In the consecutive columns we report: star's ID, first overtone period, fundamental mode period, period ratio, $\pov/\pf$, amplitudes of the first overtone, $\aov$, and of the fundamental, $\af$, modes, signal-to-noise for the detection of fundamental mode and remarks including information on the detected combination frequencies. In Fig.~\ref{fig:petrad}, we show the location of these candidates in the Petersen diagram for double-mode radial pulsation (open circles). We note that for most of our candidates, period of the fundamental mode is longer than is typical for double-mode F+1O pulsators in the OGLE-CVS. While majority of candidates in the LMC/SMC fall within the respective progression formed by the F+1O OGLE-CVS stars, or along its extension, for several stars period ratios are larger, or lower than typical in a given system. This implies that metallicity of these stars may be lower/higher than typical for Cepheids in a given system.
Amplitude ratios are usually very low; the median ratio, $\af/\aov$, is 2.5\, per cent. The most extreme case in which amplitude ratio is 44\, per cent (\idlmc{3362}) is discussed below.
\begin{table*}
\caption{Candidates for new double-mode F+1O radial pulsators. Consecutive columns contain star's ID, first overtone period, fundamental mode period, period ratio, $\pov/\pf$, amplitudes of the first overtone, $\aov$, and of the fundamental, $\af$, modes, signal-to-noise ratio for the detection of fundamental mode. Remarks in the last column can be the following: `nsO' - non stationary dominant (1O) variability, `nsX' -- non stationary additional variability, `tdp' -- additional signal firmly detected only after the time-dependent prewhitening, `al' a lower alias was selected, `0.61' -- additional variability with $\px/\pov\in(0.60,\,0.65)$ was detected, `0.68' -- additional variability with $\pov/\px\approx 0.684$ was detected, `ap' -- additional variability that does not fit the `0.61' and `0.68' schemes was detected in the star.}
\label{tab:f1o}
\begin{tabular}{lrrrrrrr}
\hline
Star & $\pov$\,(d) & $\pf$\,(d) & $\pov/\pf$ & $\aov$ (mag) & $\af$ (mag) & ${\mathrm{S/N}}$ & Additional frequencies \& Remarks\\
\hline
\idlmc{0010} & 2.565580(6) & 3.6086(7) & 0.7110 & 0.1268 & 0.0024 & 4.1 & ap\\
\idlmc{0243} & 2.343337(6) & 3.2759(5) & 0.7153 & 0.1189 & 0.0029 & 5.2 & 0.61, nsO, nsX\\
\idlmc{0297} & 4.80681(2) & 6.813(2) & 0.7059 & 0.1071 & 0.0016 & 4.4 & 0.61, nsO, nsX, tdp, al \\
\idlmc{0342} & 2.193763(5) & 3.0511(5) & 0.7190 & 0.0882 & 0.0017 & 4.2 & \\
\idlmc{0583} & 2.147148(4) & 3.0332(3) & 0.7079 & 0.0920 & 0.0023 & 6.5 & $\ff+\fmodul$, $\ff+2\fmodul$ \\
\idlmc{0839} & 2.193277(5) & 2.9919(4) & 0.7331 & 0.1212 & 0.0030 & 5.2 & nsO, nsX, 0.61 \\
\idlmc{0868} & 2.160823(3) & 2.9528(5) & 0.7318 & 0.0960 & 0.0025 & 4.4 & nsO, 0.61, ap \\
\idlmc{1134} & 3.09037(1) & 4.3426(9) & 0.7116 & 0.0819 & 0.0025 & 4.4 & nsO \\
\idlmc{1217} & 3.05071(3) & 4.2350(9) & 0.7204 & 0.0318 & 0.0017 & 4.5 & nsO, nsX \\
\idlmc{1224} & 1.912693(7) & 2.6714(4) & 0.7160 & 0.0722 & 0.0025 & 4.1 & nsO, ap, al\\
\idlmc{1312} & 2.258804(4) & 3.1587(5) & 0.7151 & 0.0783 & 0.0014 & 4.4 & \\
\idlmc{1346} & 2.462782(9) & 3.4021(6) & 0.7239 & 0.1101 & 0.0018 & 4.3 & 0.61, 0.684, ap, nsO, tdp\\
\idlmc{1425} & 2.713019(6) & 3.8192(4) & 0.7104 & 0.1209 & 0.0025 & 7.4 & $\fov+\ff$, nsO, 0.61, ap \\
\idlmc{1643} & 2.597817(5) & 3.6474(5) & 0.7122 & 0.0969 & 0.0019 & 5.6 & 0.61, ap \\
\idlmc{1747} & 2.186281(3) & 3.0866(3) & 0.7083 & 0.1084 & 0.0028 & 7.6 & nsO, ap\\
\idlmc{1918} & 1.404156(2) & 1.9494(2) & 0.7203 & 0.0832 & 0.0018 & 4.2 & \\
\idlmc{2137} & 4.21030(2) & 5.8508(9) & 0.7196 & 0.0785 & 0.0025 & 8.1 & 0.684 \\
\idlmc{2179} & 1.967818(4) & 2.7847(4) & 0.7067 & 0.1098 & 0.0024 & 4.9 & nsO, 0.61, ap\\
\idlmc{2357} & 2.324088(7) & 3.3153(5) & 0.7010 & 0.1097 & 0.0019 & 4.6 & nsO, tdp, 0.61, ap\\
\idlmc{2406} & 2.091337(4) & 2.9431(4) & 0.7106 & 0.0926 & 0.0019 & 4.5 & 0.684, ap\\
\idlmc{2409} & 2.318955(4) & 3.2890(4) & 0.7051 & 0.1159 & 0.0021 & 5.2 & nsO, ap\\
\idlmc{2540} & 2.911084(8) & 4.0548(9) & 0.7179 & 0.0951 & 0.0013 & 4.2 & nsO, tdp, 0.61\\
\idlmc{2554} & 2.581272(4) & 3.6051(4) & 0.7160 & 0.1296 & 0.0028 & 6.3 & 0.61, ap \\
\idlmc{2590} & 1.205806(2) & 1.6569(1) & 0.7278 & 0.0829 & 0.0026 & 4.6 & nsO, tdp\\
\idlmc{2617} & 2.045449(3) & 2.8046(3) & 0.7293 & 0.1095 & 0.0020 & 5.4 & 0.61 \\
\idlmc{2685} & 2.170927(4) & 3.0596(3) & 0.7096 & 0.0965 & 0.0023 & 5.3 & nsO, nsX, 0.61, ap\\
\idlmc{2719} & 2.447373(6) & 3.4178(6) & 0.7161 & 0.0855 & 0.0017 & 4.2 & 0.61, ap\\
\idlmc{2732} & 1.958648(4) & 2.6859(4) & 0.7292 & 0.0996 & 0.0016 & 4.0 & 0.61, ap\\
\idlmc{2822} & 2.214104(6) & 3.1376(3) & 0.7057 & 0.1004 & 0.0044 & 7.1 & nsO, 0.61, ap \\
\idlmc{2845} & 2.312868(4) & 3.2508(5) & 0.7115 & 0.1371 & 0.0023 & 4.0 & ap\\
\idlmc{3271} & 1.855232(3) & 2.6503(3) & 0.7000 & 0.1037 & 0.0040 & 5.8 & \\
\idlmc{3321} & 4.45108(2) & 6.331(2) & 0.7030 & 0.1076 & 0.0030 & 4.9 & 0.61, ap \\
\idlmc{3362} & 0.3069825(4)& 0.406681(2)& 0.7548 & 0.0804 & 0.0354 &12.8 & $\fov+\ff$, $\fov-\ff$ \\
\idlmc{3606} & 1.114184(2) & 1.53331(8) & 0.7267 & 0.0955 & 0.0041 & 5.3 & nsO \\
\idlmc{3669} & 2.237794(8) & 3.0948(4) & 0.7231 & 0.1180 & 0.0045 & 4.8 & \\
\idlmc{4037} & 1.916503(4) & 2.6867(3) & 0.7133 & 0.1005 & 0.0024 & 4.1 & 0.61, ap \\
\idlmc{4370} & 2.200759(7) & 3.0793(5) & 0.7147 & 0.0944 & 0.0027 & 4.5 & nsO, ap \\
\idlmc{4624} & 1.559273(5) & 2.1614(3) & 0.7214 & 0.0745 & 0.0027 & 3.8 & 0.684, al \\
\hline
\idsmc{0213} & 1.809871(4) & 2.5412(2) & 0.7122 & 0.1123 & 0.0047 & 6.3 & \\
\idsmc{1472} & 1.421955(5) & 1.9395(2) & 0.7332 & 0.0978 & 0.0039 & 4.5 & nsO, tdp \\
\idsmc{1475} & 3.26095(5) & 4.680(1) & 0.6968 & 0.090 & 0.0023 & 4.2 & nsO, tdp \\
\idsmc{1961} & 0.812985(1) & 1.10319(6) & 0.7369 & 0.100 & 0.005 & 4.2 & \\
\idsmc{3453} & 2.201674(4) & 3.0451(4) & 0.7230 & 0.1121 & 0.0021 & 4.2 & ap \\
\idsmc{3977} & 1.542793(4) & 2.1282(2) & 0.7249 & 0.1206 & 0.0036 & 4.6 & nsO, tdp, 0.61 \\
\idsmc{4845} & 1.399198(4) & 1.9355(2) & 0.7229 & 0.1014 & 0.0040 & 4.1 & al \\
\idsmc{4890} & 2.136932(5) & 3.0223(3) & 0.7071 & 0.1276 & 0.0046 & 5.5 & \\%coh., poor fit as for SMC, too low
\hline
\end{tabular}
\end{table*}
{\it Remarks on individual stars:}
\idlmc{0583}. The candidate for radial F mode, detected with ${\mathrm{S/N}}=6.5$, appears to be modulated with $\pmodul=90.3(2)$\,d, as two multiplet components are detected at $\ff+\fmodul$ and $\ff+2\fmodul$ (of higher amplitude; $\fmodul=1/\pmodul$). After prewhitening, a weak (${\mathrm{S/N}}=3.2$), but distinct signal is present at $\ff+3\fmodul$. Since all four equidistant signals (including $\ff$) are of $\sim$comparable amplitude, we cannot firmly classify this star, and conclude about possible modulation of specific frequency.
\idlmc{2137}. F-mode is prominent (${\mathrm{S/N}}=8.1$), but even more prominent is additional signal detected at $\pov/\px=0.6856$. There are three more candidates (with weaker detection of the radial F-mode) that show additional signal of the same category. These stars may be important to understand the nature of signals centred at $\pov/\px\approx 0.684$, as discussed in more detail in Sect.~\ref{ssec:686}.
\idlmc{3362}. F-mode is prominent (${\mathrm{S/N}}=12.8$); its amplitude constitutes 44\,per cent of the first overtone amplitude, which is the highest amplitude ratio in our sample. On the other hand, first overtone amplitude is among the lowest in the considered sample of F+1O candidates and both pulsation periods are the shortest in the analysed sample and among the shortest for F+1O Cepheids in the OGLE-CVS. In fact, the short-period end of the F+1O Cepheid sequence overlaps with the long-period end of the F+1O high amplitude delta Scuti stars (HADS); the discussed star fits well both sequences and so is in the transition region between HADS and classical Cepheids.
\subsubsection{Candidates for 1O+2O double-mode radial pulsation}\label{sssec:1o2o}
Inspection of Petersen diagram in Fig.~\ref{fig:petall} reveals several stars with $\px<\pov$, that fall along the progression of 1O+2O double-mode radial pulsators from OGLE-CVS, or along its extension. Based on the location in the Petersen diagram, we classify 17 stars from the LMC and 4 stars from the SMC as new candidates for double-mode radial 1O+2O pulsation. Basic data for these stars are collected in Tab.~\ref{tab:1o2o} and they are highlighted in the Petersen diagram in Fig.~\ref{fig:petrad} with large open squares. In all but a few stars amplitude of the supposed 2O is small and the detections are weak, typically with $4<{\mathrm{S/N}}<5$, and only in a few cases ${\mathrm{S/N}}$ exceeds 5. The median amplitude ratio, $\aoov/\aov$, is 5\,per cent; the highest amounts to 34\,per cent and corresponds to a star discussed below.
{\it Remarks on individual stars:}
The most prominent and interesting case is \idlmc{2575} in which second overtone is firmly detected with ${\mathrm{S/N}}=12.5$. Both frequencies are non-stationary; unresolved or barely resolved signals are detected both at $\fov$ and $\foov$ after prewhitening. In fact, clear, large amplitude modulation is well visible just by inspecting the photometric data, and a star is listed as having the Blazhko modulation in the remarks file of the OGLE-CVS \citep{o4_clouds_multimode}. A procedure of time-dependent prewhitening allows us to detect further significant frequencies, in particular, the linear combination $\fov+\foov$, and to study the variation of 1O and 2O amplitudes, visualised in Fig.~\ref{fig:2575}. The amplitude changes are clearly anti-correlated. We conclude that \idlmc{2575} is another member of the group of double-overtone Cepheids, modulated on long time scales with anti-correlated amplitude changes, studied in detail by \cite{mk09}.
Another interesting star is \idlmc{0071} as additional signal with $\px/\pov\in(0.60,\,0.65)$ is detected (see Sect.~\ref{ssec:61}) in addition to the dominant 1O and candidate 2O. Thus, this star may be a promising target for asteroseismic modelling. Admittedly the detections of additional signals in this star are weak (${\mathrm{S/N}}<5$).
\begin{table*}
\caption{Candidates for new double-mode 1O+2O radial pulsators. Structure of the table and remarks in the last column are the same as in Tab.~\ref{tab:f1o}.}
\label{tab:1o2o}
\begin{tabular}{lrrrrrrr}
\hline
Star & $\pov$\,(d) & $\poov$\,(d) & $\poov/\pov$ & $\aov$ (mag) & $\aoov$ (mag) & ${\mathrm{S/N}}$ & Additional frequencies \& Remarks\\
\hline
\idlmc{0071} & 1.385893(2) & 1.10130(6) & 0.7947 & 0.1489 & 0.0029 & 4.2 & 0.61\\
\idlmc{0232} & 0.301231(1) & 0.242359(4)& 0.8046 & 0.0358 & 0.0093 & 4.3 & \\
\idlmc{0302} & 2.343800(7) & 1.8375(2) & 0.7840 & 0.0710 & 0.0019 & 4.1 & \\
\idlmc{1334} & 2.31368(1) & 1.83350(9) & 0.7925 & 0.0440 & 0.0030 & 7.2 & ap\\
\idlmc{1345} & 2.298207(7) & 1.8078(2) & 0.7866 & 0.0608 & 0.0016 & 4.5 & \\
\idlmc{1435} & 0.2495233(3)& 0.200612(2)& 0.8040 & 0.0909 & 0.0095 & 4.3 & \\
\idlmc{1662} & 0.3151630(4)& 0.253116(3)& 0.8031 & 0.0640 & 0.0059 & 4.6 & \\
\idlmc{1710} & 0.888527(3) & 0.71657(2) & 0.8065 & 0.0759 & 0.0076 & 5.8 & nsO, nsX\\
\idlmc{2575} & 1.37710(1) & 1.10146(2) & 0.7998 & 0.0677 & 0.0231 & 12.5 & $\fov+\foov$, nsO, nsX, Blazhko\\
\idlmc{2872} & 2.038338(5) & 1.6113(1) & 0.7905 & 0.1013 & 0.0029 & 4.4 & nsO\\
\idlmc{3230} & 1.892812(6) & 1.5016(1) & 0.7933 & 0.0603 & 0.0019 & 4.0 & ap \\
\idlmc{3390} & 0.492180(1) & 0.395665(6)& 0.8039 & 0.0813 & 0.0120 & 4.6 & \\
\idlmc{3519} & 0.5050271(5)& 0.406690(6)& 0.8053 & 0.1290 & 0.0080 & 5.1 & nsO\\
\idlmc{3672} & 1.359207(3) & 1.08395(4) & 0.7975 & 0.1162 & 0.0061 & 6.4 & nsO, ap\\
\idlmc{3945} & 1.218720(2) & 0.97340(4) & 0.7987 & 0.0944 & 0.0032 & 5.0 & nsO, tdp\\
\idlmc{4391} & 2.285239(7) & 1.8273(1) & 0.7996 & 0.0777 & 0.0017 & 5.0 & nsO, tdp\\
\idlmc{4565} & 2.046588(3) & 1.6272(1) & 0.7951 & 0.1344 & 0.0024 & 4.6 & al\\
\hline
\idsmc{1003} & 0.901880(1) & 0.72517(2) & 0.8041 & 0.164 & 0.008 & 5.5 & $\fov+\foov$\\
\idsmc{2209} & 2.055730(4) & 1.6180(1) & 0.7870 & 0.1397 & 0.0029 & 4.6 & al, nsX\\
\idsmc{2988} & 2.056799(7) & 1.6437(1) & 0.7992 & 0.0777 & 0.0034 & 5.0 & nsX\\
\idsmc{4881} & 0.673569(2) & 0.54479(1) & 0.8088 & 0.130 & 0.018 & 6.7 & nsX, al\\
\hline
\end{tabular}
\end{table*}
\begin{figure}
\includegraphics[width=\columnwidth]{pap_lmc2575.eps}
\caption{Variation of amplitudes of the 1O and 2O modes in \idlmc{2575} extracted with the time-dependent Fourier analysis. Time is $\mathrm{HJD}-245\,0000$.}
\label{fig:2575}
\end{figure}
\subsubsection{Candidates for 1O+3O double-mode radial pulsation}\label{sssec:1o3o}
Four candidates for 1O+3O radial double-mode pulsation are reported in Tab.~\ref{tab:1o3o}: two stars in the LMC and two in the SMC. The additional signal in \idlmc{2561} is detected only after time-dependent prewhitening is applied due to non-stationary nature of 1O. In all stars the detections are weak (${\mathrm{S/N}}\lesssim5.0$), the additional signals are coherent and we do not detect any combination frequencies with 1O. Interestingly, in \idsmc{4092}, additional signals are detected with $\pov/\px\in (0.6,\, 0.65)$ (see Sect.~\ref{ssec:61}).
In Fig.~\ref{fig:petrad}, location of the 1O+3O candidates is plotted in the Petersen diagram (open triangles) and may be compared with $\pooov/\pov$ ratios reported in the double-mode (1O+3O) and in the triple-mode (1O+2O+3O) pulsators reported by OGLE in the LMC, SMC, Galactic bulge and Galactic disk. The 1O+3O stars already reported in the OGLE-CVS are clustered in two groups, around $\log\pov=-0.6$ and $\log\pov=-0.25$. Two new candidates are located in between these groups, while two other have a much longer pulsation period ($\log\pov\approx0.08$)
\begin{table*}
\caption{Candidates for new double-mode 1O+3O radial pulsators. Structure of the table and remarks in the last column are the same as in Tab.~\ref{tab:f1o}.}
\label{tab:1o3o}
\begin{tabular}{lrrrrrrr}
\hline
Star & $\pov$\,(d) & $\pooov$\,(d) & $\pooov/\pov$ & $\aov$ (mag) & $\aooov$ (mag) & ${\mathrm{S/N}}$ & Remarks\\
\hline
\idlmc{1274} & 0.3616755(8)& 0.246708(3) & 0.6821 & 0.052 & 0.007 & 4.2 & \\
\idlmc{2561} & 1.192038(2) & 0.80854(3) & 0.6783 & 0.0802 & 0.0026 & 4.6 & nsO, tdp\\
\hline
\idsmc{4092} & 1.0409489(7)& 0.70124(2) & 0.6737 & 0.1811 & 0.0037 & 5.1 & 0.61\\
\idsmc{4465} & 0.4216633(6)& 0.279966(4) & 0.6640 & 0.132 & 0.010 & 4.4 & \\
\hline
\end{tabular}
\end{table*}
\subsection{Double-periodic stars with $\px/\pov$ in between 0.60 and 0.65}\label{ssec:61}
Classical Cepheids with the dominant radial 1O and additional signals with $\px/\pov\in(0.60,\,0.65)$, were first reported by \cite{mk08} and \cite{o3_lmc_cep}, based on the analysis of OGLE photometry in the LMC. Later 138 1O Cepheids of the same type were detected in the SMC \citep{o3_smc_cep}. In the LMC, two clear and close sequences were reported in the Petersen diagram, while for the SMC three sequences could be distinguished. The first in-depth analysis of this form of pulsation for 138 SMC stars was presented in \cite{ss16}. It was found that signals detected at $\fx$ are often accompanied with signals centred, or located close to the sub-harmonic, $1/2\fx$. The latter signals are often non-coherent: wide bands of power excesses are detected. While the power excess is usually well centred at $1/2\fx$, the highest signal within, that had been used to characterise this structure (its frequency denoted $\fsh$), may be a bit off.
In Tab.~A1 in the on-line Appendix, section of which is shown in Tab.~\ref{tab:61} for a reference, we collect the properties of additional signals with $\px/\pov\in(0.60,\,0.65)$ for the SMC (225 stars; $12.6$\,per cent of the sample) and the LMC (291 stars, $16.5$\,per cent of the sample). The consecutive columns contain star's id, first overtone frequency, $\fov$, frequency of the additional signal, $\fx$, corresponding period ratio, $\px/\pov$, amplitude of the 1O, amplitude ratio, $\ax/\aov$, ${\mathrm{S/N}}$ for the detection at $\fx$ and remarks (described in the Table's caption).
For stars in which signals centred at $1/2\fx$ were also detected, additional data are included in Tab.~A2 in the on-line Appendix, section of which is shown in Tab.~\ref{tab:61sh} for a reference. The consecutive columns contain star's id, period ratio, $\px/\pov$, frequency of the additional signal, $\fx$, frequency of the highest signal close to sub-harmonic of $\fx$, $\fsh$, amplitude of the additional signal, $\ax$, amplitude ratio, $\ash/\ax$, ${\mathrm{S/N}}$ for the detection at $\fsh$ and remarks (described in the Table's caption).
\begin{table*}
\caption{Properties of first overtone Cepheids with period ratios, $\px/\pov$, in the $(0.60,\,0.65)$ range. Consecutive columns contain: star's id, first overtone frequency, $\fov$, frequency of the additional variability, $\fx$ , corresponding period ratio, $\px/\pov=\fov/\fx$, amplitude of the first overtone, $\aov$, and amplitude ratio, $\ax/\aov$, and remarks: `al' -- daily alias of signal at $\fx$ is higher; `nsX' -- complex appearance of the signal at $\fx$; `nsO' -- non-stationary first overtone; `cf' -- combination frequency of $\fx$ and $\fov$ detected; `sh' -- power excess at sub-harmonic frequency (around $1/2\fx$) detected; `ap'-- additional periodicity detected; `tdp' -- time-dependent analysis was conducted.}
\label{tab:61}
\begin{tabular}{lrrrrrrl}
\hline
Star & $\fov$ (d$^{-1}$) & $\fx$ (d$^{-1}$) & $\px/\pov$ & $\aov$ (mag) & $\ax/\aov$ & ${\mathrm{S/N}}$ & Remarks\\
\hline
OGLE-SMC-CEP-0088 & 0.650005(1) & 1.03295(5) & 0.6293 & 0.1521(7) & 0.024 & 4.4 & al \\
OGLE-SMC-CEP-0212 & 0.574370(1) & 0.91905(6) & 0.6250 & 0.1026(6) & 0.043 & 5.5 & sh, nsX, cf \\
OGLE-SMC-CEP-0251 & 0.5565345(9) & 0.89055(4) & 0.6249 & 0.1376(5) & 0.022 & 4.5 & sh, al\\
OGLE-SMC-CEP-0260 & 0.887240(1) & 1.45102(3) & 0.6115 & 0.1564(7) & 0.034 & 5.8 & cf, al \\
OGLE-SMC-CEP-0280 & 0.596946(1) & 0.95146(4) & 0.6274 & 0.1378(6) & 0.026 & 4.3 & sh, ap \\
\ldots & & & & & & &\\
\hline
\end{tabular}
\end{table*}
\begin{table*}
\caption{Stars with significant power excess centred at sub-harmonic frequency, $1/2\fx$. Consecutive columns contain: star's id, period ratio, $\px/\pov$,
frequency of the additional variability, $\fx$ , frequency of the highest peak detected around $1/2\fx$, $\fsh$, frequency ratio, $\fsh/\fx$, amplitude of the additional
variability, $\ax$, and amplitude ratio, $\ash/\ax$, approximate ${\mathrm{S/N}}$ for the peak at $\fsh$ and remarks: `nss' -- complex appearance of the signal at $\fsh$; `nss-broad' -- particularly broad power excess at $\fsh$;
`al' -- daily alias of signal at $\fsh$ is higher;
`tdp' -- time-dependent pre-whitening of all signals except $\fsh$ conducted.}
\label{tab:61sh}
\begin{tabular}{lrrrrrrrl}
\hline
Star & $\px/\pov$ & $\fx$ (d$^{-1}$) & $\fsh$ (d$^{-1}$) & $\fsh/\fx$ & $\ax$ (mag) & $\ash/\ax$ & ${\mathrm{S/N}}$ & Remarks\\
\hline
OGLE-SMC-CEP-0212 & 0.6250 & 0.91905(6) & 0.45285(4) & 0.4927 & 0.0044(6) & 0.82 & 4.7 & nss, al \\
OGLE-SMC-CEP-0251 & 0.6249 & 0.89055(4) & 0.44279(4) & 0.4972 & 0.0031(5) & 1.30 & 5.5 & nss, al \\
OGLE-SMC-CEP-0280 & 0.6274 & 0.95146(4) & 0.49075(4) & 0.5158 & 0.0035(6) & 1.14 & 4.8 & \\
OGLE-SMC-CEP-0348 & 0.6208 & 0.77940(4) & 0.38365(3) & 0.4922 & 0.0034(6) & 1.33 & 6.3 & nss-broad, al \\
OGLE-SMC-CEP-0477 & 0.6344 & 0.40006(4) & 0.19209(4) & 0.4802 & 0.0027(5) & 1.45 & 5.0 & nss, al\\
\ldots & & & & & & & &\\
\hline
\end{tabular}
\end{table*}
In Fig.~\ref{fig:pet61} we show the Petersen diagrams for the discussed group of stars, separately for the SMC (left) and for the LMC (right). The top panels show both the $\px/\pov$ period ratios (circles) and, for stars in which signals centred at $1/2\fx$ were detected, $\pov/\psh$ ratios (triangles), where $\psh=1/\fsh$ is a period corresponding to the highest peak within the power excess detected near $1/2\fx$. In the top panels, symbols are exactly the same as in Fig.~\ref{fig:petall}; in particular coherent signals are marked with filled symbols, while non-coherent signals are marked with open symbols. In the bottom panels, we zoom into the $\px/\pov$ period ratios only. This time, filled circles correspond to stars for which signals at $1/2\fx$ were detected, while for stars plotted with open symbols no signal at sub-harmonic frequency could be detected.
\begin{figure*}\includegraphics[width=\columnwidth]{pap_pet61smc.eps}
\includegraphics[width=\columnwidth]{pap_pet61lmc.eps}
\caption{Petersen diagrams for double-periodic stars with $\px/\pov\in(0.60,\,0.65)$ for the SMC (left panels) and the LMC (right panels). In the top panels, symbols are the same as in Fig.~\ref{fig:petall}. For stars for which power was also detected at around $1/2\fx$, we show both $\px/\pov$ (circles) and $\pov/\psh$ (triangles). In the bottom panels, we show only the $\px/\pov$ ratios. Filled symbols correspond to stars for which power was detected at around $1/2\fx$.}
\label{fig:pet61}
\end{figure*}
For the further analysis, particularly to test the model proposed to explain the nature of the discussed periodicities, it is crucial to divide the stars into sequences, based on their location in the Petersen diagram. Division solely based on $\px/\pov$ is not possible, as sequences are slanted. While in the case of the SMC nearly all stars can be assigned to a particular sequence without any ambiguity, in the LMC it is not possible; hence our procedure is based on the SMC stars. First, we divided the SMC stars into three sequences manually, in full agreement with the division lines plotted with the dashed lines in the bottom left panel of Fig.~\ref{fig:pet61} (lines were computed later on). Then, we fitted a linear function ($\px/\pov$ as a function of $\log\pov$) separately for the three sequences; results are collected in the top section of Tab.~\ref{tab:slopes}. Slopes for the three sequences are similar. Finally, we projected the location of all stars in the Petersen diagram on a line perpendicular to the reference line, for which we choose the linear fit to the middle sequence, crossing through an arbitrary reference point, at $\log\pov=0.3$ (the exact value has no impact on further analysis).
In Fig.~\ref{fig:histseq}, we show the resulting distribution of the projected period ratios in the SMC and in the LMC. The existence of three sequences is clear also in the LMC. We can select two values of the projected period ratios, marked with long arrows in Fig.~\ref{fig:histseq}, the same for the LMC and the SMC, that coincide with the minima of the distributions. These are used to separate the stars into three sequences. It is also clear that two stars in the LMC with the largest projected period ratios should not be included in the top sequence, but rather form a {\it fourth} sequence. The boundary value is arbitrarily put at the projected period ratio of 0.66 and marked with a short arrow in the bottom panel of Fig.~\ref{fig:histseq}. The corresponding division lines are plotted with dashed gray lines in the Petersen diagrams in Fig.~\ref{fig:pet61}. For the SMC, the resulting three sequences fully agree with the prior manual division.
We can now fit the linear functions into three sequences in the LMC (except for the fourth sequence with two stars only); results are collected in the bottom part of Tab.~\ref{tab:slopes}. At this point, we conclude that the slopes are all very similar; for the middle sequence they are the same, within 1$\sigma$, for the LMC and the SMC. Thus we consider our division into the sequences as final, keeping in mind that for the LMC, assignment of a few stars that fall in between the sequences is necessarily tentative.
\begin{table}
\caption{Coefficients of the linear fits, $\px/\pov=a\cdot\log\pov+b$, to the three sequences in the SMC (top section) and in the LMC (bottom section).}
\label{tab:slopes}
\begin{tabular}{lrr}
\hline
Sequence & Slope, $a$ & Intercept, $b$\\
\hline
SMC, bottom & $-0.0265\pm 0.0040$ & $0.6116\pm 0.0005$\\
SMC, middle & $-0.0307\pm 0.0042$ & $0.6319\pm 0.0013$\\
SMC, top & $-0.0331\pm 0.0059$ & $0.6537\pm 0.0027$\\
\hline
LMC, bottom & $-0.0318\pm 0.0043$ & $0.6138\pm 0.0013$\\
LMC, middle & $-0.0319\pm 0.0033$ & $0.6332\pm 0.0013$\\
LMC, top & $-0.0378\pm 0.0036$ & $0.6560\pm 0.0020$\\
\hline
\end{tabular}
\end{table}
\begin{figure}
\includegraphics[width=\columnwidth]{pap_histseq.eps}
\caption{Distributions of the projected period ratios, $\px/\pov$, for the SMC (top panel) and the LMC (bottom panel).}
\label{fig:histseq}
\end{figure}
The number of stars within each of the three sequences is given in Tab.~\ref{tab:seqprop}. Within each sequence, we have counted the stars with non-coherent signal at $\fx$ and with signals detected close to $1/2\fx$. Numbers and incidence rates (within each sequence) are also collected in Tab.~\ref{tab:seqprop}.
\begin{table}
\caption{Properties of the three sequences in the SMC and LMC: numbers of stars within each of the sequences, $N$, including stars with non-coherent signals at $\fx$ and stars with signals detected close to $1/2\fx$. Incidence rates for the last two groups are also given.}
\label{tab:seqprop}
\begin{tabular}{lrrrr}
\hline
& SMC & & LMC & \\
& $N$ & \% & $N$ & \% \\
\hline
bottom seq. & 79 & & 84 & \\
non-coh. & 25 & $31.7\pm5.2$ & 27 & $32.1\pm5.1$ \\
with sh & 1 & $1.3\pm1.3$ & 10 & $11.9\pm3.5$ \\
\hline
middle seq. & 106 & & 184 & \\
non-coh. & 58 & $54.7\pm4.8$ & 108 & $58.7\pm3.6$ \\
with sh & 86 & $81.1\pm3.8$ & 113 & $61.4\pm3.6$ \\
\hline
top seq. & 37 & & 36 & \\
non-coh. & 26 & $70.3\pm7.5$ & 18 & $50.0\pm8.3$ \\
with sh & 16 & $43.2\pm8.1$ & 19 & $52.8\pm8.3$ \\
\hline
\end{tabular}
\end{table}
Based on Figs~\ref{fig:pet61} and \ref{fig:histseq}, and Tabs~\ref{tab:slopes}, and \ref{tab:seqprop}, we observe the following.
(i) The $\px/\pov$ period ratios form three well populated sequences in the Petersen diagram, both for SMC and LMC. The sequences are slanted; in the SMC we observe a trend of increasing slope as we move from the bottom to the top sequence. In the LMC the slopes for the bottom and the middle sequence are the same, within errors, and the largest slope is for the top sequence (Tab.~\ref{tab:slopes}). In the LMC, a much larger dispersion of period ratios, as compared to the SMC is present. The three sequences are not equally populated (Tab.~\ref{tab:seqprop}). In both Clouds, middle sequence is the most populated and the top sequence is the least populated.
(ii) In 15 stars in the LMC and in 3 stars in the SMC we simultaneously detect signals corresponding to two sequences (two rows are given in Tab.~\ref{tab:61}). Assuming the \cite{wd16} model (see Sect.~\ref{ssec:discussion61}) correctly tights these signals to excitation on moderate-degree non-radial modes, such stars are interesting targets for asteroseismic modelling \citep[see][for application to RR~Lyr stars with analogous form of variability]{NS22}. The same applies to double-mode radial Cepheids with additional signals -- a few candidate stars were reported in the preceding sections.
(iii) In the LMC we tentatively identified a fourth sequence with the largest period ratios. It consists of only 2 stars. The presence of power excess centred at $1/2\fx$ in one of them indicates that we observe the same form of pulsation as for the other three sequences.
(iv) In both SMC and LMC we also observe stars with low period ratios, located below the main progression of the bottom sequences. In Fig.~\ref{fig:histseq} these stars, with projected period ratios below $0.5975$, may be considered either as a short period ratio tail of the distribution for the bottom sequence, or as a separate sequence. We adopt the former option, and include these stars in the bottom sequence.
(v) Non-coherent signals at $\fx$ are common (Tab.~\ref{tab:seqprop}). In the SMC they are most common for the top sequence ($70.3\pm7.5$ per cent), then for the middle sequence ($54.7\pm4.8$ per cent) and finally for the bottom sequence ($31.7\pm5.2$ per cent). In the LMC, the numbers are similar, except for the top sequence for which incidence rate is lower ($50.0\pm8.3$ per cent).
(vi) Since $\px/\pov$ period ratios form three sequences in the Petersen diagram, one would expect that period ratios for sub-harmonic signals, $\pov/\psh$, should also fall along three sequences. As is visible in the top panels of Fig.~\ref{fig:pet61}, this is not the case. This is because majority of signals detected at around $1/2\fx$ ($69.8\pm4.5$ per cent in the SMC and $76.2\pm3.5$ per cent in the LMC) are non-coherent, and appear in the frequency spectrum as a band of increased power, which we characterise through the highest peak within. Its frequency, $\fsh$, may be off $1/2\fx$. The point was already illustrated in Fig.~\ref{fig:fspexample} for a few stars: signals are well centred at $1/2\fx$ (marked with arrow), however the highest peak (marked with a filled circle) is off. Still, each signal detected at around $1/2f\fx$ is assigned to a specific sequence as marked in the bottom panels of Fig.~\ref{fig:pet61} (filled symbols).
In Fig.~\ref{fig:shdist} we plot the distribution of $\fsh/\fx$ ratios for the SMC and LMC stars. The distributions are centred on $0.5$, but with wide and asymmetric wings. For $55.7$ per cent of the SMC and $57.1$ per cent of the LMC stars we have $\fsh/\fx>0.5$. It indicates that more often $\fsh>0.5\fx$ for the highest peak within power excess around $1/2\fx$. This is the case for all frequency spectra illustrated in Fig.~\ref{fig:fspexample}.
\begin{figure}
\includegraphics[width=\columnwidth]{pap_shdist.eps}
\caption{Distributions of the $\fsh/\fx$ frequency ratios for the SMC (top panel) and the LMC (bottom panel).}
\label{fig:shdist}
\end{figure}
(vii) As quantified in Tab.~\ref{tab:seqprop}, signals around $1/2\fx$ are most commonly detected for the middle sequence ($81.1\pm3.8$ and $61.4\pm3.6$ per cent for the SMC and the LMC, respectively) and then for the top sequence ($43.2\pm8.1$ and $52.8\pm8.3$ per cent for the SMC and the LMC, respectively). For the bottom sequence only a single detection was reported in the SMC ($1.3\pm1.3$ per cent), but significantly more in the LMC (10 detections, $11.9\pm3.5$ per cent). In the tentatively identified fourth sequence in the LMC, one out of two stars shows signal at $1/2\fx$.
(viii) The distribution of relative amplitudes of signals centred at $1/2\fx$ with respect to amplitudes of signals at $\fx$, $\ash/\ax$, is asymmetric, as illustrated in the top panels of Fig.~\ref{fig:61shamps}. The coloured bars show relative contributions of signals with $\fsh/\fx$ larger and smaller than 0.5. We observe that in the LMC $\ash>\ax$ is more common (63\, per cent) than $\ash<\ax$, while reverse is true for the SMC, as $\ash>\ax$ holds for 35\, per cent of signals
(ix) The distribution of amplitudes of signals corresponding to specific sequences in the Petersen diagram follows the distribution outlined above -- see bottom panels of Fig.~\ref{fig:61shamps} in which $\ash$ is plotted against $\ax$, separately for both Clouds and three sequences. In the LMC we observe that typically $\ash>\ax$ for signals corresponding to the top and middle sequences. In the SMC we observe that typically $\ash<\ax$ for all sequences; in particular for nearly all signals of the top and bottom (1 signal) sequences.
\begin{figure}
\includegraphics[width=\columnwidth]{pap_61sh_amps.eps}
\caption{Distribution of amplitude ratios, $\ash/\ax$ (top panels) and plots of
$\ash$ versus $\ax$ (bottom panels). In the top panel, contributions from signals with $\fsh/\fx$ larger and smaller than 0.5 are indicated with different colors, as given in the legend. In the bottom panel, signals corresponding to different sequences are plotted with different symbols.}
\label{fig:61shamps}
\end{figure}
\subsection{Period ratios $\pov/\px$ around 0.684}\label{ssec:686}
In the Petersen diagram in Fig.~\ref{fig:petall}, an over-density of double-periodic pulsators at $\log\plong\approx 0.5$ and $P_{\rm S}/\plong\approx 0.684$ is well visible in the LMC. In all these stars $\px>\pov$ and the detected additional signal is coherent -- stars are plotted with filled triangles. A zoom into the relevant part of the Petersen diagram, showing both SMC and LMC stars, is shown in the right side of Fig.~\ref{fig:pet686}. The number of double-periodic pulsators in the SMC is much smaller; in the following stars form both Clouds are analysed together. An over-density is well captured in the histogram of period ratios for double-periodic stars with $\pov/\px$ in the $0.67-0.70$ range displayed in the bottom panel of Fig.~\ref{fig:hist686}.
The discussed double-periodic Cepheids share the properties of double-periodic RRc stars discovered by \cite{nsd15} and studied in more detail by \cite{ns19}. The Petersen diagram for double-periodic RRc stars from the latter study is displayed in the left side of Fig.~\ref{fig:pet686} and the corresponding histogram of period ratios is displayed in the top panel of Fig.~\ref{fig:hist686}. In RRc stars, the distribution is peaked around the median value, $0.6856$ (marked with red arrow), with wide wings (period ratios in between $0.67$ and $0.70$). For first overtone Cepheids, we define an equivalent group of double-periodic stars, with period ratios in between $0.68$ and $0.69$. Five stars belong to SMC ($0.3\pm0.1$\,per cent) and 23 to LMC ($1.3\pm0.3$\,per cent). These stars are marked with large colored triangles in Fig.~\ref{fig:pet686} and the corresponding histogram is plotted with black solid line in Fig.~\ref{fig:hist686}. The distribution peaks around the median value, $0.6844$ (marked with red arrow) -- a bit lower than for the RRc stars. Data for these stars are collected in Tab.~\ref{tab:1o686}. The detections are firm, with ${\mathrm{S/N}}$ usually exceeding $5.0$. Except for \idlmc{2137}, in which significant peaks at $\fov+\fx$ and $\fov-\fx$ are detected, we find no combinations with the radial 1O mode.
For RRc stars, the amplitude of the additional periodicity amounts to $3.3$ per cent of the 1O amplitude, on average. For 1O Cepheids the number is similar, amplitude of the additional signals is $3.9$ per cent of the 1O amplitude, on average.
In Fig.~\ref{fig:pet686} there are more double-periodic Cepheids with additional coherent signals and period ratios in the $0.67-0.70$ range (marked with small grey triangles in the right part of Fig.~\ref{fig:pet686}). The corresponding distribution of period ratios is plotted with grey dashed line in Fig.~\ref{fig:hist686}. Whether these are double-periodic stars of the same type, or not, is an open question.
\begin{figure}
\includegraphics[width=\columnwidth]{pap_pet686.eps}
\caption{The Petersen diagram zoomed around double-periodic stars clustered at $\pov/\px\approx0.684$. A group of double-periodic RRc stars of similar characteristics are included for a comparison.}
\label{fig:pet686}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{pap_hist686.eps}
\caption{Distribution of period ratios for double-periodic stars with dominant 1O and additional coherent periodicity with $\pov/\px$ in the $0.67-0.70$ range. Top panel shows RRc stars and bottom panel shows 1O Cepheids.}
\label{fig:hist686}
\end{figure}
\begin{table*}
\caption{Candidates for double periodic pulsators with dominant radial first overtone mode and additional periodicity with period ratios clustered around $\pov/\px=0.684$. Consecutive columns contain: star's id, first overtone period, $\pov$, additional, longer period, $\px$ , corresponding period ratio, $\pov/\px$, amplitude of the first overtone, $\aov$, amplitude of the additional periodicity, $\ax$, signal-to-noise for the detection of additional periodicity and remarks: `al' -- daily alias of a signal at $\fx$ is higher, `nsO' -- non-stationary first overtone, `cf' -- combination frequency of $\fx$ and $\fov$ detected, `ap'-- additional periodicity detected, `0.61' -- additional periodicity with $\px/\pov\in(0.60,\,0.65)$ was detected, `tdp' -- time-dependent analysis was conducted.}
\label{tab:1o686}
\begin{tabular}{lrrrrrrr}
\hline
Star & $\pov$\,(d) & $\px$\,(d) & $\pov/\px$ & $\aov$ (mag) & $\ax$ (mag) & ${\mathrm{S/N}}$ & Remarks\\
\hline
\idlmc{0113} & 1.838337(4) & 2.6856(3) & 0.6845 & 0.0780(3) & 0.0019(3) & 4.6 & \\
\idlmc{0184} & 1.793580(5) & 2.6308(3) & 0.6818 & 0.0708(3) & 0.0026(3) & 5.8 & \\
\idlmc{0193} & 1.691671(4) & 2.4717(2) & 0.6844 & 0.0961(5) & 0.0038(5) & 6.5 & ap \\
\idlmc{0676} & 2.544534(7) & 3.7275(4) & 0.6826 & 0.0887(4) & 0.0035(4) & 6.6 & ap \\
\idlmc{0817} & 1.873575(3) & 2.7420(4) & 0.6833 & 0.1210(3) & 0.0019(3) & 4.7 & ap\\
\idlmc{0925} & 2.045508(6) & 2.9970(4) & 0.6825 & 0.0786(4) & 0.0021(4) & 4.6 & ap\\
\idlmc{1346} & 2.462782(9) & 3.6047(4) & 0.6832 & 0.1101(6) & 0.0029(3) & 6.5 & 0.61, F1O, ap, nsO, tdp\\
\idlmc{1801} & 1.178183(2) & 1.7207(1) & 0.6847 & 0.0812(4) & 0.0025(4) & 5.0 & \\
\idlmc{2137} & 4.21030(2) & 6.1411(4) & 0.6856 & 0.0785(2) & 0.0068(2) & 16.5 & cf, F1O \\
\idlmc{2172} & 1.955678(4) & 2.8615(3) & 0.6834 & 0.0778(3) & 0.0023(3) & 5.9 & nsO\\
\idlmc{2406} & 2.091337(4) & 3.0413(4) & 0.6876 & 0.0926(3) & 0.0023(3) & 5.3 & ap, F1O\\
\idlmc{2432} & 1.686911(2) & 2.4637(3) & 0.6847 & 0.1118(3) & 0.0023(3) & 5.3 & \\
\idlmc{2444} & 2.232034(4) & 3.2600(3) & 0.6847 & 0.1079(3) & 0.0031(3) & 7.5 & nsO\\
\idlmc{2597} & 1.972834(4) & 2.8865(3) & 0.6835 & 0.0909(3) & 0.0022(3) & 5.3 & 0.61\\
\idlmc{2789} & 2.537724(9) & 3.7088(3) & 0.6842 & 0.0665(3) & 0.0047(3) & 8.9 & \\
\idlmc{2895} & 2.170233(5) & 3.1706(3) & 0.6845 & 0.0730(4) & 0.0030(4) & 5.5 & \\
\idlmc{3033} & 1.770260(3) & 2.5817(3) & 0.6857 & 0.0958(4) & 0.0021(4) & 4.3 & ap \\
\idlmc{3674} & 2.039138(6) & 2.9717(4) & 0.6862 & 0.0785(5) & 0.0030(5) & 4.9 & \\
\idlmc{3827} & 1.870824(4) & 2.7381(2) & 0.6833 & 0.0886(4) & 0.0033(3) & 6.5 & nsO\\
\idlmc{4494} & 4.14287(2) & 6.0405(9) & 0.6858 & 0.0792(3) & 0.0032(3) & 6.7 & nsO\\
\idlmc{4503} & 3.33323(2) & 4.8739(5) & 0.6839 & 0.0668(4) & 0.0046(4) & 7.8 & ap, nsO\\
\idlmc{4505} & 2.401212(6) & 3.5077(4) & 0.6846 & 0.0972(4) & 0.0029(4) & 5.3 & ap\\
\idlmc{4624} & 1.559273(5) & 2.2784(2) & 0.6844 & 0.0745(6) & 0.0043(6) & 5.3 & F1O\\
\hline
\idsmc{0151} & 1.532852(3) & 2.2493(1) & 0.6815 & 0.1121(5) & 0.0048(5) & 6.9 & \\
\idsmc{1598} & 2.99418(2) & 4.3778(9) & 0.6839 & 0.0653(5) & 0.0020(4) & 4.3 & nsO, tdp, al, ap\\
\idsmc{1634} & 1.288075(2) & 1.8822(2) & 0.6843 & 0.0980(6) & 0.0033(6) & 4.1 & \\
\idsmc{3158} & 1.528997(2) & 2.2275(1) & 0.6864 & 0.1348(5) & 0.0051(5) & 7.5 & \\
\idsmc{4909} & 1.84246(1) & 2.6858(3) & 0.6860 & 0.0560(7) & 0.0041(7) & 4.5 & \\
\hline
\end{tabular}
\end{table*}
\subsection{Unclassified period ratios}\label{ssec:unclassified}
Except signals that correspond to double-periodic pulsations described in the previous sections, other signals were detected in 153 1O Cepheids in the SMC ($8.6\pm0.7$\,per cent) and in 482 1O Cepheids in the LMC ($27.3\pm1.1$\,per cent). In many stars we detect more than one additional signal. In total, we have detected 168 additional signals in the SMC and 629 signals in the LMC. Period ratios for all these signals are plotted in Fig.~\ref{fig:petadd}, which is essentially Fig.~\ref{fig:petall} without candidates for double-mode radial pulsation, without double-periodic stars with $\px/\pov$ in the $0.60-0.65$ range, and without stars with period ratios centred at $\pov/\px\approx 0.684$. Basic properties of these stars are collected in Tab.~A3 in the on-line Appendix, section of which is shown in Tab.~\ref{tab:add} for a reference.
\begin{table*}
\caption{Properties of first overtone Cepheids with additional variability. The consecutive columns contain: star's id, first overtone frequency, $\fov$, frequency of the additional variability, $\fx$ , corresponding period ratio (shorter to longer), amplitude of the first overtone, $\aov$, and amplitude ratio, $\ax/\aov$, and remarks: `al' -- daily alias of signal at $\fx$ is higher; `nsx' -- complex appearance of the signal at $\fx$; `nsO' -- non-stationary first overtone; `cf' -- combination frequency of $\fx$ and $\fov$ detected; `tdp' -- time-dependent analysis was conducted.}
\label{tab:add}
\begin{tabular}{lrrrrrrl}
\hline
Star & $\fov$ (d$^{-1}$) & $\fx$ (d$^{-1}$) & $P_{\rm S}/P_{\rm L}$ & $\aov$ (mag) & $\ax/\aov$ & ${\mathrm{S/N}}$ & Remarks\\
\hline
OGLE-SMC-CEP-0021 & 0.916019(1) & 0.72656(5) & 0.7932 & 0.2030(9) & 0.025 & 4.1 & \\
OGLE-SMC-CEP-0065 & 0.575606(1) & 0.55434(4) & 0.9631 & 0.1514(7) & 0.028 & 4.8 & \\
OGLE-SMC-CEP-0114 & 0.568231(2) & 0.56159(2) & 0.9883 & 0.0988(6) & 0.064 & 7.0 & \\
OGLE-SMC-CEP-0125 & 0.540067(2) & 0.54312(2) & 0.9944 & 0.0861(6) & 0.086 & 8.5 & \\
* & 0.540067(2) & 0.41954(4) & 0.7768 & 0.0861(6) & 0.038 & 5.0 & \\
\ldots & & & & & & & \\
\hline
\end{tabular}
\end{table*}
\begin{figure*}
\includegraphics[width=\columnwidth]{pap_petaddsmc.eps}
\includegraphics[width=\columnwidth]{pap_petaddlmc.eps}
\caption{The Petersen diagram with double-periodic pulsators after filtering out the new radial double-mode candidates, stars with $\px/\pov\in(0.60,\,0.65)$ and stars with $\pov/\px$ centred at $0.684$. Double-mode radial pulsators from OGLE-CVS are still included for context. The dashed polygons indicate the range at which sub-harmonic periodicities were detected in $\px/\pov\in(0.60,\,0.65)$ stars, as defined in Fig.~\ref{fig:pet61}.}
\label{fig:petadd}
\end{figure*}
A striking difference between SMC and LMC, well visible in Fig.~\ref{fig:petadd}, and corroborated by incidence rates given at the beginning of this section, is many more additional signals detected in the latter system. In Fig.~\ref{fig:adddist} we show the distribution of first overtone periods for all 1O Cepheids and for those with additional signals detected, both for the LMC (top panel) and the SMC (bottom panel). Incidence rates are given for specific period bins with significant number of stars. In each period bin incidence rate is larger in the LMC. While the highest incidence rate in the SMC is $23.4$\,per cent (for $3.0<P_{\rm 1O}\leq 3.5$\,d), in the LMC it is $56.3$\,per cent (for $3.5<P_{\rm 1O}\leq 4.0$\,d). We stress that for first overtone periods between $3$\,d and $4$\,d $\sim$half of the 1O Cepheids in the LMC show additional signals in the frequency spectrum (and this assessment does not include the classes of signals discussed in the preceding sections).
\begin{figure}
\includegraphics[width=\columnwidth]{pap_histPadd.eps}
\caption{First overtone period distributions for all Cepheids (solid black line) and for Cepheids with additional periodicities in the frequency spectrum (dashed blue line) in the LMC (top panel) and the SMC (bottom panel). For period bins with the largest number of stars incidence rates are given. For the LMC, the incidence rates given in parenthesis result from applying the SMC's detection limit.}
\label{fig:adddist}
\end{figure}
The obvious explanation for significantly lower number of signals in the SMC may be higher detection limit, as the SMC Cepheids \citep[$\mu_{\rm SMC}=18.977$\,mag,][]{SMCdist} are further away than the LMC Cepheids \citep[$\mu_{\rm LMC}=18.477$\,mag,][]{LMCdist}. This is not the case however, which we show by applying the (period-dependent) SMC's detection limit to the LMC signals.
To estimate the detection limit in both Magellanic Clouds we follow the method used in \cite{ss16} and \cite{s17}. For every 1O Cepheid in a given Magellanic Cloud we fit an eight order Fourier series, remove severe (6-$\sigma$) outliers and conduct time-dependent prewhitening on a season-to-season basis. Fourier transform is then computed in the $(0,\,6\fov)$ range and the average noise level is computed. Data on spectra in which no strong signals remain (${\mathrm{S/N}}<5$) are used to construct four-times noise (detection limit) against pulsation period plots. Finally, low order polynomials are fitted to the date to yield the detection limit.
The resulting detection limits in the SMC and LMC are plotted with solid and dashed curves in Fig.~\ref{fig:noise}, respectively, together with amplitudes of additional signals detected. For $\pov<4.0$\,d, detection limit is clearly higher in the SMC. We can now apply the SMC detection limit to LMC stars by removing from the sample those stars, for which amplitudes of additional signals fall below SMC detection limit. As a result, we arrive at incidence rates given in parenthesis in Fig.~\ref{fig:adddist}, just below the original incidence rates. While the incidence rates are lower, they are still well above the incidence rates reported in the SMC in the corresponding period bins.
\begin{figure}
\includegraphics[width=\columnwidth]{pap_noise1.eps}
\caption{Amplitude of additional signals plotted vs. $\pov$ for the LMC (open circles) and SMC (filled circles). Average detection limits are plotted with dashed and solid lines for the LMC and SMC, respectively.}
\label{fig:noise}
\end{figure}
We conclude that significantly higher incidence rate of additional signals in the LMC, as compared to the SMC, is an intrinsic property of 1O Cepheids and does not result from different detection limits for the two galaxies (for further discussion see Sect.~\ref{ssec:propSLMC}).
In the discussion below we will often distinguish between coherent and non-coherent signals. For the former ones, after prewhitening with the additional frequency, there is no significant remnant in the frequency spectrum. The signal can be well modeled with a single sine function with constant amplitude and phase. For non-coherent signals, residual power is observed after the prewhitening, either as unresolved significant peak, or as a power excess (which is often clear even before the prewhitening). The two classes of signals seem to have different properties.
In the top panel of Fig.~\ref{fig:histadd}, we show the distribution of period ratios of additional signals with respect to first overtone period for the LMC. Note that we always plot shorter-to-longer period ratio, $\pslr$, thus either $\px/\pov$ or $\pov/\px$ is used. The distribution peaks for $\pslr>0.95$ (the fewer number of stars in the right-most bin, $\pslr>0.975$, as compared to the adjacent bin, is probably due to the fact that for the highest period ratios the signals may be unresolved with the first overtone frequency), then we observe a plateau down to $\pslr=0.75$ and decline toward lower period ratios. Green long-dash and blue short-dash lines show contributions from the coherent and non-coherent signals, respectively. In the middle and bottom panels, distributions for coherent and non-coherent signals are shown separately, this time also showing the contributions of signals for which $\px>\pov$ (green, long-dash line) and $\px<\pov$ (blue, short-dash line). We observe that the peak for high period ratios is nearly exclusively due to coherent signals. For lower period ratios, non-coherent signals appear and peak for $0.75<\pslr<0.9$. Together with coherent signals they build a plateau observed for $0.75<\pslr<0.95$. Below $\pslr=0.75$ the number of stars with both coherent and non-coherent additional periodicities gradually decreases.
\begin{figure}
\includegraphics[width=\columnwidth]{pap_histadd.eps}
\caption{Distribution of period ratios, shorter-to longer, $\pslr$, for 1O Cepheids with additional periodicities in the LMC. In the top panel contributions form stars with coherent and non-coherent signals are shown. In the middle and bottom panels, distributions for stars with coherent and non-coherent signals are shown separately, with contributions form stars with $\px<\pov$ and $\px>\pov$ plotted with different line styles.}
\label{fig:histadd}
\end{figure}
It is also clear that for majority of signals, both coherent and non-coherent $\fx<\fov$ ($\px>\pov$). While coherent signals with $\px<\pov$ are still detected, there are only a few non-coherent signals with period shorter than $\pov$.
Distributions and conclusions for the the SMC are qualitatively the same, with the exception that period ratios peak not only for high period ratios ($\pslr>0.95$), but there is a second peak within the plateau, at $0.80<\pslr<0.85$, resulting from the peak observed for non-coherent signals.
In Fig.~\ref{fig:ampadd}, we show the relative amplitude ($\ax/\aov$) of additional signals plotted against the corresponding period ratio, $\pslr$, separately for the coherent signals (top panel) and non-coherent signals (second panel). LMC and SMC stars are plotted with different symbols. In Fig.~\ref{fig:ampadd_hist}, we plot the distributions of relative amplitudes for the LMC and SMC showing contributions of coherent and non-coherent signals. Large relative amplitudes (above $10$\,per cent) are observed exclusively for coherent signals (the only exception is relative amplitude of $56$\,per cent for one non-coherent signal in the star form SMC.
For nearly all non-coherent signals relative amplitude is below $8$\,per cent. For coherent signals, very high relative amplitudes are observed for high period ratios. Also, for coherent signals, we observe that average relative amplitude is higher for period ratios above $\sim0.95$
\begin{figure}
\includegraphics[width=\columnwidth]{pap_ampadd_log.eps}
\caption{Relative amplitude ratio, $\ax/\aov$, plotted against period ratio, $\pslr$ for 1O Cepheids with coherent additional signals in the frequency spectrum (top panel) and non-coherent ones (bottom panel). LMC and SMC stars are plotted with different symbols as indicated in the key. Note the logarithmic scale on y axis.}
\label{fig:ampadd}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{pap_ampadd_hist.eps}
\caption{Distribution of relative amplitudes, $\ax/\aov$, for 1O Cepheids with additional periodicities in the frequency spectrum in the LMC (top panel) and the SMC (bottom panel). Distributions for coherent and non-coherent signals are plotted with solid and dashed lines, respectively.}
\label{fig:ampadd_hist}
\end{figure}
The observations made so far suggest that we can distinguish two classes of additional periodicities. In the first class, additional periodicities cluster around first overtone (period ratios above $0.95$, both $\px>\pov$ -- more frequent and $\px<\pov$), are coherent and may have high relative amplitudes. As we discuss later in Sect.~\ref{ssec:cnm} they may originate from non-radial modes with frequencies close to the radial mode frequency. In the second class, additional periodicities have period ratios in between $0.75$ and $0.9$ ($\px>\pov$), are non-coherent and have typically low relative amplitudes. As we argue later in Sect.~\ref{ssec:789noharm}, these signals may originate form excitation of moderate degrees, $\ell=7,\,8,\,9$ modes, in fact the same class as already discussed in Sect.~\ref{ssec:61}, but for which harmonics, that then form three distinct sequences in the Petersen diagram, are not detected.
\subsection{Periodic Modulation of Pulsation}\label{ssec:modulation}
Periodic modulation of pulsation in single-mode Cepheids was rarely reported in the literature. \cite{o4_clouds_multimode} claimed long-period, large-amplitude modulation of pulsation in a few 1O Cepheids based on the appearance of light curves only. \cite{s17} detected a periodic modulation of pulsation in OGLE photometry for a sizeable sample of F-mode Magellanic Clouds Cepheids and a single modulated 1O Cepheid was reported by \cite{Kotysz}. Here we report a sample of 24 modulated 1O Cepheids in the LMC ($1.4\pm0.3$\, per cent) and only 3 in the SMC ($0.2\pm0.1$\, per cent). In each of these stars at least two modulation peaks were detected. Exemplary frequency spectra are illustrated in Fig.~\ref{fig:fspmodul}; see also remarks on individual stars later in this section. In Tab.~\ref{tab:mod} we provide basic properties of these stars: pulsation and modulation periods, amplitudes of the first overtone and most significant modulation peaks, full list of detected modulation peaks and remarks.
The SMC sample is scarce and two of the stars appear peculiar compared to the LMC sample (nonlinear modulation in \idsmc{4241} and very long modulation period in \idsmc{4372}). Consequently, the following discussion of modulation properties is based on the LMC sample only.
In the top panel of Fig.~\ref{fig:hist_modulP}, we show the distribution of pulsation periods of modulated stars (blue dashed-line), compared with overall period distribution of 1O Cepheids in the LMC (black solid line; top panel). Bottom panel of Fig.~\ref{fig:hist_modulP} shows the distribution of modulation periods. Although the sample is rather scarce, we may conclude that distribution of pulsation periods of modulated stars follows the overall period distribution well. Most of the modulated stars have pulsation periods in between 1 and 2.5\,d. Except 5 stars with modulation periods above 300\,d, all other have modulation periods below 120\,d; for majority of stars we find $20<P_{\rm mod}<80$\,d.
Frequency spectra are scarce, however a glance at Tab.~\ref{tab:mod} reveals some regularities. First, similar to modulated F-mode Cepheids, but in contrast to Blazhko RR~Lyr stars, modulation of mean brightness (a peak at the modulation frequency) is commonly detected (in 17 out of 24 stars). The amplitude of mean brightness modulation, $A_{\rm m}$ (amplitude of the peak at $\fmodul$) is weak, in mmag range. The distribution is plotted in the top panel of Fig.~\ref{fig:hist_modulAQ}. Only in one star it exceeds 5 mmag (\idlmc{2330}).
The relative modulation amplitude of the radial 1O mode is also typically weak. We define it as $\max(A_+,\,A_-)/\aov$, where $A_+$ and $A_-$ are amplitudes of side peaks at $\fov+\fmodul$ and $\fov-\fmodul$, respectively. In all but four stars it is below 10 per cent. For the record holder, \idlmc{3985} it is 76\,per cent. Light curve changes in all four stars with relative modulation amplitude exceeding 10\,per cent are visualised with animations attached as Supporting Material. In these stars we were also able to follow the amplitude and phase changes. Panels illustrating modulation in the $A_1-\phi_1$ plane are attached to each animation. In two cases (\idlmc{1705} and \idlmc{3985}) we observe clockwise variation and in two cases (\idlmc{1474} and \idlmc{4270}) anti-clockwise variation.
In the frequency spectra we observe that doublets (single modulation peak next to $\fov$; 14 stars) are more common than triplets (two side peaks placed symmetrically around $\fov$; 10 stars). Interestingly, for doublets a side peak at $\fov+\fmodul$ is more often detected than the side peak of lower frequency $\fov-\fmodul$. For triplets, we typically find $A_+>A_-$. We illustrate these observations in the bottom panel of Fig.~\ref{fig:hist_modulAQ} in which histogram of the asymmetry parameter, $Q = (A_+-A_-)/(A_++A_-)$ \citep{Alcock2003} is plotted. For doublets $Q$ is equal $+1$ or $-1$. Clearly, positive asymmetry is dominant.
We note, that large-amplitude modulation of very long (unresolved) period is also detected in the analysed sample. An example is \idlmc{4477} which shows clear long-period and irregular modulation on a timescale of 1200-1300\,d, which is well visible already when inspecting photometric data. Nearly two cycles are covered in OGLE-IV data, but in the frequency spectrum the modulation side peaks are unresolved. More data are needed to study this interesting case in more detail.
\begin{table*}
\begin{tabular}{lrrrrrrrr}
& & & \multicolumn{4}{c}{Amplitudes of peaks at: (mag)} & & \\
ID & $\pov$\,(d) & $\pmodul$\,(d) & $\fov$ & $\fmodul$ & $\fov-\fmodul$ & $\fov+\fmodul$ & Additional modulation frequencies. & Remarks \\
\hline
\idlmc{0991} & 3.45244(3) & 59.4(1) & 0.0601 & - & 0.0028 & 0.0032 & & ap \\
\idlmc{1368} & 2.165537(6) & 46.57(9) & 0.0917 & 0.0012 & 0.0017 & 0.0017 & & ap, 0.61, nsO, tdp\\
\idlmc{1474} & 0.3022899(7)& 34.18(2) & 0.0539 & - & 0.0168 & 0.0283 & & \\
\idlmc{1626} & 1.803121(2) & 52.03(9) & 0.1341 & 0.0016 & - & 0.0016 & & \\
\idlmc{1661} & 2.372772(5) & 29.10(2) & 0.0892 & 0.0035 & - & 0.0021 & & ap \\
\idlmc{1685} & 2.667840(5) & 31.85(3) & 0.0932 & 0.0018 & - & 0.0012 & & ap \\
\idlmc{1769} & 2.080187(4) & 33.33(4) & 0.0809 & 0.0016 & - & 0.0013 & & ap \\
\idlmc{1815} & 1.927527(3) & 64.1(1) & 0.0899 & 0.0024 & - & 0.0017 & & ap \\
\idlmc{1829} & 2.162057(6) & 46.42(7) & 0.0950 & - & 0.0021 & 0.0032 & & nsO, ap \\
\idlmc{2062} & 1.469803(2) & 104.0(3) & 0.1165 & - & 0.0026 & 0.0022 & & nsO, ap\\
\idlmc{2213} & 2.417636(5) & 80.5(1) & 0.0870 & 0.0011 & - & 0.0050 & & nsO \\
\idlmc{2319} & 2.678569(7) & 42.57(4) & 0.0900 & 0.0013 & - & 0.0043 & & nsO \\
\idlmc{2330} & 0.8447634(6) & 111.7(2) & 0.1721 & 0.0052 & 0.0059 & 0.0061 & $2\fmodul$ & nsO \\
\idlmc{2344} & 4.09683(1) & 36.52(2) & 0.1146 & 0.0012 & 0.0053 & - & $2\fov-\fmodul$ & nsO\\
\idlmc{2380} & 2.007854(7) & 66.5(1) & 0.0736 & 0.0031 & - & 0.0018 & & ap \\
\idlmc{2421} & 2.285858(5) & 79.7(2) & 0.0978 & 0.0022 & - & 0.0035 & & ap, nsO \\
\idlmc{2477} & 1.940064(3) & 483(3) & 0.1047 & - & 0.0051 & 0.0020 & & \\
\idlmc{2705} & 0.2645433(1)& 330.7(5) & 0.1024 & 0.0032 & 0.0456 & 0.0092 & $2\fov-\fmodul$, $3\fov-\fmodul$ & ap \\
\idlmc{2787} & 3.39640(2) & 51.56(9) & 0.0645 & 0.0022 & 0.0014 & - & & ap \\
\idlmc{2992} & 0.4157542(4)& 473(5) & 0.1278 & - & 0.0082 & 0.0104 & & \\
\idlmc{3226} & 1.467621(3) & 36.82(3) & 0.0879 & 0.0025 & - & 0.0050 & & weak\\
\idlmc{3985} & 4.54494(6) & 13.5767(8)& 0.0232 & 0.0029 & 0.0176 & - & $2\fov-\fmodul$, $3\fov-\fmodul$, $\fov-2\fmodul$ & \\
\idlmc{4270} & 1.430489(3) & 540(2) & 0.0841 & - & 0.0086 & 0.0105 & $2\fov-\fmodul$, $2\fov+\fmodul$ & \\
\idlmc{4361} & 1.805423(5) & 335(3) & 0.0964 & 0.0031 & 0.0042 & - & & \\
\hline
\idsmc{4039} & 1.672049(4) & 132.2(6) & 0.1260 & 0.0038 & - & 0.0042 & & nsO\\
\idsmc{4241} & 3.29770(3) & 123.6(1) & 0.0295 & 0.0028 & - & 0.0022 & $2\fmodul$, $3\fmodul$, $4\fmodul$, $\fov+2\fmodul$, $\fov-2\fmodul$ & \\
\idsmc{4372} & 1.451329(2) & 943(24) & 0.1451 & - & 0.0035 & 0.0055 & $2\fov+\fmodul$ & ap\\
\hline
\end{tabular}
\caption{Properties of modulated 1O Cepheids. Consecutive columns contain: star's ID, pulsation period, modulation period, amplitudes of peaks detected at $\fov$, $\fmodul$, $\fov-\fmodul$ and $\fov+\fmodul$, and a list of other modulation peaks detected. Remarks in the last column are: `nsO' -- non-stationary first overtone, `0.61' -- additional periodicity with $\px/\pov\in(0.60,\,0.65)$ was detected, `ap'-- other additional periodicity detected, `tdp' -- time-dependent analysis was conducted.}
\label{tab:mod}
\end{table*}
\begin{figure}
\noindent\includegraphics[width=\columnwidth]{OGLE-LMC-CEP-1661_fsp.eps}\\
\noindent\includegraphics[width=\columnwidth]{OGLE-LMC-CEP-1829_fsp.eps}\\
\noindent\includegraphics[width=\columnwidth]{OGLE-LMC-CEP-2330_fsp.eps}\\
\noindent\includegraphics[width=\columnwidth]{OGLE-SMC-CEP-4241_fsp.eps}
\caption{Exemplary frequency spectra for modulated Cepheids, after prewhitening with the first overtone frequency and its harmonics (if present; dashed lines). Filled circles mark the modulation peaks, open symbols mark locations of the corresponding daily aliases.}
\label{fig:fspmodul}
\end{figure}
\begin{figure}
\noindent\includegraphics[width=\columnwidth]{pap_modulPdist.eps}
\caption{Top panel: distribution of pulsation periods of all (black solid line) and modulated (blue dotted line) first overtone Cepheids in the LMC. Bottom panel: distribution of modulation periods in the LMC.}
\label{fig:hist_modulP}
\end{figure}
\begin{figure}
\noindent\includegraphics[width=\columnwidth]{pap_modulAQdist.eps}
\caption{Top panel: distribution of amplitudes of mean brightness modulation. Bottom panel: distribution of the asymmetry parameter, $Q$. Vertical bars at $|Q|=1$ indicate the number of stars in which only one side peak was detected at first overtone frequency.}
\label{fig:hist_modulAQ}
\end{figure}
{\it Remarks on individual stars:}
\idlmc{0991}, $\pov=3.45244(3)$\,d, $\pmodul=59.4(1)$\,d. Modulation properties of this star were discussed in detail in \cite{Kotysz}.
\idlmc{1368}, $\pov=2.165537(6)$\,d, $\pmodul=46.57(9)$\,d. Peaks at $\fmodul$, $\fov-\fmodul$ and $\fov+\fmodul$ were detected with ${\mathrm{S/N}}=4.0$, $4.6$ and $4.6$, respectively, but only after time-dependent prewhitening was applied due to non-stationary nature of first overtone mode. In the same star, we detect signals from classes discussed in Sect.~\ref{ssec:61} and in Sect.~\ref{ssec:unclassified}.
\idlmc{1474}, $\pov=0.3022899(7)$\,d, $\pmodul=34.18(2)$\,d. A triplet at radial mode frequency is clear, with ${\mathrm{S/N}}=10.7$ and $7.5$ for peaks at $\fov-\fmodul$ and $\fov+\fmodul$, respectively, but this triplet represents the only significant power in the frequency spectrum. No harmonics of 1O are detected. The star holds one of the shortest periods in the whole sample and large relative modulation amplitude of 53\,per cent. Light curve changes along the modulation cycle are illustrated with an animation available as Supporting Information.
\idlmc{1661}, $\pov=2.372772(5)$\,d, $\pmodul=29.10(2)$\,d. Peaks at $\fmodul$ and at $\fov+\fmodul$ detected with ${\mathrm{S/N}}=9.5$ and $5.9$, respectively -- see Fig.~\ref{fig:fspmodul}. Clear mean brightness modulation. Two additional significant signals detected in the frequency spectrum; the one close to $\fov$ may indicate secondary modulation with $\pmodul\approx 743$\,d.
\idlmc{1829}, $\pov=2.162057(6)$, $\pmodul=46.42(7)$. Non-stationary first overtone with strong residual of ${\mathrm{S/N}}=9.2$ after prewhitening. Peaks at $\fov-\fmodul$ and at $\fov+\fmodul$ detected with ${\mathrm{S/N}}=5.9$ and $5.4$, respectively -- see Fig.~\ref{fig:fspmodul}. The signal at $\fov-\fmodul$ is non-coherent. Additional signals detected as well.
\idlmc{2330}, $\pov=0.8447634(6)$\,d, $\pmodul=111.7(2)$\,d. Triplet components at $\fov+\fmodul$ and $\fov-\fmodul$ are detected with ${\mathrm{S/N}}$ of $6.4$ and $6.1$ respectively. Then, signals at the modulation frequency and its harmonic are detected with ${\mathrm{S/N}}=5.7$ and $4.3$, respectively -- see Fig.~\ref{fig:fspmodul}. First overtone is non-stationary. A weak (${\mathrm{S/N}}=3.2$) signature of quintuplet component ($\fov+2\fmodul$) is visible in the spectrum.
\idlmc{2344}, $\pov=4.09683(1)$\,d, $\pmodul=36.52(2)$\,d. A peak at $\fov-\fmodul$ is prominent with ${\mathrm{S/N}}=11.7$. After removing low frequency power excess with two low frequency ($\nu<0.01\cd$) sines weak modulation peaks at $2\fov-\fmodul$ (${\mathrm{S/N}}=3.9$) and at $\fmodul$ (${\mathrm{S/N}}=3.4$) are detected. While for these peaks ${\mathrm{S/N}}<4$ the simultaneous presence of three modulation-related peaks prompts us to classify the star as a modulation candidate.
\idlmc{2705}, $\pov=0.2645433(1)$\,d, $\pmodul=330.7(5)$\,d. A very clear modulation of large amplitude with rich frequency spectrum. The strongest modulation signals at $\fov-\fmodul$, $\fov+\fmodul$ and $2\fov-\fmodul$ are detected with ${\mathrm{S/N}}=17.2$, $6.8$ and $5.5$, respectively. Light curve changes along the modulation cycle are illustrated with an animation available as Supporting Information.
\idlmc{3985}, $\pov=4.54494(6)$\,d, $\pmodul=13.5767(8)$\,d. Extreme case of large amplitude modulation, with peculiar light curve changes during the modulation cycle. The strongest modulation peaks are detected at $\fov-\fmodul$, $2\fov-\fmodul$, $\fmodul$ and $3\fov-\fmodul$ with ${\mathrm{S/N}}=22.8$, $10.0$, $9.7$, and $8.3$, respectively. Light curve changes along the modulation cycle are illustrated with an animation available as Supporting Information.
\idlmc{4270}, $\pov=1.430489(3)$\,d, $\pmodul=540(2)$\,d. A very clear long period amplitude modulation. Triplet components at $\fov-\fmodul$ and at $\fov+\fmodul$ detected with ${\mathrm{S/N}}=13.1$ and $14.0$, respectively. Light curve changes along the modulation cycle are illustrated with an animation available as Supporting Information.
\idlmc{4361}, $\pov=1.805423(5)$\,d, $\pmodul=335(3)$\,d. Since 1-yr alias of a signal detected at $\fov-\fmodul$ is unresolved with $\fov$ the case is a bit ambiguous. Peaks at $\fov-\fmodul$ and $\fmodul$ are firmly detected with ${\mathrm{S/N}}=5.4$ and $4.6$, respectively.
\idsmc{4241}, $\pov=3.29770(3)$\,d, $\pmodul=123.6(1)$\,d. A very peculiar star. No harmonics of first overtone are detected in the frequency spectrum. After prewhitening with $\fov$ a comb of strong peaks is visible at low frequencies, at $\fmodul$, $2\fmodul$ and $3\fmodul$ with ${\mathrm{S/N}}=6.1$, $6.4$ and $5.1$, respectively, and at $\fov+\fmodul$ (${\mathrm{S/N}}=4.8$) -- see Fig.~\ref{fig:fspmodul}. In consecutive prewhitening steps we detect additional weak modulation peaks (all with $3.5<{\mathrm{S/N}}<4$) at $4\fmodul$, $\fov+2\fmodul$ and $\fov-2\fmodul$. All signals are present when data are transformed into fluxes, which proves both pulsation and modulation frequencies originate from the same star. The star appears to have a very peculiar modulation with strongly nonlinear mean brightness variation. For ground-based data, detection of $2\fmodul$ in Blazhko RRab stars is very scarce; for \idsmc{4241} we detect harmonics up to $4\fmodul$. Light curve changes along the modulation cycle are illustrated with an animation available as Supporting Information.
\idsmc{4372}, $\pov=1.451329(2)$\,d, $\pmodul=943(24)$\,d. Star shows weakly resolved, long-period modulation. Two modulation peaks at $\fov-\fmodul$ and $\fov+\fmodul$ are detected with ${\mathrm{S/N}}=6.8$ and $7.6$, respectively. Then, after prewhitening we detect one more modulation component at $2\fov+\fmodul$ (${\mathrm{S/N}}=4.3$) and additional unclassified periodicity.
\section{Discussion}\label{sec:discussion}
\subsection{Origin of the observed additional periodicities}
In Sect.~\ref{sec:results} we have divided the detected additional periodicities into a few classes based on their period ratios with respect to radial first overtone period, nature of the additional periodicity or the appearance of additional structures in the frequency spectra. For these classes, origin of the additional variability likely differs. Before discussing possible mechanisms behind excitation of additional variability, we first note that some of the detected signals may originate due to contamination, ie., are not intrinsic to Cepheid, but originate from other variable star in its direct neighbourhood. Signals we detect form regular patterns in the Petersen diagram, or are characterised by distinct properties, which is not expected due to contamination. We may thus conclude that majority of signals we detect are intrinsic to the studied Cepheids. Possible contamination, while may add some `noise', eg., to the appearance of the Petersen diagrams, is not expected to alter statistical properties of the classes we have identified.
\subsubsection{Double-periodic stars with $\px/\pov$ in between 0.60 and 0.65}
\label{ssec:discussion61}
\cite{wd16} proposed that the three sequences observed in the Petersen diagram (Fig.~\ref{fig:pet61}) are due to harmonics of non-radial modes of moderate degrees, $\ell=7$ (top sequence), 8 (middle sequence) and 9 (bottom sequence). Harmonics should be more easy to detect due to strongly reduced geometric cancellation. Non-radial modes can also be detected; assuming intrinsic mode amplitude is the same, modes of $\ell=8$ should reach the highest amplitudes followed by modes of $\ell=7$ and $\ell=9$. Non-radial modes are also expected to have more complex appearance in the frequency spectrum due to nonlinear mode interactions within multiplets, ie., between modes of different azimuthal orders. This is what we observe (Fig.~\ref{fig:pet61}, Tab.~\ref{tab:seqprop}). In the nomenclature of Sect.~\ref{ssec:61} non-radial modes correspond to sub-harmonics, with amplitudes ($\ash$) determined by the highest peak within wide power band that is commonly observed in the frequency spectrum. Sub-harmonics are most often detected for the middle sequence, then for the top sequence and they are scarce for signals forming the bottom sequence. Differences are clear and significant for the SMC. We note that for LMC, while sub-harmonics are still more frequent for the middle sequence, the incidence rate is comparable to that of the top sequence. These observations support \cite{wd16} model. The fact that we observe so many non-radial modes directly is puzzling however, and challenging to explain, as their observed amplitudes should be strongly reduced due to cancellation. In fact in the LMC stars for which the amplitude of non-radial mode is larger than that of the harmonic, $\ash>\ax$, dominate (Fig.~\ref{fig:61shamps}). In the next section we even suggest that many of signals with period ratios in between $0.75$ and $0.90$ may also be non-radial modes of the same degrees, for which harmonics are not detected at all. This, together with significant dispersion of period ratios in the Petersen diagram in the LMC may be challenging to reconcile within \cite{wd16} model.
The tentatively identified 4th sequence would, in light of \cite{wd16} model, correspond to $\ell=6$ non-radial modes. It includes 2 stars only, however, and detailed calculations for $\ell=6$ modes were not conducted so far.
We note that even for RRc stars, {\it TESS} mission reveals double-periodic stars with period ratios significantly below 0.61, possibly indicating the excitation of $\ell=10$ modes \citep[see fig.~13 in][]{MolnarTESS}.
\subsubsection{Non-radial $\ell=7-9$ modes without harmonic?}
\label{ssec:789noharm}
In Sect.~\ref{ssec:unclassified} we have identified a broad peak ($0.75<\pslr<0.90$) in the distribution of period ratios for non-coherent signals, with $\px>\pov$ (for LMC, see Fig.~\ref{fig:histadd}). A group of double-periodic stars of similar properties was detected by \cite{sa18b} (called 1.25-mode in their paper). These authors discussed the hypothesis that these signals may correspond to non-radial modes of degrees 7, 8 and 9. We put the same hypothesis and suggest that these stars are directly coupled to stars with $\px/\pov$ in between $0.60$ and $0.65$. A glance at Fig.~\ref{fig:pet61} shows that such period ratios are expected for non-radial modes of degrees 7, 8 and 9, see preceding Section. We also note that these non-radial modes most frequently appear as non-coherent signals. In Fig.~\ref{fig:pet61} we have defined two polygons (dashed lines), separately for the SMC and the LMC, that enclose majority of stars with non-radial modes of degrees 7, 8 and 9. The same polygons are also plotted in Fig.~\ref{fig:petadd} and we can see that the broad peak in the distribution of period ratios in Fig.~\ref{fig:histadd} is mostly due to stars within these polygons. We note that coherent signals also fall within these polygons and those in Fig.~\ref{fig:pet61} do correspond to non-radial modes of degrees 7, 8 and 9.
We speculate that majority of signals that fall within the polygons in Fig.~\ref{fig:petadd} are in fact non-radial modes of degrees 7, 8 and 9, for which harmonic was not detected in the frequency spectrum. These may be both non-coherent signals (which is more frequent) and coherent ones. Among stars with $\px/\pov$ in between 0.60 and 0.65 amplitudes of non-radial modes (denoted by $\ash$ in Sect.~\ref{ssec:61}) are often larger than amplitudes of their harmonics (denoted by $\ax$ in Sect.~\ref{ssec:61}). In fact, in the LMC $\ash>\ax$ is more common -- see Fig.~\ref{fig:61shamps}. Hence, it is likely, that we may just see non-radial mode directly, but not its harmonic, eg., due to large noise level.
In Fig.~\ref{fig:polygon} we plot the distribution of relative amplitudes for signals with $\px>\pov$ that fall within the polygons plotted in the right panels of Figs~\ref{fig:pet61} and \ref{fig:petadd}. We use data on more numerous LMC sample only. In the top panel we plot the distribution for non-coherent signals, while in the bottom panel we plot the distribution for all signals. With solid line we show the distribution for stars with $\px/\pov$ in between 0.60 and 0.65 (stars within polygon in Fig.~\ref{fig:pet61}), while dotted line shows the distribution for stars for which harmonics were not detected (signals within polygon in Fig.~\ref{fig:petadd}). Despite some differences, the distributions are similar and cover the same range, which also applies to the broad maxima. It strengthens the hypothesis that all these signals arise due to the same phenomenon.
\begin{figure}
\includegraphics[width=\columnwidth]{pap_polygon.eps}
\caption{Distribution of relative amplitudes for signals within polygons plotted in Fig.~\ref{fig:pet61} (solid line) and in Fig.~\ref{fig:petadd} (dashed line). Top panel shows distribution for non-coherent signals, while bottom shows distribution for all signals.}
\label{fig:polygon}
\end{figure}
\subsubsection{Close non-radial modes?}
\label{ssec:cnm}
Another group identified in Sect.~\ref{ssec:unclassified} constitutes of stars with additional coherent variability, majority of which have large period ratios ($\pslr>0.95$), ie., their frequencies are close to the radial mode frequency. For such signals, their relative amplitude is also typically larger (Fig.~\ref{fig:ampadd}). These signals cannot be explained as due to radial modes. Two explanations are possible. Additional variability may arise due to periodic modulation of a radial mode. Then, we expect to see triplets (multiplets in general) centred on radial mode frequency, or side peaks located on one side of the radial mode frequency and its harmonics (so called doublets; see Sect.~\ref{ssec:modulation}). In the discussed stars we see only a single peak. One may speculate that other modulation peaks have too small amplitudes, below the detection limit. This explanation is not likely however. We note that the same speculation motivated \cite{Kotysz} to revisit the stars sharing the above properties, first analysed by \cite{mk09} with OGLE-II data, to look for missing modulation components. They used more extended OGLE-III/IV data with significantly lower noise levels and found a signature of weak modulation in only one out of 42 stars. The class we address also outnumbers the modulated stars we have detected. If these single peaks were due to modulation, its properties would have to be very peculiar, with extremely asymmetric side peaks.
The more likely explanation seems excitation of low-degree non-radial modes with frequencies close to the radial mode frequency. Such scenario however is difficult to reconcile with pulsation theory. While low-order non-radial modes are expected to be excited in RR~Lyr stars \citep{VanHoolst,DziembowskiCassisi}, for classical Cepheids non-radial modes with $\ell<4$ are expected to be damped \citep{wd77,Osaki77,MuletMarquis}.
\subsubsection{Stars with period ratios $\pov/\px$ around 0.684}
This group is primarily known in RR~Lyr stars. In classical Cepheids it was first reported by \cite{sa18b} (called 1.46-mode in their paper). We confirm its existence. In the Petersen diagram it is not as distinct as is the case for RR~Lyr stars, still the distribution of period ratio leaves little doubt that also in 1O Cepheids the class exists, showing a bit smaller median period ratio, $0.684$ instead of $0.686$ in RRc stars. The basic difficulty in explaining this class is long period of the additional periodicity -- longer than period of the radial fundamental mode expected for a corresponding first overtone period. Except a few comments on RRc stars \citep{wd16} no model was proposed to explain this class. A model should explain existence of this class in both RRc and 1O Cepheids, narrow distribution of period ratios and coherent nature of the additional variability.
\subsubsection{Stars with periodic modulation of pulsation}
Periodic modulation of pulsation is a well know feature of RR~Lyr stars and is called the Blazhko effect \citep{Blazhko}. It is common among fundamental mode pulsators \citep[incidence rates on order of 50\,per cent, for a review see][]{GezaBlazhkoIR}, less common in first overtone ones \citep[incidence rates of a few per cent, see eg.,][]{NetzelBlazhkoRRc} and detected also in double-mode (RRd) pulsators \citep[eg.,][]{JurcsikM3RRd,SmolecBlazhkoRRd}. Detections of periodic modulation of pulsation in classical Cepheids are scarce, however new discoveries have been appearing regularly in recent years. These include modulations of radial modes in double-mode Cepheids \citep{mk09, Rathour2021} and single-mode Cepheids \citep{s17, Rathour2021}. For single-mode overtone Cepheids, mostly long-period, large amplitude modulations, well visible already in the light curve were reported \citep[eg.,][]{MolnarSzabados,o4_clouds_multimode}. Here we reported several modulated 1O Cepheids based on the appearance of their frequency spectra. The existence of periodic modulation of pulsation in Cepheids is now firmly established. We refrain from calling the effect `Blazhko'. Certainly it is much more scarce as compared to RR~Lyr stars. In majority of cases it is a low-amplitude effect. The mechanism behind might be different.
We still lack satisfactory explanation behind Blazhko modulation in RR~Lyr stars, although the half-integer resonance model seems the most promising one -- see \cite{KollathRev} for a review. In \cite{s17} we speculated that a resonance of a different type (2:1 between radial fundamental and radial second overtone) may be crucial in bringing modulation in F-mode Cepheids, which was motivated with the highest incidence rate of modulated stars close to the resonance center. We don't see an obvious resonance, or any other mechanism that might work for 1O Cepheids.
\subsection{Properties of 1O Cepheids in the LMC and SMC}
\label{ssec:propSLMC}
A glimpse at the Petersen diagram in Fig.~\ref{fig:petall} reveals a significant difference between the two Magellanic Clouds: additional variability beyond radial modes is detected much more frequently in the LMC. For all discussed classes of pulsations, incidence rates are higher in the LMC. The difference is intrinsic and is not caused by the higher detection limit in the SMC, as we have demonstrated in Sect.~\ref{ssec:unclassified}. Thus, higher metallicity environment seems to favour excitation of additional variabilities beyond radial modes.
Except different incidence rates, different properties within the discussed classes were also reported and are most prominent in double-periodic stars with $\px/\pov$ in between 0.60 and 0.65. We observe the following. (i) Double-periodic stars in the LMC have on average longer period. This trend with increasing metallicity was already noted by \cite{Rathour2021} who also analysed classical Cepheids in the Galactic fields. (ii) The appearance of the Petersen diagram is much more `noisy' for LMC stars. The three sequences, although still well visible, are not as clearly cut as in the SMC. There are several stars in between the sequences. (iii) For stars in which signals at $\fx$ and centred at $1/2\fx$ were detected, distribution of corresponding amplitudes, $\ax$ and $\ash$, respectively, is different. While in the SMC $\ax>\ash$ is more frequent, in the LMC $\ax<\ash$ is more frequent.
Another striking difference is the appearance of the modulations in the two systems. First, the incidence rate of modulated stars is very low in the SMC: we have detected only 3 modulated stars there, while 24 were detected in the LMC. By applying the SMC's detection limit, to LMC stars, we still should detect 8 modulated stars (ie., at least two modulation side peaks would be detected).
While the statistics is low, we note that stars in the SMC have, on average, longer modulation periods.
Any model aimed at explaining the different classes of pulsations we have described, should also explain the metallicity/environmental dependencies.
\section{Conclusions}
We have analysed OGLE-IV $I$-band photometry for first overtone classical Cepheids in the Magellanic Clouds. Our most important findings are the following.
(i) We have identified new candidates for radial double-mode pulsation, including 46 candidates for F+1O pulsation, 21 candidates for 1O+2O pulsation and 4 candidates for 1O+3O pulsation.
(ii) In 225 stars in the SMC and 291 stars in the LMC we have detected additional signals that form period ratios with respect to first overtone $\px/\pov$ in the 0.60 -- 0.65 range. In the Petersen diagram these stars form three sequences -- clearly cut in the case of SMC and more diffuse in the LMC. Quantitative method to divide stars into sequences was proposed. For a significant fraction of these stars, signals centred at sub-harmonic frequencies are detected as well, which according to model proposed by \cite{wd16} are non-radial modes of moderate degrees, 7, 8 and 9. Distribution of these signals supports the Dziembowski's model. In the light of this model, the distribution of amplitudes of the detected signals is puzzling. Amplitudes of non-radial modes should be strongly reduced due to geometric cancellation, and their harmonics should be much more easy to detect. In the LMC we find that amplitudes of non-radial modes are typically larger than amplitudes of their harmonics.
(iii) Two stars were tentatively classified as belonging to the fourth sequence with highest $\px/\pov$ period ratios. Extending the \cite{wd16} model, the fourth sequence would arise because of non-radial $\ell=6$ modes.
(iv) We have identified 28 stars with additional variability of period longer than period of the radial fundamental mode expected for a given 1O period. Period ratios tightly cluster around $\pov/P=0.684$. Additional signals are coherent; typical amplitude is 3\,per cent of the radial 1O amplitude. In Cepheids, this group was first discovered by \cite{sa18b} and is analogue of double-periodic pulsation known in first overtone RR~Lyr stars.
(v) After clean-up of the sample from signals corresponding to the groups discussed above, hundreds of signals are still present across the Petersen diagram. The following regularities were spotted. For majority of signals frequencies are smaller than the frequency of the radial first overtone. Distribution of period ratios peaks for $\pslr>0.95$, then we observe a plateau down to $\pslr=0.75$ and decline further on. Coherent signals (of constant amplitude and phase) constitute the vast majority of signals close to the radial mode frequency ($\pslr>0.95$). These signals have, on average, larger relative amplitudes. The plateau is built up by both coherent and non-coherent signals. For non-coherent signals we observe a broad maximum for period ratios in between 0.75 and 0.9. We suggest that coherent signals with close to unity period ratios may be low-order non-radial modes. Signals with lower period ratios, in particular non-coherent ones, may represent non-radial modes of degrees 7, 8 and 9, as first suggested by \cite{sa18b}. Both explanations pose a challenge for theory, as low order non-radial modes are linearly damped, and for moderate-degree modes amplitudes should be strongly reduced by geometric cancellation.
(vi) In many stars various forms of multiperiodic pulsation are present simultaneously. These include eg., radial double-mode stars with additional signals with $\px/\pov$ in the $0.60-0.65$ range or clustered around $\pov/\px=0.684$. The latter two forms are also present simultaneously in two stars. All these stars are promising targets for detailed asteroseismic investigation.
(vii) In the frequency spectra of 27 stars (24 in the LMC, 3 in the SMC) we detect periodic modulation of pulsation of radial first overtone. Modulation periods are in between $\approx 13$\,d and $\approx950$\,d. Relative modulation amplitudes may be as high as 76\,per cent. Periodic modulation of pulsation is now firmly established in nearly all classes of single and double-mode pulsation in Cepheids. Still phenomenon is much more scarce than in RR Lyrae stars, in which it is known as the Blazhko effect.
(viii) Incidence rates for all discussed phenomena are larger in the more metal rich LMC. We have demonstrated that it does not result from higher detection limit in the SMC, but is intrinsic property of 1O Cepheids. Higher metallicity environment favours excitation of additional variabilities.
\section*{Acknowledgements}
This research was supported by the Polish National Science Centre, SONATA BIS grant, 2018/30/E/ST9/00598 and OPUS grant, DEC-2015/17/B/ST9/03421. RS acknowledges fruitful discussion with Pawe\l{} Moskalik.
\section*{Data Availability}
OGLE data analysed in this paper are publicly available through OGLE Collection of Variable Stars. Basic properties of the discussed stars are available through Tables published online along with the paper. Full frequency solutions for analysed stars are available on request.
\bibliographystyle{mnras}
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{
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\section{Introduction}
\setcounter{equation}{0}
\defC.\arabic{equation}{1.\arabic{equation}}
Granular media \cite{0} are by now recognised as
being paradigms of complexity \cite{comp}, especially near their
jamming limit \cite{sidjamming}.
The fact that most grains are too large
to be perturbed by the effect of room temperature leads to an `athermal'
dynamics, which is a major cause of this complexity -- configurations
once generated, are remembered, and their hysteretic effects persist in
any ensuing dynamics, and the new configurations generated therefrom.
Another
origin of complexity in such systems is their generic disorder; this leads
in particular to a random landscape of metastable states being explored if
a granular system is suitably `quenched', rather than a crystallisation into
an appropriate ordered state.
The way in which this landscape is explored depends
strongly on the driving forces applied, in the absence of any real
thermodynamics.
The work we present here is an attempt to explore many of these issues, in
particular those to do with the nature of ground states in a system near
jamming, and how these are reached.
This is the main motivation for our
focusing on the effect of `zero-temperature' dynamics on the model we will
later introduce, since zero-temperature dynamics are known to be the route
to systemic ground states.
Another aspect of interest around which our model
was designed was the exploration of spatial inhomogeneities, to answer
questions such as: how does position along a column of grains influence the
dynamics observed there? This is an important question for several reasons,
one of the earliest being that experimental measurements of density
along a column of grains
revealed wide variations \cite{sid}, depending on whether they were near
the top, the bottom or the middle -- in its turn, this implied that the
compactivity of the system \cite{edwards} was non-uniform.
A more visible manifestation of spatial inhomogeneities is the presence of
force chains \cite{bob}
and bridge networks \cite{bridges}, which are unique signatures of the
granular state.
Given the complexity of issues we wish to investigate, we have chosen
a minimal model to work with, whose ingredients are based on our experience
with several earlier ones \cite{I,II,III,IV}.
A feature that all the models
share is that of orientational disorder; every grain is allowed to occupy
one of two states, corresponding to `ordered' and `disordered'.
Disordered
orientations generate voids and waste space, whereas ordered ones do not.
Implicit in
this description is the effect of shape, which is most easily understood
in terms of the rectangular grains of aspect ratio $a$
considered in \cite{I,II}.
Grains aligned
along their long edges (length $1$) result in a fully packed column,
whereas those perched
on their short edges (length $a$) leave voids of size $1-a$.
The horizontal orientation is thus ordered, and the vertical one disordered.
Our models became progressively more sophisticated.
The earliest model of rectangular grains considered in \cite{I,II}
was strictly non-interacting, with the only effects included being due
to gravity and
excluded volume -- grains could not overlap, and those that were deeper
in a column were less free to move.
One of the major extensions in \cite{III,IV} was the introduction of
the shape parameter $\varepsilon$ to include grains of
arbitrary, i.e., non-rectangular shape -- such grains can have multiply many
orientations in reality, but the two-state description was retained
in the interests of simplicity.
The parameter $\varepsilon$ was allowed
to vary over all rational and irrational numbers, with a view to describing
regular and irregular grains, as will be explained further below.
Another difference between the two models was that the first one \cite{I,II}
interpolated between
jammed and fluidised regimes, whereas the second one was explicitly
constructed to examine the jamming regime.
In the latter case, it seemed reasonable to
assume that translational diffusion was
essentially absent, with all compaction occurring via orientational
rearrangements -- it was therefore appropriate to focus on a column, rather
a box \cite{I,II}, of grains.
Interactions were introduced in the column model of \cite{III,IV}
such that every grain now not only
felt the weight of grains above it, but was also constrained by
all of their orientations.
This modelling was appropriate as a representation
of grains in the free surface layer, which were too far from the base to
feel an undertow; in the jamming limit, however, such grains would be
expected to feel the orientational constraints arising from grains
{\it above} them.
The present model incorporates all of the above features, and generalises
them to include the presence of
orientational constraints arising from grains {\it below} a given grain.
Unlike our previous models \cite{I,II,III,IV}, where interactions propagated
downwards from the top, here they propagate {\it both upwards and downwards}.
Clearly the extent of this propagation depends on grain position
In this sense we model a column of grains with a top, a middle and a bottom.
We devote this paper to a comprehensive investigation of the ground states
of this model, and how they are reached via zero-temperature dynamics.
The main questions we will answer along the way will be related to several of
the issues mentioned above; in particular the issue of spatial inhomogeneities
will be relevant, since the dynamical regimes attainable via this model
will depend on which part of the column -- top, middle or bottom -- is being
examined.
Additionally, we will see that there is a panoply of {\it metastable}
ground states available to the system; the dynamics of their attainment
will allow us to classify the appropriate regimes as {\it ballistic},
{\it logarithmic}, {\it activated} and {\it glassy}.
The glassy regime is by far the most novel and
interesting of these regimes, and will be investigated in
greater depth than the others; its full exploration is, however,
reserved for future work.
The plan of this paper is as follows.
The definition of the model is given in Section 2.
Section 3 contains an investigation of its static properties,
with an emphasis on the ground states.
Section 4 presents a study of zero-temperature dynamics:
a rich phase diagram with four dynamical phases is revealed
and investigated thoroughly.
A discussion is presented in Section 5.
Exact results for small systems, as well as other technical results,
are presented in three appendices.
\section{The model}
\setcounter{equation}{0}
\defC.\arabic{equation}{2.\arabic{equation}}
Like the unhindered (fully directional) model described in \cite{III,IV},
the present model consists of a finite column of $N$ grains,
labelled by their depth $n=1,\dots,N$.
Each grain assumes two orientational states,
which are referred to as {\it ordered} and {\it disordered}.
We set $\sigma_n=+1$ (resp. $\sigma_n=-1$) if grain number $n$
is ordered (resp. disordered).
A configuration of the column is therefore uniquely defined
by the orientation variables $\{\sigma_n\}$.
There are $2^N$ such configurations.
The model is defined by a stochastic dynamics
which do not obey detailed balance.
In the present context, detailed balance essentially means a symmetry
in $m$ and $n$ on the dynamical effect of
grain orientation $\sigma_m$ on $\sigma_n$.
The expressions for the local fields (\ref{hdef}), (\ref{hjdef})
clearly do not obey such `action and reaction'.
Our model is therefore intrinsically out of equilibrium,
and its stationary state at finite temperature
is a genuine {\it non-equilibrium steady state}
More precisely, the model is defined as follows.
Grains are selected in a random sequential fashion
and updated with the orientation-flipping rates
\begin{equation}
\left\{\matrix{
w(\sigma_n=+1\to\sigma_n=-1)=\exp\left(-\frac{\lambda_n+H_n}{\Gamma}\right),\hfill\cr\cr
w(\sigma_n=-1\to\sigma_n=+1)=\exp\left(-\frac{\lambda_n-H_n}{\Gamma}\right),\hfill
}\right
\label{rates}
\end{equation}
where, along the lines of previous work \cite{I,II,III,IV}:
\noindent$\bullet$
$\Gamma$ is a dimensionless vibration intensity, referred to as temperature,
and related to the `fast' temperature \cite{0} in granular media.
\noindent$\bullet$
$\lambda_n$ is the activation energy felt by grain $n$.
We make the assumption that it increases linearly with the depth $n$,
but otherwise does not depend on grain orientations.
We set
\begin{equation}
\lambda_n=\frac{n\Gamma}{\xi_{\rm dyn}},
\end{equation}
so that the local frequency
\begin{equation}
\o_n=\exp\left(-\frac{\lambda_n}{\Gamma}\right)=\exp\left(-\frac{n}{\xi_{\rm dyn}}\right)
\label{odef}
\end{equation}
falls off exponentially, with a characteristic length $\xi_{\rm dyn}$.
This {\it dynamical length} corresponds to the depth of the boundary layer
beyond which grains are frozen out by the sheer weight of grains above them.
\noindent$\bullet$
$H_n$ is the local ordering field felt by grain $n$,
which determines the orientational response of grain $n$
to the orientations $\{\sigma_m\}$ of all the other grains.
In previous work \cite{III,IV} the local field $H_n$ only involved
the uniform effect of the upper grains ($m=1,\dots,n-1$).
In the present model, we also take into account the {\it back-propagation}
from grains below a given grain $n$ ($m=n+1,\dots,N$).
This effect cannot be similarly uniform.
We assume for simplicity that upward interactions are exponentially damped,
with a characteristic length $\xi_{\rm int}$, the {\it interaction length}.
We therefore set
\begin{equation}
H_n=h_n+gj_n,
\label{hdef}
\end{equation}
where we denote by $h_n$ the {\it uniform} effect of grains {\it above} $n$
($m=1,\dots,n-1$),
and by $j_n$, the {\it non-uniform} effect of grains {\it below} $n$
($m=n+1,\dots,N$), whose strength is measured by a small (positive)
coupling constant $g$.
Both components $h_n$ and $j_n$ of the local field
contain the contribution of every single grain orientation $\sigma_m=\pm1$
through the quantity $f(\sigma_m)$.
As before \cite{III,IV}, the latter assumes the following two values:
\begin{equation}
f(\sigma_n)=\left\{\matrix{
\varepsilon\hfill&\hbox{if}\hfill&\sigma_n=-1,\cr
-1\ \hfill&\hbox{if}\hfill&\sigma_n=+1.}\right
\label{fdef}
\end{equation}
A useful equivalent formula is the following:
\begin{equation}
f(\sigma_n)={\textstyle{\frac12}}(\varepsilon-1-(\varepsilon+1)\sigma_n).
\label{fdefalt}
\end{equation}
In terms of this quantity, $h_n$ and $j_n$ are:
\begin{equation}
\matrix{
\displaystyle{h_n=\sum_{m=1}^{n-1}f(\sigma_m)},\hfill\cr
\displaystyle{j_n=\sum_{m=n+1}^Nf(\sigma_m)\,\exp\left(-\frac{m-n}{\xi_{\rm int}}\right)}.}
\label{hjdef}
\end{equation}
The positive shape parameter $\varepsilon$
represents an `effective aspect ratio' for a grain of arbitrary shape.
This interpretation originated in the framework of the non-interacting model
\cite{I,II} with rectangular grains.
Rational values of $\varepsilon=p/q$ imply that the grain size is expressible
by a rectangle of sides $p$ and $q$.
Such grains are brick-like, and therefore can be packed perfectly
to build some periodic tiling.
On the other hand, when $\varepsilon$ is irrational, such tilings cannot be built,
so that the most close-packed
states are those of optimal, rather than perfect packing.
We thus continue to make the following equivalence \cite{III,IV}:
{\it rational} values of $\varepsilon$ imply grains of {\it regular} shape,
while {\it irrational} values of $\varepsilon$ imply grains of {\it irregular} shape.
The parameters of the model are therefore the number of grains $N$,
the coupling constant $g$, the shape parameter $\varepsilon$,
and most importantly the interaction and dynamical lengths $\xi_{\rm int}$ and $\xi_{\rm dyn}$.
The unhindered model of references \cite{II,III,IV}
is recovered in the absence of coupling ($g=0$).
Throughout the following, we use the notations
\begin{equation}
x_{\rm dyn}={\rm e}^{-1/\xi_{\rm dyn}},\quadx_{\rm int}={\rm e}^{-1/\xi_{\rm int}}.
\label{zdef}
\end{equation}
To close up, we mention the following recursion relations obeyed
by the components $h_n$ and $j_n$:
\begin{equation}
\matrix{
h_n=h_{n-1}+f(\sigma_{n-1}),\hfill\cr
j_n=x_{\rm int}\left(f(\sigma_{n+1})+j_{n+1}\right).}
\label{hjrel}
\end{equation}
These relations, together with the boundary values $h_1=j_N=0$,
provide a fast algorithm to evaluate the local fields,
to be used in numerical simulations.
On the other hand, the relations (\ref{hjrel}) also imply
\begin{equation}
f(\sigma_n)=h_{n+1}-h_n=\frac{j_{n-1}}{x_{\rm int}}-j_n.
\end{equation}
The latter equation determines the $j_n$ in terms of the $h_n$:
\begin{equation}
\matrix{
j_n=-x_{\rm int} h_{n+1}+j_n^{(1)},\hfill\cr
\displaystyle{j_n^{(1)}=(1-x_{\rm int})\sum_{m=n+2}^Nx_{\rm int}^{m-n-1}h_m+x_{\rm int}^{N-n}h_{N+1}}.}
\label{jexplicit}
\end{equation}
Throughout the following, we choose to impose for convenience
the boundary condition that the uppermost grain is ordered:
\begin{equation}
\sigma_1=+1.
\label{init}
\end{equation}
\section{Statics. Ground states}
\setcounter{equation}{0}
\defC.\arabic{equation}{3.\arabic{equation}}
The dynamical rules simplify in the zero-temperature limit ($\Gamma\to0$).
Indeed (\ref{rates}) yields (provided $H_n\ne0$):
\begin{equation}
\matrix{
\frac{w(\sigma_n=-1\to\sigma_n=+1)}{w(\sigma_n=+1\to\sigma_n=-1)}
=\exp\left(\frac{2H_n}{\Gamma}\right)\hfill\cr
{\hskip 37.6truemm}
\to\left\{\matrix{\infty\hfill&\hbox{if}\hfill&H_n>0,\hfill\cr
0\hfill&\hbox{if}\hfill&H_n<0.\hfill
}\right
}
\label{zero}
\end{equation}
Along the lines of references \cite{II,III,IV},
a {\it ground state} of the column is defined as a configuration
where the orientation of every grain is aligned along its local field:
\begin{equation}
\sigma_n=\mathop{\rm sign}\nolimits H_n=\left\{\matrix{
+1\ \hfill&\hbox{if}\hfill&H_n>0,\hfill\cr
-1\hfill&\hbox{if}\hfill&H_n<0.\hfill
}\right
\label{zerost}
\end{equation}
The ground states of the unhindered model ($g=0$)
have been investigated in \cite{III,IV}.
In that case, the local field $H_n=h_n$ acting on grain $n$
only depends on the grains above $n$.
Equation (\ref{zerost}) therefore yields
a recursive procedure allowing one to construct ground states:
\begin{equation}
\left\{\matrix{
h_n>0\Longrightarrow\sigma_n=+1,\ \hfill&h_{n+1}=h_n-1,\hfill\cr
h_n<0\Longrightarrow\sigma_n=-1,\ \hfill&h_{n+1}=h_n+\varepsilon.\hfill
}\right
\label{step}
\end{equation}
In a ground state, all the local fields $h_n$ lie in the range
\begin{equation}
-1\leq h_n\leq\varepsilon.
\label{range}
\end{equation}
For the present model with a non-zero coupling constant $g$,
things are more complex.
The local field $H_n$ given in (\ref{hdef}) now depends both
on the grains above $n$ (through $h_n$)
and on the grains below $n$ (through $j_n$).
The condition (\ref{zerost}) therefore couples
all the orientation degrees of freedom.
In particular the ground states admit no recursive construction
similar to (\ref{step}).
It is therefore a non-trivial task to generate all the ground states
of a finite column of $N$ grains, whose number depends on
$g$ and $\xi_{\rm int}$ in general.
The situation however simplifies in the weak-coupling regime ($g\ll1$),
where the ground states can be understood
in terms of those of the unhindered model ($g=0$)
by means of a stability analysis.
Just as in the case of the unhindered model \cite{III,IV},
rational and irrational values of $\varepsilon$ have to be considered separately,
in the case of the present hindered model.
\subsection{Rational $\varepsilon$ (regular grains)}
We first recall some facts about the ground state structure
in the unhindered model.
Using (\ref{step}) in that case, we find that
whenever the shape parameter $\varepsilon$ is a rational number (in irreducible form)
\begin{equation}
\varepsilon=\frac{p}{q},
\end{equation}
the fields $h_n$
vanish at all depths $n$ such that $n-1$ is an integer multiple of $p+q$.
The corresponding orientation $\sigma_n$ is left unspecified, and can be chosen at
will.
Ground states are therefore all the random sequences
made of two well-defined patterns, $P_1=+-\cdots$ and $P_2=-+\cdots$
These patterns have the same length $p+q$;
they are made of $p$ ordered and $q$ disordered grains,
and only differ by the orientations of their two uppermost grains.
The model therefore has an exponentially large
number of ground states, of the form $\exp(N\S)$,
where the configurational entropy \cite{sconf} reads
\begin{equation}
\S=\frac{\ln 2}{p+q}.
\end{equation}
It turns out that these ground states
are still the exact ones for the present model,
as long as the coupling constant $g$ is smaller than some threshold $g_c$.
In order to show this, we approach the problem via a stability analysis.
We assume that the $h_n$ field predominates,
consider $j_n$ as a small perturbation,
and see whether the ground states for $g=0$ are still stable
in the presence of the additional feedback of the $j_n$ field.
\noindent$\bullet$
Consider first the case when a grain is at
a depth $n=(p+q)m+1$ for some integer $m=1,2,\dots$
The local field $h_n$ vanishes, so that the orientation $\sigma_n=\eta=\pm1$
can be chosen at will for $g=0$, i.e., in the zeroth order approximation.
Once this choice is made, we have $h_{n+1}=f(\eta)$,
$\sigma_{n+1}=-\eta$, and $h_{n+2}=\varepsilon-1$.
In order to see if the ground state is stable when the effect
of the $j_n$ field is included, we have to first estimate $j_n$.
Equation (\ref{jexplicit}) can be iterated once, yielding
\begin{equation}
\matrix{
j_n=-x_{\rm int} h_{n+1}+x_{\rm int}(1-x_{\rm int})h_{n+2}+j_n^{(2)},\hfill\cr
\displaystyle{j_n^{(2)}=(1-x_{\rm int})\sum_{m=n+3}^Nx_{\rm int}^{m-n-1}h_m+x_{\rm int}^{N-n}h_{N+1}}.}
\end{equation}
First, we notice that the inequalities (\ref{range})
allow one to bound the remainder $j_n^{(2)}$
as $-x_{\rm int}^2\leq j_n^{(2)}\leqx_{\rm int}^2\varepsilon$.
Then, the values of $h_{n+1}$ and $h_{n+2}$ obtained above
yield the inequalities
$j_n>x_{\rm int}(1-x_{\rm int})\varepsilon$ if $\sigma_n=+1$ and $j_n<-x_{\rm int}(1-x_{\rm int})$ if $\sigma_n=-1$.
Therefore, the orientation $\sigma_n$ is aligned with the local field $H_n=gj_n$,
irrespective of the coupling constant $g$.
\noindent$\bullet$
Consider now a depth $n$ which is not of the form $n=(p+q)m+1$, so that
the local field $h_n$ is non-vanishing.
As $h_n$ is an integer linear combination
of terms $f(\sigma_m)$ equal either to $-1$ or to $\varepsilon=p/q$,
it therefore obeys
\begin{equation}
\abs{h_n}\ge\frac{1}{q}.
\label{hest}
\end{equation}
On the other hand, (\ref{jexplicit}) can again be used to estimate $j_n$.
The inequalities (\ref{range}) imply $-x_{\rm int}\leq j_n^{(1)}\leqx_{\rm int}\varepsilon$, and
\begin{equation}
\abs{j_n}<(1+\varepsilon)x_{\rm int}=\frac{p+q}{q}\,x_{\rm int}.
\label{jest}
\end{equation}
The inequalities (\ref{hest}) and (\ref{jest}) imply
\begin{equation}
\abs{\frac{h_n}{j_n}}>\frac{1}{(p+q)x_{\rm int}}.
\label{hjest}
\end{equation}
The full local field $H_n$ therefore
has the same sign as $h_n$ for all $n$, provided the coupling constant $g$
is smaller than some threshold $g_c$,
so that $h_n$ is the dominant field in this weak-coupling regime.
While the arguments above do not allow one to predict exact values
of $g_c$\footnote{The exact threshold coupling $g_c$ will, however,
be determined later on for $\varepsilon=1$ (see (\ref{gc1})).}
for generic rational $\varepsilon$, they do yield a lower bound:
\begin{equation}
g_c\ge\frac{1}{(p+q)x_{\rm int}}.
\label{gcdef}
\end{equation}
Also, similar arguments show that no other ground states exist in this model
in the same range of $g$, allowing us to identify, for all $g<g_c$,
the ground states of this model with those of the
unhindered model \cite{III,IV}.
\subsection{Irrational $\varepsilon$ (irregular grains)}
Once again, we review the nature of the ground states
for irregular grains (irrational $\varepsilon$) in the earlier
unhindered model \cite{III,IV}.
A unique ground state
(corresponding to optimal, rather than perfect packing) is obtained
for each value of $\varepsilon$ in that case, such that
all the fields $h_n$ generated by the recursion procedure (\ref{step})
are non-zero.
The main feature of this ground states is that it is {\it quasiperiodic}.
The nature of the ground states in the hindered model under discussion here
can be predicted by analogy with the low-temperature excitations
of the unhindered model.
Indeed it turns out that the presence of
a weak coupling ($g\ll1$) nucleates disorder
in this model, in the same way as a low but finite temperature
($\Gamma\ll1$) destroys the ground state of the unhindered model \cite{IV}.
More precisely, as long as the local field $H_n$ has the same sign as $h_n$,
i.e., for $\abs{h_n}>g\abs{j_n}$, the ground states of both unhindered
and hindered models are identical.
The depth up to which this stability condition is satisfied
can be estimated as follows.
Equation (\ref{jest}) shows that typical values of $j_n$ are of order $x_{\rm int}$.
The stability of a given ground state is determined by the grains $n$
where $h_n$ and $j_n$ are comparable, i.e., $\abs{h_n}\sim gx_{\rm int}\ll1$.
Such sites with very small $h_n$ fields are nothing but the
{\it nucleation sites} which dominate
the low-temperature behaviour of the unhindered model \cite{IV}.
The typical distance $\L(g)$ between two consecutive nucleation sites diverges
as
\begin{equation}
\L(g)\sim\frac{1}{gx_{\rm int}}
\label{lgdef}
\end{equation}
in the regime of a weak coupling ($gx_{\rm int}\ll1$).
Thus -- as one might expect -- the closer $g$ is to zero, the less chance
there is that disorder is nucleated.
As long as the size $N$ of the column is smaller than $\L(g)$, or,
equivalently, the coupling constant $g$ is smaller than
$1/(Nx_{\rm int})$, the unique ground state of the model is the quasiperiodic
ground state of the unhindered model.
For larger columns, a typical ground state can be thought of
as a sequence of independent quasiperiodic patches of mean length $\L(g)$,
pasted together end on end.
For very large sizes ($N\gg\L(g)$), there is
therefore an exponentially large number of such ground states.
The corresponding configurational entropy:
\begin{equation}
\S(g)\sim\frac{1}{\L(g)}\sim gx_{\rm int}
\end{equation}
vanishes in the weak-coupling regime ($gx_{\rm int}\ll1$).
Finally, we can provide an integrated description of the ground states
of an irrational $\varepsilon$ and of its rational approximants.
For $g$ small and $\varepsilon$ a fixed irrational, we consider the
sequence of its rational approximants $\varepsilon_k=p_k/q_k$ \cite{IV}.
The periods $p_k+q_k$ of these approximants typically grow exponentially
fast with the approximant order $k$.
Equation (\ref{gcdef}) implies the following.
For the first rational approximants in the series,
whose periods are smaller than $1/(gx_{\rm int})$,
the ground states are the same as in the unhindered model.
For all the higher approximants, whose periods exceed this threshold,
ground states are expected to be made of nearly independent patches,
whose characteristic length $\L(g)$ is given by (\ref{lgdef}),
just as for the limiting irrational.
Notice that the period of the approximant that divides these two behaviours
is of the order of $\L(g)\sim1/(gx_{\rm int})$ -- there is a reassuring consistency in
this.
\subsection{More details for $\varepsilon=1$}
We now focus on the simplest case, which is obtained when
the shape parameter equals $\varepsilon=1$.
The complex behaviour we obtain
even from this simplest of all cases is a tribute to the inherent
richness of the model.
\subsubsection*{Generic configurations}
Assume that the column size $N$ is even for definiteness.
Consider first a generic configuration.
Equation (\ref{fdef}) simplifies for $\varepsilon=1$ to $f(\sigma_n)=-\sigma_n$.
As a consequence, the components of the local fields read
\begin{equation}
h_n=-\sum_{m=1}^{n-1}\sigma_m,\quad
j_n=-\sum_{m=n+1}^Nx_{\rm int}^{m-n}\sigma_m.
\label{hjsum}
\end{equation}
The first expression shows that $h_n$ is an integer, whose parity is
opposite to $n$.
Therefore:
\noindent$\bullet$
When the depth $n=2k$ is even, $h_n$ is odd, and thus always non-vanishing.
In the weak-coupling regime ($g\ll1$), we thus obtain
$H_n\approx h_n$, irrespective of $\xi_{\rm int}$.
\noindent$\bullet$
When the depth $n=2k-1$ is odd, $h_n$ is even, and may therefore vanish.
When it does, we have $H_n=gj_n$, so that the sign of $H_n$
depends on $\xi_{\rm int}$ in general.
In order to understand the complexity which can arise from
this dependence on the interaction length $\xi_{\rm int}$, let us focus on
a column of size $N=6$, in the particular configuration $+-+-++$.
We have $\sigma_3=+1$, $h_3=0$, and $H_3=gj_3=gx_{\rm int}(1-x_{\rm int}-x_{\rm int}^2)$.
The parenthesis is a second-degree polynomial in $x_{\rm int}$.
It vanishes (in the physical range $0<x_{\rm int}<1$) when $x_{\rm int}$ equals
the inverse golden mean
\begin{equation}
\phi=\frac{\sqrt{5}-1}{2}\approx0.61803.
\label{phidef}
\end{equation}
Thus when $0<x_{\rm int}<\phi$, $H_3=gj_3$ is positive, and
the condition $\sigma_3=\mathop{\rm sign}\nolimits H_3$ is fulfilled.
This condition does not hold for $\phi<x_{\rm int}<1$.
This example is illustrative of a general property of the model.
For a finite column made of $N$ grains, the statics and dynamics
of the model depend on the relative position of $x_{\rm int}$
with respect to a finite number of threshold values,
where quantities of interest are discontinuous in general.
These threshold values are given by the following rule:
they are all the roots (in the physical range $0<x_{\rm int}<1$)
of all the reduced fields $j_n/x_{\rm int}$ with $n\ge3$ odd,
viewed as polynomials in $x_{\rm int}$, in all the configurations.
The polynomial $j_n(x_{\rm int})/x_{\rm int}$ has even degree $d=N-n-1$, so that $d\le N-4$.
These threshold values all lie in the range $1/2<x_{\rm int}<1$.
This fact can be seen as follows.
The expression (\ref{hjsum}) for the local field $j_n$ can be recast as
\begin{equation}
j_n=-x_{\rm int}\sigma_{n+1}-\sum_{m=n+2}^Nx_{\rm int}^{m-n}\sigma_m.
\label{jrecast}
\end{equation}
The sum in the right-hand side is smaller than $x_{\rm int}^2/(1-x_{\rm int})$
in absolute value.
As a consequence, the local field $j_n$ always has the sign of the leading
term, and therefore cannot vanish,
as long as $x_{\rm int}>x_{\rm int}^2/(1-x_{\rm int})$, i.e., $x_{\rm int}<1/2$.
The above property is responsible for a strikingly general result:
no dynamical quantities for arbitrary system sizes
depend on $x_{\rm int}$ for $0<x_{\rm int}<1/2$, i.e., for $\xi_{\rm int}<\xi_{{\rm int,}1}$, with
\begin{equation}
\xi_{{\rm int,}1}=\frac{1}{\ln2}\approx1.44269.
\label{xiionedef}
\end{equation}
In particular, this explains the plateau
in the data to be presented in Figure~\ref{figv}.
We have generated all the relevant polynomials up to degree $d=12$.
The numbers $A_d$ of physical roots with degree $d$,
and the numbers $B_N$ of threshold values for a column of $N$ grains,
are found to be the following:\footnote{Since some of the polynomials are
reducible, the same root can be generated several times, so that the number
$B_N$ of distinct thresholds may be smaller than
the sum $A_2+A_4+\cdots+A_{N-4}$.}
\begin{equation}
\matrix{
A_2=1,\ A_4=5,\ A_6=19,\ A_8=97,\hfill\cr
A_{10}=442,\ A_{12}=1880,\hfill\cr
B_2=0,\ B_4=0,\ B_6=1,\ B_8=6,\ B_{10}=25,\hfill\cr
B_{12}=121,\ B_{14}=563,\ B_{16}=2443.\hfill}
\end{equation}
We can extract several insights from the above numbers.
Since $B_2=B_4=0$, columns of size $N=2$ and $N=4$
exhibit no dependence on $\xi_{\rm int}$ at all.
For $N=6$, we have $B_6=1$:
the unique threshold value is the inverse golden mean $\phi$
introduced in (\ref{phidef}).
Consequently, quantities typically assume two different forms in the intervals
$0<x_{\rm int}<\phi$ or $\phi<x_{\rm int}<1$.
For $N=8$, we have $B_8=6$:
quantities typically assume seven different forms in the intervals
demarcated by the six threshold values of $x_{\rm int}$, and so on.
The interested reader is referred to
Appendix A for the explicit verification of these predictions
for columns of size $N=4$ and $N=6$.
\subsubsection*{Ground states}
We now focus on the ground states in the simple case where $\varepsilon=1$.
They consist of {\it dimerised} configurations,
made of the patterns $P_1=+-\null$ and $P_2=-+\null$.
Assuming again that the column depth $N$ is even for definiteness,
the generic ground state can be described as follows:
\begin{equation}
\sigma_{2k-1}=-\sigma_{2k}=\eta_k\quad(k=1,\dots,N/2),
\label{gs1}
\end{equation}
with the dimer variable $\eta_k=+1$ (resp. $\eta_k=-1$)
corresponding to the $P_1$ (resp. $P_2$) pattern.
The boundary condition (\ref{init}) implies $\eta_1=+1$.
The local fields in a ground state read as follows:
\begin{equation}
\matrix{
h_{2k-1}=0,\quad h_{2k}=-\eta_k,\hfill\cr
\displaystyle{j_{2k-1}=x_{\rm int}\eta_k-(1-x_{\rm int})\sum_{l=k+1}^{N/2}x_{\rm int}^{2l-2k}\eta_l},\hfill\cr
\displaystyle{j_{2k}=-(1-x_{\rm int})\sum_{l=k+1}^{N/2}x_{\rm int}^{2l-2k-1}\eta_l}.\hfill}
\label{hjgsdef}
\end{equation}
There are $2^{N/2}$ ground states in total,
or $2^{N/2-1}$ if (\ref{init}) is taken into account.
The two crystalline (uniform) ground states
\begin{equation}
\matrix{
U_+=+-+-+-+-+-\cdots,\cr U_-=-+-+-+-+-+\cdots\hfill}
\label{unidef}
\end{equation}
play a special role; intuitively, this is because any other ground state has
conflicts between successive dimer pairs, for example in a configuration
such as $+--+$ where the second and third orientations
would be in non-ideal positions.
Such conflicts would of course also be present, albeit more weakly,
between dimer pairs that were more distant from each other along
the column, e.g. $+-\cdots-+$.
This observation can be turned to a quantitative classification
of ground states by means of the {\it pseudo-energy}.
This quantity is defined for an arbitrary configuration as follows:
\begin{equation}
{\cal E}=-\sum_{n=1}^NH_n\sigma_n,
\label{edef}
\end{equation}
i.e.,
\begin{equation}
{\cal E}={\cal E}_0+g{\cal E}_1,
\end{equation}
with
\begin{equation}
{\cal E}_0=-\sum_{n=1}^Nh_n\sigma_n,\quad{\cal E}_1=-\sum_{n=1}^Nj_n\sigma_n.
\label{ecomp}
\end{equation}
This definition can be motivated as follows.
If the $\sigma_n$ were independent spins in external fields $H_n$,
(\ref{edef}) would be the corresponding Hamiltonian.
In the present model,
we recall that the local fields $H_n$ depend on the orientations $\sigma_m$
in a complex and non-symmetric way,
so that the dynamics does not obey detailed balance,
and the statics is not described by any simple Hamiltonian.
The pseudo-energy defined by (\ref{edef}), however,
provides a useful measure of the amount of configurational disorder
in the full interacting system and, in particular,
allows us to classify the ground states.
In the case of a ground state, the first component of the pseudo-energy
reads ${\cal E}_0=-N/2$
(irrespective of the ground state), whereas the second one reads
\begin{equation}
{\cal E}_1=-\frac{Nx_{\rm int}}{2}-(1-x_{\rm int})^2\sum_{1\le k<l\le N}x_{\rm int}^{2l-2k-1}\eta_k\eta_l,
\label{eone}
\end{equation}
in terms of the dimer variables $\eta_k$.
The first term in the above expression,
\begin{equation}
\mean{{\cal E}_1}=-\frac{Nx_{\rm int}}{2},
\label{emeandef}
\end{equation}
is the mean of ${\cal E}_1$, in the sense of a uniform average over all the ground
states.
The second term in (\ref{eone}) represents the fluctuation in ${\cal E}_1$
from a ground state to another, which typically grows as $N^{1/2}$.
The crystalline states $U_\pm$ introduced in (\ref{unidef}),
respectively corresponding to $\eta_k=+1$ and $\eta_k=-1$ for all $k$,
are the two absolute (global) minima of the pseudo-energy.
We find
\begin{equation}
{\cal E}_1(U_\pm)=-\frac{Nx_{\rm int}}{1+x_{\rm int}}+\frac{x_{\rm int}(1-x_{\rm int}^N)}{(1+x_{\rm int})^2}.
\end{equation}
It follows that the crystalline ground states are separated from a bulk
of roughly equivalent metastable states by an extensive pseudo-energy gap
\begin{equation}
\Delta{\cal E}=\mean{{\cal E}_1}-{\cal E}_1(U_\pm)\approx\frac{Nx_{\rm int}(1-x_{\rm int})}{2(1+x_{\rm int})}.
\end{equation}
It is remarkable that our model generates the kind
of (free) energy landscape that is familiar
in theories of glasses \cite{miguel},
and guessed to be valid for grains in the jamming limit \cite{0},
where crystalline states lie well below a band of metastable states.
Finally, the exact threshold coupling $g_c$
for ground-state stability for $\varepsilon=1$ can be evaluated as follows.
Equation (\ref{hjgsdef}) implies that the state $U_+$ is the first to be
destabilised by an increase of the coupling,
and that its weakest link is the second grain orientation $\sigma_2$.
The corresponding local field reads $H_2=-1+gj_2$,
where $j_2$ assumes its largest possible value
$j_2^\max=x_{\rm int}(1-x_{\rm int}^{N-2})/(1+x_{\rm int})$.
The threshold coupling constant
at which this first destabilisation takes place therefore reads
$g_c=1/j_2^\max$.
In the limit of an infinite column,
we therefore obtain the following exact expression for the threshold coupling:
\begin{equation}
g_c=\frac{1+x_{\rm int}}{x_{\rm int}}={\rm e}^{1/\xi_{\rm int}}+1.
\label{gc1}
\end{equation}
The threshold coupling is found to be a decreasing function
of the interaction length $\xi_{\rm int}$, blowing up exponentially at small $\xi_{\rm int}$,
and reaching a finite asymptotic value 2 in the $\xi_{\rm int}\to\infty$ limit.
It is interesting to observe that the general bound (\ref{gcdef}),
i.e., $g_c>{\rm e}^{1/\xi_{\rm int}}/2$ in the present case ($p=q=1$),
captures the qualitative features of the dependence
of the exact result on $\xi_{\rm int}$.
\section{Zero-temperature dynamics}
\setcounter{equation}{0}
\defC.\arabic{equation}{4.\arabic{equation}}
The application of zero-temperature dynamics is the canonical way
of finding the ground states of a model.
In the present model, grains are aligned one at a time with their local fields.
More precisely, grain $n$ is selected at a rate given by (\ref{odef}),
and its orientation variable $\sigma_n$
is aligned along the local field $H_n$ introduced in (\ref{hdef}),
according to the deterministic rule
\begin{equation}
\sigma_n\to\mathop{\rm sign}\nolimits H_n.
\end{equation}
This rule is well-defined for a non-zero coupling constant $g$,
because the local fields $H_n$ generically do not vanish.
This leads to {\it metastability} in
the following sense: a finite column of $N$ grains in an arbitrary initial
configuration is eventually driven
to an absorbing configuration or {\it attractor},
in a finite {\it jamming time} $T$.
This attractor is necessarily one of the ground states described earlier,
i.e., a configuration where every orientation $\sigma_n$ is aligned with $H_n$.
For a coupling constant $g$ less than the threshold $g_c$ given in (\ref{gc1}),
the attractors are the $2^{N/2}$ dimerised configurations,
whose number is halved to $2^{N/2-1}$ if
the boundary condition (\ref{init}) is taken into account.
Arbitrary initial conditions can lead to any one
of these metastable configurations being reached;
they are however {\it fragile} in the sense that a slightly different
initial condition or stochastic history generically leads to another
attractor being reached instead.
This fragility of metastable states is one of the characteristics
of granular media \cite{0}.
In what follows, we will focus on two aspects of zero-temperature dynamics:
\noindent$\bullet$ {\it Statistics of the jamming time}.
The jamming time $T$ is the random time the system takes to converge to an
attractor, it being understood that the initial configuration is disordered
and randomly chosen.
The $N$ dependence of both the mean jamming time $\mean{T}$
and of its full probability distribution are of interest.
In this respect, we introduce for further reference the reduced variance
\begin{equation}
K_T=\frac{\mathop{\rm var}\nolimits{T}}{\mean{T}^2}=\frac{\mean{T^2}}{\mean{T}^2}-1.
\label{kdef}
\end{equation}
\noindent$\bullet$ {\it Statistics of the attractors}.
The statistics of the attractors reached by stochastic dynamics
is also of special interest,
especially in relation to Edwards' flatness hypothesis \cite{edwards}.
We anticipate non-trivial results only in the glassy phase,
which will be investigated in Section 4.4.
\begin{figure}[!t]
\begin{center}
\includegraphics[angle=90,height=4.5truecm]{figa.eps}
\includegraphics[angle=90,height=4.5truecm]{figb.eps}
\includegraphics[angle=90,height=4.5truecm]{figc.eps}
\includegraphics[angle=90,height=4.5truecm]{figd.eps}
\end{center}
\caption{\small
Plots of the thickness $L(t)$ of the upper ordered layer of the column
against time $t$, for $g=0.01$, illustrating the four phases.
I: Ballistic phase (several tracks with $\xi_{\rm int}=3$, $\xi_{\rm dyn}=\infty$, $N=500$).
The slope $V=6.95$ of the thick line is taken from Figure~\ref{figv}.
II: Activated phase (one single track with $\xi_{\rm int}=50$, $\xi_{\rm dyn}=\infty$, $N=70$).
III: Logarithmic phase (several tracks with $\xi_{\rm int}=3$, $\xi_{\rm dyn}=200$, $N=500$).
The thick line shows the result (\ref{lxover}) with $V=6.95$.
IV: Glassy phase (one single track with $\xi_{\rm int}=50$, $\xi_{\rm dyn}=7$, $N=70$).}
\label{figabcd}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[angle=90,height=3.5truecm]{figdiag.eps}
\caption{\small
Schematic zero-temperature dynamical phase diagram of the model
in the $\xi_{\rm int}-\xi_{\rm dyn}$ plane,
showing the four dynamical phases revealed and investigated below.
The numbers $\xi_{{\rm int,}c}$ and $a_\infty$ will be determined later
(see (\ref{xiic}) and (\ref{ainf})).}
\label{figdiag}
\end{center}
\end{figure}
Inspired by previous work \cite{III,IV},
we monitor the various dynamical regimes of the model by means of the thickness
$L(t)$ of the upper ordered layer of the column, defined as the depth
of the uppermost grain which is not aligned with its local field:
\begin{equation}
L(t)={\rm inf}\{n\;|\;\sigma_n\ne\mathop{\rm sign}\nolimits H_n\}.
\label{ldef}
\end{equation}
Figure~\ref{figabcd} shows typical tracks $L(t)$ for four representative
choices of values of $\xi_{\rm int}$ and $\xi_{\rm dyn}$.
Here and throughout the following, we choose for definiteness
the value $g=0.01$ of the coupling constant.
This value is deep in the weak-coupling regime,
where all the results are virtually independent of the precise choice made
for the coupling constant\footnote{For comparison, we recall that the threshold
coupling $g_c$ (see (\ref{gc1})) is always larger than 2.}.
The dynamical behaviour observed turns out to be
very strongly dependent on the lengths $\xi_{\rm int}$ and $\xi_{\rm dyn}$, and
suggests the
existence of four qualitatively different dynamical phases, which
we have named {\it ballistic, logarithmic, activated} and {\it glassy}.
They will be investigated in greater detail in what follows.
The dynamical phase diagram
in the $\xi_{\rm int}-\xi_{\rm dyn}$ plane presented in Figure~\ref{figdiag} shows
already the existence of
genuine phase boundaries (where crossover phenomena become arbitrarily sharp
in the limit of an infinite column) denoted by full lines,
and crossover phenomena (which occur
when $\xi_{\rm dyn}$ is comparable with the column size $N$) indicated by dashed lines.
\subsection{\bf Phase I (Ballistic)}
This phase is illustrated by panel I of Figure~\ref{figabcd} -- it
corresponds to small $\xi_{\rm int}$ and large $\xi_{\rm dyn}$.
Physically this implies that one is looking at layers
near the free surface (large $\xi_{\rm dyn}$) of a column where grains feel
correlations from below only weakly (small $\xi_{\rm int}$).
In other words, we are looking at the `top' of a granular column.
The thickness $L(t)$ is observed to grow on average linearly with time:
\begin{equation}
\mean{L(t)}\approx Vt.
\label{vt}
\end{equation}
The (relatively small) fluctuations between different tracks
correspond to different stochastic histories.
Equation (\ref{vt}) shows that an ordered layer propagates
ballistically down the column with velocity $V$, a phenomenon which
was already encountered in the unhindered model \cite{III,IV}.
When $L(t)$ becomes equal to the column depth $N$,
an attractor is reached and the dynamics stops, so that
\begin{equation}
T\approx\frac{N}{V},
\label{fbal}
\end{equation}
again up to relatively negligible fluctuations.
\begin{figure}[!t]
\begin{center}
\includegraphics[angle=90,height=5truecm]{figv.eps}
\caption{\small
Plot of the ballistic velocity $V$ against $\xi_{\rm int}$ for $\xi_{\rm dyn}=\infty$.
Arrows indicate the value $\xi_{{\rm int,}1}$ (see (\ref{xiionedef}))
below which $V$ is constant and equal to $V_0$ given in (\ref{vzero}),
and the critical point $\xi_{{\rm int,}c}$ (see (\ref{xiic}))
at which $V$ vanishes linearly.}
\label{figv}
\end{center}
\end{figure}
Figure~\ref{figv} shows numerical values of the velocity $V$,
obtained by averaging $L(t)$ over many independent initial configurations
and histories (at least $10^4$ per point).
The first interesting feature is a plateau region observed for
$\xi_{\rm int}$ smaller than the threshold value $\xi_{{\rm int,}1}$
determined in (\ref{xiionedef}).
As predicted in Section 3.3, the velocity $V$ is found to be constant
over this region, and equal to its value in the $\xi_{\rm int}\to0$ limit:
\begin{equation}
V_0\approx9.75.
\label{vzero}
\end{equation}
As $\xi_{\rm int}$ increases, the effects of frustration begin to kick in more and more
via the back-reaction $j_n$; the resulting inefficiency of the zero-temperature
dynamics causes the velocity to decrease progressively with $\xi_{\rm int}$.
It seems strange that $V_0$ is such a large number, especially given that
it leads to the apparition of other large dimensionless numbers,
such as $\xi_{{\rm int,}c}\approx28.4$ (see (\ref{xiic}))
or $D_c\approx37$ (see (\ref{critd})).
Fortunately, we can provide a simple explanation for the high value of $V_0$,
that the naturalness principle \cite{thooft} would demand.
We recall (see the lines following (\ref{jrecast}))
that the component $j_n$ of the local field
always takes the sign of $-\sigma_{n+1}$ for $\xi_{\rm int}<\xi_{{\rm int,}1}$.
Assume that the uppermost $2k$ grains of the column are already dimerised.
So far as zero-temperature dynamics is concerned,
the next two orientations $\sigma_{2k+1}$ and $\sigma_{2k+2}$,
are entirely decoupled from the rest of the column.
Indeed, $h_{2k+1}=0$, so that $H_{2k+1}=gj_{2k+1}$ has the sign of $-\sigma_{2k+2}$,
whereas $H_{2k+2}\approx h_{2k+2}=-\sigma_{2k+1}$.
If we make the simple assumption that the four configurations
of these two orientations are equally probable,
the mean time it takes to go to one of the two possible attractors
can be shown to be $1/4$.
This newly formed dimer has a spatial size 2,
and the corresponding front velocity is $2/(1/4)=8$.
The actual velocity $V_0$ of (\ref{vzero}) is only about 21\% above this
simple-minded estimate.\footnote{We mention for comparison that in the
analogous case in the model of
\cite{III,IV}, that of an irrational $\varepsilon\to1^\pm$
in the immediate neighbourhood of $\varepsilon=1$, a similar line of reasoning yields
the value 2 for the velocity, whereas the measured velocity
$V_{1^\pm}\approx2.38$ is about 19\% above that estimate.}.
The data of Figure~\ref{figv} also show that
$V$ vanishes linearly as the borderline between Phases I and II
is approached, i.e., as $\xi_{\rm int}\to\xi_{{\rm int,}c}^-$
This behaviour fits in a natural way within
the effective description put forward in Section 4.2
in terms of biased Brownian motion (see (\ref{critv})).
\subsection{\bf Phase II (Activated)}
This phase of relatively large $\xi_{\rm int}$ and $\xi_{\rm dyn}$
is illustrated by panel II of Figure~\ref{figabcd}.
One is still considering grains that are relatively free to move,
as in the top layers of a column, but now grains are increasingly constrained
as a result of grain orientations below them.
There is, typically, only very weak order in the column
before it happens to jam: this is exemplified by a mean thickness
$\mean {L(t)}$ which is quite small compared to the column depth $N$.
Also, $L(t)$ exhibits wild fluctuations around its mean,
which look stationary over the very long time it takes for the system to jam.
After sporadic excursions to larger values, the layer thickness
suddenly jumps to $L(t)=N$, so that an attractor is reached.
This phenomenology is typical of an activated phenomenon.
We therefore expect that:
\noindent$\bullet$
The statistics of the jamming time $T$ should be approximately given
by an exponential distribution:
\begin{equation}
\rho(T)=\frac{1}{\mean{T}}\exp\left(-\frac{T}{\mean{T}}\right),
\label{expo}
\end{equation}
characterised by a single scale $\mean{T}$,
with unit reduced variance ($K_T=1$).
\noindent$\bullet$
The mean jamming time should grow exponentially with the column size:
\begin{equation}
\mean{T}\sim\exp(a(\xi_{\rm int})N),
\label{fact}
\end{equation}
at least for very large $N$,
where $a(\xi_{\rm int})$ is the reduced activation energy per grain,
i.e., the height of the entropic barrier the system has to cross
to reach the ground state.
Huge finite-size effects rule out an accurate numerical exploration
of the activated phase for generic $\xi_{\rm int}$ and $\xi_{\rm dyn}$.
In the following, setting $\xi_{\rm dyn}\to\infty$,
we examine two limits of particular interest.
We explore first the crossover between Phases I (ballistic)
and II (activated), as $\xi_{\rm int}$ approaches the critical value $\xi_{{\rm int,}c}$.
Next, we examine the regime of deep activation, when the effect
of upward interactions is maximal ($\xi_{\rm int}\gg N$).
\subsubsection*{Crossover between Phases I and II}
In order to understand the crossover between Phases I and II,
we invoke the following picture of the behaviour of the thickness $L(t)$
of the ordered layer.
In either of the two phases, it starts from the surface
and eventually propagates to the base, when an attractor is reached.
In the purely ballistic case ($\xi_{\rm int}$ very small),
the layer essentially shoots down to form an attractor.
The effect of increasing $\xi_{\rm int}$ is to `admit impediments' to this pure flow,
to cause $L(t)$ to fluctuate (diffuse) increasingly before the whole column
reaches an attractor.
The uniform effect of grains above any given grain
and the back-reaction of the grains below it are responsible for the
frustration that is increasingly encountered in the search for an attractor
as $\xi_{\rm int}$ increases.
The value of $\xi_{\rm int}$ at which both interactions balance out
is the critical point $\xi_{{\rm int,}c}$, at which the velocity $V$ vanishes,
so that the dynamics is purely diffusive.
\begin{figure}[!t]
\begin{center}
\includegraphics[angle=90,height=5truecm]{figt.eps}
\includegraphics[angle=90,height=5truecm]{figk.eps}
\caption{\small
Top: logarithmic plot of the ratio $\mean{T}/N$ against $\xi_{\rm int}$,
for a dynamical length $\xi_{\rm dyn}=\infty$, and variable column size $N$.
Bottom: plot of the reduced variance $K_T$ against $\xi_{\rm int}$,
for the same parameters.}
\label{figtk}
\end{center}
\end{figure}
The above intuitive picture is corroborated by Figure~\ref{figtk},
showing a logarithmic plot of the ratio $\mean{T}/N$
and a plot of the reduced variance $K_T$ against $\xi_{\rm int}$,
for several values of the column size $N$.
There is evidence of a continuous phase transition
between a ballistic phase for $\xi_{\rm int}<\xi_{{\rm int,}c}$ and an activated phase
for $\xi_{\rm int}>\xi_{{\rm int,}c}$.
We notice from the top panel that $\mean{T}/N$ is roughly independent of $N$
in the ballistic phase, whereas it grows fast with $N$ in the activated phase.
On the other hand, the plots of $K_T$ approximately cross at a critical value
$K_T\sim0.7$ for $\xi_{\rm int}=\xi_{{\rm int,}c}\sim26$.
This picture can be turned into the following effective model.
We treat the thickness $L(t)$ as a collective coordinate,
and model its dynamics by a biased Brownian motion on an interval,
with velocity $V$ and diffusion constant $D$.
The motion starts at time $t=0$ at an initial point $L(0)$
very near the free surface of the column,
which is considered as a reflecting boundary.
It ends at the random hitting time $t=T$
when $L(t)$ visits the base of the column, i.e., $L(t)=N$, for the first time.
Accordingly, the base is considered as an absorbing boundary.
This effective model is analysed in detail in Appendix B.
\begin{figure}[!t]
\begin{center}
\includegraphics[angle=90,height=5truecm]{figf.eps}
\includegraphics[angle=90,height=5truecm]{figg.eps}
\caption{\small
Top: logarithmic plot of $\mean{T}/N^2$ against $N(\xi_{\rm int}-\xi_{{\rm int,}c})$.
for a dynamical length $\xi_{\rm dyn}=\infty$, $\xi_{{\rm int,}c}=28.4$, and variable $N$.
Bottom: plot of the reduced variance $K_T$ against $X$,
for the same parameters.
Full lines: plots of the analytical
results (\ref{fssf}) and (\ref{fssg}) for the effective model,
rescaled according to (\ref{rescaldef}).}
\label{figfg}
\end{center}
\end{figure}
Figure~\ref{figfg} shows that both the mean jamming time $\mean{T}$
and its reduced variance $K_T$ obey finite-size scaling laws of the form
\begin{equation}
\mean{T}\approx N^2\,F(X),\quad K_T\approx G(X),
\label{fssdef}
\end{equation}
with
\begin{equation}
X=N(\xi_{\rm int}-\xi_{{\rm int,}c}).
\label{Xdef}
\end{equation}
The best data collapse is obtained for
\begin{equation}
\xi_{{\rm int,}c}\approx28.4.
\label{xiic}
\end{equation}
Furthermore, the data are observed to be in accurate agreement
with the finite-size scaling results (\ref{fssf}) and (\ref{fssg}),
derived analytically in Appendix B for the effective model.
The excellent quality of the accord suggests that the effective model predicts
the exact finite-size scaling functions of the model.
The best agreement, shown as full lines in Figure~\ref{figfg},
is obtained with the following identification:
\begin{equation}
X\approx-115(z+3.4),\quad\ln(T_0/N^2)\approx-4.3
\label{rescaldef}
\end{equation}
between, on the empirical side,
the scaling variable $X$ of the column model introduced in (\ref{Xdef}),
and, on the theoretical side, the scaling variable
\begin{equation}
z=\frac{VN}{D}
\end{equation}
of the effective model, introduced in (\ref{zeddef}),
and the diffusive time scale
\begin{equation}
T_0=\frac{N^2}{2D},
\end{equation}
introduced in (\ref{btc}).
This last relation, together with the second equation of (\ref{rescaldef}),
enable us to predict
the critical value $D_c$ of the diffusion constant for $\xi_{\rm int}=\xi_{{\rm int,}c}$.
We thus obtain $D_c\approx{\rm e}^{4.3}/2$, i.e.,
\begin{equation}
D_c\approx37.
\label{critd}
\end{equation}
The reduced variance of the jamming time at the critical point
is predicted in (\ref{bkc}) to be a universal number:
\begin{equation}
K_T=\frac{2}{3}.
\end{equation}
This prediction is again in good agreement with the apparent crossing point
of the data of the lower panel of Figure~\ref{figtk}.
Finally, we compare the predictions of the effective model in
the ballistic and activated phases with the above results.
\noindent$\bullet$
{\it Toward the ballistic phase ($\xi_{\rm int}<\xi_{{\rm int,}c}$, i.e., $V>0$)}.
In the ballistic phase, the prediction (\ref{btbal}) for the mean jamming time:
\begin{equation}
\mean{T}\approx\frac{N}{V}-\frac{D}{V^2}
\end{equation}
exhibits the observed ballistic behaviour (\ref{fbal}),
up to a finite negative correction due to diffusion.
The fluctuations of the jamming time around its mean are predicted
to be Gaussian, with a reduced variance given by (\ref{bkbal}):
\begin{equation}
K_T\approx\frac{2D}{VN}.
\end{equation}
This fall-off as $1/N$ agrees with the observation made above
that fluctuations become relatively negligible for large columns.
The critical regime of the ballistic phase corresponds
to the regime $X\to-\infty$, i.e., $z\to+\infty$,
in the finite-size scaling laws (\ref{fssdef}).
Equations (\ref{fssf}) and (\ref{rescaldef}) imply that
the scaling function $F(X)$ falls off as
$F(X)\approx A_F/\abs{X}$, with $A_F\approx115/D\approx3.1$.
This estimate implies in turn
that the velocity vanishes linearly as $\xi_{\rm int}\to\xi_{{\rm int,}c}^-$,
as $V\approx(\xi_{{\rm int,}c}-\xi_{\rm int})/A_F$, i.e.,
\begin{equation}
V\approx0.32(\xi_{{\rm int,}c}-\xi_{\rm int}).
\label{critv}
\end{equation}
The numerical value of the prefactor is in good agreement with the slope
of the extrapolation curve shown as a dashed line in Figure~\ref{figv}.
\noindent$\bullet$
{\it Toward the activated phase ($\xi_{\rm int}>\xi_{{\rm int,}c}$, i.e., $V<0$)}.
In the activated phase, the prediction (\ref{bta}) for the mean jamming time:
\begin{equation}
\mean{T}\approx\frac{D}{V^2}\;{\rm e}^{\abs{V}N/D}
\end{equation}
grows exponentially with $N$, as anticipated in (\ref{fact}).
The corresponding activation energy per unit length,
\begin{equation}
a=\frac{\abs{V}}{D},
\end{equation}
is essentially given by the negative of the velocity.
The reduced variance of the jamming time predicted by (\ref{bk}):
\begin{equation}
K_T\approx1-\frac{2(VN+3D)}{D}\;{\rm e}^{-\abs{V}N/D}
\end{equation}
converges exponentially fast to its limiting value unity,
characteristic of an exponential distribution.
The critical regime of the activated phase corresponds
to the regime $X\to+\infty$, i.e., $z\to-\infty$,
in the finite-scaling laws (\ref{fssdef}).
The effective model predicts that the scaling function $G$
has an exponential convergence toward $G(+\infty)=1$,
whereas the scaling function $F$ grows exponentially as
$F\sim\exp(-z)\sim\exp(B_FX)$ with $B_F\approx1/115\approx0.0087$.
These results corroborate our expectations,
including (\ref{expo}) and (\ref{fact}).
They also imply that the activation energy per grain
vanishes linearly as $\xi_{\rm int}\to\xi_{{\rm int,}c}^+$,
as $a(\xi_{\rm int})\approx B_F(\xi_{\rm int}-\xi_{{\rm int,}c})$, i.e.,
\begin{equation}
a(\xi_{\rm int})\approx0.0087(\xi_{\rm int}-\xi_{{\rm int,}c}).
\label{crita}
\end{equation}
Before leaving this topic, we emphasise that the simple picture of a
Brownian particle whose velocity $V$ changes sign at $\xi_{{\rm int,}c}$
appears to explain all our observations on this crossover.
\subsubsection*{Limiting behaviour for $\xi_{\rm int}\gg N$}
We now look at the slowest possible dynamics in the activated phase;
this will clearly occur when the column is at its most correlated,
where $\xi_{\rm int}$ is much larger than $N$, still keeping $\xi_{\rm dyn}=\infty$ for
simplicity.
We are thus led to investigate the doubly singular limit where
$\xi_{\rm dyn}=\xi_{\rm int}=\infty$.
The only free parameter is then the column size $N$.
\begin{figure}[!t]
\begin{center}
\includegraphics[angle=90,height=5truecm]{figlnt.eps}
\includegraphics[angle=90,height=5truecm]{figkt.eps}
\caption{\small
Top: logarithmic plot of the mean jamming time $\mean{T}$ against $N$,
for $\xi_{\rm int}=\xi_{\rm dyn}=\infty$.
Full line (hardly visible):
fit $\mean{T}=N(\ln 2)/2-6.515\ln N+15.04$ to the data for $N>\xi_{{\rm int,}c}$.
Bottom: plot of the reduced variance $K_T$ against $N$,
for the same parameters.
Arrows show the crossover scale $N=\xi_{{\rm int,}c}$ (see (\ref{xiic})).}
\label{figlntkt}
\end{center}
\end{figure}
Figure~\ref{figlntkt} shows plots of the mean jamming time $\mean{T}$
and of its reduced variance $K_T$ against $N$.
When the column size $N$ is rather small,
the system still behaves more or less ballistically;
this fast dynamics leads to the nearly linear growth of $\mean{T}$ with $N$,
and the concomitant decrease of the variance $K_T$ as a function of $N$.
When $N$ is large enough, the system is fully activated, so that the jamming
time grows exponentially with $N$, while the variance increases, rapidly
converging to its asymptotic value $K_T=1$.
The crossover between these behaviours occurs when
the column size $N$ is of the order of $\xi_{{\rm int,}c}$ (shown as arrows in both plots).
Finally, we focus on the activation energy $a(\xi_{\rm int})$ defined in (\ref{fact}).
When $\xi_{\rm int}\to\infty$, the column is at its most activated; hitting an
attractor is then a totally random process.
The jamming time is therefore expected to be simply given by the ratio between
the number $\Omega_0=2^N$ of disordered initial configurations
and the number $\Omega_\infty=2^{N/2}$ of possible attractors:
\begin{equation}
\mean{T}\sim\frac{\Omega_0}{\Omega_\infty}\sim 2^{N/2}.
\label{tentex}
\end{equation}
A similar purely entropic result is shown in Appendix C to hold
within a toy model of a Markovian dynamics on an assembly
of independent two-level systems.
The result (\ref{tentex}) implies that $a(\xi_{\rm int})$ saturates to the value
\begin{equation}
a_\infty=\frac{\ln 2}{2}\approx0.34657.
\label{ainf}
\end{equation}
This limiting value of the activation energy has been incorporated into
the fit presented in the upper panel of Figure~\ref{figlntkt}.
The good quality of the fit can be viewed as corroborating the result
(\ref{ainf}), in spite of large correction terms.
To sum up, the activation energy $a(\xi_{\rm int})$ is expected to increase
monotonically with $\xi_{\rm int}$ all over the activated phase,
and to interpolate smoothly between the linear growth (\ref{crita})
as $\xi_{\rm int}\to\xi_{{\rm int,}c}^+$ (where the given numerical value of the prefactor only
holds in the $\xi_{\rm dyn}\to\infty$ limit)
and the purely entropic limiting value (\ref{ainf}) as $\xi_{\rm int}\to\infty$.
\subsection{\bf Phase III (Logarithmic)}
This phase of relatively small $\xi_{\rm int}$ and $\xi_{\rm dyn}$
is illustrated by panel III of Figure~\ref{figabcd}.
The thickness $L(t)$ of the ordered layer follows a well-defined master curve,
growing slower than linearly with time,
again with relatively small fluctuations between different tracks.
\begin{figure}[!t]
\begin{center}
\includegraphics[angle=90,height=5truecm]{figtim.eps}
\caption{\small
Plot of $(1/N)\ln\mean{T}$ against $1/\xi_{\rm dyn}$ for $N=50$.
Empty symbols ($\xi_{\rm int}=3$) demonstrate the crossover between
Phases I (ballistic) and III (logarithmic).
Dashed line: prediction (\ref{xover}), with $V=6.95$.
Full symbols ($\xi_{\rm int}=\infty$) corroborate
the crossover (\ref{sharp}) between Phases II (activated) and IV (glassy),
in spite of large finite-size effects.
Full straight lines: crossover at $1/\xi_{\rm dyn}\approx a_{\rm eff}\approx0.155$
(see text).}
\label{figtim}
\end{center}
\end{figure}
Already encountered in the unhindered model \cite{III,IV},
this phenomenon can be explained as follows.
Equation (\ref{vt}) shows that the application of zero-temperature dynamics
causes order to propagate ballistically,
for $\xi_{\rm int}<\xi_{{\rm int,}c}$ and $\xi_{\rm dyn}$ much larger than $N$.
When $\xi_{\rm dyn}$ becomes comparable with $N$, however, grains
move progressively slowly according to their depth,
with local frequencies that scale as (\ref{odef}).
Writing the differential equation
\begin{equation}
\frac{\d L}{\d t}\approx V\,\exp\left(-\frac{L}{\xi_{\rm dyn}}\right),
\end{equation}
we find that the thickness grows according to
\begin{equation}
L(t)\approx\xi_{\rm dyn}\ln\left(1+\frac{Vt}{\xi_{\rm dyn}}\right).
\label{lxover}
\end{equation}
and the mean jamming time for a column of size $N$ reads
\begin{equation}
\mean{T}\approx\frac{\xi_{\rm dyn}}{V}\left(\exp\left(\frac{N}{\xi_{\rm dyn}}\right)-1\right).
\label{xover}
\end{equation}
This result holds all over the left of the phase diagram
of Figure~\ref{figdiag} (Phases I and III and the crossover between them).
It is confirmed quantitatively
by the data shown (empty symbols) in Figure~\ref{figtim}.
The laws (\ref{vt}) and (\ref{fbal}) are recovered for $\xi_{\rm dyn}\gg N$,
i.e., in the ballistic phase.
In the logarithmic phase, when $\xi_{\rm dyn}\ll N$,
the width of the ordered layer is predicted to grow logarithmically:
\begin{equation}
L(t)\approx\xi_{\rm dyn}\ln\frac{Vt}{\xi_{\rm dyn}},
\end{equation}
so that the mean jamming diverges exponentially fast with the column size $N$:
\begin{equation}
\mean{T}\approx\frac{\xi_{\rm dyn}}{V}\,\exp\left(\frac{N}{\xi_{\rm dyn}}\right).
\end{equation}
\subsection{\bf Phase IV (Glassy)}
The glassy phase is found when $\xi_{\rm int}$ is large and $\xi_{\rm dyn}$ is small;
this is by far the richest and most novel phase of this model.
The signal for $L(t)$, illustrated in panel IV of Figure~\ref{figabcd},
is neither nearly deterministic (as in the ballistic and logarithmic phases)
nor totally random (as in the activated phase).
The glassy phase corresponds to the `bottom' of a long column,
where grain reorientations are at their most hindered; grains in
this region are weighed down
by those above them and, additionally, feel to the fullest extent the
effect of the orientational
frustration between upper and lower grains.
This phenomenon is illustrated in Figure~\ref{figp} from different viewpoints,
using the time dependence of four observables (for the same stochastic history
which was used to illustrate Phase IV in Figure~\ref{figabcd}).
The jamming time $T\approx279\,668$ for this history
is about 2.4 times larger than the mean jamming time for the parameters
$N=70$, $\xi_{\rm int}=50$, $\xi_{\rm dyn}=7$.
The plotted observables are:
\noindent $\bullet$
the thickness $L(t)$ (see (\ref{ldef})) of the ordered upper layer,
\noindent $\bullet$
the second component ${\cal E}_1(t)$ (see (\ref{ecomp})) of the pseudo-energy,
\noindent $\bullet$
the total number $\nu(t)$ of dimers,
\noindent $\bullet$
the fraction $\nu_{+-}(t)/\nu(t)$ of $(+-)$ dimers.
\begin{figure}[!t]
\begin{center}
\includegraphics[angle=90,height=4.5truecm]{figp1.eps}
\includegraphics[angle=90,height=4.5truecm]{figp2.eps}
\includegraphics[angle=90,height=4.5truecm]{figp3.eps}
\includegraphics[angle=90,height=4.5truecm]{figp4.eps}
\caption{\small
Top to bottom:
plots of the thickness $L(t)$ of the ordered upper layer,
the second component ${\cal E}_1(t)$ of the pseudo-energy,
the numbers $\nu(t)$ of dimers,
and the fraction $\nu_{+-}(t)/\nu(t)$ of $(+-)$ dimers,
for the history illustrating Phase IV in Figure~\ref{figabcd}.
Full symbols: values of the observables in the attractor,
i.e., right at the jamming time $T$.}
\label{figp}
\end{center}
\end{figure}
The last two quantities involve the following definitions
of the numbers $\nu_{+-}$ of $(+-)$ dimers and $\nu_{-+}$ of $(-+)$ dimers:
\begin{equation}
\matrix{
\displaystyle{\nu_{+-}={\textstyle{\frac14}}\sum_{k=2}^{N/2}(1+\sigma_{2k-1})(1-\sigma_{2k})},\hfill\cr
\displaystyle{\nu_{-+}={\textstyle{\frac14}}\sum_{k=2}^{N/2}(1-\sigma_{2k-1})(1+\sigma_{2k})},\hfill\cr}
\label{nudefs}
\end{equation}
and of the total number $\nu=\nu_{+-}+\nu_{-+}$ of dimers
in a given configuration:
\begin{equation}
\nu=\nu_{+-}+\nu_{-+}={\textstyle{\frac12}}\sum_{k=2}^{N/2}(1-\sigma_{2k-1}\sigma_{2k}).
\end{equation}
The above formulae exclude the uppermost dimer,
which is fixed by the boundary condition (\ref{init}).
The four tracks shown in Figure~\ref{figp} all show
{\it strongly correlated, intermittent and non-stationary} fluctuations
at all time scales, ranging from the instantaneous to scales
of order of the jamming time $\mean{T}$ itself.
These features are commonly observed in glassy systems.
The existence of a glassy phase exhibiting this phenomenology
in a one-dimensional model represents
one of the most interesting outcomes of this work.
If we examine the dynamical history depicted in Figure~\ref{figp}, we will
notice that it can be described as an alternation between two
different kinds of periods:
\noindent$\bullet$ {\it Periods of quietude.}
Four such periods are visible in the figure.
They are characterised by quasi-stationary states with a high degree of order.
The thickness $L(t)$ and the number $\nu(t)$ of dimers fluctuate around
their maximal ground-state values of 70 and 34 respectively; the
pseudo-energy ${\cal E}_1(t)$ is correspondingly minimised,
and the fraction of $(+-)$ dimers is either close to zero or close to unity,
indicating a highly polarised column
which is close to one of the crystalline attractors $U_\pm$.
In some sense, it is as if the system has almost made its mind
up to choose one of the two global attractors, and is dawdling in its vicinity
with nearly no major fluctuations, during each of these periods of quietude.
However, these long excursions do not in any sense
anticipate the fate of the column.
In the given example,
the attractor finally chosen (full symbol) is close to $U_-$,
although the system spends most of its time in
the vicinity of the attractor $U_+$
with the fraction of $(+-)$ dimers typically close to unity during
the periods of quietude.
\noindent$\bullet$ {\it Itinerant periods.}
During these periods of confused wandering
between two consecutive periods of quietude,
all the indicators fluctuate wildly with no particular aim in sight.
All the observables monitored in Figure~\ref{figp}
are characterised by low order,
with the pseudo-energy even going positive on occasion.
We end up with a speculation based on a pictorial analogy.
The tracks in Figure~\ref{figp} are reminiscent of those obtained in
avalanche dynamics \cite{avalanches},
where periods of small random events give rise to large
system-size avalanches, which are known to be due to stress buildup and
release on the surface.
It is interesting, using this analogy,
to speculate whether the itinerant periods in Figure~\ref{figp} build
up unsustainable geometric disorder all along the column, which can only be
relieved by a systemic choice of a nearly ordered configuration (that is
close to an attractor), in which the column then lives, until disorder strikes
again in the form of the next itinerant period.
\subsubsection*{Mean jamming time}
We now focus on a quantitative analysis of several aspects of the glassy phase,
beginning with the mean jamming time $\mean{T}$.
Recall that in the activated phase, i.e., for $\xi_{\rm int}>\xi_{{\rm int,}c}$ and $\xi_{\rm dyn}$ large
enough, the jamming time grows exponentially fast with $N$ (see (\ref{fact})).
We now examine the effect of decreasing $\xi_{\rm dyn}$, to cross over into the glassy
phase.
The main effect of this is that an increasingly broad
spectrum of local frequency scales $\o_n$ kicks in, to slow down
the stochastic behaviour of the activated phase.
Interestingly, a toy model
of an assembly of independent two-level systems is able to provide
a clue to this crossover.
Its details are presented in Appendix C, but the crucial feature is that
it has two regimes -- one where entropy dominates (`entropic'),
and the other which is dominated by the slowest of the local frequencies
(`slow').
The mean jamming time is in fact what would be most naively expected
from the above competition, that is, it is the greater of the two
times that would be generated:
\begin{equation}
\mean{T}\sim\max\,\Bigl(\exp(a(\xi_{\rm int})N),\;\exp(N/\xi_{\rm dyn})\Bigr),
\label{sharp}
\end{equation}
where $\exp(a(\xi_{\rm int})N)$ is the jamming time of (\ref{fact})
in the $\xi_{\rm dyn}\to\infty$ limit,
while $1/\o_N=\exp(N/\xi_{\rm dyn})$ is the slowest microscopic time scale of the
problem.
More specifically, this implies the following:
\noindent$\bullet$
In the {\it activated} phase, when $\xi_{\rm dyn}>1/a(\xi_{\rm int})$,
the result (\ref{fact}) for the mean jamming time in the $\xi_{\rm dyn}\to\infty$ limit
is essentially unchanged -- this corresponds
to the {\it entropic} phase of the toy model of Appendix C.
\noindent$\bullet$
In the {\it glassy} phase, i.e., for $\xi_{\rm dyn}<1/a(\xi_{\rm int})$,
the jamming time grows proportionally to $1/\o_N=\exp(N/\xi_{\rm dyn})$ -- this
corresponds to the {\it slow} phase of the toy model of Appendix C.
From the above, one naturally expects there to be
a sharp transition in the $\xi_{\rm int}-\xi_{\rm dyn}$ plane, along the line defined by
\begin{equation}
\xi_{\rm dyn}(\xi_{\rm int})=\frac{1}{a(\xi_{\rm int})},
\label{tpdef}
\end{equation}
shown qualitatively in Figure~\ref{figdiag}.
For the strongest correlations, as $\xi_{\rm int}\to\infty$, the transition point
$\xi_{\rm dyn}(\xi_{\rm int})$ takes its minimum value
$\xi_{\rm dyn}(\infty)=1/a_\infty=2/(\ln 2)\approx2.8854$ (see (\ref{ainf})).
On the other hand, at the boundary of the ballistic/logarithmic
and activated regimes ($\xi_{\rm int}\to\xi_{{\rm int,}c}^+$), (\ref{crita})
predicts a divergence of the transition point of the form
\begin{equation}
\xi_{\rm dyn}(\xi_{\rm int})\approx115/(\xi_{\rm int}-\xi_{{\rm int,}c}).
\end{equation}
Despite huge finite-size effects,
our simulation data (shown as full symbols in Figure~\ref{figtim})
manifest the crossover described above.
The plateau in the left part of the data (full horizontal line)
yields the effective value $a_{\rm eff}\approx0.155$ for $\xi_{\rm int}=\infty$ and
$N=50$.
This effective value is very far from the theoretical
asymptotic value $a_\infty$ (see (\ref{ainf})),
underlining the importance of finite-size effects.
However, and reassuringly for our analysis, the crossover
does indeed take place as predicted by (\ref{tpdef}),
at a value of $1/\xi_{\rm dyn}\approx a_{\rm eff}$ (full vertical line).
\subsubsection*{Statistics of attractors}
We now turn to the statistics of attractors in the glassy phase.
The question of what they are is easily addressed.
Recall that the ground states of the model for $\varepsilon=1$ are the $2^\nu$
configurations made up of $\nu=N/2-1$ dimers, which satisfy
the boundary condition (\ref{init}).
By construction, these are the possible attractors of zero-temperature dynamics.
The next question, which relates to their {\it dynamical attainability},
is less easy to answer.
A precise formulation of this question is: What is the probability
$Q({\cal C})$ that the application of zero-temperature dynamics leaves the column
in a given attractor ${\cal C}$,
starting from a uniformly chosen random initial configuration?
Or, more physically: {\it how and where does a constrained system,
starting from random initial conditions, attain jamming?}
This question has held centre stage in theoretical \cite{compcoop,gl} and
experimental \cite{sidnature}
explorations of granular media and many other complex systems,
ever since Edwards postulated that
the entropic landscape of granular systems was flat \cite{edwards}.
Edwards' {\it flatness} hypothesis (in the strong sense)
implies that the attractors are sampled uniformly by the dynamics,
so that $Q({\cal C})$ is independent of the attractor ${\cal C}$,
and therefore equal to the reciprocal of the total number of attractors.
Our reason for introducing these issues at such a late stage
in this paper is that the statistics of attractors
are likely to be non-trivial only in the glassy phase.
All the other phases indeed manifest sufficiently stochastic behaviour that
one would expect the entropic landscape to be at least approximately flat.
A central quantity in this framework is therefore the dynamical entropy
\begin{equation}
S=-\sum_{\cal C} Q({\cal C})\ln Q({\cal C}).
\end{equation}
In the case where the attractors are sampled uniformly,
according to Edwards' hypothesis,
the dynamical entropy assumes its maximal value:
\begin{equation}
S_\max=\nu\ln 2,
\label{sedw}
\end{equation}
where $\nu=N/2-1$.
Measuring entropies directly via numerical simulations is known
to be a very difficult task.
Instead, we resort to an inspired guess.
Since it seems likely that the crystalline attractors $U_\pm$
introduced in (\ref{unidef}) will play a special role in the dynamics,
we use them implicitly to define quantities
of interest on the attractors reached by the dynamics:
\noindent$\bullet$
A {\it global} indicator is provided
by the probability distribution $p(\nu_{+-})$ of the number of $(+-)$ dimers.
By using the dimer variables $\eta_k$ introduced in (\ref{gs1}),
the definitions (\ref{nudefs}) can be simplified as:
\begin{equation}
\nu_{+-}={\textstyle{\frac12}}\sum_{k=2}^{N/2}(1+\eta_k),\quad
\nu_{-+}={\textstyle{\frac12}}\sum_{k=2}^{N/2}(1-\eta_k),
\end{equation}
so that $\nu_{+-}+\nu_{-+}=\nu=N/2-1$.
If Edwards' hypothesis holds,
i.e., if the $2^\nu$ attractors are all equally likely to occur
as attractors, the distribution of $\nu_{+-}$ is binomial:
\begin{equation}
p(\nu_{+-})=\frac{1}{2^\nu}\bin{\nu}{\nu_{+-}}
=\frac{\nu!}{2^\nu(\nu_{+-})!(\nu_{-+})!}.
\label{binlaw}
\end{equation}
In such a binomial distribution, the extremal values $\nu_{+-}=0$ and
$\nu_{+-}=\nu$,
corresponding to the crystalline attractors $U_\pm$
(where all the dimers are of the same kind) are the least probable.
On the other hand, if the actual distributions obtained deviate from
the binomial law (\ref{binlaw}), this would indicate strongly
that all the attractors are {\it not} equally likely,
the entropic landscape is {\it not} flat,
and thus of course that Edwards' hypothesis does not hold.
\noindent$\bullet$
A {\it local} indicator of attractor structure is the correlation function
\begin{equation}
\chi_k=\frac{\mean{(\nu_{+-}-\nu_{-+})\eta_k}-1}{\nu-1}.
\end{equation}
This correlation measures the trend for the $k$-th dimer
to be aligned with every other dimer.
It vanishes identically if Edwards' flatness hypothesis holds.
In the extreme opposite situation where only the crystalline
attractors $U_\pm$ are reached by the dynamics,
the above correlation takes its maximal value $\chi_k=1$ for all $k$.
\begin{figure}[!t]
\begin{center}
\includegraphics[angle=90,height=5truecm]{figb2.eps}
\includegraphics[angle=90,height=5truecm]{figr2.eps}
\caption{\small
Histogram plots of the probability distribution $p(\nu_{+-})$
for $N=50$ (hence $\nu=24$).
Top: $\xi_{\rm dyn}=10$ and variable $\xi_{\rm int}$.
Bottom: $\xi_{\rm int}=\infty$ and variable $\xi_{\rm dyn}$.
The binomial distribution (\ref{binlaw}) is shown as thick full lines.}
\label{fig2}
\end{center}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[angle=90,height=5truecm]{figb3.eps}
\includegraphics[angle=90,height=5truecm]{figr3.eps}
\caption{\small
Plot of the dimer correlation function $\chi_k$ against depth $n=2k$.
Parameters are as in Figure~\ref{fig2}.}
\label{fig3}
\end{center}
\end{figure}
Our numerical results for the probability distribution $p(\nu_{+-})$
and the dimer correlation function
$\chi_k$ are shown in Figures~\ref{fig2} and \ref{fig3}.
All the data were taken for a system of size $N=50$,
which has $\nu=24$ dimers that are free to reorient.
Figure~\ref{fig2} shows the variation
of the form of $p(\nu_{+-})$ with, first, fixed $\xi_{\rm dyn}=10$ and variable
$\xi_{\rm int}$, and next, fixed $\xi_{\rm int}=\infty$ and variable $\xi_{\rm dyn}$.
Figure~\ref{fig3}
shows the variation of $\chi_k$ along the same diagnostic lines.
In Figure~\ref{fig2}, the binomial distribution (corresponding to Edwards'
flatness hypothesis) is shown by a thick full line, with which the
data for the lowest value of $\xi_{\rm int}$ are
almost completely aligned.
The statistics of attractors is thus very close to being uniform
in the ballistic and logarithmic phases, which correspond to the top layers
of a column.
As $\xi_{\rm int}$ increases, there is a gradual crossover
to a non-trivial two-peaked distribution;
the same trend is visible in the lower panel,
when $\xi_{\rm dyn}$ decreases for infinite $\xi_{\rm int}$.
At the beginning of the crossover,
with its small deviations from uniform sampling, one recognises
the activated phase, which corresponds to the middle of a column.
By the time that the two-peaked distribution is obtained
in both parts of Figure~\ref{fig2}, the parameters -- $\xi_{\rm int}$ large, and
$\xi_{\rm dyn}$ small -- correspond clearly to the glassy phase.
Here, attractors in the neighbourhoods
of the crystalline states $U_\pm$ are eventually
favoured, after long periods of systemic wandering.
The above observations are reinforced by Figure~\ref{fig3}.
In the upper panel,
the correlation function $\chi_k$ is essentially zero for low $\xi_{\rm int}$,
increasing progressively as $\xi_{\rm int}$ is increased.
In the lower panel, i.e., for infinite $\xi_{\rm int}$,
the correlation function is never quite zero even for high $\xi_{\rm dyn}$, and
only manifests a stronger depth-dependence as $\xi_{\rm dyn}$ decreases.
These results reinforce those found in an independent model which
uses random graphs to model grains near jamming, where entropic deviations
from Edwards' flatness occur in certain regions of parameter space
\cite{johannes}.
To recapitulate, the salient feature that emerges is that the
system prefers increasingly to live in the neighbourhood of its two global
minima, the attractors $U_\pm$, as one goes deep into the glassy phase.
As a consequence, the dynamical entropy decreases from a value
close to its maximal value (\ref{sedw})
at the boundary between Phases II and IV,
to zero in the deepest part of the glassy phase ($\xi_{\rm int}\gg N$, $\xi_{\rm dyn}\ll N$).
Furthermore, Edwards' flatness is well obeyed overall in three out of
four phases in this model; it is massively violated only
deep in the glassy phase of this model, where the configurational
landscape is completely rough.
Our success in constructing this glassy phase via such a minimal model
relied on the inclusion of two crucial ingredients:
\noindent$\bullet$
Long-range interactions ($\xi_{\rm int}>N$), in order for the dimers to jam
cooperatively, rather than independently of each other;
\noindent$\bullet$
A broad spectrum of local frequencies ($\xi_{\rm dyn}$ small) to slow down
the relaxation, and thus prevent a purely activated mechanism driven by entropy.
It is noteworthy that both these minimal ingredients rely on {\it collective}
effects -- one to do with interactions in {\it space}, the other
to do with degrees of freedom in {\it time}.
This model-specific conclusion
at once agrees with, and reinforces, general
notions \cite{miguel} of cooperativity in glassy dynamics.
\section{Discussion}
\setcounter{equation}{0}
\defC.\arabic{equation}{5.\arabic{equation}}
The motivation for this model came from a mental image of grains
in a box under shaking: what
could be at once so simple, or -- as we came to see in time -- so complex? Our
first, simplest, model \cite{I,II} involved only the effects of gravity
on non-interacting grains -- deeper grains carried the weight of grains above
them, so were less free to move.
This was modelled by using a single dynamical length $\xi_{\rm dyn}$,
representing the thickness of the dynamical boundary layer:
grains at a depth $n$ much less than $\xi_{\rm dyn}$
(which can be examined by setting $\xi_{\rm dyn}\to\infty$) are free to move, whereas
those where $n\sim\xi_{\rm dyn}$ have lower frequencies of motion, as normal in
non-Newtonian fluids.
Three phases were found; in the `fluidised' phase,
grains flew as well as moved along the surface, with a relatively quick
propagation of order down the sandbox.
Grain disorder was essentially frozen in, in the `glassy' phase, with
a very slow propagation of order from the free surface.
The `intermediate' phase
was in some ways the most interesting, with a true competition between
fast and slow dynamics.
In hindsight, it is astonishing that these diverse behaviours
-- especially the shape-dependent `ageing' effects of the glassy regime --
were manifested in a totally non-interacting model.
A more realistic, interacting model
of the glassy regime was presented in \cite{III,IV}.
Since close-packed grains can typically not diffuse spatially, it was
sufficient to model a column, rather than a box, of grains.
In the model of \cite{III,IV}, grain motions were constrained not just
by the masses, but also by the orientations of grains above them,
thus generating directional long-range interactions.
The effect of compaction around jamming was modelled by a single
local field $h_n$, representing the excess void space \cite{br}
for grain $n$, which could be minimised
by a suitable choice of grain orientation.
Additionally, in this model,
grains were allowed to have arbitrary shapes, so that the
the disordered orientation of a grain could occupy any volume $\varepsilon$,
and, correspondingly generate any void space.
The propagation of order in this model proceeded from the free surface to the
base, and was `causal in space' --
in that while upper grains constraint lower ones, the converse was not true.
It took us some time and several explorations to realise that while the
model of \cite{III,IV} had at least the flavour of the interactions needed
to model a jammed glassy phase -- e.g. the constraining effect of long-range
grain correlations -- its
lack of slow dynamics (except those arising from the trivial effect
of grain masses) was a direct result of its spatial causality.
Essentially, provided a grain was not blocked down by the weight of other
grains, it was free to orient itself subject only to the orientations
of grains {\it above} itself -- that is, we were modelling the behaviour of the
top layers of a jammed column of grains, which never felt the undertow of the
base.
It was small surprise, therefore, that the ordering dynamics
for $\xi_{\rm dyn}\to\infty$ were ballistic.
To model a column of grains with spatial inhomogeneities -- that
is, a column with a top, a middle, and a bottom --
we discovered that orientational constraints needed to be inserted in a
{\it non-directed} way.
This enabled us to model frustration -- in this context, the need of a given
grain to balance
the effects of two competing local fields $h_n$ and $j_n$ -- which led
in its turn, to slow dynamics.
Still keeping
the orientational constraints of previous models \cite{I,II,III,IV} via
the field $h_n$, as well as the effect of `gravitational slowing down' via
$\xi_{\rm dyn}$,
we therefore introduced here the notion that grains were also constrained
by grains {\it below} them, via the field
$j_n$ which propagated over a correlation length $\xi_{\rm int}$.
From this very heuristic and pictorial modelling
has emerged a model column of grains that manifests all the complexity
of earlier models \cite{III,IV}, and adds
some more via the introduction of the activated and glassy phases.
Our main success is of course in the realisation of a glassy phase
for which a minimal combination of two physical
ingredients -- {\it strong, bi-directional, orientational correlations and a
broad spectrum of local frequencies arising from a natural depth-dependence} --
appears to be necessary.
The richness of this phase is worthy of further
exploration, especially to do with issues concerning higher-order correlations
and ageing.
Finally, we mention that our model provides
some rather interesting and general insights into the nature
of optimisation -- the granular column modelled
here reaches its ground states in
strikingly different ways, in the four dynamical phases mentioned above.
It is tempting to think of these phases as representing different spatial
parts -- `top', `middle' and `bottom' -- of a column, and to connect their
different routes to compaction with the issue of
inhomogeneities in real granular media \cite{0}.
While this picture is an
appealing one, one should remember that the four phases of this model
were obtained by varying $\xi_{\rm int}$ and $\xi_{\rm dyn}$; the translation of our results
to apply to a real column
would involve the natural apparition of such variations as a function of depth.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,747
|
The user only enters model, height, and width of a window, and system makes the required calculations to produce the cutting list of the window parts.
The ability to add new windows models easily.
Calculates the exact project cost by calculating of the cost of all windows parts plus labor and waste percentage.
Reduces time spent on projects pricing significantly.
Prints quotations that include windows shapes.
The ability to print quotations using MS Word.
Projects report by customer or project date.
Quotations Query by customer or quotation number.
Quotations details are saved for future reference.
Security system for users based on passwords.
System is available in English and Arabic Language.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,264
|
UK Box Office Predictions: Disney Vs. Disney! Dumbo or Captain Marvel?
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UK Box Office Predictions: Will comedy-horror Us scare off Captain Marvel?
UK Box Office Predictions w/c 18th Mar: Will comedy-horror Us scare off Captain Marvel?
Read More UK Box Office Predictions: Will comedy-horror Us scare off Captain Marvel?
UK Box Office Predictions: What Men Want Is To See Captain Marvel!
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UK Box Office Predictions: Time For Captain Marvel To Soar!
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UK box office predictions w/c 18th Feb: Will UK cinema goers forgive Liam Neeson?
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UK box office predictions w/c 11th Feb: Will The Kid Who Will Be King also rule the box office?
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 668
|
Chrysis inaequalis ist eine Art aus der Familie der Goldwespen (Chrysididae).
Merkmale
Die Wespen erreichen eine Körperlänge von 5 bis 10 Millimetern. Die Pleuren des Mesonotums besitzen unterseits zwei Zähnchen. Das Gesicht ist deutlich konkav eingebuchtet. Das zweite und dritte Tergit besitzen einen Mittelkiel, am Rand des dritten befinden sich zudem vier Zähnchen. Der Thorax ist blau, mit einigen grünlichen Bereichen, der Hinterleib ist golden-rot gefärbt. Beide Bereiche glänzen metallisch. Die Art ist bei genauer Betrachtung gut von den anderen, zum Teil schwer zu unterscheidenden Arten der Gattung Chrysis zu unterscheiden, ähnelt diesen jedoch sehr.
Vorkommen
Die Art kommt in Nordafrika, Süd- und Mitteleuropa, bis zur Mandschurei vor. Sie besiedelt verschiedene wärmebegünstigte Lebensräume. Die Tiere fliegen von Ende Juni bis Mitte September. Sie sind in Mitteleuropa selten.
Lebensweise
Chrysis inaequalis parasitiert vermutlich an Bienen der Gattung Osmia, konnte aber in der Zucht auch aus einer Lehmzelle von Eumenes coarctatus gezogen werden. Die Imagines fliegen an sonnenexponierten Lehmwänden, Felsen und gelegentlich auch an Totholz.
Belege
Literatur
Rolf Witt: Wespen. Beobachten, Bestimmen. Naturbuch-Verlag, Augsburg 1998, ISBN 3-89440-243-1.
Einzelnachweise
Chrysidoideen
Chrysidoidea
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,279
|
Each of our handmade pottery goblets are a great example of the beauty of art for every day. Thrown in one piece on a potter's wheel, goblets are then (when firm) trimmed to an elegant stem and bowl. Afterwards, each of our stoneware goblets is decorated with an abstract painterly design using clay slips. Visit our Door County pottery studio in Ellison Bay, or order yours online!
Our clay and glazes are food safe. Care instructions are included.
Expect variations in hand thrown pottery.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,656
|
Međupodsavezna liga Zagreb je bila liga četvrtog stupnja nogometnog prvenstva Jugoslavije u sezoni 1963./64.
Sudjelovalo je 12 klubova, a prvak je bila "Duga Resa".
Ljestvica
Kostajnica tadašnji naziv za Hrvatsku Kostajnicu
Rezultatska križaljka
Izvori:
Unutarnje poveznice
Zagrebačka zona 1963./64.
Podsavezna liga Karlovac 1963./64.
Podsavezna liga Koprivnica 1963./64.
Vanjske poveznice
Izvori
Stjepan Kaurić: Nogometni klub Duga Resa : 1929. – 2009., Duga Resa, 2009., (Grad Duga Resa), str. 28
Krešimir Perušić i dr.: 100 godina nogometa u Karlovačkoj županiji : 1903. – 2003., Karlova, 2004., , str. 310
1963-64
MP Zagreb
1963-64
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,360
|
#ifndef BOOST_MULTI_INDEX_DETAIL_RND_INDEX_LOADER_HPP
#define BOOST_MULTI_INDEX_DETAIL_RND_INDEX_LOADER_HPP
#if defined(_MSC_VER)&&(_MSC_VER>=1200)
#pragma once
#endif
#include <boost/config.hpp> /* keep it first to prevent nasty warns in MSVC */
#include <algorithm>
#include <boost/detail/allocator_utilities.hpp>
#include <boost/multi_index/detail/auto_space.hpp>
#include <boost/multi_index/detail/prevent_eti.hpp>
#include <boost/multi_index/detail/rnd_index_ptr_array.hpp>
#include <boost/noncopyable.hpp>
#include <cstddef>
namespace boost{
namespace multi_index{
namespace detail{
/* This class implements a serialization rearranger for random access
* indices. In order to achieve O(n) performance, the following strategy
* is followed: the nodes of the index are handled as if in a bidirectional
* list, where the next pointers are stored in the original
* random_access_index_ptr_array and the prev pointers are stored in
* an auxiliary array. Rearranging of nodes in such a bidirectional list
* is constant time. Once all the arrangements are performed (on destruction
* time) the list is traversed in reverse order and
* pointers are swapped and set accordingly so that they recover its
* original semantics ( *(node->up())==node ) while retaining the
* new order.
*/
template<typename Allocator>
class random_access_index_loader_base:private noncopyable
{
protected:
typedef typename prevent_eti<
Allocator,
random_access_index_node_impl<
typename boost::detail::allocator::rebind_to<
Allocator,
char
>::type
>
>::type node_impl_type;
typedef typename node_impl_type::pointer node_impl_pointer;
typedef random_access_index_ptr_array<Allocator> ptr_array;
random_access_index_loader_base(const Allocator& al_,ptr_array& ptrs_):
al(al_),
ptrs(ptrs_),
header(*ptrs.end()),
prev_spc(al,0),
preprocessed(false)
{}
~random_access_index_loader_base()
{
if(preprocessed)
{
node_impl_pointer n=header;
next(n)=n;
for(std::size_t i=ptrs.size();i--;){
n=prev(n);
std::size_t d=position(n);
if(d!=i){
node_impl_pointer m=prev(next_at(i));
std::swap(m->up(),n->up());
next_at(d)=next_at(i);
std::swap(prev_at(d),prev_at(i));
}
next(n)=n;
}
}
}
void rearrange(node_impl_pointer position,node_impl_pointer x)
{
preprocess(); /* only incur this penalty if rearrange() is ever called */
if(position==node_impl_pointer(0))position=header;
next(prev(x))=next(x);
prev(next(x))=prev(x);
prev(x)=position;
next(x)=next(position);
next(prev(x))=prev(next(x))=x;
}
private:
void preprocess()
{
if(!preprocessed){
/* get space for the auxiliary prev array */
auto_space<node_impl_pointer,Allocator> tmp(al,ptrs.size()+1);
prev_spc.swap(tmp);
/* prev_spc elements point to the prev nodes */
std::rotate_copy(
&*ptrs.begin(),&*ptrs.end(),&*ptrs.end()+1,&*prev_spc.data());
/* ptrs elements point to the next nodes */
std::rotate(&*ptrs.begin(),&*ptrs.begin()+1,&*ptrs.end()+1);
preprocessed=true;
}
}
std::size_t position(node_impl_pointer x)const
{
return (std::size_t)(x->up()-ptrs.begin());
}
node_impl_pointer& next_at(std::size_t n)const
{
return *ptrs.at(n);
}
node_impl_pointer& prev_at(std::size_t n)const
{
return *(prev_spc.data()+n);
}
node_impl_pointer& next(node_impl_pointer x)const
{
return *(x->up());
}
node_impl_pointer& prev(node_impl_pointer x)const
{
return prev_at(position(x));
}
Allocator al;
ptr_array& ptrs;
node_impl_pointer header;
auto_space<node_impl_pointer,Allocator> prev_spc;
bool preprocessed;
};
template<typename Node,typename Allocator>
class random_access_index_loader:
private random_access_index_loader_base<Allocator>
{
typedef random_access_index_loader_base<Allocator> super;
typedef typename super::node_impl_pointer node_impl_pointer;
typedef typename super::ptr_array ptr_array;
public:
random_access_index_loader(const Allocator& al_,ptr_array& ptrs_):
super(al_,ptrs_)
{}
void rearrange(Node* position,Node *x)
{
super::rearrange(position?position->impl():node_impl_pointer(0),x->impl());
}
};
} /* namespace multi_index::detail */
} /* namespace multi_index */
} /* namespace boost */
#endif
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,265
|
Q: why does there is no update in Node *root? This is a very simple example for inserting a number.
typedef struct node {
int data;
struct node *left, *right;
} node;
node * newNode(int val) {
node* n = malloc(sizeof(node));
n->data=val;
n->left=NULL;
n->right=NULL;
return n; }
void insert(node* node, int key) {
if (node == NULL)
node = newNode(key);
}
int main() {
node *root = NULL;
insert(root, 5);
printf("%d\n", root->data);
return 0;
}
The problem is, Why does the printf print nothing when I insert 5 in root?
A: In C all arguments are passed by value. That means the value is copied into the argument variable, and when you do the assignment node = newNode(key); you only assign to the local node variable.
There are two solutions to this problem:
*
*Return the new node instead:
node* insert(node* the_node, int key) {
if (the_node == NULL)
the_node = newNode(key);
return the_node;
}
...
root = insert(root, 5);
*Emulate pass by reference, which can be done by passing a pointer to the variable using the address-of operator &:
void insert(node** the_node, int key) {
if (*the_node == NULL)
*the_node = newNode(key);
}
...
insert(&root, 5);
A: You passed the pointer root to the function insert by value. You can imagine the function definition and its call the following way
node *root = NULL;
insert(root, 5);
//...
void insert( /* node* node, int key */ ) {
node* node = root;
int key = 5;
if (node == NULL)
node = newNode(key);
}
As you see the function deals with a copy of the pointer root. Changing the copy that is the value in the local variable node does not influence on the value stored in the pointer root.
You have to pass the pointer root by reference. For example
void insert(node** node, int key) {
if (*node == NULL)
*node = newNode(key);
}
int main() {
node *root = NULL;
insert(&root, 5);
//...
In the terms of C passing by reference means passing an object indirectly through a pointer to it. In this case the object will be accessible in the function through the passed pointer.
Pay attention to that it would be more safe to define the function newNode the following way
node * newNode( int val )
{
node *n = malloc( sizeof( node ) );
if ( n != NULL )
{
n->data = val;
n->left = NULL;
n->right = NULL;
}
return n;
}
In this case the function insert can look like
int insert( node **node, int key )
{
while ( *node != NULL )
{
if ( key < ( *node )->data )
{
node = &( *node )->left;
}
else
{
node = &( *node )->right;
}
}
*node = newNode( key );
return *node != NULL;
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 252
|
Q: regular expression to match nested brackets I work in notepad++ and need to replace occurrences like
new int[(cw[0] - index) * 2];
It starts with "new int" followed by square brackets which may contain another pair of square brackets.
What is the regular expression for this?
A: In notepad++ the following should match occurrences:
new int(\[(?>[^\[\]]|(?1))*\])
This is leveraging recursion into a capture group to match the nested brackets as explained on "matching balanced constructs" section
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,493
|
<?php
namespace Application\Form;
use Phalcon\Forms\Element\Hidden;
use Phalcon\Forms\Element\Check;
use Phalcon\Forms\Element\File;
use Application\Form\Element\Image;
abstract class Form extends \Phalcon\Forms\Form
{
protected $helper;
public function renderDecorated($name)
{
if (!$this->has($name)) {
return "form element '$name' not found<br />";
}
$this->helper = $this->getDI()->get('helper');
$element = $this->get($name);
$messages = $this->getMessagesFor($element->getName());
$html = '';
if (count($messages)) {
$html .= '<div class="ui error message">';
$html .= '<div class="header">' . $this->helper->translate('Error form validation') . '</div>';
foreach ($messages as $message) {
$html .= '<p>' . $message . '</p>';
}
$html .= '</div>';
}
if ($element instanceof Hidden) {
echo $element;
} else {
switch (true) {
case $element instanceof Check:
$html .= '<div class="form-group">';
$html .= '<div class="checkbox">';
$html .= '<label>';
$html .= $this->render($name, ['type' => 'checkbox']);
if ($element->getLabel()) {
$html .= $element->getLabel();
}
$html .= '</label>';
$html .= '</div>';
$html .= '</div>';
break;
case $element instanceof Image:
$html = $this->renderImage($element);
break;
case $element instanceof File:
$html .= '<div class="form-group">';
$html .= $this->makeLabel($element);
$html .= $this->render($name, ['type' => 'file']);
$html .= '</div>';
break;
default:
$html .= '<div class="form-group">';
$html .= $this->makeLabel($element);
$html .= $this->render($name, ['class' => 'form-control']);
$html .= '</div>';
}
}
return $html;
}
public function renderAll()
{
$html = '';
if ($this->getElements()) {
foreach ($this->getElements() as $element) {
$html .= $this->renderDecorated($element->getName());
}
}
return $html;
}
private function makeLabel($element)
{
if ($element->getLabel()) {
return '<label for="' . $element->getName() . '">' . $this->helper->translate($element->getLabel()) . '</label>';
} else {
return '';
}
}
/**
* @param Image $element
* @return string $html
*/
private function renderImage($element)
{
$html = '<div class="form-group">';
if ($element->getLabel()) {
$html .= '<label>' . $element->getLabel() . '</label>';
}
if ($element->getValue()) {
$html .= '<section onclick="selectText(this);">' . $element->getValue() . '</section>';
} else {
$html .= '<br>';
}
$html .= '<div class="fileinput fileinput-new" data-provides="fileinput">
<div class="fileinput-preview thumbnail" data-trigger="fileinput" style="width: 200px; min-height: 100px">';
if ($element->getValue()) {
$url = $this->getDI()->get('url');
$html .= '<img src="' . $url->path() . $element->getValue() . '" width="200">';
}
$html .= '</div>
<div>
<span class="btn btn-default btn-file">
<span class="fileinput-new">Select image</span>
<span class="fileinput-exists">Change</span>
<input type="file" name="'.$element->getName().'">
</span>
<a href="#" class="btn btn-default fileinput-exists" data-dismiss="fileinput">Remove</a>
</div>
</div>
</div>';
return $html;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,689
|
We couldn't find anyone willing to accept responsibility for the expansion of a Welsh coalmine
Peadar O'Cearnaigh
Following the UN climate summit COP26 in Glasgow in November last year, 190 countries and organisations, including the UK, agreed to:
both phase out coal power and end support for new coal power plants
So it's somewhat hypocritical that Energybuild's coal mine in Aberpergwm, near Glynneath in Wales, could soon get an extension licence. Environmental group the Coal Action Network (CAN) opposes this extension and it's calling on both the Welsh and UK governments to block it.
However, there appears to be confusion as to which government or department makes the decision to grant or reject the extension. The Canary reveals there is disagreement between the Welsh and UK governments over responsibility for granting this extension. We also highlight CAN's rebuttal of Energybuild's "clean" mining claims.
Aren't we phasing out coal?
Britain has committed to phasing out coal from its energy system by October 2024. The UK government also wants to decarbonise steel production in the UK by 2035. According to CAN, granting a licence extension at Aberpergwm undermines this:
The people we have spoken with are shocked that the UK is embarking on a new commitment to mine up to 40 million tonnes of coal until 2039, emitting around 100 million tonnes of CO2—as well as methane—into our atmosphere.
It's written an open letter to both the Welsh and UK governments demanding action. Additionally, over 4,000 letters have been written to Wales's deputy climate change minister Lee Waters and Michael Gove, secretary of state for levelling up, housing and communities, to demand a block on this extension. CAN said it's:
Horn of Africa climate-crisis-fuelled drought drives 22 million to hunger
A senior Tory has received a blue plaque for his commitment to filth
seeking to directly engage with relevant Ministers such as Lee Waters and Michael Gove with our recent open letter demanding a response to the over 4000 emails sent to them by people dismayed that this climate wrecking project may slip through, in the shadow of COP26.
A knock-on effect?
CAN also believes the granting of this extension licence could have a knock-on effect:
Aberpergwm coal mine extension could set a dangerous precedent. Bryn Bach Coal Ltd, which operates the Glan Lash opencast coal mine, is currently awaiting a planning permission outcome on its application for an extension. They each claim to be the only suppliers of high quality anthracite coal, so have a similar (if conflicting) platform, and permission for one could bolster the chances of the other gaining permission if Ministers don't step in.
So it believes the Welsh and UK governments need to put a moratorium on all new commercial coal mining. This, according to CAN, would send the message out that the UK really is "leading the world in phasing down coal".
Green mining?
Energybuild claims the coal from its mine "fits seamlessly into the green economic revolution". It adds that the mine is:
the only producer of high-grade anthracite in Western Europe. The area's geology contributes to the unique characteristics of the anthracite we produce – clean burn, low emission, low sulphur, high efficiency.
The operation has vast in-situ resources of High-Grade Anthracite (HGA) – measured to international JORC 2012 standards. Unique characteristics, low impurities, ideal specific gravity for water filtration and other processes.
The company also promises to move away from supplying the steelworks at Port Talbot. But CAN dismisses Energybuild's claims as greenwash. It has also written to the Welsh government to rebut the company's claims and to state that the UK already has a sufficient supply of coal to manage a coal-free steel transition. Additionally CAN wrote:
Increasing the supply and likely reducing the price in the process, is likely to jeopardise decarbonisation as it acts as a disincentive for companies to invest in alternative equipment for lower-carbon steel, and if they make the decision to replace e.g. a basic oxygen furnace, they're locked into using coal for many years to come due to the size of investment and longevity of the equipment.
It added:
Mining in the UK won't reduce the same amount of mining in Russia, Canada, or Colombia – it will add to it, as explained by LSE Prof. Ekins (OBE) in this interview we conducted.
It seems nobody wants to take responsibility
The Welsh government appears committed to moving away from fossil fuels:
The draft coal policy is part of the Welsh Government's decisive shift away from the use of fossil fuels in order to tackle the climate emergency. The Welsh Government's policy objective is to avoid the continued extraction and consumption of fossil fuels.
And its deputy climate change minister Lee Waters said:
We want to keep this coal in the ground
However, while it told The Canary it opposes the fossil fuel extraction, it says it can do nothing about the Aberpergwm extension:
We have been clear that we do not support the extraction of fossil fuels and are focused on the climate emergency.
As the original licence here was issued before licensing powers were devolved, Welsh Ministers are not able to intervene in the licensing process and appropriately apply Welsh policy.
The Canary also contacted Michael Gove's Department for Levelling Up, Housing and Communities. Gove's department is due to make a decision on a controversial coalmine in West Cumbria. But Gove's department referred us to the Department for Business, Energy & Industrial Strategy (BEIS). The BEIS then said it was a matter for the coal authority and that it was Welsh ministers who needed to give their approval to the application:
The Coal Authority is responsible for licensing coal mines, including for Aberpergwm, and BEIS ministers have no formal role in the licensing process. The Coal Industry Act 1994 states, where a coal operator wants to mine in Wales, it must seek the approval of the Welsh Ministers as part of its application for a licence to do so.
It then added its position on the future of coal:
Our Net Zero Strategy makes it clear that coal has no part to play in our future power generation which is why we're phasing it out of our electricity by 2024 – a year earlier than planned.
When we contacted the Coal Authority, it said:
Policy for coal mining in Great Britain is set by the UK, Scottish and Welsh Governments through planning policy and legislation and the UK Government through coal licensing policy and legislation.
To operate a coal mine an operator needs relevant rights and permissions including planning permission, a licence from the Coal Authority and to notify the Health and Safety Executive.
Under current legislation, when considering a coal licence application the Coal Authority needs to consider:
Whether the applicant can finance coal mining operations and related liabilities
The nature of the land or property that may be impacted by subsidence and that damage can be properly compensated by the operator
That the operation will be carried out by properly experienced people
If these tests are met and the other permissions are in place then the Coal Authority has to issue a licence. In Wales additional powers were created in the Wales Act 2017 to enable Welsh Ministers to make the final decision on coal licensing matters.
The process for determining the Aberpergwm licence applications has not yet been concluded. This means that we cannot comment further at this time.
A just transition for the community
However, it should also be noted that some people in Glynneath support the mine. They say their livelihoods depend on it. So, if we're as serious about transitioning away from fossil fuels and tackling the climate emergency as we need to be, we need governments to stop delaying and act. They also need to support this community's transition away from coal.
Just as communities need to be consulted before we start mining, they also need to be consulted before we stop mining. The environment, income, and livelihoods of people living in these communities depends on it. As CAN said, the support for the mine from the local community comes from:
a historical lack of support for creating alternative jobs that date back to the Thatcher era
Giving the local community a genuine choice means investment in infrastructure, retraining in desirable and viable jobs, and financial support for small and medium sized enterprises, particularly cooperative and social-interest companies that build and reinvest in their communities.
Communities have already begun workable energy alternatives to help the transition to sustainable energy. So it's time the Welsh and UK governments stopped the blame game and started organising a sustainable transition for Glynneath and the wider community.
Featured image via Just The Job – YouTube Screengrab & Energybuild Screengrab
Tracy Keeling , 31st January 2023
Vast iceberg breaks off Antarctic ice shelf
The 'Johnson' defence
Concerns raised after a Black prisoner at HMP Long Lartin spends over a year in segregation
Aberpergwm
Glynneath
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,374
|
The Cook Islands national under-20 football team is the national U-20 team of the Cook Islands and is controlled by the Cook Islands Football Association. With a population of around 24,000 people it remains one of the smallest FIFA teams.
History
Cook Islands have only won two games so far. A 2–0 win awarded against Samoa in 2001 and a 5–0 win against American Samoa in 2016.
Competition Record
OFC
The OFC Under 20 Qualifying Tournament is a tournament held once every two years to decide the qualification spots for Oceania Football Confederation (OFC) and representatives at the FIFA U-20 World Cup.
Current squad
The following players were called up for the 2022 OFC U-19 Championship from 7 to 20 September 2022. Names in italics denote players who have been capped for the Senior team.
Caps and goals as of 14 September 2022 after the game against American Samoa.
2018 squad
The following players were called up for the 2018 OFC U-19 Championship from 26 May to 2 June 2018. Names in italics denote players who have been capped for the Senior team.
Caps and goals as of 2 June 2018 after the game against American Samoa.
Squad for the 2016 OFC U-20 Championship
Caps and goals as of 10 September 2016 after the game against Tahiti.
Fixtures and results
2016
2018
List of coaches
Tuka Tisam (-2016)
Matt Calcott (2016-2017)
Alan Taylor (2018-)
See also
Cook Islands men's national football team
Cook Islands men's national under-17 football team
Cook Islands women's national football team
Cook Islands women's national under-17 football team
References
External links
Cook Islands Football Federation official website
U
Oceanian national under-20 association football teams
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,264
|
package vcl
import (
. "github.com/ying32/govcl/vcl/api"
. "github.com/ying32/govcl/vcl/types"
"unsafe"
)
type TListView struct {
IWinControl
instance uintptr
// 特殊情况下使用,主要应对Go的GC问题,与LCL没有太多关系。
ptr unsafe.Pointer
}
// 创建一个新的对象。
//
// Create a new object.
func NewListView(owner IComponent) *TListView {
l := new(TListView)
l.instance = ListView_Create(CheckPtr(owner))
l.ptr = unsafe.Pointer(l.instance)
return l
}
// 动态转换一个已存在的对象实例。
//
// Dynamically convert an existing object instance.
func AsListView(obj interface{}) *TListView {
instance, ptr := getInstance(obj)
if instance == 0 { return nil }
return &TListView{instance: instance, ptr: ptr}
}
// -------------------------- Deprecated begin --------------------------
// 新建一个对象来自已经存在的对象实例指针。
//
// Create a new object from an existing object instance pointer.
// Deprecated: use AsListView.
func ListViewFromInst(inst uintptr) *TListView {
return AsListView(inst)
}
// 新建一个对象来自已经存在的对象实例。
//
// Create a new object from an existing object instance.
// Deprecated: use AsListView.
func ListViewFromObj(obj IObject) *TListView {
return AsListView(obj)
}
// 新建一个对象来自不安全的地址。注意:使用此函数可能造成一些不明情况,慎用。
//
// Create a new object from an unsecured address. Note: Using this function may cause some unclear situations and be used with caution..
// Deprecated: use AsListView.
func ListViewFromUnsafePointer(ptr unsafe.Pointer) *TListView {
return AsListView(ptr)
}
// -------------------------- Deprecated end --------------------------
// 释放对象。
//
// Free object.
func (l *TListView) Free() {
if l.instance != 0 {
ListView_Free(l.instance)
l.instance, l.ptr = 0, nullptr
}
}
// 返回对象实例指针。
//
// Return object instance pointer.
func (l *TListView) Instance() uintptr {
return l.instance
}
// 获取一个不安全的地址。
//
// Get an unsafe address.
func (l *TListView) UnsafeAddr() unsafe.Pointer {
return l.ptr
}
// 检测地址是否为空。
//
// Check if the address is empty.
func (l *TListView) IsValid() bool {
return l.instance != 0
}
// 检测当前对象是否继承自目标对象。
//
// Checks whether the current object is inherited from the target object.
func (l *TListView) Is() TIs {
return TIs(l.instance)
}
// 动态转换当前对象为目标对象。
//
// Dynamically convert the current object to the target object.
//func (l *TListView) As() TAs {
// return TAs(l.instance)
//}
// 获取类信息指针。
//
// Get class information pointer.
func TListViewClass() TClass {
return ListView_StaticClassType()
}
func (l *TListView) AddItem(Item string, AObject IObject) {
ListView_AddItem(l.instance, Item , CheckPtr(AObject))
}
func (l *TListView) AlphaSort() bool {
return ListView_AlphaSort(l.instance)
}
// 清除。
func (l *TListView) Clear() {
ListView_Clear(l.instance)
}
// 清除选择。
func (l *TListView) ClearSelection() {
ListView_ClearSelection(l.instance)
}
// 删除选择的。
func (l *TListView) DeleteSelected() {
ListView_DeleteSelected(l.instance)
}
func (l *TListView) GetHitTestInfoAt(X int32, Y int32) THitTests {
return ListView_GetHitTestInfoAt(l.instance, X , Y)
}
func (l *TListView) GetItemAt(X int32, Y int32) *TListItem {
return AsListItem(ListView_GetItemAt(l.instance, X , Y))
}
func (l *TListView) GetNearestItem(Point TPoint, Direction TSearchDirection) *TListItem {
return AsListItem(ListView_GetNearestItem(l.instance, Point , Direction))
}
func (l *TListView) GetNextItem(StartItem *TListItem, Direction TSearchDirection, States TListItemStates) *TListItem {
return AsListItem(ListView_GetNextItem(l.instance, CheckPtr(StartItem), Direction , States))
}
func (l *TListView) IsEditing() bool {
return ListView_IsEditing(l.instance)
}
// 全选。
func (l *TListView) SelectAll() {
ListView_SelectAll(l.instance)
}
// 自定义排序,ASortProc参数无效,仅仅用来兼容Delphi的。
//
// Custom sorting, ASortProc parameter is invalid, Only used to be compatible with Delphi.
func (l *TListView) CustomSort(SortProc PFNLVCOMPARE, lParam int) bool {
return ListView_CustomSort(l.instance, SortProc , lParam)
}
// 是否可以获得焦点。
func (l *TListView) CanFocus() bool {
return ListView_CanFocus(l.instance)
}
// 返回是否包含指定控件。
//
// it's contain a specified control.
func (l *TListView) ContainsControl(Control IControl) bool {
return ListView_ContainsControl(l.instance, CheckPtr(Control))
}
// 返回指定坐标及相关属性位置控件。
//
// Returns the specified coordinate and the relevant attribute position control..
func (l *TListView) ControlAtPos(Pos TPoint, AllowDisabled bool, AllowWinControls bool, AllLevels bool) *TControl {
return AsControl(ListView_ControlAtPos(l.instance, Pos , AllowDisabled , AllowWinControls , AllLevels))
}
// 禁用控件的对齐。
//
// Disable control alignment.
func (l *TListView) DisableAlign() {
ListView_DisableAlign(l.instance)
}
// 启用控件对齐。
//
// Enabled control alignment.
func (l *TListView) EnableAlign() {
ListView_EnableAlign(l.instance)
}
// 查找子控件。
//
// Find sub controls.
func (l *TListView) FindChildControl(ControlName string) *TControl {
return AsControl(ListView_FindChildControl(l.instance, ControlName))
}
func (l *TListView) FlipChildren(AllLevels bool) {
ListView_FlipChildren(l.instance, AllLevels)
}
// 返回是否获取焦点。
//
// Return to get focus.
func (l *TListView) Focused() bool {
return ListView_Focused(l.instance)
}
// 句柄是否已经分配。
//
// Is the handle already allocated.
func (l *TListView) HandleAllocated() bool {
return ListView_HandleAllocated(l.instance)
}
// 插入一个控件。
//
// Insert a control.
func (l *TListView) InsertControl(AControl IControl) {
ListView_InsertControl(l.instance, CheckPtr(AControl))
}
// 要求重绘。
//
// Redraw.
func (l *TListView) Invalidate() {
ListView_Invalidate(l.instance)
}
// 绘画至指定DC。
//
// Painting to the specified DC.
func (l *TListView) PaintTo(DC HDC, X int32, Y int32) {
ListView_PaintTo(l.instance, DC , X , Y)
}
// 移除一个控件。
//
// Remove a control.
func (l *TListView) RemoveControl(AControl IControl) {
ListView_RemoveControl(l.instance, CheckPtr(AControl))
}
// 重新对齐。
//
// Realign.
func (l *TListView) Realign() {
ListView_Realign(l.instance)
}
// 重绘。
//
// Repaint.
func (l *TListView) Repaint() {
ListView_Repaint(l.instance)
}
// 按比例缩放。
//
// Scale by.
func (l *TListView) ScaleBy(M int32, D int32) {
ListView_ScaleBy(l.instance, M , D)
}
// 滚动至指定位置。
//
// Scroll by.
func (l *TListView) ScrollBy(DeltaX int32, DeltaY int32) {
ListView_ScrollBy(l.instance, DeltaX , DeltaY)
}
// 设置组件边界。
//
// Set component boundaries.
func (l *TListView) SetBounds(ALeft int32, ATop int32, AWidth int32, AHeight int32) {
ListView_SetBounds(l.instance, ALeft , ATop , AWidth , AHeight)
}
// 设置控件焦点。
//
// Set control focus.
func (l *TListView) SetFocus() {
ListView_SetFocus(l.instance)
}
// 控件更新。
//
// Update.
func (l *TListView) Update() {
ListView_Update(l.instance)
}
// 将控件置于最前。
//
// Bring the control to the front.
func (l *TListView) BringToFront() {
ListView_BringToFront(l.instance)
}
// 将客户端坐标转为绝对的屏幕坐标。
//
// Convert client coordinates to absolute screen coordinates.
func (l *TListView) ClientToScreen(Point TPoint) TPoint {
return ListView_ClientToScreen(l.instance, Point)
}
// 将客户端坐标转为父容器坐标。
//
// Convert client coordinates to parent container coordinates.
func (l *TListView) ClientToParent(Point TPoint, AParent IWinControl) TPoint {
return ListView_ClientToParent(l.instance, Point , CheckPtr(AParent))
}
// 是否在拖拽中。
//
// Is it in the middle of dragging.
func (l *TListView) Dragging() bool {
return ListView_Dragging(l.instance)
}
// 是否有父容器。
//
// Is there a parent container.
func (l *TListView) HasParent() bool {
return ListView_HasParent(l.instance)
}
// 隐藏控件。
//
// Hidden control.
func (l *TListView) Hide() {
ListView_Hide(l.instance)
}
// 发送一个消息。
//
// Send a message.
func (l *TListView) Perform(Msg uint32, WParam uintptr, LParam int) int {
return ListView_Perform(l.instance, Msg , WParam , LParam)
}
// 刷新控件。
//
// Refresh control.
func (l *TListView) Refresh() {
ListView_Refresh(l.instance)
}
// 将屏幕坐标转为客户端坐标。
//
// Convert screen coordinates to client coordinates.
func (l *TListView) ScreenToClient(Point TPoint) TPoint {
return ListView_ScreenToClient(l.instance, Point)
}
// 将父容器坐标转为客户端坐标。
//
// Convert parent container coordinates to client coordinates.
func (l *TListView) ParentToClient(Point TPoint, AParent IWinControl) TPoint {
return ListView_ParentToClient(l.instance, Point , CheckPtr(AParent))
}
// 控件至于最后面。
//
// The control is placed at the end.
func (l *TListView) SendToBack() {
ListView_SendToBack(l.instance)
}
// 显示控件。
//
// Show control.
func (l *TListView) Show() {
ListView_Show(l.instance)
}
// 获取控件的字符,如果有。
//
// Get the characters of the control, if any.
func (l *TListView) GetTextBuf(Buffer *string, BufSize int32) int32 {
return ListView_GetTextBuf(l.instance, Buffer , BufSize)
}
// 获取控件的字符长,如果有。
//
// Get the character length of the control, if any.
func (l *TListView) GetTextLen() int32 {
return ListView_GetTextLen(l.instance)
}
// 设置控件字符,如果有。
//
// Set control characters, if any.
func (l *TListView) SetTextBuf(Buffer string) {
ListView_SetTextBuf(l.instance, Buffer)
}
// 查找指定名称的组件。
//
// Find the component with the specified name.
func (l *TListView) FindComponent(AName string) *TComponent {
return AsComponent(ListView_FindComponent(l.instance, AName))
}
// 获取类名路径。
//
// Get the class name path.
func (l *TListView) GetNamePath() string {
return ListView_GetNamePath(l.instance)
}
// 复制一个对象,如果对象实现了此方法的话。
//
// Copy an object, if the object implements this method.
func (l *TListView) Assign(Source IObject) {
ListView_Assign(l.instance, CheckPtr(Source))
}
// 获取类的类型信息。
//
// Get class type information.
func (l *TListView) ClassType() TClass {
return ListView_ClassType(l.instance)
}
// 获取当前对象类名称。
//
// Get the current object class name.
func (l *TListView) ClassName() string {
return ListView_ClassName(l.instance)
}
// 获取当前对象实例大小。
//
// Get the current object instance size.
func (l *TListView) InstanceSize() int32 {
return ListView_InstanceSize(l.instance)
}
// 判断当前类是否继承自指定类。
//
// Determine whether the current class inherits from the specified class.
func (l *TListView) InheritsFrom(AClass TClass) bool {
return ListView_InheritsFrom(l.instance, AClass)
}
// 与一个对象进行比较。
//
// Compare with an object.
func (l *TListView) Equals(Obj IObject) bool {
return ListView_Equals(l.instance, CheckPtr(Obj))
}
// 获取类的哈希值。
//
// Get the hash value of the class.
func (l *TListView) GetHashCode() int32 {
return ListView_GetHashCode(l.instance)
}
// 文本类信息。
//
// Text information.
func (l *TListView) ToString() string {
return ListView_ToString(l.instance)
}
func (l *TListView) AnchorToNeighbour(ASide TAnchorKind, ASpace int32, ASibling IControl) {
ListView_AnchorToNeighbour(l.instance, ASide , ASpace , CheckPtr(ASibling))
}
func (l *TListView) AnchorParallel(ASide TAnchorKind, ASpace int32, ASibling IControl) {
ListView_AnchorParallel(l.instance, ASide , ASpace , CheckPtr(ASibling))
}
// 置于指定控件的横向中心。
func (l *TListView) AnchorHorizontalCenterTo(ASibling IControl) {
ListView_AnchorHorizontalCenterTo(l.instance, CheckPtr(ASibling))
}
// 置于指定控件的纵向中心。
func (l *TListView) AnchorVerticalCenterTo(ASibling IControl) {
ListView_AnchorVerticalCenterTo(l.instance, CheckPtr(ASibling))
}
func (l *TListView) AnchorSame(ASide TAnchorKind, ASibling IControl) {
ListView_AnchorSame(l.instance, ASide , CheckPtr(ASibling))
}
func (l *TListView) AnchorAsAlign(ATheAlign TAlign, ASpace int32) {
ListView_AnchorAsAlign(l.instance, ATheAlign , ASpace)
}
func (l *TListView) AnchorClient(ASpace int32) {
ListView_AnchorClient(l.instance, ASpace)
}
func (l *TListView) ScaleDesignToForm(ASize int32) int32 {
return ListView_ScaleDesignToForm(l.instance, ASize)
}
func (l *TListView) ScaleFormToDesign(ASize int32) int32 {
return ListView_ScaleFormToDesign(l.instance, ASize)
}
func (l *TListView) Scale96ToForm(ASize int32) int32 {
return ListView_Scale96ToForm(l.instance, ASize)
}
func (l *TListView) ScaleFormTo96(ASize int32) int32 {
return ListView_ScaleFormTo96(l.instance, ASize)
}
func (l *TListView) Scale96ToFont(ASize int32) int32 {
return ListView_Scale96ToFont(l.instance, ASize)
}
func (l *TListView) ScaleFontTo96(ASize int32) int32 {
return ListView_ScaleFontTo96(l.instance, ASize)
}
func (l *TListView) ScaleScreenToFont(ASize int32) int32 {
return ListView_ScaleScreenToFont(l.instance, ASize)
}
func (l *TListView) ScaleFontToScreen(ASize int32) int32 {
return ListView_ScaleFontToScreen(l.instance, ASize)
}
func (l *TListView) Scale96ToScreen(ASize int32) int32 {
return ListView_Scale96ToScreen(l.instance, ASize)
}
func (l *TListView) ScaleScreenTo96(ASize int32) int32 {
return ListView_ScaleScreenTo96(l.instance, ASize)
}
func (l *TListView) AutoAdjustLayout(AMode TLayoutAdjustmentPolicy, AFromPPI int32, AToPPI int32, AOldFormWidth int32, ANewFormWidth int32) {
ListView_AutoAdjustLayout(l.instance, AMode , AFromPPI , AToPPI , AOldFormWidth , ANewFormWidth)
}
func (l *TListView) FixDesignFontsPPI(ADesignTimePPI int32) {
ListView_FixDesignFontsPPI(l.instance, ADesignTimePPI)
}
func (l *TListView) ScaleFontsPPI(AToPPI int32, AProportion float64) {
ListView_ScaleFontsPPI(l.instance, AToPPI , AProportion)
}
func (l *TListView) AutoSort() bool {
return ListView_GetAutoSort(l.instance)
}
func (l *TListView) SetAutoSort(value bool) {
ListView_SetAutoSort(l.instance, value)
}
func (l *TListView) AutoSortIndicator() bool {
return ListView_GetAutoSortIndicator(l.instance)
}
func (l *TListView) SetAutoSortIndicator(value bool) {
ListView_SetAutoSortIndicator(l.instance, value)
}
func (l *TListView) AutoWidthLastColumn() bool {
return ListView_GetAutoWidthLastColumn(l.instance)
}
func (l *TListView) SetAutoWidthLastColumn(value bool) {
ListView_SetAutoWidthLastColumn(l.instance, value)
}
func (l *TListView) SmallImagesWidth() int32 {
return ListView_GetSmallImagesWidth(l.instance)
}
func (l *TListView) SetSmallImagesWidth(value int32) {
ListView_SetSmallImagesWidth(l.instance, value)
}
func (l *TListView) SortColumn() int32 {
return ListView_GetSortColumn(l.instance)
}
func (l *TListView) SetSortColumn(value int32) {
ListView_SetSortColumn(l.instance, value)
}
func (l *TListView) SortDirection() TSortDirection {
return ListView_GetSortDirection(l.instance)
}
func (l *TListView) SetSortDirection(value TSortDirection) {
ListView_SetSortDirection(l.instance, value)
}
func (l *TListView) LargeImagesWidth() int32 {
return ListView_GetLargeImagesWidth(l.instance)
}
func (l *TListView) SetLargeImagesWidth(value int32) {
ListView_SetLargeImagesWidth(l.instance, value)
}
func (l *TListView) StateImagesWidth() int32 {
return ListView_GetStateImagesWidth(l.instance)
}
func (l *TListView) SetStateImagesWidth(value int32) {
ListView_SetStateImagesWidth(l.instance, value)
}
func (l *TListView) ToolTips() bool {
return ListView_GetToolTips(l.instance)
}
func (l *TListView) SetToolTips(value bool) {
ListView_SetToolTips(l.instance, value)
}
func (l *TListView) ScrollBars() TScrollStyle {
return ListView_GetScrollBars(l.instance)
}
func (l *TListView) SetScrollBars(value TScrollStyle) {
ListView_SetScrollBars(l.instance, value)
}
func (l *TListView) ColumnCount() int32 {
return ListView_GetColumnCount(l.instance)
}
func (l *TListView) Action() *TAction {
return AsAction(ListView_GetAction(l.instance))
}
func (l *TListView) SetAction(value IComponent) {
ListView_SetAction(l.instance, CheckPtr(value))
}
// 获取控件自动调整。
//
// Get Control automatically adjusts.
func (l *TListView) Align() TAlign {
return ListView_GetAlign(l.instance)
}
// 设置控件自动调整。
//
// Set Control automatically adjusts.
func (l *TListView) SetAlign(value TAlign) {
ListView_SetAlign(l.instance, value)
}
func (l *TListView) AllocBy() int32 {
return ListView_GetAllocBy(l.instance)
}
func (l *TListView) SetAllocBy(value int32) {
ListView_SetAllocBy(l.instance, value)
}
// 获取四个角位置的锚点。
func (l *TListView) Anchors() TAnchors {
return ListView_GetAnchors(l.instance)
}
// 设置四个角位置的锚点。
func (l *TListView) SetAnchors(value TAnchors) {
ListView_SetAnchors(l.instance, value)
}
func (l *TListView) BiDiMode() TBiDiMode {
return ListView_GetBiDiMode(l.instance)
}
func (l *TListView) SetBiDiMode(value TBiDiMode) {
ListView_SetBiDiMode(l.instance, value)
}
// 获取窗口边框样式。比如:无边框,单一边框等。
func (l *TListView) BorderStyle() TBorderStyle {
return ListView_GetBorderStyle(l.instance)
}
// 设置窗口边框样式。比如:无边框,单一边框等。
func (l *TListView) SetBorderStyle(value TBorderStyle) {
ListView_SetBorderStyle(l.instance, value)
}
// 获取边框的宽度。
func (l *TListView) BorderWidth() int32 {
return ListView_GetBorderWidth(l.instance)
}
// 设置边框的宽度。
func (l *TListView) SetBorderWidth(value int32) {
ListView_SetBorderWidth(l.instance, value)
}
func (l *TListView) Checkboxes() bool {
return ListView_GetCheckboxes(l.instance)
}
func (l *TListView) SetCheckboxes(value bool) {
ListView_SetCheckboxes(l.instance, value)
}
// 获取颜色。
//
// Get color.
func (l *TListView) Color() TColor {
return ListView_GetColor(l.instance)
}
// 设置颜色。
//
// Set color.
func (l *TListView) SetColor(value TColor) {
ListView_SetColor(l.instance, value)
}
func (l *TListView) Columns() *TListColumns {
return AsListColumns(ListView_GetColumns(l.instance))
}
func (l *TListView) SetColumns(value *TListColumns) {
ListView_SetColumns(l.instance, CheckPtr(value))
}
func (l *TListView) ColumnClick() bool {
return ListView_GetColumnClick(l.instance)
}
func (l *TListView) SetColumnClick(value bool) {
ListView_SetColumnClick(l.instance, value)
}
// 获取约束控件大小。
func (l *TListView) Constraints() *TSizeConstraints {
return AsSizeConstraints(ListView_GetConstraints(l.instance))
}
// 设置约束控件大小。
func (l *TListView) SetConstraints(value *TSizeConstraints) {
ListView_SetConstraints(l.instance, CheckPtr(value))
}
// 获取设置控件双缓冲。
//
// Get Set control double buffering.
func (l *TListView) DoubleBuffered() bool {
return ListView_GetDoubleBuffered(l.instance)
}
// 设置设置控件双缓冲。
//
// Set Set control double buffering.
func (l *TListView) SetDoubleBuffered(value bool) {
ListView_SetDoubleBuffered(l.instance, value)
}
// 获取设置控件拖拽时的光标。
//
// Get Set the cursor when the control is dragged.
func (l *TListView) DragCursor() TCursor {
return ListView_GetDragCursor(l.instance)
}
// 设置设置控件拖拽时的光标。
//
// Set Set the cursor when the control is dragged.
func (l *TListView) SetDragCursor(value TCursor) {
ListView_SetDragCursor(l.instance, value)
}
// 获取拖拽方式。
//
// Get Drag and drop.
func (l *TListView) DragKind() TDragKind {
return ListView_GetDragKind(l.instance)
}
// 设置拖拽方式。
//
// Set Drag and drop.
func (l *TListView) SetDragKind(value TDragKind) {
ListView_SetDragKind(l.instance, value)
}
// 获取拖拽模式。
//
// Get Drag mode.
func (l *TListView) DragMode() TDragMode {
return ListView_GetDragMode(l.instance)
}
// 设置拖拽模式。
//
// Set Drag mode.
func (l *TListView) SetDragMode(value TDragMode) {
ListView_SetDragMode(l.instance, value)
}
// 获取控件启用。
//
// Get the control enabled.
func (l *TListView) Enabled() bool {
return ListView_GetEnabled(l.instance)
}
// 设置控件启用。
//
// Set the control enabled.
func (l *TListView) SetEnabled(value bool) {
ListView_SetEnabled(l.instance, value)
}
// 获取字体。
//
// Get Font.
func (l *TListView) Font() *TFont {
return AsFont(ListView_GetFont(l.instance))
}
// 设置字体。
//
// Set Font.
func (l *TListView) SetFont(value *TFont) {
ListView_SetFont(l.instance, CheckPtr(value))
}
func (l *TListView) FlatScrollBars() bool {
return ListView_GetFlatScrollBars(l.instance)
}
func (l *TListView) SetFlatScrollBars(value bool) {
ListView_SetFlatScrollBars(l.instance, value)
}
func (l *TListView) FullDrag() bool {
return ListView_GetFullDrag(l.instance)
}
func (l *TListView) SetFullDrag(value bool) {
ListView_SetFullDrag(l.instance, value)
}
func (l *TListView) GridLines() bool {
return ListView_GetGridLines(l.instance)
}
func (l *TListView) SetGridLines(value bool) {
ListView_SetGridLines(l.instance, value)
}
// 获取隐藏选择。
func (l *TListView) HideSelection() bool {
return ListView_GetHideSelection(l.instance)
}
// 设置隐藏选择。
func (l *TListView) SetHideSelection(value bool) {
ListView_SetHideSelection(l.instance, value)
}
func (l *TListView) HotTrack() bool {
return ListView_GetHotTrack(l.instance)
}
func (l *TListView) SetHotTrack(value bool) {
ListView_SetHotTrack(l.instance, value)
}
func (l *TListView) IconOptions() *TIconOptions {
return AsIconOptions(ListView_GetIconOptions(l.instance))
}
func (l *TListView) SetIconOptions(value IObject) {
ListView_SetIconOptions(l.instance, CheckPtr(value))
}
func (l *TListView) Items() *TListItems {
return AsListItems(ListView_GetItems(l.instance))
}
func (l *TListView) SetItems(value *TListItems) {
ListView_SetItems(l.instance, CheckPtr(value))
}
func (l *TListView) LargeImages() *TImageList {
return AsImageList(ListView_GetLargeImages(l.instance))
}
func (l *TListView) SetLargeImages(value IComponent) {
ListView_SetLargeImages(l.instance, CheckPtr(value))
}
func (l *TListView) MultiSelect() bool {
return ListView_GetMultiSelect(l.instance)
}
func (l *TListView) SetMultiSelect(value bool) {
ListView_SetMultiSelect(l.instance, value)
}
func (l *TListView) OwnerData() bool {
return ListView_GetOwnerData(l.instance)
}
func (l *TListView) SetOwnerData(value bool) {
ListView_SetOwnerData(l.instance, value)
}
func (l *TListView) OwnerDraw() bool {
return ListView_GetOwnerDraw(l.instance)
}
func (l *TListView) SetOwnerDraw(value bool) {
ListView_SetOwnerDraw(l.instance, value)
}
// 获取只读。
func (l *TListView) ReadOnly() bool {
return ListView_GetReadOnly(l.instance)
}
// 设置只读。
func (l *TListView) SetReadOnly(value bool) {
ListView_SetReadOnly(l.instance, value)
}
func (l *TListView) RowSelect() bool {
return ListView_GetRowSelect(l.instance)
}
func (l *TListView) SetRowSelect(value bool) {
ListView_SetRowSelect(l.instance, value)
}
// 获取使用父容器颜色。
//
// Get parent color.
func (l *TListView) ParentColor() bool {
return ListView_GetParentColor(l.instance)
}
// 设置使用父容器颜色。
//
// Set parent color.
func (l *TListView) SetParentColor(value bool) {
ListView_SetParentColor(l.instance, value)
}
// 获取使用父容器双缓冲。
//
// Get Parent container double buffering.
func (l *TListView) ParentDoubleBuffered() bool {
return ListView_GetParentDoubleBuffered(l.instance)
}
// 设置使用父容器双缓冲。
//
// Set Parent container double buffering.
func (l *TListView) SetParentDoubleBuffered(value bool) {
ListView_SetParentDoubleBuffered(l.instance, value)
}
// 获取使用父容器字体。
//
// Get Parent container font.
func (l *TListView) ParentFont() bool {
return ListView_GetParentFont(l.instance)
}
// 设置使用父容器字体。
//
// Set Parent container font.
func (l *TListView) SetParentFont(value bool) {
ListView_SetParentFont(l.instance, value)
}
// 获取以父容器的ShowHint属性为准。
func (l *TListView) ParentShowHint() bool {
return ListView_GetParentShowHint(l.instance)
}
// 设置以父容器的ShowHint属性为准。
func (l *TListView) SetParentShowHint(value bool) {
ListView_SetParentShowHint(l.instance, value)
}
// 获取右键菜单。
//
// Get Right click menu.
func (l *TListView) PopupMenu() *TPopupMenu {
return AsPopupMenu(ListView_GetPopupMenu(l.instance))
}
// 设置右键菜单。
//
// Set Right click menu.
func (l *TListView) SetPopupMenu(value IComponent) {
ListView_SetPopupMenu(l.instance, CheckPtr(value))
}
func (l *TListView) ShowColumnHeaders() bool {
return ListView_GetShowColumnHeaders(l.instance)
}
func (l *TListView) SetShowColumnHeaders(value bool) {
ListView_SetShowColumnHeaders(l.instance, value)
}
// 获取显示鼠标悬停提示。
//
// Get Show mouseover tips.
func (l *TListView) ShowHint() bool {
return ListView_GetShowHint(l.instance)
}
// 设置显示鼠标悬停提示。
//
// Set Show mouseover tips.
func (l *TListView) SetShowHint(value bool) {
ListView_SetShowHint(l.instance, value)
}
func (l *TListView) SmallImages() *TImageList {
return AsImageList(ListView_GetSmallImages(l.instance))
}
func (l *TListView) SetSmallImages(value IComponent) {
ListView_SetSmallImages(l.instance, CheckPtr(value))
}
func (l *TListView) SortType() TSortType {
return ListView_GetSortType(l.instance)
}
func (l *TListView) SetSortType(value TSortType) {
ListView_SetSortType(l.instance, value)
}
func (l *TListView) StateImages() *TImageList {
return AsImageList(ListView_GetStateImages(l.instance))
}
func (l *TListView) SetStateImages(value IComponent) {
ListView_SetStateImages(l.instance, CheckPtr(value))
}
// 获取Tab切换顺序序号。
//
// Get Tab switching sequence number.
func (l *TListView) TabOrder() TTabOrder {
return ListView_GetTabOrder(l.instance)
}
// 设置Tab切换顺序序号。
//
// Set Tab switching sequence number.
func (l *TListView) SetTabOrder(value TTabOrder) {
ListView_SetTabOrder(l.instance, value)
}
// 获取Tab可停留。
//
// Get Tab can stay.
func (l *TListView) TabStop() bool {
return ListView_GetTabStop(l.instance)
}
// 设置Tab可停留。
//
// Set Tab can stay.
func (l *TListView) SetTabStop(value bool) {
ListView_SetTabStop(l.instance, value)
}
func (l *TListView) ViewStyle() TViewStyle {
return ListView_GetViewStyle(l.instance)
}
func (l *TListView) SetViewStyle(value TViewStyle) {
ListView_SetViewStyle(l.instance, value)
}
// 获取控件可视。
//
// Get the control visible.
func (l *TListView) Visible() bool {
return ListView_GetVisible(l.instance)
}
// 设置控件可视。
//
// Set the control visible.
func (l *TListView) SetVisible(value bool) {
ListView_SetVisible(l.instance, value)
}
func (l *TListView) SetOnAdvancedCustomDraw(fn TLVAdvancedCustomDrawEvent) {
ListView_SetOnAdvancedCustomDraw(l.instance, fn)
}
func (l *TListView) SetOnAdvancedCustomDrawItem(fn TLVAdvancedCustomDrawItemEvent) {
ListView_SetOnAdvancedCustomDrawItem(l.instance, fn)
}
func (l *TListView) SetOnAdvancedCustomDrawSubItem(fn TLVAdvancedCustomDrawSubItemEvent) {
ListView_SetOnAdvancedCustomDrawSubItem(l.instance, fn)
}
// 设置改变事件。
//
// Set changed event.
func (l *TListView) SetOnChange(fn TLVChangeEvent) {
ListView_SetOnChange(l.instance, fn)
}
// 设置控件单击事件。
//
// Set control click event.
func (l *TListView) SetOnClick(fn TNotifyEvent) {
ListView_SetOnClick(l.instance, fn)
}
func (l *TListView) SetOnColumnClick(fn TLVColumnClickEvent) {
ListView_SetOnColumnClick(l.instance, fn)
}
func (l *TListView) SetOnCompare(fn TLVCompareEvent) {
ListView_SetOnCompare(l.instance, fn)
}
// 设置上下文弹出事件,一般是右键时弹出。
//
// Set Context popup event, usually pop up when right click.
func (l *TListView) SetOnContextPopup(fn TContextPopupEvent) {
ListView_SetOnContextPopup(l.instance, fn)
}
func (l *TListView) SetOnCustomDraw(fn TLVCustomDrawEvent) {
ListView_SetOnCustomDraw(l.instance, fn)
}
func (l *TListView) SetOnCustomDrawItem(fn TLVCustomDrawItemEvent) {
ListView_SetOnCustomDrawItem(l.instance, fn)
}
func (l *TListView) SetOnCustomDrawSubItem(fn TLVCustomDrawSubItemEvent) {
ListView_SetOnCustomDrawSubItem(l.instance, fn)
}
func (l *TListView) SetOnData(fn TLVDataEvent) {
ListView_SetOnData(l.instance, fn)
}
func (l *TListView) SetOnDataFind(fn TLVDataFindEvent) {
ListView_SetOnDataFind(l.instance, fn)
}
func (l *TListView) SetOnDataHint(fn TLVDataHintEvent) {
ListView_SetOnDataHint(l.instance, fn)
}
// 设置双击事件。
func (l *TListView) SetOnDblClick(fn TNotifyEvent) {
ListView_SetOnDblClick(l.instance, fn)
}
func (l *TListView) SetOnDeletion(fn TLVDeletedEvent) {
ListView_SetOnDeletion(l.instance, fn)
}
func (l *TListView) SetOnDrawItem(fn TLVDrawItemEvent) {
ListView_SetOnDrawItem(l.instance, fn)
}
func (l *TListView) SetOnEdited(fn TLVEditedEvent) {
ListView_SetOnEdited(l.instance, fn)
}
func (l *TListView) SetOnEditing(fn TLVEditingEvent) {
ListView_SetOnEditing(l.instance, fn)
}
// 设置停靠结束事件。
//
// Set Dock end event.
func (l *TListView) SetOnEndDock(fn TEndDragEvent) {
ListView_SetOnEndDock(l.instance, fn)
}
// 设置拖拽结束。
//
// Set End of drag.
func (l *TListView) SetOnEndDrag(fn TEndDragEvent) {
ListView_SetOnEndDrag(l.instance, fn)
}
// 设置焦点进入。
//
// Set Focus entry.
func (l *TListView) SetOnEnter(fn TNotifyEvent) {
ListView_SetOnEnter(l.instance, fn)
}
// 设置焦点退出。
//
// Set Focus exit.
func (l *TListView) SetOnExit(fn TNotifyEvent) {
ListView_SetOnExit(l.instance, fn)
}
// 设置拖拽下落事件。
//
// Set Drag and drop event.
func (l *TListView) SetOnDragDrop(fn TDragDropEvent) {
ListView_SetOnDragDrop(l.instance, fn)
}
// 设置拖拽完成事件。
//
// Set Drag and drop completion event.
func (l *TListView) SetOnDragOver(fn TDragOverEvent) {
ListView_SetOnDragOver(l.instance, fn)
}
func (l *TListView) SetOnInsert(fn TLVDeletedEvent) {
ListView_SetOnInsert(l.instance, fn)
}
// 设置键盘按键按下事件。
//
// Set Keyboard button press event.
func (l *TListView) SetOnKeyDown(fn TKeyEvent) {
ListView_SetOnKeyDown(l.instance, fn)
}
// 设置键键下事件。
func (l *TListView) SetOnKeyPress(fn TKeyPressEvent) {
ListView_SetOnKeyPress(l.instance, fn)
}
// 设置键盘按键抬起事件。
//
// Set Keyboard button lift event.
func (l *TListView) SetOnKeyUp(fn TKeyEvent) {
ListView_SetOnKeyUp(l.instance, fn)
}
// 设置鼠标按下事件。
//
// Set Mouse down event.
func (l *TListView) SetOnMouseDown(fn TMouseEvent) {
ListView_SetOnMouseDown(l.instance, fn)
}
// 设置鼠标进入事件。
//
// Set Mouse entry event.
func (l *TListView) SetOnMouseEnter(fn TNotifyEvent) {
ListView_SetOnMouseEnter(l.instance, fn)
}
// 设置鼠标离开事件。
//
// Set Mouse leave event.
func (l *TListView) SetOnMouseLeave(fn TNotifyEvent) {
ListView_SetOnMouseLeave(l.instance, fn)
}
// 设置鼠标移动事件。
func (l *TListView) SetOnMouseMove(fn TMouseMoveEvent) {
ListView_SetOnMouseMove(l.instance, fn)
}
// 设置鼠标抬起事件。
//
// Set Mouse lift event.
func (l *TListView) SetOnMouseUp(fn TMouseEvent) {
ListView_SetOnMouseUp(l.instance, fn)
}
// 设置大小被改变事件。
func (l *TListView) SetOnResize(fn TNotifyEvent) {
ListView_SetOnResize(l.instance, fn)
}
func (l *TListView) SetOnSelectItem(fn TLVSelectItemEvent) {
ListView_SetOnSelectItem(l.instance, fn)
}
func (l *TListView) SetOnItemChecked(fn TLVCheckedItemEvent) {
ListView_SetOnItemChecked(l.instance, fn)
}
// 设置启动停靠。
func (l *TListView) SetOnStartDock(fn TStartDockEvent) {
ListView_SetOnStartDock(l.instance, fn)
}
// 获取画布。
func (l *TListView) Canvas() *TCanvas {
return AsCanvas(ListView_GetCanvas(l.instance))
}
func (l *TListView) DropTarget() *TListItem {
return AsListItem(ListView_GetDropTarget(l.instance))
}
func (l *TListView) SetDropTarget(value *TListItem) {
ListView_SetDropTarget(l.instance, CheckPtr(value))
}
func (l *TListView) ItemFocused() *TListItem {
return AsListItem(ListView_GetItemFocused(l.instance))
}
func (l *TListView) SetItemFocused(value *TListItem) {
ListView_SetItemFocused(l.instance, CheckPtr(value))
}
func (l *TListView) SelCount() int32 {
return ListView_GetSelCount(l.instance)
}
func (l *TListView) Selected() *TListItem {
return AsListItem(ListView_GetSelected(l.instance))
}
func (l *TListView) SetSelected(value *TListItem) {
ListView_SetSelected(l.instance, CheckPtr(value))
}
func (l *TListView) TopItem() *TListItem {
return AsListItem(ListView_GetTopItem(l.instance))
}
func (l *TListView) VisibleRowCount() int32 {
return ListView_GetVisibleRowCount(l.instance)
}
func (l *TListView) ItemIndex() int32 {
return ListView_GetItemIndex(l.instance)
}
func (l *TListView) SetItemIndex(value int32) {
ListView_SetItemIndex(l.instance, value)
}
// 获取依靠客户端总数。
func (l *TListView) DockClientCount() int32 {
return ListView_GetDockClientCount(l.instance)
}
// 获取停靠站点。
//
// Get Docking site.
func (l *TListView) DockSite() bool {
return ListView_GetDockSite(l.instance)
}
// 设置停靠站点。
//
// Set Docking site.
func (l *TListView) SetDockSite(value bool) {
ListView_SetDockSite(l.instance, value)
}
// 获取鼠标是否在客户端,仅VCL有效。
//
// Get Whether the mouse is on the client, only VCL is valid.
func (l *TListView) MouseInClient() bool {
return ListView_GetMouseInClient(l.instance)
}
// 获取当前停靠的可视总数。
//
// Get The total number of visible calls currently docked.
func (l *TListView) VisibleDockClientCount() int32 {
return ListView_GetVisibleDockClientCount(l.instance)
}
// 获取画刷对象。
//
// Get Brush.
func (l *TListView) Brush() *TBrush {
return AsBrush(ListView_GetBrush(l.instance))
}
// 获取子控件数。
//
// Get Number of child controls.
func (l *TListView) ControlCount() int32 {
return ListView_GetControlCount(l.instance)
}
// 获取控件句柄。
//
// Get Control handle.
func (l *TListView) Handle() HWND {
return ListView_GetHandle(l.instance)
}
// 获取父容器句柄。
//
// Get Parent container handle.
func (l *TListView) ParentWindow() HWND {
return ListView_GetParentWindow(l.instance)
}
// 设置父容器句柄。
//
// Set Parent container handle.
func (l *TListView) SetParentWindow(value HWND) {
ListView_SetParentWindow(l.instance, value)
}
func (l *TListView) Showing() bool {
return ListView_GetShowing(l.instance)
}
// 获取使用停靠管理。
func (l *TListView) UseDockManager() bool {
return ListView_GetUseDockManager(l.instance)
}
// 设置使用停靠管理。
func (l *TListView) SetUseDockManager(value bool) {
ListView_SetUseDockManager(l.instance, value)
}
func (l *TListView) BoundsRect() TRect {
return ListView_GetBoundsRect(l.instance)
}
func (l *TListView) SetBoundsRect(value TRect) {
ListView_SetBoundsRect(l.instance, value)
}
// 获取客户区高度。
//
// Get client height.
func (l *TListView) ClientHeight() int32 {
return ListView_GetClientHeight(l.instance)
}
// 设置客户区高度。
//
// Set client height.
func (l *TListView) SetClientHeight(value int32) {
ListView_SetClientHeight(l.instance, value)
}
func (l *TListView) ClientOrigin() TPoint {
return ListView_GetClientOrigin(l.instance)
}
// 获取客户区矩形。
//
// Get client rectangle.
func (l *TListView) ClientRect() TRect {
return ListView_GetClientRect(l.instance)
}
// 获取客户区宽度。
//
// Get client width.
func (l *TListView) ClientWidth() int32 {
return ListView_GetClientWidth(l.instance)
}
// 设置客户区宽度。
//
// Set client width.
func (l *TListView) SetClientWidth(value int32) {
ListView_SetClientWidth(l.instance, value)
}
// 获取控件状态。
//
// Get control state.
func (l *TListView) ControlState() TControlState {
return ListView_GetControlState(l.instance)
}
// 设置控件状态。
//
// Set control state.
func (l *TListView) SetControlState(value TControlState) {
ListView_SetControlState(l.instance, value)
}
// 获取控件样式。
//
// Get control style.
func (l *TListView) ControlStyle() TControlStyle {
return ListView_GetControlStyle(l.instance)
}
// 设置控件样式。
//
// Set control style.
func (l *TListView) SetControlStyle(value TControlStyle) {
ListView_SetControlStyle(l.instance, value)
}
func (l *TListView) Floating() bool {
return ListView_GetFloating(l.instance)
}
// 获取控件父容器。
//
// Get control parent container.
func (l *TListView) Parent() *TWinControl {
return AsWinControl(ListView_GetParent(l.instance))
}
// 设置控件父容器。
//
// Set control parent container.
func (l *TListView) SetParent(value IWinControl) {
ListView_SetParent(l.instance, CheckPtr(value))
}
// 获取左边位置。
//
// Get Left position.
func (l *TListView) Left() int32 {
return ListView_GetLeft(l.instance)
}
// 设置左边位置。
//
// Set Left position.
func (l *TListView) SetLeft(value int32) {
ListView_SetLeft(l.instance, value)
}
// 获取顶边位置。
//
// Get Top position.
func (l *TListView) Top() int32 {
return ListView_GetTop(l.instance)
}
// 设置顶边位置。
//
// Set Top position.
func (l *TListView) SetTop(value int32) {
ListView_SetTop(l.instance, value)
}
// 获取宽度。
//
// Get width.
func (l *TListView) Width() int32 {
return ListView_GetWidth(l.instance)
}
// 设置宽度。
//
// Set width.
func (l *TListView) SetWidth(value int32) {
ListView_SetWidth(l.instance, value)
}
// 获取高度。
//
// Get height.
func (l *TListView) Height() int32 {
return ListView_GetHeight(l.instance)
}
// 设置高度。
//
// Set height.
func (l *TListView) SetHeight(value int32) {
ListView_SetHeight(l.instance, value)
}
// 获取控件光标。
//
// Get control cursor.
func (l *TListView) Cursor() TCursor {
return ListView_GetCursor(l.instance)
}
// 设置控件光标。
//
// Set control cursor.
func (l *TListView) SetCursor(value TCursor) {
ListView_SetCursor(l.instance, value)
}
// 获取组件鼠标悬停提示。
//
// Get component mouse hints.
func (l *TListView) Hint() string {
return ListView_GetHint(l.instance)
}
// 设置组件鼠标悬停提示。
//
// Set component mouse hints.
func (l *TListView) SetHint(value string) {
ListView_SetHint(l.instance, value)
}
// 获取组件总数。
//
// Get the total number of components.
func (l *TListView) ComponentCount() int32 {
return ListView_GetComponentCount(l.instance)
}
// 获取组件索引。
//
// Get component index.
func (l *TListView) ComponentIndex() int32 {
return ListView_GetComponentIndex(l.instance)
}
// 设置组件索引。
//
// Set component index.
func (l *TListView) SetComponentIndex(value int32) {
ListView_SetComponentIndex(l.instance, value)
}
// 获取组件所有者。
//
// Get component owner.
func (l *TListView) Owner() *TComponent {
return AsComponent(ListView_GetOwner(l.instance))
}
// 获取组件名称。
//
// Get the component name.
func (l *TListView) Name() string {
return ListView_GetName(l.instance)
}
// 设置组件名称。
//
// Set the component name.
func (l *TListView) SetName(value string) {
ListView_SetName(l.instance, value)
}
// 获取对象标记。
//
// Get the control tag.
func (l *TListView) Tag() int {
return ListView_GetTag(l.instance)
}
// 设置对象标记。
//
// Set the control tag.
func (l *TListView) SetTag(value int) {
ListView_SetTag(l.instance, value)
}
// 获取左边锚点。
func (l *TListView) AnchorSideLeft() *TAnchorSide {
return AsAnchorSide(ListView_GetAnchorSideLeft(l.instance))
}
// 设置左边锚点。
func (l *TListView) SetAnchorSideLeft(value *TAnchorSide) {
ListView_SetAnchorSideLeft(l.instance, CheckPtr(value))
}
// 获取顶边锚点。
func (l *TListView) AnchorSideTop() *TAnchorSide {
return AsAnchorSide(ListView_GetAnchorSideTop(l.instance))
}
// 设置顶边锚点。
func (l *TListView) SetAnchorSideTop(value *TAnchorSide) {
ListView_SetAnchorSideTop(l.instance, CheckPtr(value))
}
// 获取右边锚点。
func (l *TListView) AnchorSideRight() *TAnchorSide {
return AsAnchorSide(ListView_GetAnchorSideRight(l.instance))
}
// 设置右边锚点。
func (l *TListView) SetAnchorSideRight(value *TAnchorSide) {
ListView_SetAnchorSideRight(l.instance, CheckPtr(value))
}
// 获取底边锚点。
func (l *TListView) AnchorSideBottom() *TAnchorSide {
return AsAnchorSide(ListView_GetAnchorSideBottom(l.instance))
}
// 设置底边锚点。
func (l *TListView) SetAnchorSideBottom(value *TAnchorSide) {
ListView_SetAnchorSideBottom(l.instance, CheckPtr(value))
}
func (l *TListView) ChildSizing() *TControlChildSizing {
return AsControlChildSizing(ListView_GetChildSizing(l.instance))
}
func (l *TListView) SetChildSizing(value *TControlChildSizing) {
ListView_SetChildSizing(l.instance, CheckPtr(value))
}
// 获取边框间距。
func (l *TListView) BorderSpacing() *TControlBorderSpacing {
return AsControlBorderSpacing(ListView_GetBorderSpacing(l.instance))
}
// 设置边框间距。
func (l *TListView) SetBorderSpacing(value *TControlBorderSpacing) {
ListView_SetBorderSpacing(l.instance, CheckPtr(value))
}
func (l *TListView) Column(Index int32) *TListColumn {
return AsListColumn(ListView_GetColumn(l.instance, Index))
}
// 获取指定索引停靠客户端。
func (l *TListView) DockClients(Index int32) *TControl {
return AsControl(ListView_GetDockClients(l.instance, Index))
}
// 获取指定索引子控件。
func (l *TListView) Controls(Index int32) *TControl {
return AsControl(ListView_GetControls(l.instance, Index))
}
// 获取指定索引组件。
//
// Get the specified index component.
func (l *TListView) Components(AIndex int32) *TComponent {
return AsComponent(ListView_GetComponents(l.instance, AIndex))
}
// 获取锚侧面。
func (l *TListView) AnchorSide(AKind TAnchorKind) *TAnchorSide {
return AsAnchorSide(ListView_GetAnchorSide(l.instance, AKind))
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,918
|
\section{Introduction}
\baselineskip15pt Suppose that a finite group $G$ admits an
automorphism $\varphi$.
It follows from the classification
of finite simple groups that if $\varphi$ is fixed-point-free, that is,
$C_G(\varphi )=1$, then $G$ is soluble \cite{row}, and when in
addition $|\varphi |$ is a prime, $G$ is nilpotent by Thompson's
theorem \cite{tho59} (which does not use the classification but
rather lies in its foundation). Extending the Brauer--Fowler
theorem, using the classification Hartley \cite{har} proved that if $|C_G(\varphi )|=m$, then
$G$ has a soluble subgroup of $(|\varphi |,m)$-bounded index. (Henceforth
we write, say, ``$(a,b,\dots )$-bounded'' to abbreviate
``bounded above in terms of $a, b,\dots
$ only''.)
Now let $G$ be soluble from the outset; further results were
obtained about the Fitting height (the length of a shortest normal
series with nilpotent factors). When $C_G(\varphi )=1$, by a special
case of Dade's theorem \cite{dad} the Fitting height of $G$ is
bounded in terms of $\alpha (|\varphi |)$ --- the number of prime factors
of $|\varphi |$ counting multiplicities; the coprime case $(|G|,|\varphi
|)=1$ of this result was proved earlier as a special case of
Thompson's theorem \cite{tho64}. In the general situation, when
$|C_G(\varphi )|=m$, it is conjectured that $G$ has a subgroup of
$(|\varphi |,m)$-bounded index with Fitting height bounded in terms of
$\alpha (|\varphi |)$. This conjecture was proved in the coprime case by
Hartley and Isaacs \cite{ha-is} using Turull's results \cite{tu}, and in the case where $|\varphi |$ is a
prime-power by Hartley and Turau \cite{ha-tu}. A weaker bound for the Fitting height,
in terms of $|\varphi |$ and $m$ was also obtained in the case
where $|\varphi |$ is a product of two prime-powers in an unpublished
note by Hartley \cite{ha-unp}. (The aforementioned results of Thompson \cite{tho64}, Hartley--Issacs \cite{ha-is}, and Turull \cite{tu} are actually about any, not necessarily cyclic, soluble groups of automorphisms, and Dade's theorem \cite{dad} is about any Carter subgroup.)
When there is a bound for the Fitting height, further studies are
naturally reduced to nilpotent groups. It is conjectured that if
$C_G(\varphi )=1$, then the derived length of $G$ is bounded in terms
of $|\varphi |$. So far this is proved only when $|\varphi |$ is a prime due
to Higman \cite{hi} (and Kreknin--Kostrikin \cite{kr,kr-ko} with
an explicit bound), or $|\varphi |=4$ due to Kov\'acs \cite{kov}. In
these two cases even the `almost fixed-point-free' theorems were
proved by Khukhro \cite{khu90}, and Khukhro and Makarenko
\cite{khu-mak-4}, which give a subgroup of $|\varphi |$-bounded
nilpotency class or derived length with $(|\varphi |,|C_G(\varphi
)|)$-bounded index.
Another area where definitive results of this kind were proved is the case where $G$ is a finite $p$-group and $|\varphi |$ is a power of $p$
(Alperin \cite{alp}, Khukhro \cite{khu85}, Shalev \cite{sha93}, Khukhro \cite{khu93}, Medvedev \cite{med}, Jaikin-Zapirain \cite{jai}).
All these results on nilpotent groups are based on the
corresponding theorems on automorphisms of Lie rings. In
particular, by Kreknin's theorem \cite{kr} a Lie ring $L$ with a fixed-point-free
automorphism of finite order $n$ is soluble of $n$-bounded derived length.
Khukhro and Makarenko \cite{khu-mak04} also proved almost
solubility of a Lie algebra (or a Lie ring, under some additional conditions, which hold, for example, for finite Lie rings) with an almost
fixed-point-free automorphism $\varphi$ of finite order, with a
`strong' bound, in terms of $|\varphi |$ only, for the derived length
of a subalgebra (or a subring) of bounded codimension (or index in the additive
group). But group-theoretic analogues of
these results remain open conjectures, except for the cases where $|\varphi |$ is a prime or 4, as described above.
Therefore it makes sense to obtain results in this direction under
additional conditions. One such result was obtained by Shumyatsky \cite{shu01}: if a finite group $G$ admits a
fixed-point-free automorphism $\varphi$ of order $2^n$ such that the
fixed-point subgroup $C_G(\varphi ^{2^{n-1}})$ of the involution $\varphi
^{2^{n-1}}$ is nilpotent of class $c$, then $G$ is soluble of
$(n,c)$-bounded derived length. The purpose of the present paper
is an `almost fixed-point-free' generalization of this result.
\begin{theorem}\label{t1} Suppose that a finite group $G$ admits an
automorphism $\varphi$ of order $2^n$ such that the fixed-point subgroup $C_G(\varphi ^{2^{n-1}})$ of the involution
$\varphi ^{2^{n-1}}$ is nilpotent of class~$c$. Let $m=|C_G(\varphi
)|$ be the number of fixed points of $\varphi$. Then $G$ has a characteristic soluble
subgroup of $(m,n,c)$-bounded index that has $(n,c)$-bounded
derived length. \end{theorem}
In fact, the condition in the theorem that $C_G(\varphi ^{2^{n-1}})$
is nilpotent of class $c$ can be weakened to requiring all Sylow subgroups of
$C_G(\varphi ^{2^{n-1}})$ to be nilpotent of class at most~$c$; see Remark~\ref{r1}.
The standard inverse limit argument yields a consequence for
locally finite groups.
\begin{corollary}\label{c1}
Suppose that a locally finite group $G$ contains an
element $g$ of order $2^n$ with finite centralizer of order
$m=|C_G(g )|$ such that the centralizer
$C_G(g ^{2^{n-1}})$ of the involution $g ^{2^{n-1}}$ is nilpotent
of class $c$. Then $G$ has a characteristic soluble subgroup of finite
$(m,n,c)$-bounded index that has $(n,c)$-bounded derived length.
\end{corollary}
Here, too, the condition that $C_G(g ^{2^{n-1}})$
is nilpotent of class $c$ can be weakened to requiring all nilpotent subgroups of
$C_G(g ^{2^{n-1}})$ to be nilpotent of class at most $c$.
In our recent paper \cite{kms145} we also used Theorem~\ref{t1} to prove that if a
locally finite group $G$ has a $2$-element $g$ with Chernikov centralizer such that
the involution in $\langle g\rangle$ has nilpotent centralizer, then $G$ has a soluble subgroup of finite index.
By the aforementioned results the proof of Theorem~\ref{t1} reduces to the case
of nilpotent groups, where a Lie ring method of `graded
centralizes' developed in \cite{khu90,khu-mak04} is used in
conjunction with ideas of the proof in \cite{shu01}.
We state separately the corresponding Lie ring result, which is used in the
proof of Theorem~\ref{t1}.
\begin{theorem}\label{t2} Suppose that a finite
Lie ring $L$ admits an automorphism $\varphi$ of order $2^n$ such
that the fixed-point subring $C_L(\varphi ^{2^{n-1}})$ of the
involution $\varphi ^{2^{n-1}}$ is nilpotent of class $c$. Let
$m=|C_L(\varphi )|$ be the number of fixed points of $\varphi$. Then $L$ has
ideals $M_1\geqslant M_2$ such that $M_1$ has $(m,n)$-bounded index in
the additive group $L$, the quotient $M_1/M_2$ is nilpotent of class at most $c+1$,
and $M_2$ is nilpotent of $(n,c)$-bounded class. \end{theorem}
Theorem~\ref{t2} can be extended to Lie algebras over a field and
to other classes of Lie rings admitting such an automorphism of
order $2^n$. Here we confine ourselves to the case of finite
Lie rings, since this is sufficient for the
purpose of proving Theorem~\ref{t1}. Even in view of the aforementioned general
Khukhro--Makarenko theorem \cite{khu-mak04}, Theorem~\ref{t2} still makes sense, since it gives a stronger `metanilpotent' conclusion (of course, under stronger assumptions).
In \S\,\ref{s-prelim} we give definitions, introduce notation, and
list several results that are used in the sequel. In
\S\,\ref{s-rings} we prove Theorem~\ref{t2} on Lie rings using a
modification of the method of graded centralizers developed in
\cite{khu90,khu-mak04} for studying almost fixed-point-free
automorphisms. Theorem~\ref{t1} is proved in
\S\,\ref{s-groups}. Known
results reduce the proof to the case of a nilpotent
group. Then we firstly apply the Lie ring method similarly to \cite{shu01} to obtain a `weak' bound, depending on $m,n,c$, for the nilpotency class of $[G,\varphi ^{2^{n-1}}]$. Finally, Theorem~\ref{t2}, or rather one of the
propositions in its proof, is used to obtain the required `strong' bound, in terms of $n,c$ only, for the nilpotency class of $[H,\varphi ^{2^{n-1}}]$ for a certain subgroup $H$ of $(m,n,c)$-bounded index. When a subgroup of $(m,n,c)$-bounded index and of $(n,c)$-bounded derived length is constructed, we obtain a characteristic subgroup of $(m,n,c)$-bounded index and of the same derived length due to the general result \cite{khu-mak-char} on subgroups of finite index satisfying a multilinear commutator law; see Theorem~\ref{t-char}.
\section{Preliminaries} \label{s-prelim}
First we recall some definitions and notation. Products in a Lie
ring are called commutators. A
simple commutator $[a_1,a_2,\dots ,a_s]$ of weight (length)
$s$ is the commutator $[...[[a_1,a_2],a_3],\dots ,a_s]$. The Lie subring and the ideal
generated by a subset~$S$ are denoted by $\langle S\rangle $ and
${}_{{\rm id}}\!\left< S \right>$, respectively. For additive subgroups $U,V$ of a Lie ring, $[U,V]$ denotes the additive subgroup generated by all commutators $[u,v]$, $u\in U$, $v\in V$. Terms of the lower central series of a Lie ring $L$ start from
$\gamma_1(L)=L$, and by induction, $\gamma_{i+1}(L)=[\gamma_i(L),L]$. A Lie ring
$L$ is nilpotent of class at most~$h$ if $\gamma_{h+1}(L)=0$. Terms of the derived series start from $L=L^{(0)}$, and by induction, $L^{(i+1)}=[L^{(i)},L^{(i)}]$. A Lie ring $L$ is soluble of derived length at most $d$ if $L^{(d)}=0$.
Let $A$ be an additively written abelian group. A Lie ring $L$ is
\textit{$A$-graded} if
$$L=\bigoplus_{a\in A}L_a\qquad \text{ and }\qquad[L_a,L_b]\subseteq L_{a+b},\quad a,b\in A,$$
where the grading components $L_a$ are subgroups of the additive group of~$L$.
Elements of the $L_a$ are called \textit{homogeneous} (with
respect to this grading), and commutators in homogeneous elements
\textit{homogeneous commutators}. A subgroup
$H$ of the additive group of $L$ is said to be \textit{homogeneous}
if $H=\bigoplus_{a\in A} (H\cap L_a)$; then we set $H_a=H\cap
L_a$. Obviously, any subring or an ideal generated by homogeneous
additive subgroups is
homogeneous. A homogeneous subring and the
quotient ring by a homogeneous ideal can be regarded as $A$-graded
rings with the induced gradings.
\begin{Index Convention} For a homogeneous element of a $({\Bbb Z}/n{\Bbb Z})$-graded Lie ring $L$ we use a small letter
with an index that only indicates the grading component to which
this element belongs: $x_i\in L_i$. Thus, different elements can
be denoted by the same symbol, since it will only matter to which
component these elements belong. For example, $x_1$ and
$x_1$ can be different elements of $L_1$, so that $[x_1,\, x_1]$
can be a nonzero element of $L_2$. These indices are considered
modulo~$n$; for example, $a_{-i}\in L_{-i}=L_{n-i}$.
\end{Index Convention}
Note that under the Index Convention a homogeneous commutator
belongs to the component $L_s$, where $s$ is the sum modulo $n$ of the indices of all the elements occurring in this commutator.
Suppose that a Lie ring $L$ admits an automorphism
$\varphi $ of order $n$. Let $\omega$ be a
primitive $n$-th root of unity. We extend the ground ring by
$\omega$ and denote by $\widetilde L$ the ring $L\otimes _{{\Bbb
Z} }{\Bbb Z} [\omega ]$. Then $\varphi $ naturally acts on
$\widetilde L$ and, in particular, $C_{\widetilde L}(\varphi ) =
C_L(\varphi )\otimes _{{\Bbb Z} }{\Bbb Z} [\omega ]$.
We define the analogues of eigenspaces $L_k$ for
$k=0,\,1,\,\ldots ,n-1$ as
$$
L_k=\big\{ a\in \widetilde L\mid a^{\varphi}=\omega ^{k}a\big\} .
$$
If $n$ is invertible in the ground ring of $L$ (for example,
when $L$ is finite of order coprime to $n$), then
$$
\widetilde L= L_0 \oplus L_1 \oplus \dots \oplus L_{n-1}
$$
(see, for example,~\cite[Ch.~10]{hpbl}). This is a $({\Bbb
Z}/n{\Bbb Z})$-grading because
$$
[L_s,\, L_t]\subseteq L_{s+t\,({\rm mod}\,n)}\quad \text{for all}\;\, s,t.$$
\begin{notat}
Whenever we say that $L_0\oplus L_1\oplus \cdots\oplus L_{n-1}$ is a $({\Bbb Z} /n{\Bbb Z} )$-graded Lie ring, we mean that the $L_i$ are the grading components, so that $[L_s,\, L_t]\subseteq L_{s+t\;({\rm
mod}\;n)}$.
\end{notat}
We now state the `graded' version of the Khukhro--Makarenko theorem \cite{khu-mak04} on Lie rings with an almost fixed-point-free automorphism of finite order.
\begin{theorem}[{\cite[Corollary~2]{khu-mak04}}]\label{t3} Suppose that $L=
L_0\oplus L_1\oplus \cdots\oplus L_{n-1}$ is a $({\Bbb Z} /n{\Bbb
Z} )$-graded Lie ring. If the component $L_0$ is finite of order $m$, then
$L$ has a soluble homogenous
ideal $M$ of $n$-bounded derived length and of finite $(m,n)$-bounded index in the additive group of~$L$.
\end{theorem}
We now introduce specialized notation for our case of an automorphism of order $2^n$. Let $L= L_0 \oplus L_1\oplus \dots \oplus L_{2^n-1} $ be a $({\Bbb Z}/2^n{\Bbb Z})$-graded Lie ring.
\begin{notat}
Let $L_{\rm odd}$ denote the set of all `odd' grading
components $L_j$ with odd $j$, and let $L^-$ be their sum.
Similarly, let $L_{\rm even }$ denote the set of all `even'
components $L_i$ with even $i$, and let $L^+$ be their sum.
We also abuse this notation by letting $L_{\rm odd}$ and $L_{\rm even }$ denote the unions of the corresponding
components. We use similar notation for any homogeneous additive subgroup $X$ and its components.
\end{notat}
For $L_i,L_j\in L_{\rm odd}$ and $L_k,L_l\in L_{\rm even}$ we clearly have $[L_i,L_j]\in L_{\rm even}$, $[L_i,L_k]\in L_{\rm odd}$, and $[L_k,L_l]\in L_{\rm even}$. Therefore $L^+$ is a subring of $L$, while $L^-$ is not. Note also that the subring generated by $L_{\rm odd}$ is an ideal of $L$.
The next theorem is essentially a reformulation of Shumyatsky's theorem for Lie rings
\cite{shu01}.
\begin{theorem}[{\cite[Proposition~2.6]{shu01}}]\label{t-shu} Suppose that $L= L_0 \oplus L_1 \oplus \dots
\oplus L_{2^n-1}$ is a $({\Bbb Z}/2^n{\Bbb Z})$-graded Lie ring such that the subring $L^+$ is nilpotent of class
$c$ and $L_0=0$. Then the subring generated by $L^-$ is nilpotent
of $(n,c)$-bounded nilpotency class $f(n,c)$.
\end{theorem}
Since $\langle L^-\rangle ={}_{\rm id}\langle L^-\rangle$, under the hypotheses of Theorem~\ref{t-shu} the Lie ring $L$ satisfies
$$
\gamma_{f(n,c)+1}(\gamma_{c+1}(L))=0.
$$
For dealing with a Lie ring whose additive group
is a finite $2$-group we need the following `combinatorial' corollary of Theorem~\ref{t-shu}. Slightly abusing notation, we use the same symbols $\gamma _i$ (as those denoting terms of the lower central series) to denote Lie polynomials that are simple multilinear commutators and write
\begin{align*}
(\gamma_{i}\circ \gamma_{j})(x_1,x_2,\ldots, x_{ij})&=\gamma_{i} \big(\gamma_{j}(x_1, \ldots, x_{j}),\dots , \gamma_{j} (x_{(i-1)j+1},\ldots,
x_{ij})\big)\\ & =\big[[x_1, x_2,\ldots, x_{j}],
[x_{j+1},\ldots, x_{2j}],\ldots, [x_{(i-1)j+1},\ldots,
x_{ij}]\big].
\end{align*}
\begin{corollary}\label{cor-shu}
Let $n, c$ be positive integers, and $f=f(n,c)$ the value of the function given by
Theorem~$\ref{t-shu}$. For $r=(c+1)(f+1)$, the following holds. If
we arbitrarily and formally assign lower indices $i_1, i_2, \ldots
, i_r$ to elements $y_{i_1}, y_{i_2}, \dots, y_{i_r}$ of an
arbitrary Lie ring, then the commutator
$(\gamma_{f+1}\circ \gamma_{c+1})(y_{i_1},y_{i_2},\ldots,
y_{i_r})$ can be represented as a linear combination of
commutators in the same elements $y_{i_1}, y_{i_2}, \dots,
y_{i_r}$ each of which contains either a subcommutator with zero
modulo $2^n$ sum of indices or a subcommutator of weight $c+1$ of the form $[g
_{2u_1}, g_{2u_2},\dots ,g_{2u_{c+1}} \,]$ with even indices, where every element $ g_{2j}$ is a commutator in
$y_{i_1}, y_{i_2}, \dots, y_{i_r}$ such that the sum of indices of
all the elements involved in $g_{2j}$ is congruent to $2j$
modulo~$2^n$.
\end{corollary}
\begin {proof}
Let $M$ be a free Lie ring freely generated by $x_{i_1},
x_{i_2}, \dots, x_{i_{r}}$. For each
$i=0,\,1,\,\dots ,2^n-1$, let $M_i$ be the additive subgroup of $M$
generated by all commutators in the
generators $x_{i_j}$ with the sum of indices congruent to $i$
modulo $2^n$. Then, obviously, $M=M_0\oplus M_1\oplus \cdots
\oplus M_{2^n-1}$ and $ [M_i,M_j]\subseteq M_{i+j\,({\rm mod\,
2^n)}}$, so this is a $({\Bbb Z} /2^n{\Bbb Z})$-grading.
By Theorem~\ref{t-shu} we obtain
$$
(\gamma_{f+1}\circ \gamma_{c+1})(x_{i_1},x_{i_2},\ldots,
x_{i_r}) \in {}_{{\rm
id}}\!\left<M_0\right>+
\gamma _{c+1}\big(M_0+M_2+\dots +M_{2^{n}-2}\big).
$$
By the definition of the $M_i$ this inclusion
is equivalent to the required equality for $y_{i_j}=x_{i_j}$. Since the elements
$x_{i_1}, x_{i_2}, \dots x_{i_r}$ freely generate the Lie ring
$M$, the same equality holds for any elements $y_{i_j}$ in any Lie ring.
\end{proof}
The following theorem was proved by P.~Hall \cite{hall} for
groups; the assertion for Lie rings is proved by essentially the
same (even simpler) arguments. The bound for the nilpotency class
was later improved by other authors, up to the best possible bound
in \cite{stew}.
\begin{theorem}[{P.~Hall \cite{hall}}]\label{t-hall}
If a Lie ring $L$ has a nilpotent ideal $K$ of nilpotency
class $k$ such that the quotient $L/[K,K]$ is nilpotent of class
$l$, then $L$ is nilpotent of $(k,l)$-bounded class.
\end{theorem}
The following lemmas are well-known properties of fixed-point
subgroups. As a rule, the induced automorphism of a quotient group
by an invariant normal subgroup is denoted by the same letter.
\begin{lemma}\label{l-fp} Let $\alpha$ be an automorphism of a finite group
$G$, and $N$ a normal $\alpha$-invariant subgroup of $G$.
{\rm (a)} Then $|C_{G/N}(\alpha )|\leqslant |C_{G}(\alpha )|$.
{\rm (b)} If in addition $(|N|, |\alpha |)=1$, then $C_{G/N}(\alpha )=
C_{G}(\alpha )N/N$. \end{lemma}
The following lemma follows from the consideration of the Jordan normal form of the automorphism regarded as a linear transformation of invariant elementary abelian sections.
\begin{lemma}\label{l-fpp2} Let $p$ be a prime number and suppose
that a
finite abelian group~$A$ of exponent~$p^a$ admits an automorphism of order~$p^k$ with exactly $p^b$ fixed points. Then $|A|\leqslant p^{abp^k}$.
\end{lemma}
Recall that a \emph{multilinear} (or \emph{outer})
\emph{commutator} is any commutator $\varkappa $ of weight $w$ in $w$ distinct group variables; in other words, $\varkappa $ is obtained by nesting commutators, but using always
different variables. Laws $\varkappa =1$ for multilinear commutators $\varkappa $ define many popular soluble group varieties, including those of nilpotent groups of given class, and of soluble groups of given derived length. The following Khukhro--Makarenko theorem \cite{khu-mak-char} greatly facilitates working with subgroups of finite index satisfying a multilinear commutator law. In the special case of nilpotency laws this result was obtained by Bruno and Napolitani \cite{brna}. (Further generalizatons and improvements of this theorem were obtained in \cite{kh-kl-ma-me,kl-me,kl-mi}.)
\begin{theorem}[{\cite[Theorem~1]{khu-mak-char}}]\label{t-char}
If a group $G$ has a subgroup $H$ of finite
index $k$ satisfying the law $\varkappa (H)=1$, where
$\varkappa$ is a multilinear commutator of weight $w$, then $G$
also has a characteristic subgroup $C$ of finite $(k,w)$-bounded index
satisfying the same law $\varkappa (C)=1$.
\end{theorem}
\section{Lie rings}\label{s-rings}
The bulk of the proof of Theorem~\ref{t2} is about $({\Bbb Z}/2^n{\Bbb Z})$-graded Lie rings considered in the following proposition.
\begin{proposition}\label{p1} Suppose that $L= L_0 \oplus L_1 \oplus \dots
\oplus L_{2^n-1}$ is a $({\Bbb Z}/2^n{\Bbb Z})$-graded Lie ring.
Suppose that the subring $L^+$ is nilpotent of class $c$, while the component $L_0$ is finite of order $m$. Then $L$ contains
a homogeneous nilpotent ideal $M$ of $(n,c)$-bounded nilpotency class such that $M\cap L^-$ has $(m,n)$-bounded index in the additive group $L^-$.
\end{proposition}
By Theorem~\ref{t3} the Lie ring $L$ contains a soluble
homogeneous ideal of $n$-bounded derived length and of
$(m,n)$-bounded index in $L$. Therefore
Proposition~\ref{p1} will be proved if we prove the following proposition, taking
advantage of induction on the derived length.
\begin{proposition}\label{p2}
Suppose that under the hypotheses of Proposition~\ref{p1} the Lie ring $L$ has a soluble homogeneous ideal
$A$ of derived length $d$ such that $A\cap L^-$ has index $l$ in the additive group $L^-$.
Then
$L$ contains a homogeneous nilpotent ideal $B$ of $(d,n,c)$-bounded nilpotency class such that $B\cap L^-$
has $(d,l,m,n)$-bounded index in $L^-$.
\end{proposition}
\begin{proof} The sought-for ideal $B$ is constructed by using certain
additive subgroups $L_j(t)\leqslant L_j$ of the components $L_{j}$,
so-called graded centralizers of levels $t=1,2,3$. We also use induction of $d$. Clearly, if $d=1$, then $A$ is abelian and we can put $B=A$. So we assume that $A$ is not
abelian.
Let $R=A^{(d-2)}$ be the penultimate (metabelian) term of the derived series
of $A$. First we construct graded centralizers
$R_j(1)\leqslant R_j$ in $R$, which are additive subgroups of $(m,n)$-bounded
index in the grading components $R_j=R\cap L_j$, and fix certain
elements called $r$-representatives, whose total
number is $(m,n)$-bounded.
\begin{definition*} The \emph{pattern\/} of a homo\-geneous
commutator is its bracket structure together with the arrangement
of the indices under the Index Convention. The \emph{weight\/} of a
pattern is the weight of the commutator. The commutator is then
called the \emph{value\/} of its pattern on the given elements. For
example, $[a_2,[b_1,b_1]]$ and $[x_2,[z_1,y_1]]$ are values of the
same pattern of weight~3.
\end{definition*}
\paragraph{\bf \boldmath Definition of representatives
in $R$.} For every $i\ne 0$ and for every pair $({\bf P},a_0)$
consisting of the pattern ${\bf P}$ of a
simple commutator of weight $2^n$ with one and the same index
$i\ne 0$
(repeated $2^n$ times) and a commutator $a_0\in R_0$ that is the
value of this pattern on homo\-geneous elements of~$R_{i}$ we fix
one such representation. (The same element $a_0\in R_0$ may appear
in different pairs if it is equal to values of different patterns;
the same pattern may appear in different pairs if different
commutators are the values of this pattern.) The elements of
$R_{j}$,\, $j\ne 0$, involved in these fixed representations are
called \emph{$r$-representatives} and
denoted by $r_j(0)\in R_j$ under the Index
Convention: recall that the same symbol can denote different
elements. Thus, the commutator $a_0$ mentioned above is equal to
$[\underbrace{r_i(0),\ldots ,r_{i}(0)}_{2^n}]$.
Since the total number of patterns ${\bf P}$ under
consideration is equal to $2^n-1$ and the number of elements in
$R_0$ is at most $m$, the number of $r$-representatives
is $(m,n)$-bounded.
\medskip
The definition of $r$-representatives implies the following.
\begin{lemma} \label{l-freezer}
Every simple homo\-geneous commutator in elements of $R$ of length $2^n$ with one and
the same index $i\ne 0$ repeated $2^n$ times can be represented
as a commutator of the same pattern in
$r$-representatives. \end{lemma}
Before defining graded centralizers, we introduce the
following homomorphisms.
\begin{definition} \label{d-th}
Let $\vec z=(z_{i_1},\ldots ,z_{i_k})$ be an ordered tuple of
elements $z_{i_s}\in L_{i_s}$, \, $i_s\ne 0$, such that
$i_1+\cdots + i_k\not\equiv 0\, (\mbox{mod}\, 2^n)$. We put $j=-
i_1-\cdots - i_k\, (\mbox{mod}\, 2^n)$ and define
the mapping
$$
\vartheta _{\vec z}:\, \, y_j\rightarrow
[y_j,\, z_{i_1},\,\ldots ,\, z_{i_k}].\label{e1}
$$
By linearity this is a homomorphism of the additive group $L_j$
into $L_0$
by
the choice of~$j$.
Since $|L_0|\leqslant m$, we have $|L_j:\mbox{Ker}\,
\vartheta _{\vec z}|\leqslant m$. Clearly, we also have
$|R_j:\mbox{Ker}\, \eta _{\vec z}|\leqslant m$ for the restriction
$\eta _{\vec z}$ of $\vartheta _{\vec z}$ to $R_j$.
\end{definition}
\paragraph{\bf \boldmath Definition of graded centralizers in $R$.}
We define the \emph{graded
centralizers in $R$}
by setting for each $i\ne 0$
$$
R_i(1)=\bigcap_{\vec r}\,\mbox{Ker}\,\eta _{\vec r},
$$
where $ \eta _{\vec r}$ is defined in Definition~\ref{d-th} with
$\vec r=\left(r_{i}(0), \ldots , r_{i}(0)\right) $ running over
all possible ordered tuples of length $2^n-1$ consisting of
(possibly different) $r$-representatives with the same index
$i$. Elements of $R_i(1)$ for $i=1,\dots ,2^n-1$ are also
called \emph{centralizers in $R$} for short and are denoted by
$r_i(1)$ (under the Index Convention). The number of
$r$-representatives is $(m,n)$-bounded and $|R_i:\mbox{Ker}\, \eta
_{\vec r}|\leqslant m$ for all $\vec r$. Hence this is an intersection
of $ (m,n) $-boundedly many subgroups of $m$-bounded index
in~$R_i$ and therefore $R_i(1)$
also has $(m,n)$-bounded index in the additive
group~$R_{i}$.
\medskip
By construction, we have the following centralizer property:
\begin{equation}\label{e-cr} \big[ r_i(1), \underbrace{r_{i}(0),\,\ldots ,\,
r_{i}(0)}_{2^n-1} \big]=0 \end{equation}
for any centralizer $r_i(1)\in
R_i(1)$ in $R$ and any $r$-representatives $r_i(0)$ with the same
index $i\ne 0$. (Here, as always under the Index Convention,
the elements $r_{i}(0)$ can be different.)
We also need to introduce another set of representatives in
$R$.\medskip
\paragraph{\bf \boldmath Coset representatives in $R$} For each $j\ne 0$ we fix an arbitrary system of coset representatives of the subgroup $R_j(1)$ in the additive
group~$R_j$. These elements are denoted by $ q_j\in R_j$ (under
the Index Convention) and called \emph{coset repre\-sen\-ta\-tives
in $R$}. The total number of coset repre\-sen\-ta\-tives is
$(m,n)$-bounded, since the indices $|R_j:R_j(1)|$ are
$(m,n)$-bounded for all $j\ne 0$ by construction.
\medskip
Our next construction is of representatives and graded
centralizers in the whole ring $L$.
\smallskip
\paragraph{\bf \boldmath Definition of level $1$ for $L$.}
We define the \emph{graded
centralizers of level $1$} in $L$ by setting for each $i\ne 0$
$$
L_i(1)=\bigcap_{\vec z}\,\mbox{Ker}\,\vartheta _{\vec z},
$$
where the $ \vartheta _{\vec z}$ are defined in
Definition~\ref{d-th} and $\vec z=( q_{j}, \ldots , q_{j}) $
runs over all possible ordered tuples of length $k\leqslant 2^n-1$
consisting of coset representatives in $R$ with the same index
$j\ne 0$ such that
$$
i+ kj\equiv 0\, (\mbox{mod}\, 2^n).
$$
(Under the Index Convention the tuple $\vec
z=( q_{i}, \ldots , q_{i}) $ may consist of different coset
representatives $q_i$.) Elements of the $L_i(1)$ are also
called \emph{centralizers of level $1$ in $L$} and are denoted by
$y_i(1)$ (under the Index Convention). The number of coset
representatives in $R$ is $(m,n)$-bounded and $|L_i:\mbox{Ker}\,
\vartheta _{\vec z}|\leqslant m$ for all $\vec z$. Hence this is an
intersection of $ (m,n) $-boundedly many subgroups of $m$-bounded
index in~$L_i$ and therefore $L_i(1)$
also has $(m,n)$-bounded index in the additive
group~$L_{i}$.
For each
$j\ne 0$ we also fix an arbitrary system of coset representatives of
the subgroup $L_j(1)$ in the additive group $L_j$. These elements
are denoted by $ b_j(1)$ (under the Index Convention) and called
\emph{coset repre\-sen\-ta\-tives of level 1 in $L$}. The total
number of coset repre\-sen\-ta\-tives is $(m,n)$-bounded, since
the indices $|L_j:L_j(1)|$ are $(m,n)$-bounded for all
$j=1,\,2,\ldots ,2^n-1$.
\medskip
\paragraph{\bf \boldmath Definition of level $2$ in $L$.}
We define the \emph{graded centralizers of
level $2$ in $L$} by setting for each $j\ne 0$
$$
L_j(2)=L_j(1)\cap \bigcap_{\vec z}\,\mbox{Ker}\,\vartheta _{\vec
z},
$$
where the $ \vartheta _{\vec z}$ are defined in
Definition~\ref{d-th} and $\vec z=\left( b_{i_1}(1),\, \ldots ,\,
b_{i_k}(1)\right) $ runs over all possible ordered tuples of all
lengths $k\leqslant 2^{3n+1}$ consisting of coset representatives
of level 1 in $L$ such that
$$
j+ i_1+\cdots + i_k\equiv 0\, (\mbox{mod}\, 2^n).
$$
Elements of the $L_j(2)$ are called \emph{centralizers of level $
2$} and are denoted by $y_j(2)$ (under the Index Convention). The
number of coset representatives of level $1$ in $L$ is
$(m,n)$-bounded and $|L_j:\mbox{Ker}\, \vartheta _{\vec z}|\leqslant m$
for all $\vec z$. Hence this is an intersection of $ (m,n)
$-boundedly many subgroups of $m$-bounded index in~$L_j$ and
therefore $L_j(2)$
also has $(m,n)$-bounded index in the additive
group~$L_{j}$.
For each $j\ne 0$, we now fix an
arbitrary system of coset representatives of the subgroup $L_j(2)$
in the additive group $L_j$. These elements are
denoted by $ b_j(2)$ (under the Index Convention) and called {\it
coset repre\-sen\-ta\-tives of level~$2$ in $L$}. The total number
of coset repre\-sen\-ta\-tives of level $2$ is $(m,n)$-bounded,
since the indices $|L_j:L_j(2)|$ are $(m,n)$-bounded for all
$j=1,\,2,\ldots ,2^n-1$.
\medskip
\paragraph{\bf \boldmath Definition of level $3$ in $L$.}
We define the \emph{graded centralizers of level $3$ in $L$} by
setting for each $j\ne 0$
$$
L_j(3)=L_j(2)\cap \bigcap_{\vec z}\,\mbox{Ker}\,\vartheta _{\vec
z},
$$
where the $ \vartheta _{\vec z}$ are defined in
Definition~\ref{d-th} and $\vec z=\left( b_{i_1}(2),\, \ldots ,\,
b_{i_k}(2)\right) $ runs over all possible ordered tuples of all
lengths $k\leqslant 2^{3n+1}$ consisting of coset representatives of
level 2 in $L$ such that
$$
j+ i_1+\cdots + i_k\equiv 0\, (\mbox{mod}\, 2^n).
$$
Elements of the $L_j(3)$ are called \emph{centralizers of level $
3$} and are denoted by $y_j(3)$ (under the Index Convention). The
number of coset representatives of level $2$ in $L$ is
$(m,n)$-bounded and $|L_j:\mbox{Ker}\, \vartheta _{\vec z}|\leqslant m$
for all $\vec z$. Hence this is an intersection of $ (m,n)
$-boundedly many subgroups of $m$-bounded index in~$L_j$ and
therefore $L_j(3)$
also has $(m,n)$-bounded index in the additive
group~$L_{j}$.
The construction of centralizers and coset representatives of
levels $\leqslant 3$ in $L$ is complete.
\medskip
Note that by construction we have
\begin{equation}\label{e-incl}
L_j(k+1)\leqslant L_j(k)
\end{equation}
for all $j$ and $k$.
The definition of centralizers $y_v(1)$ of level 1 implies the
following centralizer property with respect to coset
representatives in $R$:
\begin{equation}\label{e-clr}
[ y_i(1),
\underbrace{q_{j},\ldots , q_{j}}_{k}\,]=0,
\end{equation}
for any $k\leqslant
2^{n}-1$ for any (possibly different) coset representatives
$q_j$ in $R$ with the same index $j$ such that $i+kj\equiv
0\,({\rm mod}\,2^n)$.
The definitions of centralizers $y_v(t)$ of levels $t=2,3$ imply
the following centralizer property with respect to coset
representatives in $L$ of the preceding level:
\begin{equation}\label{e-cll}
[y_j(t), b_{i_1}(t-1),\ldots , b_{i_k}(t-1)]=0,
\end{equation}
for any $k\leqslant
2^{3n+1}$ for any coset representatives in $L$ of level $t-1$ such
that $j+i_1 +\dots +i_k\equiv 0\,({\rm mod}\,2^n)$.
The following two lemmas are similar to
\cite[Lemma~3]{khu-mak04} and a special case of
\cite[ Lemma~9]{khu-mak04}, but we have to reproduce the proofs, since the
definitions of representatives and graded centralizers here are somewhat different.
\begin{lemma} \label{l-3}
Any commutator of the form $[a_{-i},\,y_i(k)]$,
where $y_i(k)$ is a centralizer of level $k=2,3$, is equal to a
commutator of the form $[y_{-i}(k-1),\,y_i(k)]$, where
$y_{-i}(k-1)$ is a centralizer of level $k-1$. \end{lemma}
\begin{proof} We have $a_{-i}=b_{-i}(k-1)+y_{-i}(k-1)$ for some coset representative $b_{-i}(k-1)$ and a centralizer
$y_{-i}(k-1)$ of level $k-1$. Then
$$
[a_{-i},\,y_i(k)]=[b_{-i}(k-1),\,y_i(k)]+[y_{-i}(k-1),\,y_i(k)]=[y_{-i}(k-1),\,y_i(k)],
$$
since $[b_{-i}(k-1),\,y_i(k)]=0$ by~\eqref{e-cll}. \end{proof}
\begin{lemma}\label{l-9} For any $j\ne 0$ any commutator \begin{equation}\label{e-91}
[y_j(3),\,a_{k_1},\,a_{k_2},\ldots ,a_{k_s}]\in L_0 \end{equation}
(under the
Index Convention), of any length, for any indices
$k_i\in\{0,1,\dots ,2^{n-1}-1\}$ such that $ j+k_1+\cdots
+k_s\equiv 0\;({\rm mod\;}2^n)$ is equal to a linear combination
of elements of the form $[y_{-k}(1),y_k(1)]$ for various $ k\ne
0$. \end{lemma}
\begin{proof} We use induction on $s$. If $s=0$ there is nothing to prove,
since $ j\ne 0$.
If $s=1$, this follows from Lemma~\ref{l-3}, by which $[y_j(3),\,a_{-j}]=
[y_j(3),\, y_{-j}(2)]$, and from the inclusions \eqref{e-incl}.
For $s>1$ by the Jacobi identity we can permute the elements
$a_{k_u}$ in the commutator \eqref{e-91} modulo
$$
\sum \limits_{t=1}^{s-1}\;\sum \limits_{j+i_1+\cdots +i_t\equiv
0\;({\rm mod\;}2^n)}[L_j(3),L_{i_1},\ldots ,L_{i_{t}}].
$$
By the induction hypothesis all elements in this sum can be
expressed in the required form. Therefore we may freely permute
the $a_{k_u}$ in \eqref{e-91} in order
to express our commutator in the
required form.
We express every element $a_{k_u}$ in \eqref{e-91} with non-zero
index $k_u\ne 0$ in the form $b_{k_u}(2)+y_{k_u}(2)$ and
substitute all these expressions into the commutator \eqref{e-91}.
We obtain a linear combination of commutators
$$
[y_j(3),\,z_{k_1},\,z_{k_2},\ldots ,z_{k_s}],
$$
where the $z_{k_u}$ are either $b_{k_u}(2)$, or $y_{k_u}(2)$, or
$a_0$, and $ j+k_1+\cdots +k_s\equiv 0\;({\rm mod\;}2^n)$. If among the $z_{k_u}$ there is
at least one $y_{k_u}(2)$, then we transfer it to the right end of
the commutator, denote by $a_{-k_u}$ the preceding initial
segment, and apply Lemma~\ref{l-3}: $ [a_{-k_u},
y_{k_u}(2)]=[y_{-k_u}(1), y_{k_u}(2)]$, which is of required form
by the inclusions \eqref{e-incl}.
Hence it remains to consider the case of a commutator
\begin{equation}\label{e-9}
[y_j(3),\,z_{k_1},\,z_{k_2},\ldots ,z_{k_s}],
\end{equation}
where all the $z_{k_u}$ are either $b_{k_u}(2)$ with $k_u\ne 0$ or
$a_0$, and $ j+k_1+\cdots +k_s\equiv 0\;({\rm mod\;}2^n)$.
(Note that the $z_i$ cannot all be $a_0$, since $ j\ne 0$.) We now prove that such a
commutator is actually equal to~0. We do this by showing that
some of the entries $b_{k_u}(2)$ can be placed at the beginning
after $y_j(3)$ producing an initial segment of bounded weight
with zero sum of indices modulo $2^n$, which is equal to $0$ by
\eqref{e-cll}.
For each index $u\ne 0$ that occurs less than $2^{2n}$ times we
transfer all the $b_u(2)$ (if any) to the left to place them right after
$y_j(3)$ (in any order). Let $\hat y_{t}\in L_t$ denote the
initial segment of length $<2^{3n}$ formed in this way. If there
are no other indices, that is, indices $k\ne 0$ for which there
are at least $2^{2n}$ elements $ b_{k}(2)$ in the commutator, then
the only elements outside $\hat y_{t}$ are $a_0$ and we must have
$t=0$, since the original sum of indices was $0$ modulo~$2^n$.
Then $\hat y_{t}=0$ by \eqref{e-cll} and the proof is
complete.
Thus, we can assume that there are non-zero indices $v_1,\ldots ,
v_r$, where $1\leqslant r\leqslant 2^n-1$, such that for each $v_i$ there
are at least $2^{2n}$ elements $ b_{v_i}(2)$ in the commutator \eqref{e-9}.
Let $v={\rm gcd}(v_1,\ldots ,v_r)$ be the greatest common divisor
of $v_1, \ldots ,v_r$. Since the sum of all indices is $ 0$ modulo
$2^n$, the number ${\rm gcd}(v,2^n)$ must divide~$t$.
By the Chinese
remainder theorem there exist integers $u_i$ such that $ v= u
_1v_1+\cdots +u_rv_r$. Replacing the $u_i$ by their residues
modulo $2^n$ and changing notation we have $ v=u _1v_1+\cdots
+u_rv_r+u2^n$, where $ u_i\in \{ 0,\,1,\ldots ,2^n-1\}$ for all $
i$ and $u$ is an integer.
Since ${\rm gcd}(v,2^n)$ divides~$t$, there is $w\in \{ 0, 1,\dots , 2^n - 1\}$ such that $t + wv \equiv 0 \;({\rm mod}\; 2^n)$. Substituting the expression for $v$ we obtain
\begin{equation}\label{e-congr}
0\equiv t + wv \equiv t + wu_1v_1 + \dots + wu_rv_r \;({\rm mod}\; 2^n).
\end{equation}
We now arrange an initial segment of the commutator by placing after
$\hat y_{t}$ exactly $wu_1$ elements $b_{v_1}(2)$, then exactly
$wu_2$ elements $ b_{v_2}(2)$, and so on, up to exactly $wu_r$
elements $b_{v_r}(2)$. This is possible because $wu_i\leqslant 2^{2n}$
for each $i$, and there are at least $2^{2n}$ elements $
b_{v_i}(2)$ outside $\hat y_{t}$.
The resulting initial segment
has zero sum of indices modulo $2^n$ by \eqref{e-congr} and has length
$\leqslant 2^{3n}+
2^{3n}$. Hence
it is equal to $0$ by \eqref{e-cll}. \end{proof}
We now proceed with the proof of Proposition~\ref{p2}. Consider
the ideal $I={}_{\rm id}\big\langle L_{\rm odd}(3)\big\rangle \cap
A$.
Clearly, $I\cap L^-$ has $(l,n,m)$-bounded index in $L^-$,
since $A\cap L^-$ has index $l$ in $L^-$, and each component in
$L_{\rm odd}(3)$ has $(n,m)$-bounded index in the corresponding
component in $L_{\rm odd}$. Let $S=I^{(d-2)}$ be the $(d-2)$-nd
term of the derived series of $I$. Note that, clearly, $S\leqslant
R=A^{(d-2)}$.
\begin{lemma}\label{l-r1} The ideal $S$ is nilpotent of $(n,c)$-bounded
class. \end{lemma}
\begin{proof}
Recall that we write $S^-=\sum S_{\rm odd}$ and $S^+=\sum S_{\rm
even}$, and similarly for $[S,S]$. We represent $S$ as the sum of
two ideals $S= J_1+J_2$, where $J_1=[S,S]+S^-$ and $J_2=[S,
S]+S^+$. Since $S$ is metabelian, $[S, S]^-$ is an ideal of
$J_2$. By hypothesis, $\gamma_{c+1}(J_2)\leqslant [S, S]^-$.
We claim that $J_1$ is nilpotent of $n$-bounded class. For
that, we need to show that any simple commutator
$$
[a_{i_1},a_{i_2},a_{i_3},\dots ]
$$
of large enough $n$-bounded length in homogeneous elements of
$J_1$ is equal to~0. Since $S$ is metabelian, we can assume that
all the entries starting from the third one are from $S^-$, so the
commutator is a linear combination of commutators of the form
\begin{equation}\label{e-j1}
[[a_{i_1},a_{i_2}],a_{\rm odd}, a_{\rm odd}, \dots ],
\qquad a_{i_j}\in S_{i_j},\quad a_{\rm odd}\in S_{\rm odd},
\end{equation}
and all
the entries $a_{\rm odd}$ can be freely permuted without changing the
commutator. When the length is large enough, we can rearrange
these entries in such a way that there will be an initial segment
in $[S,S]_0$ of $n$-bounded length, which we denote by $w_0$. With
large enough length of \eqref{e-j1} there will remain at least
$2^{n-1}(2^{n+1}+2^n-4)+1$ elements $a_{\rm odd}$ outside the initial
segment $w_0$, and therefore at least $2^{n+1}+2^n-3$ of them with
the same (odd) index, say, $j$. These entries $a_j$ can be
moved to be placed at the beginning after $w_0$. Therefore it suffices to prove that
the commutator
\begin{equation}\label{e-j2} [w_0,\underbrace{a_{j}, a_{j},
\dots ,a_j }_{2^{n+1}+2^n-3}\,], \qquad \text{where }2\nmid j, \end{equation}
is equal to zero.
Since $S\leqslant R$ we can represent all the entries $a_{j}$ in
\eqref{e-j2} in the form
$a_{j}=r_{j}(1)+ q_{j}$, where the $r_{j}(1)$ are centralizers in $R$, and the $q_{j}$ are coset representatives in $R$. (Note that these elements may no longer be in $I$). After expanding all brackets, we obtain a linear combination of commutators
\begin{equation}\label{e-j3}
[w_0,\underbrace{z_{j}, z_{j}, \dots ,z_j}_{2^{n+1}+2^n-3}\,],
\qquad \text{where }2\nmid j, \end{equation}
and
each $z_{j}$ is either $r_{j}(1)$ or $q_{j}$. In each of such
commutators there are either
at least $2^n+1$ entries $r_{j}(1)$, or
at least $2^{n+1}-3$ entries $q_{j}$.
By permuting the entries $r_{j}(1)$ and $q_{j}$ (we can freely
permute these elements, since $R$ is metabelian and $w_0 \in [R, R]$, $r_{j}(1), q_{j}\in R$), we obtain from \eqref{e-j3} either a
commutator with an initial segment
\begin{equation}\label{e-j4}
[w_0,\underbrace{r_{j}(1), r_{j}(1), \dots
,r_{j}(1)}_{2^{n}+1}\,], \qquad \text{where }2\nmid j, \end{equation}
or a
commutator with an initial segment
\begin{equation}\label{e-j5}
[w_0,\underbrace{q_{j}, q_{j}, \dots ,q_{j}}_{2^{n+1}-3}\,],
\qquad \text{where }2\nmid j. \end{equation}
Thus, it suffices to show that
both commutators \eqref{e-j4} and \eqref{e-j5} are equal to~0.
In the commutator \eqref{e-j4} we regard the initial segment of
the first two entries $ a_j= [ w_0, r_{j}(1)]$ simply as an
element of the ideal $R$ which belongs to $R_j$. By
Lemma~\ref{l-freezer} we can represent the initial segment
$$
[a_j , \underbrace{r_{j}(1), r_{j}(1), \dots
,r_{j}(1)}_{2^{n}-1}\,]\in L_0
$$
in terms of $r$-representatives of level 0, so that the commutator
\eqref{e-j4} becomes equal to
$$
[ [\underbrace{r_j(0), \dots ,r_j(0)}_{2^{n}}\,], r_{j}(1)].
$$
This commutator in turn is equal to a linear combination of
commutators of the form
$$
[ r_{j}(1), \underbrace{r_j(0), \dots ,r_j(0)}_{2^{n}}\,],
$$
in each of which the initial segment of length $2^n$ is equal to
0 by \eqref{e-cr}.
We now consider the commutator \eqref{e-j5}. Its initial segment
$w_0$, being in $I$, also belongs to ${}_{\rm id}\langle
L_{\rm odd}(3)\rangle$ and therefore is a linear combination of
elements of the form
\begin{equation} \label{e-long} [l_{\rm odd}(3), u_{i_1},
u_{i_2}, \dots , u_{i_s}]
\end{equation}
with zero sum of indices modulo $2^n$, where $u_{i_k}\in L_{i_k}$ are arbitrarily homogeneous elements, in any number. By
Lemma~\ref{l-9} an element of the form \eqref{e-long} can be
represented as a linear combination of elements of the form
$[y_{-k}(1), y_{k}(1)]$, for various, not necessarily odd, $k$
(and these elements are not necessarily contained in $I$).
Therefore the commutator \eqref{e-j5} is a linear combination
of commutators of the form
\begin{equation}\label{e-s} \big[ [y_{-k}(1),
y_{k}(1)], \underbrace{q_j, q_{j}, \dots
,q_{j}}_{2^{n+1}-3}\,\big]. \end{equation}
By the Jacobi identity, the
commutator \eqref{e-s} is equal to a linear combination of
commutators of the form
$$
\big[ [y_{-k}(1), \underbrace{q_j, q_{j}, \dots ,q_{j}}_{k}\, ]
,\, [y_{k}(1), \underbrace{q_j, q_{j}, \dots
,q_{j}}_{2^{n+1}-3-k}\,]\big].
$$
In such a commutator, one of the two subcommutators contains a
subcommutator of the form
$$
[y_{\pm k}(1), \underbrace{q_j, q_{j}, \dots ,q_{j}}_{2^n-1}\, ].
$$
Since $j$ is odd, there is an initial segment in $L_0$, which is
equal to 0 by \eqref{e-clr}.
Thus, we have proved that the ideal $J_1$ of $S$ is nilpotent of $n$-bounded class $c_1$.
As a result,
\begin{align*}
\gamma_{c+1+c_1+1}(S)&=\gamma_{c+1+c_1+1}(J_1+J_2)\\
&\leqslant \gamma_{c_1+1}(J_1)+\gamma_{c+1}(J_2)\\
&\leqslant [S, S]^-.
\end{align*}
But $S$ is an ideal of $L$, and hence $\gamma_{c+1+c_1+1}(S)$ is
also an ideal of $L$. Since $[[S, S]^-, L_{\rm odd}]\leqslant L^+$, the
inclusion $\gamma_{c+1+c_1+1}(S)\leqslant [S, S]^-$ implies that
$[\gamma_{c+c_1+2}(S), L^-]=0$. This means that $L^-$ is contained
in the centralizer of the ideal $\gamma_{c+c_1+2}(S)$, and then
$[\gamma_{c+c_1+2}(S), {}_{\rm id}\langle L^-\rangle ]=0$. In
particular, $[\gamma_{c+c_1+2}(S), I]=0$, and therefore
$\gamma_{c+c_1+3}(S)=0$.
The lemma is proved. \end{proof}
We now complete the proof of Proposition~\ref{p2}. The quotient
$L/[S,S]$ contains the homogeneous ideal $I/[S,S]$ of derived
length at most $d-1$ and its intersection with the image of $L^-$
has $(l,n,m)$-bounded index $t$ in the image of $L^-$. By the
induction hypothesis, there is a homogeneous nilpotent ideal
$J/[S,S]$ of $(d-1,n,c)$-bounded class whose intersection with the
image of $L^-$ has $(d-1,t,m,n)$-bounded index in the image of
$L^-$. Then the quotient $(S+J)/[S,S]$ is also nilpotent of
$(d-1,n,c)$-bounded class. Since $S$ is nilpotent of
$(n,c)$-bounded class by Lemma~\ref{l-r1}, we obtain that the
full inverse image $B=J+S$ of $(J+S)/[S,S]$ is nilpotent of
$(d,n,c)$-bounded class by Hall's
Theorem~\ref{t-hall}.
This is a required ideal, since its intersection with $L^-$
has $(d,l,n,m)$-bounded index in $L^-$ (recall that $t$ is an
$(l,n,m)$-bounded number). \end{proof}
\begin{proof}[Proof of Proposition~\ref{p1}]
Proposition~\ref{p1} follows from Proposition~\ref{p2} and Theorem~\ref{t3}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{t2}] Recall that $L$ is a finite Lie
ring admitting an automorphism $\varphi$ of order $2^n$ such that the
fixed-point subring $C_L(\varphi ^{2^{n-1}})$ of the involution $\varphi
^{2^{n-1}}$ is nilpotent of class $c$, and $m=|C_L(\varphi )|$ is the
number of fixed points of $\varphi$. We wish to prove that $L$ has
homogeneous ideals $M_1\geqslant M_2$ such that $M_1$ has $(m,n)$-bounded index in the
additive group $L$, the quotient $M_1/M_2$ is nilpotent of class at most $c+1$, and $M_2$ is nilpotent of
$(n,c)$-bounded class.
First we extend the ground ring by a $2^n$-th root of unity $\omega$
forming $\tilde L=L\otimes _{{\Bbb Z} }{\Bbb Z} [\omega ]$. Then $|C_{\tilde L}(\varphi
)|\leqslant m^{2^n}$ and $C_{\tilde L}(\varphi ^{2^{n-1}})=C_L(\varphi
^{2^{n-1}})\otimes _{{\Bbb Z} }{\Bbb Z} [\omega ]$ is also nilpotent of class $c$.
Therefore it is clearly sufficient to prove the theorem for
$\tilde L$, so we assume that $L=\tilde L$ in what follows.
We begin with the case where $L$ has odd order. Then
$$
L=\bigoplus _{i=0}^{2^n-1}L_i\qquad \text{and}\qquad [L_i,L_j]\leqslant L_{i+j\,(\rm mod\,2^n)}
$$
for the analogues of eigenspaces of $\varphi$
$$
L_i=\{x\in L\mid x^{\varphi }=\omega ^ix\}, \qquad i=0,1,\dots ,2^n-1.
$$
Thus, this is a $({\Bbb Z} /2^n{\Bbb Z} )$-grading of $L$ and $L$ satisfies
the hypotheses of Proposition~\ref{p1}. By that proposition, $L$
contains a homogeneous nilpotent ideal $M_2$ of $(n,c)$-bounded
nilpotency class such that $M_2\cap L^-$ has $(m,n)$-bounded index
in the additive group $L^-$. The latter means that in the
inherited grading of the quotient $\bar L=L/M_2$ the order of
$\bar L^-$ is $(m,n)$-bounded.
For any fixed homogeneous element $a_j\in \bar L ^-$ (so that $j$
is odd) and for any even $k$ the map $b_k\to [a_j,b_k]$ from $\bar
L_k$ to $\bar L_{j+k}\in \bar L_{\rm odd}$ is linear with
$(m,n)$-bounded image. Therefore its kernel has $(m,n)$-bounded
index in $\bar L_k\in \bar L_{\rm even}$. As a result, $C_{\bar
L^+}(a_j)$ has $(m,n)$-bounded index in $\bar L^+$. Taking the
intersection over an $(m,n)$-bounded number of homogeneous
elements generating $\bar L^-$ we obtain that $|\bar L^+:C_{\bar
L^+}(\bar L^-)| $ is $(m,n)$-bounded and therefore $|\bar L :
C_{\bar L}(\bar L^-)| $ is also $(m,n)$-bounded, since $|\bar
L^-|$ is $(m,n)$-bounded.
Let $K=\langle \bar L^-\rangle$ be the subring of $\bar L$
generated by $\bar L^-$. Recall that $K$ is an ideal of $\bar L$, and therefore $C_{\bar L}(K)$ is also an ideal of $\bar L$.
The index $|\bar L : C_{\bar L}(K)| $ is also $(m,n)$-bounded
since $C_{\bar L}(K) =C_{\bar L}(\bar L^-)$. We prove that the
ideal $C_{\bar L}(K)$ is nilpotent of class at most $c+1$. Indeed,
$\bar L /K=(\bar L^++K)/K$ is nilpotent of class at most $c$
by hypothesis, and $C_{\bar L}(K)\cap K$ is central in $K$. Hence,
$$
[\underbrace{C_{\bar L}(K),\ldots, C_{\bar L}(K)}_{c+1}, C_{\bar
L}(K) ]\subseteq [K, C_{\bar L}(K)]=0
$$
and therefore
$\bar M_1=C_{\bar L}(K)$ is a nilpotent ideal
of class at most $c+1$. Then its full inverse image $M_1$ and the
aforementioned ideal $M_2$ satisfy the conclusion of
Theorem~\ref{t2}.
We now consider the case where the additive group of $L$ is a
finite $2$-group. Although we no longer have a direct sum, it is well known (see, for example, \cite[Ch. 10]{hpbl}) that
$$
2^nL\leqslant L_0+L_1+\cdots +L_{2^n-1}\qquad \text{and}\qquad [L_i,L_j]\leqslant L_{i+j\,(\rm mod\,2^n)}.
$$
By Corollary~\ref{cor-shu} applied to the subring $M=L_0+L_1+\cdots
+L_{2^n-1}$ we have
$$
\gamma_{f(n,c)+1}\big(\gamma_{c+1}(M)\big)
\leqslant {}_{{\rm id}}\!\left<L_0\right>.
$$
It follows that
$$
m \gamma_{f(n,c)+1}\big(\gamma_{c+1}(M)\big) \leqslant m\,\,{}_{{\rm
id}}\!\left<L_0\right>=0,
$$
since $mL_0=0$ by Lagrange's theorem.
Hence,
$$
\gamma_{f(n,c)+1}\big(\gamma_{c+1}(mM)\big)=0.
$$
By Lemmas
\ref{l-fp}(a) and \ref{l-fpp2} the index of the additive subgroup
$2^nmL \leqslant mM$ in $L$ is $(n,m)$-bounded, and hence $M_1=2^nmL$ and $M_2=\gamma_{c+1}(M_1)$ are
the required ideals.
In the case of an arbitrary finite Lie ring, $L$ is a direct
sum of two ideals $L=T_2\oplus T_{2'}$, where the additive group $T_2$ is the Sylow $2$-subgroup of $L$, and $T_{2'}$ is the Hall $2'$-subgroup of $L$. As shown above,
$T_2$ and $T_{2'}$ contain ideals $I_1$ and $I_2$, respectively, of
$(m,n)$-bounded indices such that
$$
\gamma_{g(n,c)}\big(\gamma_{c+2}(I_k)\big)=0, \qquad k=1,2
$$
for some $(n,c)$-bounded number $g(n,c)$. Since $[T_2, T_{2'}]=0$, it follows that $I_1$ and $I_2$ are commuting ideals of $L$.
The sum $M_1=I_1+ I_2$ and $M_2=\gamma_{c+2}(M_1)$ are the sought-for ideals of $L$.
\end{proof}
\section{Groups}\label{s-groups}
Here we prove the main group theoretic result. Known results reduce the proof to the case where $G$ is a nilpotent group of odd order. Then we firstly apply the Lie ring method similarly to \cite{shu01} to obtain a `weak' bound, depending on $m,n,c$, for the nilpotency class of $[G,\varphi ^{2^{n-1}}]$. Then Theorem~\ref{t2} is used to obtain the required `strong' bound, in terms of $n,c$ only, for the nilpotency class of $[H,\varphi ^{2^{n-1}}]$ for a certain subgroup $H$ of $(m,n,c)$-bounded index.
\begin{proof}[Proof of Theorem~\ref{t1}] Recall that we have a finite group
$G$ admitting an automorphism $\varphi$ of order $2^n$ such that
$C_G(\varphi ^{2^{n-1}})$ is nilpotent of class $c$, and $m=|C_G(\varphi )|$ is the
number of fixed points of $\varphi$. We need to prove that $G$ has a
soluble subgroup of $(m,n,c)$-bounded index that has
$(n,c)$-bounded derived length.
We begin with reduction to the case where $G$ is a nilpotent group
of odd order. The group $G$ has a soluble subgroup of
$(m,n)$-bounded index by Hartley's theorem \cite{har}. Therefore
we can assume from the outset that $G$ is soluble. The quotient
$G/O_{2',2}(G)$ acts faithfully by conjugation on the Frattini
quotient $V=T/\Phi (T)$ of the $2$-group $T=O_{2',2}(G)/
O_{2'}(G)$. By Lemma~\ref{l-fp}(a) we have $|C_V(\varphi )|\leqslant m$.
Therefore the order of $V$ is $(m,n)$-bounded by
Lemma~\ref{l-fpp2}. As a result, the order of $G/O_{2',2}(G)$ is
also $(m,n)$-bounded.
By Lemma~\ref{l-fp}(a) we have $|C_T(\varphi )|\leqslant m$. By Khukhro's
theorem \cite{khu93} on $p$-automorphisms of finite $p$-groups, the group $T$ contains a subgroup $U$ of
$(m,n)$-bounded index that has $n$-bounded derived length. By
Theorem~\ref{t-char}
this subgroup
$U$ can be assumed to be characteristic in $G/O_{2'}(G)$ and therefore normal and $\varphi$-invariant.
By the Hartley--Turau theorem \cite{ha-tu}, the index of the
$n$-th Fitting subgroup $F_{n}(G)$ in $G$ is $(m,n)$-bounded. By
Lemma~\ref{l-fp}(b) every factor
$Q_i=F_i(O_{2'}(G))/F_{i-1}(O_{2'}(G))$ of the Fitting series of
$F_n(O_{2'}(G))$ admits the action (not necessarily faithful) of
the automorphism $\varphi$ such that $|C_{Q_i}(\varphi )|\leqslant m$ and
$C_{Q_i}(\varphi ^{2^{n-1}})$ is nilpotent of class at
most $c$.
Suppose that Theorem~\ref{t1} is already proved for the case where
$G$ is a nilpotent group of odd order. Then every $Q_i$ has a
subgroup $R_i$ of $(m,n,c)$-bounded index that is soluble of
$(n,c)$-bounded derived length; by Theorem~\ref{t-char}
this
subgroup can be assumed to be characteristic. Let $\tilde
T=O_{2',2}(G)$, $\tilde U$, $\tilde Q_i=F_i(O_{2'}(G))$, and
$\tilde R_i$ denote the inverse images in $G$ of the sections
$T$, $U$, $Q_i$, and $R_i$, respectively. We can set
$$
H=O_{2',2}(G)\cap C_G(\tilde T/\tilde U)\cap
C_G\big(O_{2'}(G)/\tilde Q_n\big)\cap \bigcap _{i=1}^{n-1}
C_G(\tilde Q_i/\tilde R_i). $$
(Here the centralizer of a section $A/B$ is defined naturally as
$C_G(A/B)=\{g\in G\mid [A,g]\leqslant B\}$.) Then $H$ is a subgroup of
$(m,n,c)$-bounded index, since all the quotients $G/C_G(\tilde T/ \tilde
U)$, $G/ C_G\big(O_{2'}(G)/\tilde Q_n\big)$, $G/ C_G(\tilde Q_i/\tilde R_i)$ embed into the automorphism groups of sections of $(m,n,c)$-bounded order. The intersections
of the images of $H$ with the sections $\tilde T/\tilde U$, $O_{2'}(G)/\tilde R_n$, and
$\tilde Q_i/\tilde R_i$ are central in $H$ by construction. Let
$g$ be the derived length of $U$, and $f_i$ the derived length
of $R_i$. Then
\begin{align*}
[H,H]^{(g)}&\leqslant [O_{2',2}(G), \,C_G(\tilde T/\tilde U)]^{(g)}\cap H\\
&\leqslant \tilde U^{(g)}\cap H\leqslant O_{2'}(G)\cap H,
\end{align*}
\begin{align*}
[O_{2'}(G)\cap H, \,O_{2'}(G)\cap H] &\leqslant [O_{2'}(G),
C_G\big(O_{2'}(G)/\tilde Q_n\big)]\cap H
\\
&\leqslant \tilde Q_n
\cap H,
\end{align*}
and
\begin{align*}
\big[\tilde Q_i\cap H, \, \tilde Q_i\cap H\big]^{(f_i)} &\leqslant \big[\tilde Q_i,\,
C_G(\tilde Q_i/\tilde R_i)\big]^{(f_i)}\cap H\\
&\leqslant \tilde R_i^{(f_i)}\cap H\leqslant \tilde Q_{i-1}\cap H,
\end{align*}
where $i=1,2, \ldots, n$ and $\tilde Q_0=1$. It follows
that $H$ is soluble of derived length at most
$$
1+g+1+\sum_{i=1}^n
(1+f_i),
$$
which is an $(n,c)$-bounded number. Thus, $H$ satisfies the
conclusion of Theorem~\ref{t1}, which completes our reduction.
Therefore in what follows we can assume from the outset that $G$
is a nilpotent group of odd order.
We now obtain a `weak' bound, in terms of $m,n,c$,
for the nilpotency class of the subgroup $[G,\varphi ^{2^{n-1}}]$.
For that we consider the
associated Lie ring of $[G,\varphi ^{2^{n-1}}]$, but preliminary lemmas are
stated in terms of abstract Lie rings. To
lighten the notation we denote $\psi =\varphi ^{2^{n-1}}$.
We denote by $[L, \psi ]$ the additive subgroup generated by $\{-l+l^{\psi}\mid l\in L\}$.
\begin{lemma}\label{l-2.3} If $L$ is a finite metabelian and nilpotent Lie
ring of odd order admitting an automorphism $\varphi$ of order $2^n$
such that $|C_L(\varphi )|=m$, then the ideal $[L, \psi ]+[L,L]$
is nilpotent of $(m,n)$-bounded class.
\end{lemma}
\begin{proof} We actually show that the ideal $[L, \psi ]+[L,L]$
is nilpotent of class at most $1+(m+1)(2^n-1)$.
We can assume from the outset that the ground
ring contains a primitive $2^n$-th root of 1, since the extension of the ground ring by this root may only
increase the size of the fixed-point subring in terms
of $n$. As in
\S\,\ref{s-prelim}, we decompose $L$ into the direct sum of
analogues of
eigenspaces $L_i$, which serve as components of a
$({\Bbb Z}/2^n{\Bbb Z})$-grading. Then $[L,\psi ]=L^-$. Therefore we need to
show that any simple homogeneous commutator of length $2+(m+1)(2^n-1)$ in
elements of $L_{\rm odd}$ and $[L,L]$ is trivial. Since $[L,L]$ is
abelian, we can assume that starting from the third place all
entries are in $L_{\rm odd}$, and all these entries can be freely
permuted without changing the commutator. By \cite[Lemma~2.2]{shu01} any
sequence of $2^n-1$ odd numbers can be rearranged to produce
an initial segment with any pre-assigned sum modulo $2^n$. Therefore we can rearrange the $(m+1)(2^n-1)$ entries of our commutator, starting from the third one, so as to produce $m+1$ initial segments in $L_0$. As a result, since $|L_0|=m$,
there will be two different initial segments equal to the same
element in $L_0$. The longer of these two segments can be
substituted instead of the shorter one, then again in the
resulting longer commutator, and so on. Thus the commutator
becomes equal to an ever longer commutator. Since $L$ is nilpotent
by hypothesis, the commutator is equal to 0. \end{proof}
\begin{lemma}\label{l-2.5} Suppose that a finite metabelian and nilpotent
Lie ring $L$ of odd order admits an automorphism $\varphi$ of order
$2^n$ such that $|C_L(\varphi )|=m$ and $C_L(\psi )$ is nilpotent of
class $c$. Then $\gamma _{g}(L)\leqslant [[L,L], \psi ]$ for some $(m, n, c)$-bounded number $g$. \end{lemma}
\begin{proof} As is Lemma~\ref{l-2.3} we can assume that the ground ring
contains a primitive $2^n$-th root of 1 and $L$ is graded by
analogues of eigenspaces of $\varphi$, so that $[[L,L], \psi
]=[L,L]^-$. Consider the ideals $J_1=[L,L]+L^-$ and
$J_2=[L,L]+L^+$; then $L=J_1+J_2$. By Lemma~\ref{l-2.3} we have
$\gamma _{f}(J_1)=0$ for some $(m, n)$-bounded number $f$. Since
$ [L,L]^-$ is an ideal of $J_2$, we have $\gamma _{c+1}(J_2)\leqslant
[L,L]^-$ by hypothesis. We now obtain
\begin{align*}
\gamma _{f+c+1}(J_1+J_2)&\leqslant \gamma _{f}(J_1) + \gamma
_{c+1}(J_2)\\
&\leqslant 0+[L,L]^-,
\end{align*}
as required.
\end{proof}
\begin{proposition}\label{p-2.6} Suppose that a finite Lie ring $L$ of odd order admits an
automorphism $\varphi$ of order $2^n$ such that $|C_L(\varphi )|=m$ and
$C_L(\psi )$ is nilpotent of class $c$. Then the Lie ring
generated by $[L,\psi ]$ is nilpotent of $(m, n, c)$-bounded class.
\end{proposition}
\begin{proof} As before we can assume that the ground ring contains a
primitive $2^n$-th root of 1 and $L$ is graded by analogues of
eigenspaces $L_i$ of $\varphi$. We can obviously assume that
$
L=\langle L^-\rangle =\langle [L,\psi ]\rangle
$.
Consider the lower central series of $L$. The
fixed points of $\varphi$ in its factors are images of the fixed points
in $L$ by Lemma~\ref{l-fp}(b). Therefore there are at most $m$
factors where $\varphi$ is not fixed-point-free. We obtain a series of
$\varphi$-invariant ideals of length at most $2m+1$ each factor of
which either is central or admits $\varphi$ as a fixed-point-free
automorphism. The fixed-point subrings of $\psi$ in these factors
are images of subrings of $C_L(\psi )$ by Lemma~\ref{l-fp}(b) and
therefore are nilpotent of class at most $c$. By
Theorem~\ref{t-shu} the factors with fixed-point-free action of $\varphi$
are soluble of $(n,c)$-bounded derived length. As a result, $L$
is soluble of $(m,n,c)$-bounded derived length. Therefore it is
sufficient to prove by induction on the derived length $d$ of $L=\langle L^-\rangle$ that $L$ is nilpotent of $(d,m,n,c)$-bounded class. If $d=1$, there is nothing to prove, so let $d\geqslant 2$.
Let $R=L^{(d-2)}$ be the penultimate (metabelian) term of the
derived series of $L$. By the induction hypothesis, $L/[R,R]$ is
nilpotent of $(d-1,m,n,c)$-bounded class. By Lemma~\ref{l-2.5},
$\gamma _{g}(R)\leqslant [R,R]^-$ for an $(m,n,c)$-bounded number $g$. But
$\gamma _{g}(R)$ is an ideal of $L$, and therefore $[\gamma
_{g}(R), L]=0$, since $[[R, R]^-, L_{\rm odd}]\leqslant L^+$ and $L=\langle
L^-\rangle$. Therefore, in particular, $\gamma _{g+1}(R)=0$. It
remains to apply Hall's Theorem \ref{t-hall},
by which $L$ is
nilpotent of $(d,m,n,c)$-bounded class, as required. \end{proof}
We now complete the proof of Theorem~\ref{t1}.
Recall that we already have a reduction to the case of a finite
nilpotent group $G$ of odd order admitting an automorphism $\varphi$ of
order $2^n$ such that the fixed-point subgroup $C_G(\psi )$ of the involution $\psi=\varphi ^{2^{n-1}}$ is nilpotent of
class $c$. For $m=|C_G(\varphi )|$ being the number of fixed points of
$\varphi$, we need to prove that $G$ has a soluble subgroup of
$(m,n,c)$-bounded index that has $(n,c)$-bounded derived length.
Recall that the associated Lie ring $L(D)$ of a group $D$ is defined on the direct sum of lower central factors $L(D)=\bigoplus_i \gamma _i(D)/\gamma _{i+1}(D)$. For $a\in g _{i} (D)$, $b\in \gamma _{j}(D)$, the Lie products are defined by $[a+\gamma _{i+1} (D),\,b+\gamma _{j+1}(D)]=[a,b]+\gamma _{i+j+1}(D)$ via the group commutator $[a,b]$ on the right and extended to $L(D)$ by linearity. This definition is correct because of the inclusions $[\gamma _i(D),\gamma _j(D)]\leqslant \gamma _{i+j}(D)$.
These inclusions also imply that for any $k$ and any $a_i\in D$,
\begin{equation}\label{e-ass}
[a_1,\dots ,a_k]\gamma _{k+1}(D)=[\bar a_1,\dots \bar a_k],
\end{equation}
where the left-hand side is the image of the group commutator in $\gamma _{k}(D)/\gamma _{k+1}(D)$ and the right-hand side is the commutator in $L(D)$ of the images of $a_i$ in $D/\gamma _2(D)$. In particular, if $D$ is a nilpotent group, then $L(D)$ is a nilpotent Lie ring and its nilpotency class is exactly the same as that of $D$.
Consider the associated Lie ring $L([G,\psi ])$ of $[G,\psi ]$.
By Lemma~\ref{l-fp}(b) the induced automorphism $\varphi$ denoted by
the same letter has the same number $|C_{L([G,\psi ])}(\varphi
)|=|C_{[G,\psi ]}(\varphi )|\leqslant m$ of fixed points. Since
$C_{L([G,\psi ])}(\psi )$ is the sum of the images of subgroups
of $C_{[G,\psi ]}(\psi )$ by Lemma~\ref{l-fp}(b), it is easy to
see that $C_{L([G,\psi ])}(\psi )$ is also nilpotent of class at
most $c$. By Proposition~\ref{p-2.6} we obtain that $L([G,\psi
])$, and therefore also $[G,\psi ]$, is nilpotent of
$(m,n,c)$-bounded class $k$. However, our aim is a subgroup of
bounded index with derived length `strongly' bounded, in terms of
$n$ and $c$ only, independently of $m=|C_G(\varphi )|$. We will
achieve this goal by applying Theorem~\ref{t2} to find a subgroup
$H$ of $(m,n,c)$-bounded index such that $[H,\varphi ^{2^{n-1}}]$ is
nilpotent of $(n,c)$-bounded class.
We extend the ground ring by a primitive $2^n$-th root of unity
$\omega$ forming $L=L([G,\psi ])\otimes _{{\Bbb Z} }{\Bbb Z} [\omega ]$. Then $L= L_0
\oplus L_1 \oplus \dots \oplus L_{2^n-1}$ is a $({\Bbb Z}/2^n{\Bbb Z})$-graded
Lie ring with grading components $L_i$ --- analogues of eigenspaces of
$\varphi$ --- satisfying $[L_s, L_t]\subseteq L_{s+t\,({\rm mod}\,2^n)}$. As
usual, the Lie ring $L([G,\psi ])$ is considered to be embedded in
$L$ as $L([G,\psi ])\otimes 1$. The Lie ring $L$ is nilpotent of
the same nilpotency class $k$. We also have $|C_L(\varphi )|\leqslant
m^{2^n}$ and $C_L(\psi )$ is nilpotent of class at most $c$.
By Proposition~\ref{p1} the Lie ring $L$ has a nilpotent ideal $B$
of $(n,c)$-bounded class $h$ such that $B\cap L^-$ has
$(m,n)$-bounded index in $L^-$. Since $[G,\psi]$ is generated by
elements $x$ such that $x^{\psi}=x^{-1}$, it follows that $L$ is
generated by elements $l$ such that $l^{\psi}=-l$, that is,
$L=\langle L^-\rangle=L^-+[L,L]$. Then $M=B +[L,L]$ is an ideal
of $(m,n)$-bounded index in $L$. The nilpotency class of $M=B +[L,L]$ is
strictly smaller than the nilpotency class $k$ of $L$, as long as $k$ was higher than $h$. Indeed,
consider any commutator of weight $k$ in elements of $B\cup [L,L]$. If it involves at least one element of $[L,L]$, then it clearly belongs to $\gamma _{k+1}(L)=0$; otherwise it belongs to $\gamma _{k}(B)$, which is trivial when $k>h$.
Consider $T=M\cap L([G,\psi ])$, which is an
ideal of the Lie ring $L([G,\psi ])$ containing $\gamma _2(L([G,\psi ]))$.
Taking the `full inverse image' of $T$ modulo $\gamma _2([G,\psi ])$ we obtain a subgroup $G_1$ of $(m,n)$-bounded
index in $[G,\psi ]$. As long as $h<k$, the nilpotency class of $G_1$ is strictly
smaller than the nilpotency class $k$ of $[G,\psi ]$. Indeed,
consider any commutator $[a_1,\dots ,a_k]$ of weight $k$ in elements $a_i\in G_1$. Since $\gamma _{k+1}([G,\psi ])=1$, by formula \eqref{e-ass} we have
$$
[a_1,\dots ,a_k]=[\bar a_1,\dots ,\bar a_k],
$$
where $\bar a_i$ is the image of $a_i$ in $[G,\psi ]/\gamma _2([G,\psi ])$. By construction, $\bar a_i\in T$ and therefore the Lie ring commutator on the right is equal to $0$ if $k>h$, which also means that $
[a_1,\dots ,a_k]=1$.
By the Bruno--Napolitani theorem \cite[Lemma~3]{brna} (see also Theorem~\ref{t-char}), there is also a characteristic subgroup of $[G,\psi ]$ that has $(m,n)$-bounded index in $[G,\psi ]$ and is nilpotent of class at most $k-1$. Changing notation we denote this
subgroup again by $G_1$, which is now normal in $G$ and $\varphi$-invariant. Then the product
$G_2=G_1C_G(\psi )$ is a $\varphi$-invariant subgroup of $G$ of
$(m,n)$-bounded index (the latter because $G=[G,\psi ]C_G(\psi )$ by Lemma~\ref{l-fp}(b)), and the nilpotency class
of $[G_2, \psi]\leqslant G_1$ is strictly smaller than $k$. We can now apply
the same arguments to $G_2$ and so on, at each step obtaining a
$\varphi$-invariant subgroup $G_{2i}$ containing $C_G(\psi )$ and having
$(m,n)$-bounded index in $G_{2i-2}$ such that $[G_{2i},\psi ]$
has nilpotency class strictly smaller than that of $G_{2i-2}$ --- as
long as the latter remains greater than the $(n,c)$-bounded number
$h$ given by Propositon~\ref{p1}. The number of these steps is
$(m,n,c)$-bounded, since the nilpotency class of $[G,\psi ]$ is
$(m,n,c)$-bounded. As a result, we arrive at a subgroup $H$ of
$(m,n,c)$-bounded index in $G$ such that $[H,\psi ]$ is nilpotent
of $(n,c)$-bounded class at most $h$. Since $C_H(\psi )$ is nilpotent of class
at most $c$ by hypothesis, this subgroup $H$ is soluble of
$(n,c)$-bounded derived length. By Theorem~\ref{t-char} there is also a characteristic subgroup of $(m,n,c)$-bounded index in $G$ which has the same derived length as $H$. \end{proof}
\begin{remark}\label{r1}
The condition in the theorem that $C_G(\varphi ^{2^{n-1}})$ is nilpotent of class $c$ can be weakened to requiring all Sylow subgroups of
$C_G(\varphi ^{2^{n-1}})$ to be nilpotent of class at most $c$. Indeed, that condition is not used in the reduction at the beginning of the section to the case $G=O_{2'}(G)$. After that, as we saw, it is sufficient to consider the factors $Q_i$ of the Fitting series of $O_{2'}(G)$. If all Sylow subgroups of
$C_G(\varphi ^{2^{n-1}})$ are nilpotent of class at most $c$, then $C_{Q_i}(\varphi ^{2^{n-1}})$ is nilpotent of class at most $c$ for every $i$ and we find ourselves under the hypotheses of Theorem~\ref{t1}.
Similarly, in Corollary~\ref{c1} the condition that $C_G(g ^{2^{n-1}})$ is nilpotent of class $c$ can be weakened to requiring all nilpotent subgroups of
$C_G(g ^{2^{n-1}})$ to be nilpotent of class at most $c$, because when applying the inverse limit argument to a system of finite subgroups containing $g$, we would be able to use the aforementioned stronger version of Theorem~\ref{t1}.
\end{remark}
\section*{Acknowledgments} This work was supported by CNPq-Brazil. The first author thanks CNPq-Brazil
and the University of Brasilia for support and hospitality that he
enjoyed during his visits to Brasilia. The second
author was supported by the Russian Foundation for Basic
Research, project no. 13-01-00505.
The authors thank the referee for careful reading of the paper and several helpful comments.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,935
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\section{Introduction}
\label{intro}
The problem of geodesic mappings of pseudo-Riemannian manifold was first started by Levi-Civita [12]. There exists many monographs and papers devoted to the theory of geodesic mappings and transformations
\cite{1,2,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,mid,mi80r,v1,v2,v3}.
Geodesic mappings play an important role in the general theory of relativity \cite{7,23}.
Let $\an=(\mn,\na)$ be a $n$-dimensional manifolds \mn\ with affine connection~$\na$ without torsion. We denote the ring of smooth functions on \mn\ by $f(\mn)$, the Lie algebra of smooth vector fields on \mn\ by $X(\mn)$ and arbitrary smooth vector fields on \mn\ by $X,Y,Z$.
A diffeomorphism $f\colon\an\to\ann$ is called a {\it geodesic mapping} of \an\ onto \ann\ if $f$ maps any geodesic curve on \an\ onto a geodesic curve on \ann\ \cite{21,30}.
A manifold \an\ admits a geodesic mapping onto \ann\ if and only if the equation~\cite{21,30
$$
\bar\na_XY=\na_XY+\psi(X)Y+\psi(Y)X
$$
holds for any vector fields $X,Y$ and where $\psi$ is a differential form on $\mn (=\bar M_n)$.
If $\psi$ then geodesic mapping is called {\it trivial} and {\it nontrivial} if $\psi\neq0$.
Let $\vn=(\mn,g)$ be an $n$-dimensional pseudo-Riemannian manifolds with metric tensor $g$ and $\na$ be a {\it Levi-Civita connection}.
A pseudo-Riemannian manifold \vn\ admits a geodesic mapping onto pseudo-Riemannian manifold \vnn\ if and only if there exists a differential form on \vn\ such that the {\it Levi-Civita equation} \cite{21,30}
\be{1}
(\na_Z\bar g)(X,Y) = 2\psi(Z)\bar g(X,Y)+\psi(X)\bar g(Y,Z)+\psi(Y)\bar g(X,Z)
\ee
holds for any vector fields $X,Y,Z$. Or in the coordinate form
\be{2}
\bar g_{ij,k}=2\psi_k\bar g_{ij}+\psi_i\bar g_{jk}+\psi_j\bar g_{ik},
\ee
where $\psi_i=\na_i\Psi$, $\Psi$ is a scalar field.
The Levi-Civita equations \re{1} is not linear so that is not convenient for investigations. Sinyukov \cite{21,30} proved that a pseudo-Riemannian manifold \vn\ admits a geodesic mapping if and only if there exist a differential form $\la(X)$ and a regular symmetric bilinear form $a(X,Y)$
on~\vn\ such that the equation
\be{3}
(\na_Z a)(X,Y) = \la(X) g(Y,Z)+\la(Y) g(X,Z)
\ee
holds for any vector fields $X,Y,Z$. Or in the coordinate form
\be{4}
a_{ij,k}=\la_ig_{jk}+\la_j g_{ik},
\ee
where $\la_i=-a^s_i\psi_s$, $a^s_i=g^{st}a_{ti}$, $g^{st}$ are the contravariant components of the metric $g$.
Note that $\la_i=\na_i\Lambda$, $\Lambda$ is a scalar field.
If \vn\ $(n>2)$ admits two linearly independent solutions not proportional to the metric tensor $g$ then \cite{21
\be{5}
(\na_Y\la)(X)=K\,a(X,Y)+\mu\,g(X,Y),
\ee
\\[-10mm]
\be{6}
\na_X\mu=2K\,\la(X),
\ee
where $K$ is a constant and $\mu$ is a scalar field on \vn. Or in the coordinate form
\be{7}
\na_j\la_i=K\,a_{ij}+\mu\,g_{ij},
\ee
\\[-10mm]
\be{8}
\na_k\mu=2K\,\la_k.
\ee
A pseudo-Riemannian manifold satisfying the equations \re{3}, \re{5}, \re{6} is called a {\it \vnk-space}.
This spaces for Riemannian manifolds were introduced by Solodovnikov \cite{31}
as $V(K)$-space and with another problem for pseudo-Riemannian manifolds were introduced by Mikes \cite{mid,21} as $V_n(B)$-space (in this case $B=-K$).
A vector field $\ph$ on a pseudo-Riemannian manifold \vn\ is called a {\it concircular}~if
\be{9}
(\na_Y\ph)X=\r\,g(X,Y),
\ee
where $\r$ is a scalar field on \vn, see~Yano \cite{yaco}.
If $\r\neq0$ a concircular field belongs to the {\it basic type} and belongs to the {\it exceptional type} otherwise.
A pseudo-Riemannian manifold \vn\ admitting a concircular field is called an {\it equidistant space} \cite{21,30}.
The equidistant space belongs to the basic type if it admits a concircular field of the mane type and belongs to the exceptional type if it admits concircular fields only the exceptional type.
Concircular fields play an important role in the theories of conformal and geodesic mappings and transformations. They were studied by a number of geometers: Brinkmann \cite{bri}, Fialkow \cite{6}, Yano \cite{yaco}, Sinyukov \cite{30}, Aminova \cite{2},
Mike\v s \cite{mi80r,13}, Shandra \cite{25,26,27,28}, etc.
The linear space of all concircular fields on \vn\ denotes by \conv.
If $\nad{1}\ph,\dots,\nad{m}\ph$ is a basis in \conv\ then the tensor field
$$
a=\sum_{\a,\b=1}^m \pod{\a\b}C\ (\nad{\a}\ph\otimes\nad{\b}\ph)
$$
is a solution of the system \re{2}, where \smash{$\pod{\a\b}C\ (=\pod{\b\a}C)$} are some constants. So \vn\ admits the geodesic mapping.
Pseudo-Riemannian manifolds admitting concircular fields form the class of manifolds is closed with respect to the geodesic mappings \cite{21,30}. Let pseudo-Riemannian manifold \vn\ admits a geodesic mapping onto pseudo-Riemannian manifold \vnn\ if there exists a concircular field $\ph$ on \vn\ then there exists a concircular field $\bar\ph$ on \vnn\ such that
\be{10}
\bar\r = \exp(\Psi)\ (\r+g^{ij}\ph_i\psi_j).
\ee
A concircular field $\ph$ is said to be {\it special} if \cite{21,28}
\be{11}
Z(\r) = K\,g(Z,\ph),
\ee
where $\ph$ is a constant, and is said to be is said to be {\it convergent} \cite{29} if $\r$ is a constant.
A pseudo-Riemannian manifold \vn\ admitting a convergent field is called a {\it Shirokov space}.
If there exist two linearly independent concircular field on \vn\ then all concircular fields on \vn\ are special with the same constant $K$, see \cite{21}. A pseudo-Riemannian manifold \vn\ admitting a special concircular field is a \vnk-space. On a \vnk-space any concircular field is special.
\section{Shirokov spaces and \vnk\ spaces $(K\neq0)$}
\begin{lemma
Let pseudo-Riemannian manifold $V_{n+1}=(M_{n+1},G)$ admits a convergent fields $\tilde \ph$ such that
\be{12}
\hbox{a) \ \ } \|\tilde \ph\|<0 \hbox{\ \ and \ \ \ b) \ \ }
(\tilde\nabla_{\tilde Y}\tilde \ph )\tilde X=K\,G(\tilde X,\tilde Y),
\ee
for any vector fields $\tilde X,\tilde Y$ on $M_{n+1}$, where $K\ (\neq0)$ is a constant. Then there exists the adapted coordinate system $(x^I)=(x^0,x^i)$ in which the components $G_{IJ}$ of the metric $G$ reduce to the form
\be{13}
G_{IJ}=\exp(2\,K\,x^0)\ \left(
\begin{array}{cc} -1 & 0 \\[2mm]
0 & \displaystyle\frac{g_{ij}(x^k)}{K}
\end{array}
\right)
\ee
where $g_{ij}(x^k)$ is the components of the metric of some $\vn=(\mn,g)$, $I,J,\dots=1,\dots,n+1$, $i,j,\dots=1,\dots,n$.
\end{lemma}
\noindent {\it Proof.} Let $\tilde {\ph}{}^I$ be the components of the vector fields $\tilde {\ph}$ $g$-conjugate with a convergent fields $\tilde {\ph}$ in a coordinate system $(x^I)$ on $V_{n+1}=(M_{n+1},G)$. Then due to (\ref{12}b) they satisfy
\be{14}
\tilde \na_J\tilde \ph^I=K\,\delta^I_J .
\ee
Let $D$ be the linear space of all vector fields on $V_{n+1}$ which are orthogonal to $\nad*\ph$. It easy to check that $D$ is involutive. So if we use as a natural basis of $X(M_{n+1})$ the basis $\{e_I\}=\{\nad*\ph,e_i\}$, where $\{e_i\}$, is the basis in $D$, we get the coordinate system $(x^I)=(x^0,x^i)$ in which
\be{15}
\hbox{a) \ \ } \tilde \ph{}^I=\delta^I_0; \hbox{\ \ \ b) \ \ }
G_{i0}=0.
\ee
In these coordinates the equations \re{14} are equivalent to
\be{16}
\tilde \G^I_{0J}=K\ \delta^I_J ,
\ee
where $\tilde \G{}^I_{JK}$ are the components of the Levi-Civita connection of the metric $G$.
Let us consider the conditions \re{16}. If $I=0, J=j$ we have
\be{17}
\partial_jG_{00}=0.
\ee
If $I=0, J=0$ we get
\be{18}
\partial_jG_{00}=2K\,G_{00}.
\ee
It follows from \re{17} and \re{18} that
\ $G_{00}=C\cdot\exp{(2K\,x^0)}$,
where $C$ is a constant. Due to (\ref{12}a) that $C<0$. We can take it such that $C=-1$. So
\be{19}
G_{00}=- \exp{(2K\,x^0)}.
\ee
If $I=i,J=j$ we obtain \
$
\partial_0G_{ij}=2K\,G_{ij}
$. \
So
\be{20}
G_{ij}=\exp{(2K\,x^0)}\ \frac{g_{ij}(x^k)}{K}.
\ee
It follows from (\ref{15}b), \re{19}, \re{20} that in the coordinate system $(x^I)=(x^0,x^i)$ components $G_{IJ}$ reduces to the form \re{13}.
Conversely, if the components $G_{IJ}$ of the metric $G$ in the coordinate system $(x^I)=(x^0,x^i)$ reduce to the form \re{13} then the components $\tilde \G^I_{JK}$ of the Levi-Civita connection reduce to the form:
\be{21}
\tilde \G^0_{00}=K, \ \
\tilde \G^0_{0j}=0, \ \
\tilde \G^i_{0j}=\delta^i_j, \ \
\tilde \G^0_{ij}=g_{ij}, \ \
\tilde \G^k_{ij}=\G^k_{ij},
\ee
where $\G^k_{ij}$ are the components of the Levi-Civita connection of the metric $g$. Using direct calculations it easy to verify that vector field with components
$\tilde \ph{}^I_0=\delta^I_0$
by virtue \re{21} satisfies the conditions (\ref{12}a), \re{14}.\smallskip
\noindent {\bf Remark 1} \ The components $G^{IJ}$ of the inverse metric $G$ in the adapted coordinate system $(x^I)=(x^0,x^i)$ reduce to the form
\be{22}
G^{IJ}=\exp(-2\,K\,x^0)\ \left(
\begin{array}{cc} -1 & 0 \\[2mm]
0 & K\,g^{ij}(x^k)
\end{array}
\right)
\ee
\begin{lemma}\label{le2}
The pseudo-Riemannian manifold $V_{n+1}=(M_{n+1},G)$ with the metric defined by the conditions \re{13} admits an absolutely parallel convector field $\tilde \ph$ if and only if its components in the adapted coordinate system $(x^I)=(x^0,x^i)$ reduce to the form
\be{23}
\tilde \ph_ I=\exp(Kx^0)\left(\r(x^k),\ph_i(x^k)\right),
\ee
where $\r(x^k)$ and $\ph_i(x^k)$ satisfy the following equations on $\vn=(\mn,g)$:
\be{24}
\na_j\ph_i=\r\,g_{ij},
\ee
\be{25}
\na_j\r=K\,\ph_j.
\ee
\end{lemma}
{\it Proof.} Let $\tilde \ph_I$ be the components of an absolutely parallel covector field $\tilde \ph$ in the adapted coordinate system
$(x^I)=(x^0,x^i)$ on $V_{n+1}=(M_{n+1},G)$. So
\be{26}
\tilde \na_J\tilde \ph_I=0
\ee
If $I=0, J=0$ we get from \re{26} by virtue \re{21}
$$
\partial_0\tilde \ph_0-K\ \tilde \ph_0 =0.
$$
Thus
\be{27}
\tilde \ph_0=\exp(Kx^0)\,\r(x^k).
\ee
If $I=i, J=0$:
$$
\partial_0\tilde \ph_i-K\ \tilde \ph_i =0.
$$
Hence,
\be{28}
\tilde \ph_i=\exp(Kx^0)\,\tilde \ph_i(x^k).
\ee
If $I=0, J=j$:
$$
\partial_j\tilde \ph_0-K\ \tilde \ph_j =0.
$$
Due to \re{27}, \re{28} we have
$$
\na_j\r=K\,\ph_j
$$
If $I=i, J=j$:
$$
\partial_j\tilde \ph_i-g_{ij}\tilde \ph_0-\G^a_{ij}\tilde \ph_a=0.
$$
Thus,
$$
\na_j\ph_i=\r\,g_{ij}.
$$
Conversely, using direct calculations it easy to check that if the covector field $\tilde \ph$ has components
$
\tilde \ph_i=\exp(Kx^0)\ (\r(x^k),\ph_i(x^k))
$
in the adapted coordinate system $(x^I)=(x^0,x^i)$ on
$V_{n+1}=(M_{n+1},G)$ with metric \re{13}, where and satisfy the equations \re{24}, \re{25} on , then due to \re{21} is absolutely parallel. \smallskip
\noindent{\bf Remark 2} \ The equations \re{24}, \re{25} are the coordinate forms of the equations \re{9}, \re{11} defining a special concircular field. So the conditions \re{23} establish a one-to-one correspondence between absolutely parallel covector fields on the Shirokov space $V_{n+1}=(M_{n+1},G)$ and special concircular fields on the \vnk-space $K\neq0$.
In a similar way, it is possible to prove the following statement.
\begin{lemma}\label{le3}
The pseudo-Riemannian manifold $V_{n+1}=(M_{n+1},G)$ with the metric defined by the conditions \re{13} admits an absolutely parallel symmetric bilinear form $\tilde a$ if and only if its components in the adapted coordinate system $(x^I)=(x^0,x^i)$ reduce to the form
\be{29}
\tilde a_{IJ}=\exp(2\,K\,x^0)\ \left(
\begin{array}{cc} \mu(x^k) & \la_i(x^k) \\[2mm]
\la_j(x^k) & a_{ij}(x^k)
\end{array}
\right)
\ee
where $a_{ij}(x^k)$, $\la_i(x^k)$, and $\mu(x^k)$ satisfy the equations \re{4}, \re{7}, \re{8} on $\vn=(\mn,g)$.
\end{lemma}
{\bf Remark 3} \ The equations \re{4}, \re{7}, \re{8} define a \vnk-space. So the conditions \re{29} establish a one-to-one correspondence between absolutely parallel symmetric bilinear forms on the Shirokov space $V_{n+1}=(M_{n+1},G)$ and solutions of the system \re{4}, \re{7}, \re{8} defining geodesic mappings of the \vnk-space $(K\neq0)$.\smallskip
\noindent{\bf Remark 4} \ The set of absolutely parallel symmetric bilinear forms on $\vn=(\mn,g)$ is special Jordan algebra $J_0$ with the operation of multiplication
$\nad1A*\nad2A=\{\nad1A;\nad2A\}$, where $A$ is the linear operator $g$-conjugate with a bilinear form $a$, defined by
$g(AX,Y)=a(X,Y)$, and $\{\nad1A;\nad2A\}$ is a Jordan brackets
\be{30}
\{\nad1A;\nad2A\}=\frac12\ \left( \nad1A\,\nad2A +\nad2A\,\nad1A \right).
\ee
The condition \re{30} can be rewritten in the vector form as
\be{31}
2\,\{\nad1a;\nad2a\}(X,Y)= \nad1a\left( \nad2A X,Y \right)+\nad1a\left( \nad2A Y,X \right).
\ee
Or in the coordinate form
\be{32}
2\,\{\nad1a;\nad2a\}_{ij}= g^{ab}\ \nad1a\left(\nad1a_{ai}\nad2a_{bj} +\nad1a_{aj}\nad2a_{bi} \right).
\ee
This statement follows from the Lemma \ref{le2}.
\begin{theorem}\label{th1}
The set of solutions of the system \re{4}, \re{7}, \re{8} on a
\vnk-space $(K\neq0)$ forms a special Jordan algebra $J$ with the operation of multiplication
$\left\{
(\nad1a,\nad1\la,\nad1\mu);(\nad2a,\nad2\la,\nad2\mu)=
(\nad3a,\nad3\la,\nad3\mu)
\right\}$, where
\be{33}
2\,\nad3a(X,Y)= K\left(\nad1a ( \nad2A X,Y )+\nad1a( \nad2A Y,X )\right)-
\left( \nad1\la\otimes\nad2\la + \nad2\la\otimes\nad1\la \right) (X,Y),
\ee
\be{34}
2\,\nad3\la(X)= K\left(\nad1\la (\nad2A X)+\nad2\la( \nad1A X)\right)-
\left( \nad1\mu\nad2\la(X) + \nad2\mu\nad1\la(X) \right) ,
\ee
\be{35}
\nad3\mu= K\,g^{-1}\left(\nad1\la \,\nad2\la\right)-
\nad1\mu\,\nad2\mu.
\ee
The algebra $J$ is isomorphic to the special Jordan algebra $J_0$ of absolutely parallel symmetric bilinear forms on the Shirokov space $V_{n+1}=(M_{n+1},G)$ with the metric \re{13}.
\end{theorem}
Proof of the theorem immediately follows from the Lemma \ref{le2} and \re{22}, \re{29},~\re{32}.\\[2mm]
{\bf Remark 5} \ Due to \re{31} the unit of the algebra $J_0$ is $G$ so the unit of the algebra~$J$ is
$\displaystyle\left(\frac gK,0,-1\right)$.
\\
{\bf Remark 6} \ If there exists a convergent fields $\tilde \ph$ on
$V_{n+1}=(M_{n+1},G)$ such that $\|\tilde \ph\|>0$. Then there exists the adapted coordinate system $(x^I)=(x^0,x^i)$ in which the components $G_{IJ}$ of the metric $G$ reduce to the form
$$
G_{IJ}=\exp(2\,K\,x^0)\ \left(
\begin{array}{cc} 1 & 0 \\[2mm]
0 & \displaystyle\frac{-g_{ij}(x^k)}{K}
\end{array}
\right),
$$
where $g_{ij}(x^k)$ is the components of the metric of some
$\vn=(\mn,g)$. Using this metric and \re{31} we can define new operation of multiplication $\{\cdot,\cdot\}_2$. It is obvious that $\{\nad1A;\nad2A\}=-\{\nad1A;\nad2A\}_2$.
\begin{corollary}
Let $\vn=(\mn,g)$ be a \vnk-space $(K\neq0)$ then there exists the solution $(a,\la,\mu)$ of the system \re{4}, \re{7}, \re{8} satisfying the following conditions:
\be{36}
K\,a(AX,Y)-(\la\otimes\la)(X,Y)=\smash{\frac{e\,g(X,Y)}{K}},
\ee
\be{37}
K\,\la(AX)-\mu\,\la(X)=0,
\ee
\be{38}
K\,g^{-1}(\la,\la)-\mu^2=-e,
\ee
where $e$ takes values $\pm1,0$.
\end{corollary}
{\it Proof.} Let $\tilde b$ be an absolutely parallel symmetric bilinear form on the Shirokov space $V_{n+1}=(M_{n+1},G)$ with the metric \re{13}. Then as it has shown in \cite{10} there exists the absolutely parallel symmetric bilinear form $\tilde a$ on
$V_{n+1}=(M_{n+1},G)$ such that $\tilde A{}^2=e$ or in the equivalent form
\be{39
\tilde a(\tilde A\tilde X,\tilde Y)=e\,G(\tilde X,\tilde Y).
\ee
The equation \re{39} means that $\{\tilde a,\tilde a\}=e\,G$. Hence if $(a,\la,\mu)$ is the corresponding solution of the system \re{4}, \re{7}, \re{8} on the \vnk-space $(K\neq0)$ then taking into account \re{33}, \re{34}, \re{35} we get \re{36}, \re{37}, \re{38}. \smallskip
As mentioned above concircular fields generate a solution of the equation~\re{2}. Denote this set of solutions by $J_c$.
\begin{theorem}\label{th2}
$J_c$ is an ideal of $J$.
\end{theorem}
{\it Proof.} To prove that $J_c$ is an ideal of $J$ on $\vn=(\mn,g)$ is equivalent to prove that $J_{0c}$ is an ideal of $J_0$ on $V_{n+1}=(M_{n+1},G)$, where $J_{0c}$ is the set of absolutely parallel symmetric bilinear forms generated by absolutely parallel convector fields.
Let $\nad1\ph,\dots,\nad m\ph$ be a basis of the linear space
Conv($V_{n+1}$) of absolutely parallel convector fields on $V_{n+1}=(M_{n+1},G)$. Then any absolutely parallel symmetric bilinear forms generated by absolutely parallel convector fields has the components
$$
\tilde b_{IJ}=\sum_{\a,\b=1}^m \pod{\a\b}C (\nad\a\ph_I\nad\b\ph_J),
$$
where $\pod{\a\b}C\ (=\pod{\b\a}C)$ are some constants.
Let
$\tilde a_{IJ}$ be the components of arbitrary absolutely parallel symmetric bilinear form $\tilde a$. We should prove that
\hbox{$\{\tilde a,\tilde b\}\in J_{0c}$}. We have
\be{40
2\{\tilde a,\tilde b\}=\!G^{DT}\!\!\!\sum_{\a,\b=1}^m \!\!\pod{\a\b}C (\nad\a\ph_I\nad\b\ph_D\,\tilde a_{TJ}+\nad\a\ph_J\nad\b\ph_D\,\tilde a_{TI})=\!\!\!\!
\sum_{\a,\b=1}^m \!\!\pod{\a\b}C (\nad\a\ph_I\nad\b\Phi_J+\nad\a\ph_J\nad\b\Phi_I),
\ee
where $\nad\b\Phi_I=\nad\b\ph_D\, \tilde a_{TI} \,G^{DT}$ is an absolutely parallel convector field. Therefore,
\be{41
\nad\b\Phi_{I}=\sum_{\g=1}^m F^\b_\g\ \nad\g\ph_I
\ee
where $F^\b_\g$ are some constants. It follows from \re{40}, \re{41} that
$$
2\,\{\tilde a,\tilde b\}_{IJ}=\sum_{\a,\b,\g=1}^m
\left(F^\g_\b \pod{\a\g}C + F^\g_\a \pod{\b\g}C\right)\, \nad\a\ph_I\nad\b\ph_J.
$$
Thus, $\{\tilde a,\tilde b\}\in J_{0c}$.
\section{$V_n(0)$-spaces}
Let $(\mn.g)$ be a $V_n(0)$-space then there exists a solution of the system
\be{42
\na_k a_{ij}=\la_i g_{jk} + \la_j g_{ik},
\ee
\be{43
\na_k \la_i=\mu\, g_{ik},
\ee
where $\mu$ is a constant, and $\la_i=\na_i\Lambda$. Thus, a $V_n(0)$-space is a Shirokov space.
\begin{lemma}\label{le4}
If the $V_n(0)$-space does not admit any convergent fields of the basic type and $\ph$ is an absolutely parallel convector field on it. Then there exists the sequence of absolutely parallel covector fields $\left\{\nad\a\ph\right\}\ (\a\in \N)$ such that
\be{44
\hbox{a) \ \ }
\nad{\a+1}\ph(X)=\nad{\a}\ph(AX)-\nad\a f\,\la(X),
\hbox{\ \ \ \ b) \ \ }
\nad\a\ph(\la^*)=0, \ \ \forall\a\in \N,
\ee
where $\nad1\ph=\ph$, $d\nad\a f=\nad\a\ph$, $\la^*$ is the vector field $g$-conjugate with $\la$.
\end{lemma}
{\it Proof.} Taking into account that the $V_n(0)$ does not admit any convergent fields of the basic type we obtain from \re{43} that
\be{45
\na_k\la_i=0.
\ee
Let $\ph_i$ be the components of an absolutely parallel convector field $\ph$ on a $V_n(0)$. Denote $\nad1\ph=\ph$. Consider the covector field
\be{46
\nad2\ph_i=a^t_i\,\nad1\ph-\nad1f\la_i
\ee
where $a^t_i$ are components of the linear operator $A \ (a^j_i=g^{jl}a_{il})$. It follows from \re{46} due to \re{42}, \re{45}
\be{47
\na_k\nad2\ph_i = \nad1\ph_t\la^t g_{ik},
\ee
where $\la^t=g^{ti}\la_i$. According to our assumption it follows from \re{47} that
$$
\nad1\ph_t\la^t=0
\hbox{ \ \ and \ \ }
\na_k\nad2\ph_i=0.
$$
Applying now similar argumentation to the covector $\nad2\ph_i$ and continuing the process in this way, we obtain the desired sequence.\smallskip
\noindent{\bf Remark 7} \ The equation (\ref{44}b) due to (\ref{44}a) can be rewritten as
\be{48
\ph(\nad{\a-1}\la^*)=0, \ \ \forall\a\in \N,
\ee
where $\nad\a A$ is the $\a$-s power of linear operator $A$.
\begin{theorem}\label{th3}
Let a pseudo-Riemannian manifold \vn\ be a $V_n(0)$-space. Then there exists a convergent field of the basic type on \vn\ or there exists the sequence of linearly independent absolutely parallel convector fields $\{\nad\a\la\}$, $(\a=1,2\dots,p\leq n-1)$ such that
\be{49
a) \ \ \nad{\a+1}\la(X)=\nad\a\la(AX)-\nad\a\Lambda\,\a(X), \ \ \
b) \ \ \la(\nad{\a-1}A\,\la^*)=0,\ \ \forall\a\in A,
\ee
\be{50
\nad p\la(AX)=\nad p\Lambda\,\la(X),
\ee
where $\nad1\la=\la$, $\la^*$ is the vector field $g$-conjugate with $\la$.
\end{theorem}
\noindent{\it Proof.} \ 1) It follows from \re{43} that if $\mu\neq0$ then $\la$ is a convergent field of the basic type on $V_n(0)$.
2) Let $\mu=0$ then $\na\la=0$. According to the Lemma \ref{le4} and the Remark 7 we can construct the sequence of absolutely parallel convector fields $\{\nad\a\la\}\ (\a\in \N)$ such that
$$
a) \ \ \nad{\a+1}\la(X)=\nad\a\la(AX)-\nad\a\Lambda\ \la(X), \ \ \ \
b) \ \ \la(\nad{\a-1}A\la^*)=0, \ \ \forall\a\in\N.
$$
This sequence contains no more than $p\ (\leq n-1)$ linearly independent covectors. Otherwise, $V_n(0)$ will be locally flat and so it will admit a convergent field of the basic type. Thus,
$$
\nad{p+1}\la=\sum_{\a=1}^p C_\a\nad\a\la,
$$
where $C_\a$ are constants and $\nad1\la,\dots,\nad p\la$ are linearly independent. Changing $\nad\a\Lambda$ (defined to a constant) we can make $\nad{p+1}\la=0$. So we get \re{50}.
\begin{corollary}\label{co2}
If the $V_n(0)$-space does not admit any converging fields of the basic type and $\ph$ is an absolutely parallel convector field on it. Then
\be{51
\nad{\a-1}\la(\ph^*)=0, \ \ \forall\a\in\N
\ee
where $\ph^*$ is the vector field $g$-conjugate with $\ph$.
\end{corollary}
{\it Proof.} \ We get from \re{48}
$$
(\nad{\a-1}A\la^*)=\nad{\a-1}A\la(\ph^*)=\nad{\a-1}\la(\ph^*)=0.
$$
The following statement holds.
\begin{theorem}\label{th4}
Let pseudo-Riemannian manifold \vn\ admits a geodesic mapping onto pseudo-Riemannian manifold \vnn\ if there exists a concircular field of the basic type on \vnn\ then there exists a concircular field of the basic type on \vn.
\end{theorem}
{\it Proof.} \ Let $\bar\ph$ be a concircular field of the basic type on \vnn\ ($\bar\r\neq0$) then there exists a concircular field $\ph$ on \vn. Let us suppose the contrary that \vn\ does not admit concircular fields of the basic type. It means that $\r=0$. So $\ph$ is an absolutely parallel convector field and, therefore, \vn\ is a $V_n(0)$-space \cite{27}. So according to Theorem \ref{th3} there exists \vn\ on the sequence of linearly independent absolutely parallel convector fields
$\{\nad\a\la\}\ (\a=1,2,\dots,p\leq n-1)$
satisfying \re{49}, \re{50}. The equation \re{50} in the coordinate form can be written as
\be{52
a^t_i\nad p\la_t=\nad p\Lambda\la_i.
\ee
Contracting \re{52} with $\bar a{}^i_j$ (the inverse operator to $a^i_j$) by $i$ and taking into account that
$\la_i=-a^t_i\psi_t$
we get
\be{53
\nad p\la_j=-\nad p\Lambda\psi_j .
\ee
The condition \re{51} means that $\ph^t\nad p\la_t=0$. Hence, due to $\nad p\Lambda\neq0$ it follows from \re{53} that
$
\ph^t\psi_t=0.
$
From another hand since $\bar\r\neq0$ and $\r=0$ the equation
\re{10} gives us
$
\ph^t\psi_t\neq 0.
$
This contradiction proves the theorem.
\\[1mm]
{\bf Remark 8} \ The Theorem \ref{th4} shows that pseudo-Riemannian manifolds admitting a concircular field of the basic type (i.e. equidistant spaces of the basic type) form the class of manifolds closed with respect to the geodesic mappings. The same properties have spaces of constant curvature \cite{21,30}, Einstein spaces \cite{14,21}, and \vnk-spaces \cite{21}.
\begin{corollary}\label{co3}
Let an equidistant space of the exeptional type \vn\ admits a geodesic mapping onto a pseudo-Riemannian manifold \vnn\ then \vnn\ is a equidistant space of the exeptional type.
\end{corollary}
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\end{document}
\endinput
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{"url":"https:\/\/www.hpmuseum.org\/forum\/showthread.php?tid=3076&pid=28472&mode=threaded","text":"Nice GPS case for the WP34s\n02-23-2015, 09:09 PM (This post was last modified: 02-23-2015 10:27 PM by rprosperi.)\nPost: #15\n rprosperi Senior Member Posts: 3,675 Joined: Dec 2013\nRE: Nice GPS case for the WP34s\nI found this case which looks like it would work for the HP-20b\/WP-34s conversion. (and cheap $9.99). I will let you know how well the calculator fits when it arrives. (It should fit nicely, as other cases for the 35s have already been used for the WP-34s.) here I have one of these, I can check tonite; they're padded nicely and well-made. I got it for my Prime, as I don't like the hard slider Prime cover - my college-age son immediately pointed out I am certainly a grup (any trekkies out there?) for switching to a case like this. Update: As expected, a WP-34S fits just fine in this case. The case is obviously bigger than needed for a 34S (intended for 33S, 35S and Prime), however it is not so large that the 34S is at all loose in the case. It is snug enough that the 34S fits without moving vertically or horizontally, but the case would also accept the Quick Reference Guide or Pocket Reference described earlier. Don't let the$9.99 price scare you, it's a very nice case for all these machines.","date":"2019-12-11 07:49:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8295319676399231, \"perplexity\": 11106.468821527664}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575540530452.95\/warc\/CC-MAIN-20191211074417-20191211102417-00224.warc.gz\"}"}
| null | null |
Q: Reclassifying Landcover Raster in ENVI 5.3? I have produced a landcover raster using Landsat 8 OLI, Pansharpened image. The derived landcover classified image is placed below:
Now, I am intending to reclassify this image to change its class values (1,2,3...) in accordance with my ranking values. Secondly, resampling will also be required to change its pixel size from 15x15 to 30x30m resolution.
I have searched ENVI 5.3 tools but didn't find such options which are although available in ArcGIS. Can anyone suggest to me how I can edit and save class values and resample pixel size in ENVI 5.3?
A: To change the class values of (1,2,3...) based on specific criteria using ENVI, I think you can do it by editing ENVI header file, but first you need to have a copy of the land cover image first before adjusting the header file.
Go to Preprocessing -> Manage Raster Data -> Edit ENVI Header, as you can see in the link. You can then adjust the class name based on the criteria you want.
However, if I were you, I will not follow this process by editing the ENVI header as there is a risk of corrupting the file. If you do not have Spatial Analyst to reclassify the raster image to the specific criteria that you want, you can use open source software like SAGA under QGIS, it has a reclassify tool and you can do the same thing. You can refer to this answer on how to use the reclassify tool in SAGA.
Regarding the second question on how to resample your image using ENVI and change the image spatial resolution from 15m to 30m, go Preprocessing -> Manage Raster Data -> Resize Data. Then set the x,y factor to 0.5 to increase your pixel size by 2x. Choose a resampling scheme, either nearest neighbor or pixel aggregate.
A: One way of reclassifying a raster in ENVI is to rely on 'Band Math'.
Example:
(float(b1) EQ 1)*30+(float(b1) EQ 2)*50+(float(b1) EQ 3)*70 ...
The above will allow you to assign the new values, in place of the old. The EQ-function is short for equal, and will result in 1s where float(b1) is equal to the specified value, and 0s everywhere else. Making a sum of such statements will cover your entire image.
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Q: Why does my rails update method using the same strong params as my create method, give me an argument error? I am using Rails 5.2 and building an ecomm site.
My new.html.erb page has a simple_form with a combination of Product fields as well as other params which are handled outside the strong params and the creation of the new instance.
One of the features of the form is a eventListener which creates the :sku automatically based on four input values. The product number, the sku and the live status are hidden.
Here is the simplified version of the html.erb:
<%= simple_form_for @new_product do |f| %>
<%= f.error_notification %>
<%= f.input :name%>
<%= f.association :size, collection: @all_sizes.map{ |s| [s.name, s.id, {'data-code-num' => s.code_num}]}, input_html:{class: 'sku-suffix-component'} %>
<%= f.association :color, collection: @all_colors.map{ |s| [s.name, s.id, {'data-code-num' => s.code_num}]}, input_html:{class: 'sku-suffix-component'} %>
<%= f.association :pattern, collection: @all_patterns.map{ |s| [s.name, s.id, {'data-code-num' => s.code_num}]}, input_html:{class: 'sku-suffix-component'} %>
<%= f.input :product_number, as: :hidden, input_html:{ value: @new_product.product_number, class: "sku-suffix-component" } %>
<%= f.input :sku,as: :hidden, html:{id: "new-product-form-sku-input"} %>
<%= f.input :live_status, as: :hidden, :input_html => { :value => @new_product.live_status } %>
<%= f.input :description, as: :text %>
<%= f.association :brand%>
<%= f.association :style %>
<%= f.input :price %>
<%= f.input :quantity, input_html:{value: 1}%>
<%= f.association :segment %>
<%= f.association :main_category %>
<%= f.association :category %>
<%= f.association :country, class: '' %>
<!-- Here are some inputs for adding records to a material join table -->
<!-- And the names of the inputs are dynamically created -->
<% 5.times.with_index do |_, i| %>
<% num = i + 1 %>
<label class="form-control-label integer" for="material_percent_id_<%= num < 10 ? "0" + num.to_s : num.to_s %>">Percent</label>
<input class="form-control numeric integer required" type="number" step="1" name="material_percent_<%= num < 10 ? "0" + num.to_s : num.to_s %>" id="material_percent_id_<%= num < 10 ? "0" + num.to_s : num.to_s %>">
<label class="form-control-label select" for="material_id_id_<%= num < 10 ? "0" + num.to_s : num.to_s %>">Material Component #<%= num %> </label>
<select class="form-control select" name="material_id_<%= num < 10 ? "0" + num.to_s : num.to_s %>" id="material_id_id_<%= num < 10 ? "0" + num.to_s : num.to_s %>">
<option value=""></option>
<% @all_materials.each do |material| %>
<option value="<%= material.id %>"><%= material.name %></option>
<% end %>
</select>
<% end %>
<!-- Here are some inputs for adding multiple photos to the products using active_storage -->
<% (1..8).each do |i| %>
<%= f.label "Photo [#{i}]" %>
<%= f.file_field :photos, multiple: true %>
<% end %>
<%= f.button :submit %>
<% end %>
Creating new products using this simple_form works fine, along with instances in the join table, through the create method, shown here:
def create
@all_sizes = Size.all
@all_colors = Color.all
@all_patterns = Pattern.all
@new_product = Product.new(product_params)
@all_materials = Material.all
if @new_product.save
5.times.with_index do |_, i|
if params["material_id_0#{(i + 1)}"] != ""
ProductMaterial.create!(product_id: @new_product.id,
material_id: params["material_id_0#{(i + 1)}"].to_i,
percent: params["material_percent_0#{(i + 1)}"].to_i)
else
break
end
end
redirect_to @new_product
else
render :new
end
end
Using nearly the exact same form (some code added to dynamically render the join table inputs and the photos inputs correctly), but all inputs present; and using the exact same strong params in the controller; going through the update method in the controller produces an argument error. Here is the update method:
def update
@product = Product.find(params[:id])
@product = Product.update(product_params) # here is where the error happens
@all_sizes = Size.all
@all_colors = Color.all
@all_patterns = Pattern.all
@all_materials = Material.all
if @product.save
5.times.with_index do |_, i|
if params["material_id_0#{(i + 1)}"] != ""
ProductMaterial.create!(product_id: @product.id,
material_id: params["material_id_0#{(i + 1)}"].to_i,
percent: params["material_percent_0#{(i + 1)}"].to_i)
else
break
end
end
redirect_to @product
else
render :edit
end
end
Here is the exact syntax of the error as seen in the server:
ArgumentError (wrong number of arguments (given 1, expected 2)):
app/controllers/products_controller.rb:72:in `update'
A: Here
@product = Product.update(product_params)
you're trying to call instance method update on the Product class itself. new is a class method, so it works good in create action. How it should be:
def update
@product = Product.find(params[:id])
# here you define different @all instances
# you don't need to update and save separately, because instance is saved already
# if you call update on it and update goes well
if @product.update(product_params)
# here goes the rest controller code
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 88
|
@interface SLActiveOffer : SLOffer {
}
@property (nonatomic, retain) NSNumber* redistributable;
@property (nonatomic, retain) NSNumber* redistributeProcessing;
@property (nonatomic, retain) NSNumber* isProcessing;
+ (NSArray*)offersArrayfromDictionary:(NSDictionary*)data;
- (void)populateFromDictionary:(NSDictionary*)data;
@end
@interface SLActiveOfferTableItem : TTTableLinkedItem {
SLActiveOffer* _offer;
}
@property(nonatomic,retain) SLActiveOffer* offer;
+ (id)itemWithOffer:(SLActiveOffer*)offer URL:(NSString*)URL;
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,044
|
module CrunchApi
module Default
ENDPOINT = 'https://crunch.co.uk' unless defined? CrunchApi::Default::ENDPOINT
class << self
def options
Hash[CrunchApi.keys.map { |key| [key, send(key)] }]
end
def consumer_key
ENV['CRUNCH_CONSUMER_KEY']
end
def consumer_secret
ENV['CRUNCH_CONSUMER_SECRET']
end
def oauth_token
ENV['CRUNCH_OAUTH_TOKEN']
end
def oauth_token_secret
ENV['CRUNCH_OAUTH_TOKEN_SECRET']
end
def endpoint
ENDPOINT
end
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,379
|
{"url":"https:\/\/www.physicsforums.com\/threads\/quaternion-for-3d-fps-camera.346977\/","text":"# Quaternion for 3D FPS Camera\n\n1. Oct 19, 2009\n\n### sarah2529\n\nI'm trying to look how the formula made on this http:\/\/www.gamedev.net\/reference\/programming\/features\/quatcam\/page2.asp\" [Broken]. I want to know how this formula was derived. I tried to look on Quaternions and Rotation Sequences book but they use sin\/cos which is so different on this.\n\nN.w = a.w*b.w - a.x*b.x - a.y*b.y - a.z*b.z\nN.x = a.w*b.x + a.x*b.w + a.y*b.z - a.z*b.y\nN.y = a.w*b.y - a.x*b.z + a.y*b.w + a.z*b.x\nN.z = a.w*b.z + a.x*b.y - a.y*b.x + a.z*b.w\n\nor maybe I'm lost?\n\nLast edited by a moderator: May 4, 2017\n2. Oct 19, 2009\n\n### D H\n\nStaff Emeritus\nThe quaternions form a number system under which addition and multiplication are each closed and associative. Addition is commutative in the quaternions but multiplication is not. There are many ways to look at quaternions as a whole. One view is that a quaternion comprises four real numbers under which multiplication is defined as in the original post. Another view, Hamilton's view, is that a quaternion comprises a real part and three distinct imaginary parts. Yet another is that a quaternion comprise a scalar real part and a vectorial (three vector) imaginary part: $Q=(q_s, \\mathbf q_v)$.\n\nDenoting two quaternions as $Q_1=(q_{1s}, \\mathbf q_{1v}), Q_2 = (q_{2s}, \\mathbf q_{2v})$ then the quaternion product $Q_1Q_2$ can be written using the vector dot and cross products as\n\n$$Q_1Q_2 = (q_{1s}q_{2s} - \\mathbf q_{1v}\\cdot\\mathbf q_{2v},\\;\\; q_{1s}\\mathbf q_{2v} + q_{2s}\\mathbf q_{1v} + \\mathbf q_{1v}\\times\\mathbf q_{2v})$$\n\nThe book Quaternions and Rotation Sequences primarily focuses on unit quaternions. While the unit quaternions are useful (very useful) for representing rotations and transformation, there is a lot more to the quaternions than just the unit quaternions.\n\n3. Oct 20, 2009\n\n### sarah2529\n\nRead it many times but I still didn't understand some of the things you've said. I just don't get it how it is made.\n\nAnyway, do you know what's the pre-requisite for quaternion topics? Is it Linear Algebra? Because I don't understand most of the topics in the book Quaternions and Rotation Sequences. Maybe I jump ahead so fast.\n\n4. Oct 20, 2009\n\n### D H\n\nStaff Emeritus\n\nSome more detailed questions:\nDo you understand complex numbers?\nDo you understand Euler's identity $\\exp(i\\theta) = \\cos \\theta + i\\sin\\theta$?\nHow much do you know about real and complex analysis?\nHow much do you know about linear algebra?\n\n5. Oct 20, 2009\n\n### sarah2529\n\nI guess so.\n\nNope. What's that?\n\nYes, I'm taking Calculus 2 class right now.\n\nSo little. But I know how to use row reduced in matrices. Thanks to the opencourseware.\n\nBasing on what you've asked, I really did jump so fast. So what would you recommend to me? I really want to understand those quaternions for my 3D First Person Camera Engine.\n\n6. Oct 20, 2009\n\n### D H\n\nStaff Emeritus\nThat response means you do not really understand complex numbers.\n\nWhoa, Sarah!\n\nHere's a sequence in an undergrad math curriculum: Calculus I & II, Ordinary differential equations, Multivariable calculus, Introduction to analysis, Linear algebra, Real analysis, Complex analysis, Calculus of variations, Differential geometry, Abstract algebra (I, II, & III !!), General topology, Geometric topology, Algebraic topology, ...\n\nYou'll run across the quaternions in the second or third abstract algebra class. You'll find out why they are useful for representing rotations when you take algebraic topology. Trying to understand quaternions when you have not yet finished Calc 2 is jumping the gun. Without the requisite intermediate knowledge all you will have will be a bunch of formulae that just don't make much sense.\n\nAs a test, see how much sense you can make of this Wikipedia article: http:\/\/en.wikipedia.org\/wiki\/Charts_on_SO(3)\n\nIf you insist on jumping the gun, I suggest checking out a text on abstract algebra or modern algebra. Mine gets to the quaternions on page 503.\n\nLast edited: Oct 20, 2009\n7. Oct 20, 2009\n\n### sarah2529\n\nOMG. I'm jumping so far. Do I really need to take \"Introduction to analysis->......->Differential geometry\" for me to understand Abstract algebra very much? Well I hope I can do that by myself because my curriculum only has until Multivariable calculus (I'm studying Multivariable calculus some time. Vectors... Matrices.. 18.02 in MIT opencourseware helps me so much.).\n\nCurriculum here sucks so bad. I'm doom! I'm 18 and I barely know Multivariate calculus. Am I a failure? lol\n\nHonestly, I really thought Calculus 2 is enough to learn quaternions. I guess I need to finish the lecture notes I found for multivariate so I can move to the next math.\n\nBtw, thanks for the help. Now I realized that my knowledge in math is not enough for my target thesis project.\n\nLast edited: Oct 20, 2009\n8. Oct 20, 2009","date":"2018-01-16 16:20:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.6494426727294922, \"perplexity\": 1107.3252766656692}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084886437.0\/warc\/CC-MAIN-20180116144951-20180116164951-00543.warc.gz\"}"}
| null | null |
\section{Introduction}
\vspace{10pt}
The globular cluster $\omega$ Centauri (NGC~5139) is regarded to be
the largest and most massive member of the Galactic cluster system
with a tidal radius of 69 parsecs \citep{har96}, an estimated mass of
$5.1\times10^6 M_\odot$, and a measured central velocity dispersion of
$22 \pm 4$ $\rm {km}~\rm s^{-1}$\ \citep {mey95}. The cluster presents a large scale
global rotation, measured with radial velocities, of 8 $\rm {km}~\rm s^{-1}$\ at a
radius of 11pc from the center \citep{mer97} and confirmed with proper
motions \citep{van00}, which makes it one of the most flattened
galactic globular clusters \citep{whi87}. A rotating flattened model
including proper motion and radial velocity datasets by \citet{ven06}
calculate a lower total mass of $2.5\times10^6 M_\odot$ and confirm
the central line of sight velocity dispersion value of 20 $\rm {km}~\rm s^{-1}$. They
measure a dynamical distance of $4.8\pm0.3$ kpc (which we adopt for
this paper). $\omega$ Cen has a peculiar highly bound retrograde orbit
\citep{din01}. It also has a stellar population that makes it stand
out from the rest of the Galactic globular clusters due to its
complexity. It shows a broad metallicity distribution
\citep{bed04,nor96}, as well as a kinematical and spatial separation
between the different subpopulations \citep{pan03, nor97}.
All the above results have led to the hypothesis that $\omega$ Cen is
not a classical globular cluster, but instead is the nucleus of an
accreted galaxy \citep{fre03,bek03,mez05}. The scenarios of it being
the product a merger of two globular clusters \citep{ick88} and of
self-enrichment \citep{iku00} have also been proposed to explain the
stellar populations.
The high measured velocity dispersion together with the possibility of
being a stripped galaxy make $\omega$ Cen an interesting candidate for
harboring a black hole in its center. The extrapolation of
$M_\bullet-\sigma$ relation for galaxies \citep{geb00a,fer00,tre02}
predicts a $1.3\times10^4 M_{\odot}$ black hole for this cluster. The
sphere of influence of such black hole for a star cluster at the
distance of $\omega$ Cen with a velocity dispersion of 20 $\rm {km}~\rm s^{-1}$\ is
$\sim$ 5\arcsec, making it an excellent target for ground-based
observations.
Two globular clusters have been suggested for harboring an
intermediate mass black hole in their nucleus. One is the galactic
cluster M15 \citep{geb00,ger02,ger03} and the other is G1, a giant
globular cluster around M31 \citep{geb02,geb05}. M15 is assumed to be
in a post-core collapse state, therefore its dynamical state has been
debated between harboring a black hole or containing a large number of
compact remnants in its center \citep{bau03a,bau03b}. Unfortunately,
observational constraints between these two hypothesis remain
inconclusive \citep{bos06}. G1 on the other hand, has a core with
characteristics closer to those of $\omega$ Cen, and observations
support the black hole interpretation for G1. \citet{bau03c} propose
an alternative interpretation for G1 in which they match the
observations with a model of two colliding globular clusters. The G1
black hole models are preferred since the M/L profile is expected to
be flat in its core, so any rise in the velocity dispersion is
unlikely to be due to remnants that concentrated there from mass
segregation. The black hole interpretation for G1 is strongly
supported by radio \citep{ulv07} and x-ray \citep {poo06} detections
centered on the nucleus. $\omega$ Cen and G1 have similar properties
in both their photometric and kinematic profiles. In this paper we
report photometric and kinematical measurements that suggest the
presence of a central black hole in $\omega$ Cen.
\begin{figure}
\plotone{f1.eps}
\caption{Surface brightness profiles for $\omega$ Cen. The circles
show our measured photometric points from the ACS (filled) and
H-alpha (open) images. The triangles show photometric points
obtained from ground based images by Trager et al. The dashed line
is Trager's Chebychev fit. The solid line is our smooth fit to the
combination of the ACS points inside 40\arcsec\ and Trager's
Chebychev fit outside.}
\label{sb}
\end{figure}
\section{Surface Brightness Profile}
\vspace{10pt}
The surface density profile at large radii for $\omega$ Cen has been
measured from a combination of star counts and aperture photometry
from ground based images \citep{mey87,tra95,van00}. We measure the
central part of the profile taking advantage of
$\it{Hubble~Space~Telescope}$ ($\it{HST}$) spatial resolution. We
measure integrated light from an ACS F435W image (340 sec) applying
the technique described in detail in \citet{noy06}, which uses a
robust statistical estimator, the bi-weight, to calculate number of
counts per pixel on a given annulus around the center of the
cluster. As a test, we also measure the profile from a narrow band
H-alpha image from the Rutgers Fabry-Perot \citep{xie06} with lower
spatial resolution. Since both images have a limited radial coverage,
we use the Chebychev fit of \citet{tra95} for the surface brightness
profile to cover the full, radial extent of the cluster. All profiles
are normalized to the Trager profile.
Having accurate coordinates for the center of the cluster is crucial
when measuring density profiles. Using the wrong center typically
produces a shallower inner profile. We use a technique where we take
an initial guess center, divide the cluster in eight concentric
sectors around this center, and calculate the standard deviation of
the sum of stars for the eight sectors. The radius of the sectors is
chosen to be as large as the image will allow, in this case it is
$\sim$2\arcmin. We repeat the procedure for a grid of center
coordinates and use the one that has the minimum standard
deviation. Details about the technique can be found in
\citet{noy06}. The coordinates for our center are RA $13:26:46.043$ and
DEC $-47:28:44.8$ on the ACS dataset J6LP05WEQ using its WCS zeropoint.
The measured profiles from the B-band and H-alpha images are
consistent as can be seen on Figure \ref{sb}. The H-alpha profile
follows the turnover around the core radius very well up to
100\arcsec~and it also shows the rise toward the center, but it is
noisier than the ACS profile. The solid line is a smooth fit made to
the combination of the photometric points from ACS inside
40\arcsec~and Trager et al. Chebychev fit outside 40\arcsec. For
comparison, we include the \citet{tra95} photometric points in the
plot. The surface brightness profile shows a continuous rise toward
the center with a logarithmic slope of $-0.08\pm0.03$, which is in
contrast to the common notion that $\omega$ Cen has a flat
core. \nocite{van00} Van Leeuwen et al (2000) perform star counts for
giant stars and notice that they are more concentrated than previously
thought. Our result is consistent with their finding.~\citet{bau05}
perform N-body models of star clusters with initial King profiles and
containing a central black hole. They predict the formation of a
shallow cusp of $-0.1$ to $-0.3$ logarithmic slope after $1.5-4$
relaxation times.~\citet{tre07} perform similar N-body models and
conclude that clusters containing central IMBHs evolve to
configurations that have extended cores, with $r_c/r_h\sim0.3$. The
$r_c/r_h$ ratio for $\omega$ Centauri is 0.33, which is in very good
agreement with the predictions from this models. Our observed surface
brightness profile is intriguing considering that it follows the two
specific predictions from these N-body models, but of course, only
kinematical measurements can determine the mass profile, and test the
existence of a central black hole.
\section{Kinematic Measurements}
\vspace{10pt}
Obtaining kinematic information of the central regions of Galactic
globular clusters is a challenging task since the brightest stars
dominate with typical ground-based conditions and the extreme crowding
produces confusion. Measuring individual radial velocities requires a
spatial resolution that can only be achieved with adaptive optics or
from space. On the other hand, measuring velocity dispersion from an
integrated spectrum is subject to shot noise due to giant stars whose
contribution dominates the light. \citet{dub97} calculate the relative
contribution to integrated light by different stellar groups. They
find that the contribution from the few brightest stars is equal in
weight to that of the much larger numbers of fainter stars. Therefore,
the only way to obtain accurate radial velocity dispersion
measurements from an integrated spectrum is if the participation of
the brightest stars can somehow be avoided or at least minimized. One
way to do this is by observing crowded regions with an integral field
unit (IFU) which produces individual spectra of subsections in the
region (typically $\sim$0.2\arcsec\ in size). One can exclude the
spectra affected by the brightest stars when measuring the integrated
background light and thus decrease the shot-noise contribution to the
uncertainty. The Gemini telescopes operate primarily using a queue
scheduling, which makes them an excellent tool to measure integral
field spectra of globular clusters since observing constraints (such
as excellent seeing) can be specified in advance, and data is only
taken when the required observing constraints are met.
\begin{figure}
\plotone{f2.ps}
\caption{The central field observed with the GMOS-IFU. The green box
represents the GMOS field of view. Top left: ACS image of the
observed region. The red circle marks the center of the cluster. Top
right: Convolved ACS image to reproduce the reported seeing during
observations. Bottom left: GMOS acquisition image. Bottom right:
Reconstructed GMOS-IFU image.}
\label{frame59}
\end{figure}
\begin{figure}
\plotone{f3.ps}
\caption{Same as fig 2 for the field 14\arcsec\ away.}
\label{frame52}
\end{figure}
As part of the Science Verification program for the Gemini GMOS-South
IFU, we obtained nod-and-shuffle observations on February 29 2004
(program ID: GS-2003B-SV-208). We use the IFU in 2-slit
nod-and-shuffle mode, which gives a field of view of
$5\arcsec\times5\arcsec$, comprised of 700 individual lenslets plus
fiber elements, each of which covers approximately 0\arcsec.2 on the
sky. We use the R600 grating, yielding a resolving power R=5560, which
we measure from the lamp spectral lines, along with the Calcium
Triplet filter to give a wavelength coverage of 7900-9300\AA. Three
fields are observed, each for a total integration time of 900 sec on
source and 900 sec on sky. The observations are made using the
nod-and-shuffle technique with 30 sec sub-integrations observed in a
B-A-A-B pattern, where A is the on-source position and B is the sky
position, located 498\arcsec\ away. The nod-and-shuffle technique
improves the sky subtraction, especially in the presence of CCD
fringing, by sampling the object and sky on exactly the same CCD
pixels, with exactly the same light path, on timescales comparable to
those of the sky emission line variability. The first of the three
fields is located at the cluster center, and the second field is
centered 14\arcsec\ away. The third field appears to have been
pointing somewhere else but, despite much efforts, we cannot determine
exactly where the IFU observations are pointed (they do not match
anything in the acquisition image for this field). The reconstructed
IFU image for this third field contains fewer stars and the PSF is
obviously broader than for the other two. It is clear that the
exposure was taken during much worse seeing conditions than the other
two fields, so shot-noise effects are likely to be important; for this
reasons we exclude the third field from further analysis. Using the
standard tasks from the IRAF-GEMINI package we sky subtract,
flat-field, and extract the spectra for each fiber and apply a
wavelength calibration.
\begin{figure}
\plotone{f4.eps}
\caption{Velocity dispersion profile for $\omega$ Cen with various
central black hole models. Filled squares are the dispersions and
uncertainties from the GMOS-IFU and open circles are from individual
radial velocity measurements. A set of isotropic spherical models of
varying black hole masses is shown for comparison. The thick line is
the no black hole model and the thin lines represent models with
black holes as labeled. The dashed (red) line is the velocity
dispersion profile for the best-fit orbit-based model (in Section
4).}
\label{models}
\end{figure}
The standard flat subtraction does not remove all of the fringing
pattern in the image. As a result, a constant number of counts have to
be subtracted from the data frames before flattening in order to
reduce fringing problems. Relative to the bias frame, the amount of
additional counts from scattered light is about 8\%. Even after this
procedure, there is some residual fringing that can only be alleviated
by combining individual fibers over the full field into one
spectrum. To combine individual fibers we first divide by the
continuum. The intention is to de-weight the bright stars with respect
to the fainter ones, which helps to lessen the problems due to shot
noise. For the continuum fit, we run a boxcar of dimension
$111\times1$ over the reduced image, and then divide the central pixel
by the median of the pixels in the box. This procedure brings all
spectra to the same continuum level. We then combine every individual
fiber with the six adjacent ones, since this represents one seeing
disk for the observations
Figure \ref{frame59} shows the reconstructed image from the IFU fibers
for the central frame and the acquisition image as well as the same
region on the ACS image. We also show a convolved image (with the
reported seeing for the observations) of the ACS frame. The same match
is performed for the field 14\arcsec~away (Fig \ref{frame52}). Both
ACS fields contain $\sim100$ resolved stars. We construct a luminosity
function for the detected stars for each field and compare it to the
luminosity function of the entire cluster core. The luminosity
function is consistent between the two fields. The brightest stars
detected in both fields are two magnitudes fainter than the brightest
stars detected in the core of the cluster. This excludes the
possibility of the integrated spectra being artificially broadened by
the presence of more blue straggler stars in the central field
compared to the field 14\arcsec\ away. Using the photometric
measurements of individual stars together with the reported seeing, we
calculate how many stars contribute to each fiber. Excluding the
fibers which are dominated by a single star we estimate that the
integrated spectrum of the background unresolved light represent about
60 stars in both fields.
We focus on the Ca triplet region (8450\AA -8700\AA) for our
analysis. We measure the relative velocities between each fiber for
the two fields and obtain velocity distributions from the individual
fiber velocities. We fit a Gaussian to the velocity distributions and
observe that the one for the central field is clearly broader than for
the one 14\arcsec\ away. The largest relative velocity between two
fibers is 80 $\rm {km}~\rm s^{-1}$\ for the central frame, and 60 $\rm {km}~\rm s^{-1}$\ for the other
one. Using the dispersion of the individual fiber velocities as a
measure of the cluster velocity dispersion will be biased. Since
multiple stars, in general, provide light to an single fiber, the
measured velocity in that fiber will be pulled toward the cluster mean
as opposed to representing one star. Thus, the dispersions of the
fiber velocities will be biased significantly low. This is what we
find although the central frame does have a obviously larger spread in
fiber velocities.
To properly estimate the velocity dispersion we have to rely on the
integrated light, and require template stars in this
case. Unfortunately, we do not have isolated stars that are free from
the fringe problems mentioned earlier, so we cannot accurately use
template stars observed with the same instrument. We rely on the
template stars observed by \citet{wal05}, from VLT-UVES observations
at around R=35000. We convolve the spectra to our measured resolving
power. To extract the velocity dispersion from the integrated light we
utilize the non-parametric, pixel-based technique as described in
\citep{geb00b,pin03}. We choose an initial velocity profile in bins,
convolve it with the template (or set of templates), and calculate
residuals to the integrated spectrum. The parameters for the line of
sight velocity distribution (either velocity bin values or, if
desired, a parametric Gauss-Hermite expansion) are varied to provide
the best match. Monte Carlo simulations determine the uncertainties,
and use the measured noise in the spectrum.
The dispersion fitting routine allows for a mismatch in the equivalent
width between the object and template. In this case, there is a 30\%
difference in the equivalent width of the calcium triplet lines. We
do not know whether this is caused by the scattered light (unlikely
given the amplitude), the specific templates we used, omega Cen's
particular composition, or a combination of all three. We have run a
variety of tests to determine whether stars of difference equivalent
widths would cause a bias in dispersions, and find no such bias. We
have also measured the dispersions using template stars from the same
IFU data, since there is at least one star that is fairly
isolated. The uncertainties are larger due to the scattered light
problems, but the value of the dispersion remains the same. Thus, we
conclude that template issues are not a significant source of bias in
the dispersion estimate.
\begin{figure}
\plotone{f5.eps}
\caption{$\chi^2$ vs. black hole mass. The minimum is found for a
black hole mass of $4.0\times10^4 M_\odot$, with 68\% confidence
limit at $3$ and $4.75\times10^4 M_\odot$ marked by the dashed
line. For our model assumptions, the no black hole model is excluded
at greater than the 99\% confidence}
\label{chibh}
\end{figure}
We combine the spectra from individual fibers using a biweight
estimator. Different sets of fibers for each frame are combined in
order to test for consistency in our results. First, we combine every
fiber on the frame, then we exclude the 25\%, 50\% and 75\% brightest
fibers. We measure the velocity dispersion for these four spectra for
each frame. The measured velocity for the central frame is always
higher than the one for the frame 14\arcsec\ away for every equivalent
pair of combined spectra. We measure velocity dispersions from 21.8 to
25.2 $\rm {km}~\rm s^{-1}$\ for the central field, and 18.2 to 19.1 $\rm {km}~\rm s^{-1}$\ for the field
14\arcsec\ away. We adopt $23.0\pm2.0$$\rm {km}~\rm s^{-1}$\ for the central field and
$18.6\pm1.6$$\rm {km}~\rm s^{-1}$\ for the other. The latter measurement coincides with
the central velocity dispersion value measured for $\omega$ Cen by
various authors. Van den Ven et al. (2006) measure a line of sight
velocity dispersion profile by combining various datasets
\citep{sun96,may97,rei06,xie06}. They use 2163 individual radial
velocity measurements divided into polar apertures to obtain the final
velocity dispersion profile. We use dispersion estimates as presented
in their table 4. The average radius for the spectra that contribute
to the central values is 2.5\arcsec\ and 14\arcsec\ for the
second. Figure \ref{models} presents the velocity dispersion data.
\section{Models}
\vspace{10pt}
As discussed in Section 2, the central shape of the surface brightness
profile of $\omega$ Cen resembles that found by \citet{bau05} in star
clusters harboring black holes. The presence of an intermediate black
hole at the center of this cluster is one of the possibilities for
explaining the observed rise in velocity dispersion. We have run two
types of modeling: 1) spherical and isotropic, and 2) flattened,
orbit-based models that allow for general anisotropy. Although the
orbit-based models are more general, they do not consider dynamical
stability. For a system with a half light relaxation time shorter than
its age, like $\omega$ Cen, the dynamical evolution cannot be ignored.
Therefore, while the two models give similar results we quote results
from the isotropic analysis and use the orbit-based models to explore
possible velocity anisotropies.
\begin{figure}
\plotone{f6.eps}
\caption{Inferred density profiles. The solid line is the
deprojected density profile for the luminous component. The dashed
line represents the required dark component to reproduce the
observed kinematics.}
\label{dens}
\end{figure}
For the isotropic analysis, we create a series of models using the
non-parametric method described in \citet{geb95b}. As a first step we
apply a reddening correction to the observed surface brightness
profile. \citet{har96} reports a 0.1 reddening for this cluster,
which, although being relatively low, it is important for the proper
$M/L$ determination of the models. The reddening correction will only
affect the $M/L$ value of the models, but not the shape of the
profiles. We deproject the surface brightness profile using Abel
integrals assuming spheroidal symmetry. The integral involves a
derivative of the profile, therefore, any amount of noise present is
amplified. We apply a spline smoother to the surface brightness
profile before deprojecting and thus obtain a luminosity density
profile as discussed in \citet{geb96}. By assuming an M/L ratio, we
calculate a mass density profile, from which the potential and the
velocity dispersion can be derived. We repeat the calculation adding a
variety of central point masses ranging from 0 to
$7.5\times10^4M_\odot$ while keeping the global $M/L$ value fixed. Van
de Ven et al (2006) measure a fairly constant stellar $M/L$ profile
for $\omega$ Cen of $2.5\pm0.1$. We find a constant mass luminosity
ratio of 2.7, since this provides the best match to the observed
velocity dispersion profile outside the core.
Figure \ref{models} shows the comparison between the different models
and the measured dispersion profile. The most relevant part of the
comparison is the rise inside the core radius, in particular the rise
between the two innermost measurements. As it can be seen, an
isotropic model with no black hole present predicts a slight decline
in velocity dispersion toward the center, instead we observe a clear
rise. The predicted central velocity for the no black hole model is
14.6 $\rm {km}~\rm s^{-1}$\ which is well below any line of sight velocity dispersion
measured inside 1\arcmin. The calculated $\chi^2$ values for each
model are plotted in Figure \ref{chibh}, as well as a line showing a
$\Delta\chi^2=1$. The $\chi^2$ curve implies a best-fitted black hole
mass of $4^{+0.75}_{-1}\times10^4M_\odot$. Even the original velocity
dispersion profile without our two innermost measurement already
points to an intriguing discrepancy, but the central measurements
confirm an important rise in $M/L$ from the core radius to the center
of this cluster. The central $M/L$ value is 6.7, which is a
considerable rise from the value of 2.7 just inside the core
radius. Our best fit model implies a central density of
$5.6\times10^7M_\odot/pc^3$ the largest measured in a globular
cluster.
\begin{figure}
\plotone{f7.ps}
\caption{Contours of $\chi^2$ as a function of black hole mass and
mass-to-light ratio. Each point represents a particular model. The
contours represent the 68, 90, 95, and 99\% confidence for one
degree-of-freedom, implying $\Delta\chi^2=1.0, 2.7, 4.0,$ and 6.6. The
circled point is the model that has the minimum value.}
\label{2dchi}
\end{figure}
We also construct axisymmetric orbit-based dynamical models. The
models are based on the formulation by \citet{sch79} and are
constructed as in \citet{geb00b,geb03}. These models provide the most
freedom possible of the distribution function for an axisymmetric
system. We use the same deprojection as describe above, except we also
include the observed flattening. Assuming an $M/L$ ratio and a BH
mass, the mass distribution of the cluster is obtained and from it,
the potential can be computed. Using this potential, we generate about
$10^4$ representative orbits. The best match to the observed
photometric and kinematical data provide the orbital weights for a
given potential. The process is repeated for various $M/L$ values and
BH masses until the minimum $\chi^2$ model is found. The kinematical
observations matched here are the individual radial velocities from
van den Ven et al. (2006), and the two integrated measurements
described above. We plan to perform more detailed orbit-based models
including proper motion measurements in the future.
Figure \ref{2dchi} plots $\chi^2$ as a function of black hole mass and
stellar M/L. The best fit model requires a black hole mass of
$3\pm1\times10^4~M_\odot$, with 1-sigma of the isotropic result. As
expected, since orbit-based models are more general, the difference in
$\chi^2$ compared to the no black hole case is smaller than in the
isotropic case. The $\Delta\chi^2$ between the best-fitted black hole
model and no black hole model is 4 (marginalized over M/L), implying a
95\% significance. In comparison, the $\chi^2$ difference for the
isotropic models is 25. As discussed below, the reason for the
difference is that the orbit-based models allow for significant radial
anisotropy in the core for the no black hole model, while the best-fitted
mass produces a nearly isotropic distribution consistent with previous
measurements \citep{ven06}.
We also plot the velocity dispersion profile as measured in the
best-fit orbit based model in Figure \ref{models}, given as the dashed
(red) line. The dispersion profile for the orbit-based model with no
black hole is very similar, with no obvious correlated
differences. The reason there are no obvious differences is because
the overall change in $\chi^2$ is small and the orbit-based models
tend to redistribute orbital weights to spread $\chi^2$ over the full
radial range. As discussed in Gebhardt et al. (2003), it is difficult
to see differences in radial profiles of projected kinematics between
models with black holes and without, even for galaxies where the
overall $\chi^2$ difference is large. This is understood since the
orbit models have the freedom to change the orbital properties in such
a way to even out the $\chi^2$ differences over the full spatial
extent of the observations. For omega Cen, the black hole model only
provides an increase in $\chi^2$ of 4; thus with 23 data points, the
average difference in terms of the measurement uncertainty is 0.42. For
these reasons, the orbit-based models provide a modest significance
for a black hole, and the argument is significantly strengthened when
considering the need for the strong amount of radial anisotropy when a
black hole is not included.
The observations we use in the dynamical modeling rely on only the
first and second moments of the velocity distribution. Since we have
individual velocities (except for the central two radial points), we
can utilize the full velocity profile. Furthermore, proper motion data
exists for data at larger radii (van den Ven et al. 2006). Including
both effects, full velocity profiles and proper motions, will be the
subject of a more detailed paper on omega Cen.
\subsection{Alternative to a Black Hole: Dark Remnants}
A possible alternative to explain the observed rise in $M/L$ toward
the center is a concentration of dark stellar remnants, e.g. neutron
stars, stellar mass black holes, or massive white dwarfs. Using the
observed velocity dispersion profile, we calculate the total enclosed
mass and from this, the mass density profile. We then compare this
with the enclosed mass implied by the luminosity density profile,
assuming the same M/L as for the models in the previous section. From
these two profiles, we can estimate the density profile of the implied
extended dark component, if we assume this was the cause of the
velocity dispersion rise towards the center. Figure \ref{dens} shows
the estimated density profile for the dark and luminous components. It
is clear that, if the velocity rise is due to an extended distribution
of dark stellar remnants, the density profile of this dark component
needs to be very concentrated and steep, with a logarithmic slope of
$\sim-2.0$ (the slope of the dashed line in the figure), resembling a
cluster undergoing core collapse. The relaxation time for $\omega$ Cen
implies a much slower dynamical evolution than the one necessary to
reach such a configuration. Core-collapse models have shown that when
a cluster has reached such a high degree of mass segregation, the
observable core to half light radius gets very small, with values
below 0.05 \citep{bre94,mak96}, while this ratio is 0.3 for
$\omega$~Cen. Also, the concentration value for $\omega$~Cen is
$c=1.6$ \citep{har96}, which is too low a value for the cluster to
have undergone core-collapse. No evolutionary model predicts such a
concentrated distribution of dark remnants inside a cluster with a
shallow extended core for the visible stars. Furthermore, the
required number of dark remnants is around 1\% of the cluster mass.
While this number is expected from stellar evolution, it is not
expected to have all of the remnants to be concentrated inside of the
core.
\begin{figure}
\plotone{f8.eps}
\caption{Radial over tangential anisotropy vs. radius from
orbit-based models. The solid line is for the best fit model
containing a black hole. The dashed line is for a model with no
black hole present. The vertical line marks the location of the
core radius.}
\label{anis}
\end{figure}
\subsection{Alternative to a Black Hole: Orbital Anisotropy}
Another possibility is that the observed rise in velocity dispersion
is due to velocity anisotropy in the cluster. The orbit-based models
explore this possibility. The two main results from the orbit analysis
are that 1) the lowest $\chi^2$ model is the one with a central black
hole and 2) the model with no black hole requires a substantial amount
of radial anisotropy. Figure \ref{anis} shows that without the
presence of a central black hole, a large degree of radial anisotropy
-- $\sigma_r/\sigma_t$=1.5 -- is required inside $0.3~r_c$. At
$r>28$\arcsec, the models with and without a black hole are close to
isotropic, in agreement with the results of van den Ven et
al. (2006). For a system as dense as $\omega$~Cen, such a degree of
anisotropy as measured in the no black hole case is expected to be
quickly erased through relaxation processes. However, even with such a
strong amount of radial anisotropy, the no black hole case is a poorer
fit than the best-fitted black hole. One of the disadvantages of the
orbit-based models is that we cannot include the dynamical stability
arguments in the $\chi^2$ analysis, and for a system with a short
relaxation time such as a globular cluster, this may be important. We,
therefore, adopt results from the isotropic models, in particular
since the analysis of van den Ven et al. (2006) find an isotropic
distribution.
\section{Discussion}
\vspace{10pt}
We measure the surface brightness profile for the globular cluster
$\omega$ Centauri (NGC~5139) from an ACS image in the central
40\arcsec. The profile shows a continuous rise toward the center with
a logarithmic slope of $-0.08\pm0.03$, in contrast with previous
measurements which found a flat core. The shape of the profile is
similar to that obtained from numerical models of star clusters
containing black holes in their centers. We measure a line of sight
velocity dispersion for two 5\arcsec$\times$5\arcsec~regions, one at
the center of the cluster and the other 14\arcsec~away. We detect a
rise in velocity dispersion from 18.6 $\rm {km}~\rm s^{-1}$\ for the outer field to 23
$\rm {km}~\rm s^{-1}$\ for the central one. We combine these two measurements with
previously measured velocity dispersion at larger radii.
When we compare the observed velocity dispersion profile with a series
of isotropic models containing black holes of various masses, we find
that a black hole of $4.0 ^{+0.75}_{-1.0}\times~10^4 M_\odot$ is
necessary to match the observations. We explore alternative
explanations for the observed rise in our central velocity dispersion
measurements. First we consider the possibility that the observed
$M/L$ rise is due to the presence of an extended component composed of
dark remnants such as neutron stars or faint white dwarfs. The density
profile of the dark component is required to be extremely concentrated
toward the center, with a configuration practically decoupled from the
luminous component. $\omega$ Cen has a weak cusp in the central
luminosity density profile, implying that the gravitational potential
is very shallow inside the core and therefore mass segregation is only
a weak effect. \citet{fer06} confirm the lack of segregation by
measuring the radial distribution of blue straggler stars, which are
heavy stars and should sink to the center of the cluster if there is
mass segregation. They find a flat radial distribution of blue
stragglers with respect to lighter stellar populations. Also, the
formation channel of the blue stragglers is not collisional, as it
would be expected if there was a considerable amount of mass
segregation. With this evidence in hand, there is no reason to expect
a large variation of $M/L$ inside the core due to stellar content, so
a detected rise in $M/L$ is likely to come from the presence of a
concentrated massive object.
\begin{figure}
\plotone{f9.eps}
\caption{$M_\bullet-\sigma_{vel}$ relation for elliptical galaxies
and bulges. The solid line is the relation in \citet{tre02}.
$\omega$ Cen lies on the low mass extrapolation and suggests a
similarity between it and the galaxies. Different types of systems
such as star clusters and low luminosity AGN appear to populate the
low mass end of the diagram.}
\label{bhsigma}
\end{figure}
There is also a stability argument against a dense compact cluster of
dark remnants. The central density as measured from the 23~$\rm {km}~\rm s^{-1}$\
dispersion estimate at 1.8\arcsec\ is $5.6\times10^7
M_\odot/pc^3$. This is the largest measured for a globular cluster and
it would be difficult to maintain using stellar remnants. Obviously,
if the density is due to solar mass remnants, over $10^4$ remnants
would be required inside of 0.05 parsecs. Using the arguments of Maoz
(1998) and Miller (2006), this mass and density makes $\omega$~Cen one
of the better examples where stellar remnants can be ruled out due to
evaporation. Maoz estimates that for these numbers, any cluster of
remnants will have evaporated within the age of the cluster.
An observed velocity dispersion rise toward the center of a cluster
can also occur if a degree of anisotropy is present. If more radial
orbits are present, those stars pass near the center at higher
velocities than they would in an isotropic case. This possibility is
evaluated with our orbit-based models. Our results agree with the
model by van den Ven et al. (2006) in showing no anisotropy in the
central 10 arcmin. Their models show a degree of tangential anisotropy
at large radius, but no radial anisotropy. Models without a central
black hole but having a large degree of anisotropy inside 28\arcsec\
are not as as good as models including a black hole. Furthermore, the
high degree of radial anisotropy is highly unstable in dense systems
like $\omega$~Cen and therefore it is an unlikely explanation for the
observed kinematics.
Figure \ref{bhsigma} shows the known $M_\bullet-\sigma_v$ relation for
black holes in elliptical galaxies and bulges \citep{geb00a,fer00}. The
galaxies used to determine the relation \citep{tre02} are plotted
along with objects containing smaller black holes in low luminosity
quasars \citep{bar05}, two nearby low luminosity AGN (NGC~4595 and
Pox~52), and three globular clusters (G1, M15 and $\omega$ Cen). We
also plot the upper limit for the black hole mass in the nucleus of
M33 \citep{geb01}, which does not lie on the correlation. The black
hole in $\omega$ Cen lies above the relation, but it is consistent
with the scatter observed a larger masses. The measured black hole
mass is 1.6\% of the total mass of the cluster, which is much larger
than the canonical value of $\sim0.3$\% for larger spheroids
\citep{mag98}. If $\omega$ Cen is indeed the nucleus of an accreted
galaxy it is expected that it's original mass was considerably larger
than what we measure now. \citet{bek03} reproduce the current mass and
orbital characteristics of $\omega$ Cen with a model of an accreted
$10^7M_\odot$ dwarf galaxy. A mass of $4\times10^7M_\odot$ for the
original spheroid would put the black hole near the 0.3\% value.
The two pieces of observational evidence that $\omega$ Cen could
harbor a central black hole come from the photometry and the
kinematics. From the HST image of $\omega$ Cen, we measure a central
logarithmic surface brightness slope of $-0.08\pm0.03$. This value is
very similar to that claimed by the N-body simulations of Baumgardt et
al. (2005) that are most likely explained by a central black hole.
Standard core-collapse does not lead to such a large core with a
shallow central slope. The black hole tends to prevent core collapse
while leaving an imprint of a shallow cusp. It will be important to
run models tailored to $\omega$~Cen to see if one can cause and
maintain a shallow cusp without invoking a central black hole.
However, the main observational evidence for the central mass comes
from the increase in the central velocity dispersion, where we detect
a rise from 18.6 to 23~$\rm {km}~\rm s^{-1}$\ from radii of 14 to 2.5\arcsec. In fact,
even excluding the Gemini data presented here, the previous
ground-based data suggest a central mass concentration as well. The
core of $\omega$~Cen is around 155 \arcsec\ (about 2.5 \arcmin), so
the dispersion rise is seen well within the core.
\acknowledgments
\vspace{5pt}
E.N. would like to thank Tim de Zeeuw for very valuable discussions
and Glenn van de Ven for kindly sharing his data. K.G. acknowledges
NSF CAREER grant AST 03-49095. We thank Carl Jakob Walcher for
promptly making his data available to us. This publication is based on
observations made with the NASA/ESA Hubble Space Telescope, which is
operated by the Association of Universities for Research in Astronomy,
Inc., under NASA contract NAS 5-26555, and observations obtained at
the Gemini Observatory, which is operated by the Association of
Universities for Research in Astronomy, Inc, under cooperative
agreement with the NSF on behalf of the Gemini partnership: the
National Science Foundation (United States), the Particle Physics and
Astronomy Research Council (United Kingdom), the National Research
Council (Canada), CONACYT (Chile), the Australian Research Council
(Australia), CNPq (Brazil) and CONICET (Argentina). We acknowledge the
technical support from the Canadian Astronomy Data Centre, which is
operated by the Herzberg Institute of Astrophysics, National Research
Council of Canada, and the support by CONACYT.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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{"url":"https:\/\/cran.itam.mx\/web\/packages\/pathfindR\/vignettes\/pathfindr_vignette.html","text":"# pathfindR - An R Package for Pathway Enrichment Analysis Utilizing Active Subnetworks\n\n#### 2018-11-20\n\npathfindR is an R package for pathway enrichment analysis of gene-level omics data utilizing active subnetworks. The package also enables hierarchical clustering of the enriched pathways. The method is described in detail in Ulgen E, Ozisik O, Sezerman OU. 2018. pathfindR: An R Package for Pathway Enrichment Analysis Utilizing Active Subnetworks. bioRxiv. https:\/\/doi.org\/10.1101\/272450\n\nOur motivation to develop this package was that direct pathway enrichment analysis of differential RNA\/protein expression or DNA methylation results may not provide the researcher with the full picture. That is to say; pathway enrichment of only the list of significant genes may not be informative enough to explain the underlying disease mechanisms.\n\nAn active subnetwork is defined as a group of interconnected genes in a protein-protein interaction network (PIN) that contains most of the significant genes. Therefore, these active subnetworks define distinct disease-associated sets of genes, whether discovered through differential expression analysis or discovered because of being in interaction with a significant gene.\n\nTherefore, we propose to leverage information from a PIN to identify distinct active subnetworks and then perform pathway enrichment analyses on these subnetworks. Briefly, this workflow first maps the significant genes onto a PIN and finds active subnetworks. Next, pathway enrichment analyses are performed using each gene set of the identified active subnetworks. Finally, these enrichment results are summarized and returned as a data frame. This workflow is implemented as the function run_pathfindR() and further described in the \u201cEnrichment Workflow\u201d section of this vignette.\n\nThis process usually yields a great number of enriched pathways with related biological functions. We therefore implemented a pairwise distance metric based on kappa statistics (as proposed by Huang et al. 1) between terms and based on this distance metric, also implemented hierarchical clustering and fuzzy clustering 2 of the pathways. Details of clustering and partitioning of pathways are presented in the \u201cPathway Clustering\u201d section of this vignette.\n\n# Enrichment Workflow\n\nThe overview of the enrichment workflow is presented in the figure below:\n\nFor this workflow, the wrapper function run_pathfindR() can be used. This function takes in a data frame consisting of Gene Symbol, log-fold-change (optional) and adjusted-p values. The first 6 rows of an example input dataset (of rheumatoid arthritis differential-expression) can be found below:\n\nsuppressPackageStartupMessages(library(pathfindR))\ndata(\"RA_input\")\nknitr::kable(head(RA_input))\nILMN_1755092 FAM110A -0.6939359 0.0000034\nILMN_1730628 RNASE2 1.3535040 0.0000101\nILMN_1729801 S100A8 1.5448338 0.0000347\nILMN_1714991 S100A9 1.0280904 0.0002263\nILMN_1762037 TEX261 -0.3235994 0.0002263\nILMN_1718610 ARHGAP17 -0.6919330 0.0002708\n\nExecuting the workflow is straightforward (but takes several minutes):\n\nRA_output <- run_pathfindR(RA_input)\n\nThe user may want to change certain arguments of the function:\n\n# to change the output directory\nRA_output <- run_pathfindR(RA_input, output = \"new_directory\")\n\n# to change the PIN (default = Biogrid)\nRA_output <- run_pathfindR(RA_input, pin_name = \"IntAct\")\n# to use an external PIN of user's choice\nRA_output <- run_pathfindR(RA_input, pin_name = \"\/path\/to\/myPIN.sif\")\n\n# available gene sets are KEGG, Reactome, BioCarta, GO-BP, GO-CC and GO-MF\n# default is KEGG\n# to change the gene sets used for enrichment analysis\nRA_output <- run_pathfindR(RA_input, gene_sets = \"BioCarta\")\n\n# to change the active subnetwork search algorithm (default = \"GR\", i.e. greedy algorithm)\n# for simulated annealing:\nRA_output <- run_pathfindR(RA_input, search_method = \"SA\")\n\n# to change the number of iterations (default = 10)\nRA_output <- run_pathfindR(RA_input, iterations = 5)\n\n# to manually specify the number processes used during parallel loop by foreach\n# defaults to the number of detected cores\nRA_output <- run_pathfindR(RA_input, n_processes = 2)\n\n# to report the non-DEG active subnetwork genes\nRA_output <- run_pathfindR(RA_input, list_active_snw_genes = TRUE)\n\nFor a full list of arguments, see ?run_pathfindR.\n\nThe workflow consists of the following steps :\n\nAfter input testing, the program attempts to convert any gene symbol that is not in the PIN to an alias symbol that is in the PIN. Next, active subnetwork search is performed via the selected algorithm. The available algorithms for active subnetwork search are:\n\n\u2022 Greedy Algorithm (based on Ideker et al. 3),\n\u2022 Simulated Annealing Algorithm (based on Ideker et al. 4) and\n\u2022 Genetic Algorithm (based on Ozisik et al. 5).\n\nNext, pathway enrichment analyses are performed using the genes in each of the active subnetworks. For this, up-to-date information on human gene sets from KEGG, Reactome, BioCarta and Gene Ontology were retrieved and is available for use within the package. The user may specify custom gene sets, including gene sets for non-human organisms, as described in the section \u201cpathfindR Analysis with Custom Gene Sets\u201d.\n\nDuring enrichment analyses, pathways with adjusted-p values larger than the enrichment_threshold (an argument of run_pathfindR(), defaults to 0.05) are discarded. The results of enrichment analyses over all active subnetworks are combined by keeping only the lowest adjusted-p value for each pathway.\n\nThis process of active subnetwork search and enrichment analyses is repeated for a selected number of iterations (indicated by the iterations argument of run_pathfindR()), which is performed in parallel via the R package foreach.\n\nThe wrapper function returns a data frame that contains the lowest and the highest adjusted-p values for each enriched pathway, as well as the numbers of times each pathway is encountered over all iterations. The first two rows of the example output of the pathfindR-enrichment workflow (performed on the rheumatoid arthritis data RA_output) is shown below:\n\ndata(\"RA_output\")\nknitr::kable(head(RA_output, 2))\nID Pathway Fold_Enrichment occurrence lowest_p highest_p Up_regulated Down_regulated\nhsa00190 Oxidative phosphorylation 71.86252 10 3e-07 3e-07 NDUFB3, NDUFA1, COX7C, COX7A2, UQCRQ, COX6A1, ATP6V0E1, ATP6V1D ATP6V0E2\nhsa05012 Parkinson\u2019s disease 63.72714 10 4e-07 4e-07 NDUFA1, NDUFB3, UQCRQ, COX6A1, COX7A2, COX7C SLC25A5, VDAC1, UBE2G1\n\nThe function also creates an HTML report results.html that is saved in a directory, by default named pathfindr_Results but can be changed by changing the argument output_dir, under the current working directory. This report contains links to two other HTML files:\n\n## 1. all_pathways.html\n\nThis document contains a table of the active subnetwork-oriented pathway enrichment results. Each enriched pathway name is linked to the visualization of that pathway, with the gene nodes colored according to their log-fold-change values. This table contains the same information as the returned data frame. Columns are:\n\n\u2022 ID: KEGG ID of enriched pathway\n\u2022 Pathway: Description the pathway\n\u2022 Fold_Enrichment: Fold enrichment value for the pathway.\n\u2022 occurrence: The number of times the pathway was found to be enriched over all iterations\n\u2022 lowest_p: the lowest adjusted-p value of the pathway over all iterations\n\u2022 highest_p: the highest adjusted-p value of the pathway over all iterations\n\u2022 Up_regulated: the up-regulated genes involved in the pathway\n\u2022 Down_regulated: the down-regulated genes involved in the pathway\n\n## 2. genes_table.html\n\nThis document contains a table of converted gene symbols. Columns are:\n\n\u2022 Old Symbol: the original gene symbol\n\u2022 Converted Symbol: the alias symbol that was found in the PIN\n\u2022 Change: the provided change value\n\u2022 p-value: the provided adjusted p value\n\nThe document contains a second table of genes for which no interactions were identified (after checking for alias symbols).\n\n# Pathway Clustering\n\nFor this workflow, the wrapper function cluster_pathways() is used. This function first calculates the pairwise kappa statistics between the terms in the input data frame. By default, the function performs hierarchical clustering of the terms using this kappa matrix, automatically determines the optimal number of clusters by maximizing the average silhouette width and returns a data frame with cluster assignments:\n\ndata(\"RA_output\")\nRA_clustered <- cluster_pathways(RA_output)\n#> The maximum average silhouette width was 20.35 for k = 8\n\n## First 2 rows of clustered terms data frame\nknitr::kable(head(RA_clustered, 2))\nID Pathway Fold_Enrichment occurrence lowest_p highest_p Up_regulated Down_regulated Cluster Status\nhsa00190 Oxidative phosphorylation 71.86252 10 3e-07 3e-07 NDUFB3, NDUFA1, COX7C, COX7A2, UQCRQ, COX6A1, ATP6V0E1, ATP6V1D ATP6V0E2 1 Representative\nhsa05012 Parkinson\u2019s disease 63.72714 10 4e-07 4e-07 NDUFA1, NDUFB3, UQCRQ, COX6A1, COX7A2, COX7C SLC25A5, VDAC1, UBE2G1 1 Member\n## The 8 representative terms\nknitr::kable(RA_clustered[RA_clustered$Status == \"Representative\", ]) ID Pathway Fold_Enrichment occurrence lowest_p highest_p Up_regulated Down_regulated Cluster Status 1 hsa00190 Oxidative phosphorylation 71.86252 10 0.0000003 0.0000003 NDUFB3, NDUFA1, COX7C, COX7A2, UQCRQ, COX6A1, ATP6V0E1, ATP6V1D ATP6V0E2 1 Representative 3 hsa03040 Spliceosome 49.39033 10 0.0000005 0.0000005 SF3B6, LSM3, BUD31 SNRPB, SF3B2, U2AF2, PUF60, HNRNPA1, PCBP1, SRSF5, SRSF8, SNU13, DDX23, EIF4A3 2 Representative 5 hsa03010 Ribosome 39.02933 10 0.0000011 0.0000063 RPS24, RPL26, RPL39, RPL31, MRPL33, MRPS18C RPLP2 3 Representative 9 hsa04064 NF-kappa B signaling pathway 67.75926 10 0.0000025 0.0000025 LY96 IKBKB, PRKCQ, CARD11, TICAM1, CSNK2A2, PARP1, UBE2I 4 Representative 10 hsa04714 Thermogenesis 39.27370 10 0.0000069 0.0000069 COX6A1, COX7A2, COX7C, NDUFA1, NDUFB3, UQCRQ CREB1, ADCY7, ACTB, ACTG1, SMARCA4, ARID1A, KDM1A, MTOR 5 Representative 15 hsa04130 SNARE interactions in vesicular transport 79.54348 10 0.0000664 0.0000664 STX10, STX6 BET1L, SNAP23, STX2 6 Representative 38 hsa03050 Proteasome 85.09302 10 0.0018696 0.0018696 PSMD7, PSMB10 7 Representative 48 hsa03420 Nucleotide excision repair 59.65761 10 0.0053843 0.0053843 GTF2H5, POLE4 POLD2, RPA1, XPC 8 Representative # to display the heatmap of kappa statistics RA_clustered <- cluster_pathways(RA_output, plot_hmap = TRUE, plot_clusters_graph = FALSE) #> The maximum average silhouette width was 20.35 for k = 8 # to display the dendrogram and optimal clusters RA_clustered <- cluster_pathways(RA_output, plot_dend = TRUE, plot_clusters_graph = FALSE) #> The maximum average silhouette width was 20.35 for k = 8 # to change agglomeration method (default = \"average\") RA_clustered <- cluster_pathways(RA_output, hclu_method = \"centroid\") #> The maximum average silhouette width was 6.9 for k = 7 Alternatively, the fuzzy clustering method (as described by Huang et al.6) can be used: RA_clustered <- cluster_pathways(RA_output, method = \"fuzzy\") # Pathway Scores per Sample The function calculate_pw_scores can be used to calculate the pathway scores per sample. This allows the user to individually examine the scores and infer whether a pathway is activated or repressed in a given sample. For a set of pathways $$P = \\{P_1, P_2, ... , P_k\\}$$, where each $$P_i$$ contains a set of genes, i.e. $$P_i = \\{g_1, g_2, ...\\}$$, the pathway score matrix $$PS$$ is defined as: $$PS_{p,s} = \\frac{1}{k} \\sum_{g \\in P_p} GS_{g,s}$$ for each pathway $$p$$ and for each sample $$s$$. $$GS$$ is the gene score per sample matrix and is defined as: $$GS_{g,s} = (EM_{g,s} - \\bar{x}_g) \/ sd_g$$ where $$EM$$ is the expression matrix (columns are samples, rows are genes), $$\\bar{x}_g$$ is the mean expression value of the gene and $$sd_g$$ is the standard deviation of the expression values for the gene. An example application is provided below: ## Pathway data frame pws_table <- pathfindR::RA_clustered # selecting \"Representative\" pathways for clear visualization pws_table <- pws_table[pws_table$Status == \"Representative\", ]\n\n## Expression matrix\nexp_mat <- pathfindR::RA_exp_mat\n\n## Vector of \"Case\" IDs\ncases <- c(\"GSM389703\", \"GSM389704\", \"GSM389706\", \"GSM389708\",\n\"GSM389711\", \"GSM389714\", \"GSM389716\", \"GSM389717\",\n\"GSM389719\", \"GSM389721\", \"GSM389722\", \"GSM389724\",\n\"GSM389726\", \"GSM389727\", \"GSM389730\", \"GSM389731\",\n\"GSM389733\", \"GSM389735\")\n\n## Calculate pathway scores and plot heatmap\nscore_matrix <- calculate_pw_scores(pws_table, exp_mat, cases)","date":"2019-08-23 15:55:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.45172813534736633, \"perplexity\": 10751.876482831756}, \"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\/1566027318894.83\/warc\/CC-MAIN-20190823150804-20190823172804-00411.warc.gz\"}"}
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A Cornish entrepreneur has been shortlisted for two accolades at a national awards for women in business.
Katherine George, founder of social media marketing agency Oh So Social, has been shortlisted in two categories at this year's Best Business Women Awards, which aims to recognise dedicated and successful female entrepreneurs from across the UK.
George is a finalist in both the Best Business Woman in Marketing & PR, and Best Young Entrepreneur categories.
She said: "It is an absolute honour to be recognised at a national level and shortlisted for not one, but two awards at the Best Business Women Awards.
The winners of all the categories will be announced next month.
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{
"redpajama_set_name": "RedPajamaC4"
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Viewing Topics 9 - 12 of 396
What syslog-ng is
Introduction to syslog-ng > What syslog-ng is
The syslog-ng application is a flexible and highly scalable system logging application that is ideal for creating centralized and trusted logging solutions. Among others, syslog-ng OSE allows you the following.
Secure and reliable log transfer
The syslog-ng OSE application enables you to send the log messages of your hosts to remote servers using the latest protocol standards. You can collect and store your log data centrally on dedicated log servers. Transfer log messages using the <?XML:NAMESPACE PREFIX = "MadCap" NS = "http://www.madcapsoftware.com/Schemas/MadCap.xsd" />TCP protocol ensures that no messages are lost.
Disk-based message buffering
To minimize the risk of losing important log messages, the syslog-ng OSE application can store messages on the local hard disk if the central log server or the network connection becomes unavailable. The syslog-ng application automatically sends the stored messages to the server when the connection is reestablished, in the same order the messages were received. The disk buffer is persistent – no messages are lost even if syslog-ng is restarted.
Secure logging using TLS
Log messages may contain sensitive information that should not be accessed by third parties. Therefore, syslog-ng OSE supports the Transport Layer Security (TLS) protocol to encrypt the communication. TLS also allows you to authenticate your clients and the logserver using X.509 certificates.
Flexible data extraction and processing
Most log messages are inherently unstructured, which makes them difficult to process. To overcome this problem, syslog-ng OSE comes with a set of built-in parsers, which you can combine to build very complex things.
Filter and classify
The syslog-ng OSE application can sort the incoming log messages based on their content and various parameters like the source host, application, and priority. You can create directories, files, and database tables dynamically using macros. Complex filtering using regular expressions and boolean operators offers almost unlimited flexibility to forward only the important log messages to the selected destinations.
Parse and rewrite
The syslog-ng OSE application can segment log messages to named fields or columns, and also modify the values of these fields. You can process JSON messages, key-value pairs, and more.
To get the most information out of your log data, syslog-ng OSE allows you to correlate log messages and aggregate the extracted information into a single message. You can also use external information to enrich your log data.
Big data clusters
The log data that your organization has to process, store, and review increases daily, so many organizations use big data solutions for their logs. To accomodate this huge amount of data, syslog-ng OSE natively supports storing log messages in HDFS files and Elasticsearch clusters.
Message queue support
Large organizations increasingly rely on queuing infrastructure to transfer their data. syslog-ng OSE supports Apache Kafka, the Advanced Message Queuing Protocol (AMQP), and the Simple Text Oriented Messaging Protocol (STOMP).
SQL, NoSQL, and monitoring
Storing your log messages in a database allows you to easily search and query the messages and interoperate with log analyzing applications. The syslog-ng application supports the following databases: MongoDB, MSSQL, MySQL, Oracle, PostgreSQL, and SQLite.
syslog-ng OSE also allows you to extract the information you need from your log data, and directly send it to your Graphite, Redis, or Riemann monitoring system.
Wide protocol and platform support
syslog protocol standards
syslog-ng not only supports legacy BSD syslog (RFC3164) and the enhanced RFC5424 protocols but also JavaScript Object Notation (JSON) and journald message formats.
Heterogeneous environments
The syslog-ng OSE application is the ideal choice to collect logs in massively heterogeneous environments using several different operating systems and hardware platforms, including Linux, Unix, BSD, Sun Solaris, HP-UX, Tru64, and AIX.
IPv4 and IPv6 support
The syslog-ng application can operate in both IPv4 and IPv6 network environments, and can receive and send messages to both types of networks.
What syslog-ng is not
Introduction to syslog-ng > What syslog-ng is not
The syslog-ng application is not log analysis software. It can filter log messages and select only the ones matching certain criteria. It can even convert the messages and restructure them to a predefined format, or parse the messages and segment them into different fields. But syslog-ng cannot interpret and analyze the meaning behind the messages, or recognize patterns in the occurrence of different messages.
Why is syslog-ng needed?
Introduction to syslog-ng > Why is syslog-ng needed?
Log messages contain information about the events happening on the hosts. Monitoring system events is essential for security and system health monitoring reasons.
The original syslog protocol separates messages based on the priority of the message and the facility sending the message. These two parameters alone are often inadequate to consistently classify messages, as many applications might use the same facility, and the facility itself is not even included in the log message. To make things worse, many log messages contain unimportant information. The syslog-ng application helps you to select only the really interesting messages, and forward them to a central server.
Company policies or other regulations often require log messages to be archived. Storing the important messages in a central location greatly simplifies this process.
What is new in syslog-ng Open Source Edition 3.17?
Introduction to syslog-ng > What is new in syslog-ng Open Source Edition 3.17?
Version 3.17 of syslog-ng Open Source Edition includes the following main features.
linux-audit() source driver
A new source driver, linux-audit(), has been added. The linux-audit() source reads and automatically parses the Linux audit logs. For details, see Administration Guide.
exclude-kmsg() system source option
A new system source option, exclude-kmsg() makes it possible to avoid duplicate collection of kernel logs or errors in kernel log collection (for example, in scenarios where the log management on the host system and the containerized solution are collecting the kernel logs simultaneously). When set to yes, syslog-ng OSE will omit kernel logs on platforms where they are available separately. For details, see Administration Guide
SCL syntax updates
You can now refer to any additional parameters at the end of the argument in a block by adding three dots to it (…). It tells syslog-ng OSE that this macro accepts `__VARARGS__`, therefore any name-value pair can be passed without validation. For details, see Administration Guide.
You can now make parameters mandatory in block definitions by defining them with empty brackets (). For details, see Administration Guide.
The failover() option allows you to specify what happens after syslog-ng OSE fails over to a secondary server. Additionally, the failover-servers() option has been deprecated and removed from the document. For more information about the failover() option, see Administration Guide.
The default value of the --skip-tokens parameter of the loggen application has been changed to 0. For details, see Administration Guide.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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{"url":"https:\/\/testbook.com\/question-answer\/rs-240-is-divided-equally-among-n-child--60b5ea137264bba52199bd89","text":"# Rs. 240 is divided equally among 'n' children. If there were 10 more children then each child would get Rs. 4 less, then find n.\n\nFree Practice With Testbook Mock Tests\n\n## Options:\n\n1. 16\n\n2. 20\n\n3. 18\n\n4. 14\n\n### Correct Answer: Option 2 (Solution Below)\n\nThis question was previously asked in\n\nDSSSB TGT Social Studies General Section Male - 9 Sept 2018\n\n## Solution:\n\nGiven:\n\nTotal amount = Rs. 240\n\nCalculation:\n\nn children will get = $$\\frac{240}{n}$$\n\nIf 10 more children, total number of children = n + 10\n\nThe share decreases by 4 = $$\\frac{240}{n} - 4$$\n\nNow,\n\n$$(n + 10 )\u00d7(\\frac{240}{n} - 4) = 240$$\n\n\u21d2 (n +10) \u00d7 (240 - 4n) = 240n\n\n\u21d2 240n + 2400 - 4n2 - 40n = 240n\n\n\u21d2 4n2 + 40n - 2400 = 0\n\n\u21d2 4n2 +120n - 80n - 2400n = 0\n\n\u21d2 4n(n + 30) - 80(n + 30) = 0\n\n\u21d2 (4n - 80)(n + 30) = 0\n\n\u21d2 n = 80\/4 = 20\n\n\u21d2 n = - 30\n\nn cannot be negative, so n = 20\n\n\u2234 The value of n is 20.","date":"2021-07-31 13:19:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5853406190872192, \"perplexity\": 5610.879835576532}, \"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-2021-31\/segments\/1627046154089.6\/warc\/CC-MAIN-20210731105716-20210731135716-00613.warc.gz\"}"}
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{"url":"https:\/\/www.physicsforums.com\/threads\/if-f-g-cont-then-g-o-f-cont.262949\/","text":"# If f,g cont. , then g o f cont.?\n\nIf f,g cont. , then \"g o f\" cont.?\n\n## Homework Statement\n\nLet X,Y,Z be spaces and f:X-->Y and g:Y-->Z be functions.\nIf f,g are continuous,\nthen so is g o f.\n\n## The Attempt at a Solution\n\nThis is my proof so far. I would be appreciative if someone could point out the weak points.\n\n\"Proof\". Let f,g be continuous. Show \"g o f\" continuous. Show g(f(x)) continuous.\n\nSince f is continuous, and since f:X-->Y, we know that x is an element of W such that f[W]$$\\subseteq$$V for V open, V$$\\subseteq$$Y.\n\nSince g is continuous, and since g:Y-->Z, we know that f(x) is an element of V such that g[V]$$\\subseteq$$M, for M open, M$$\\subseteq$$Z.\n\nTherefore, g[ f(x) ] $$\\in$$ M $$\\subseteq$$ Z.\n\nThus, \"g o f\" is continuous.\n\nLet me try to reiterate it. Say we have a neighborhood of f(x), V. Then we can find a neighborhood of x, U in X such that f(U) is in V. In the same way, if we have a neighborhood of g(f(x)) W in Z, we can find a neighborhood of f(x) T in Y, such that g(T) is in W. So, we should be able to find a neighborhood of x, S in X such that g(f(S)) is a subset of W.\n\nLast edited:\n\nIt sounds alot better your way. I understand it more now.","date":"2021-06-22 02:26:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8996955752372742, \"perplexity\": 1011.2697090496347}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623488504969.64\/warc\/CC-MAIN-20210622002655-20210622032655-00080.warc.gz\"}"}
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\section{Introduction}
Recall that a Hodge structure $V_{\mathbb C} = \bigoplus_{p+q=n} V^{p,q}$ is said to be effective if $V^{p,q} = 0$ unless $p,q \geq 0$, and, it is said to be geometric if it is isomorphic to a Hodge substructure of the cohomology of a smooth, projective variety over $\mathbb C$.
For $m \in \mathbb Z$, the Tate twist $V(m)$ is defined by $V(m)^{p,q} = V^{p+m,q+m}$.
A geometric Hodge structure must be effective, though not conversely.
The general Hodge conjecture as formulated by Grothendieck \cite{Grothendieck} implies that any effective Tate twist of a geometric Hodge structure is again geometric.
A Hodge structure $V$ is said to be of \emph{CM-type} if it is polarizable, and, its Hodge group $G$ is abelian.
In this paper we show (Theorem \ref{main}) that any effective Hodge structure of CM-type is geometric.
Moreover, such a Hodge structure is isomorphic (without having to take a Tate twist) to a Hodge structure appearing in the cohomology of a CM abelian variety.
That this is true up to a Tate twist is due to Serre (see \cite{MilneShih}*{(1.7), p.~234}).
In a series of papers \cites{Abdulali1997, Abdulali2000, Abdulali2001, Abdulali2002, Abdulali2004}, we have shown that for certain abelian varieties $A$ over $\mathbb C$, the general Hodge conjecture for $A$ is implied by the usual Hodge conjecture for a certain class of abelian varieties.
Chad Schoen and Fumio Hazama have kindly pointed out to me an error in the proof of Prop.~4.4.1 of \cite{Abdulali1997}.
Consequently, this proposition, Theorem 6.1 of \cite{Abdulali1997}, and, Theorem 3.1 of \cite{Abdulali2001}, are false as stated.
Correcting these errors, we prove in an elementary manner that the usual Hodge conjecture for all CM abelian varieties implies the general Hodge conjecture for the same class.
This result has been independently obtained by Hazama \cites{Hazama2002, Hazama2003} using different methods.
Finally, we give several examples of CM abelian varieties for which the general Hodge conjecture can be proved by these methods.
\section{Hodge structures}
By a \emph{rational Hodge structure} of \emph{weight} $n$ we mean a finite dimensional vector space $V$ over $\mathbb Q$, and a decomposition
$V_{\mathbb C} = \bigoplus_{p+q=n} V^{p,q}$
such that $\overline{V^{p,q}} = V^{q,p}$ for all $p,q$.
It is \emph{effective} if $V^{p,q} = 0$ unless $p,q \geq 0$.
A Hodge structure $V$ determines a morphism
$$\lambda_V : S^1 \to GL(V_{\mathbb R}),$$
where $S^1$ is the unit circle in the complex plane, such that $\lambda_V(e^{i\theta})$ acts on $V^{p,q}$ as multiplication by $e^{(p-q)i\theta}$.
The element $C = \lambda_V(i)$ is called the \emph{Weil operator}.
A \emph{polarization} of $V$ is a morphism of Hodge structures $\psi : V \otimes V \to \mathbb Q(-n)$ such that $\psi (x,Cy)$ is symmetric and positive definite on $V_{\mathbb R}$.
The \emph{Hodge group} (or Special Mumford-Tate group) of $V$ is defined to be the smallest algebraic subgroup $G$ of $GL(V)$ such that $G(\mathbb R)$ contains the image of $\lambda_V$.
It is a reductive algebraic group over $\mathbb Q$ if $V$ is polarizable \cite{900}*{Prop.~3.6, p.~44}.
Let $A$ be an abelian variety over $\mathbb C$.
The Hodge group of $A$ is the Hodge group of $H^1(A,\mathbb Q)$.
$A$ is of CM-type if and only if its Hodge group is abelian (Mumford \cite{Mumford1969}*{p.~347}).
A rational Hodge structure $V$ is said to be of \emph{CM-type} if it is polarizable, and, its Hodge group $G$ is abelian.
It is said to be \emph{trivial} if $G$ is trivial, and \emph{nontrivial} otherwise.
We shall now collect together some facts about CM Hodge structures.
For the convenience of the reader we include proofs of some \lq\lq{well-known}\rq\rq\ facts.
The main references are Deligne \cite{900} and Schappacher \cite{Schappacher}.
Let $V$ be a Hodge structure with Hodge group $G$.
The endomorphism algebra of $V$ is given by $D := \End_G V$.
If $E$ is a subfield of $D$, then, $V$ may be considered as a vector space over $E$.
We say that $V$ is of \emph{type} $E$ if $V$ is $1$-dimensional over $E$.
Assume that $V$ is an irreducible Hodge structure of CM-type.
Then $D$ is a division algebra.
Let $E$ be the center of $D$, and let $T_E$ be the torus $R_{E/\mathbb Q} \mathbb G_{m,E}$.
Since $G$ is abelian, it is contained in $T_E$.
Consider $V$ as a vector space over $E$.
Any $E$-subspace of $V$ is also a $G$-submodule of $V$.
Since $V$ is irreducible, it must be $1$-dimensional over $E$.
Therefore $D = \End_G V = E$.
Further, from the irreducibility of $V$, we see that $G(\mathbb Q)$ generates $E$ as an algebra over $\mathbb Q$.
We have $E_{\mathbb C} := E \otimes_{\mathbb Q} \mathbb C = \oplus_{\sigma\in S} E^{\sigma}$, where $E^{\sigma} = E \otimes_{E,\sigma} \mathbb C$, and, $S$ is the set of embeddings of $E$ into $\mathbb C$.
We consider $E$ embedded in $E_{\mathbb C}$ via $e \mapsto (\sigma(e))$.
Then the trace map $\Tr_{E_{\mathbb C}/\mathbb C}: E_{\mathbb C} \to \mathbb C$ given by
$$\bigoplus_{\sigma\in S} E^{\sigma} \ni (a_{\sigma}) \mapsto \sum_{\sigma \in S} a_{\sigma}$$
is compatible with $\Tr_{E/{\mathbb Q}}: E \to \mathbb Q$.
\begin{lemma}
If $V$ is a nontrivial irreducible Hodge structure of CM-type with Hodge group $G$, then, $E = \End_G V$ is a CM-field.
\end{lemma}
\begin{proof}
Let $\psi: V \times V \to \mathbb Q$ be a polarization of $V$.
Since $G$ is a torus contained in $\Aut \psi$, which is either an orthogonal group or a symplectic group, we have that $G(\mathbb R)$ is compact.
It follows that $E$ is totally imaginary, for a real embedding of $E$ would induce a nontrivial character of $G(\mathbb R)$ with noncompact image.
The polarization $\psi$ determines an involution $e \mapsto e^{\star}$ of $E$ by the rule
$$\psi(x,ey) = \psi(e^{\star}x,y).$$
Let $F$ be the fixed field of this involution.
Then $[E:F]$ is either $1$ or $2$, depending on whether the involution is trivial or not.
For $g \in G$ we have $\psi(x,y) = \psi(gx,gy) = \psi(g^{\star}gx,y)$, so $g^{\star}g = 1$.
Thus $\star$ is nontrivial, and, $[E:F] = 2$.
We extend $\star$ to an involution of $E \otimes_{\mathbb Q} \mathbb R$; the set of elements fixed by the extended involution is $F \otimes_{\mathbb Q} \mathbb R$.
Let $n$ be the weight of the Hodge structure.
Since the Weil operator $C$ is an element of $G(\mathbb R)$ such that $C^2 = (-1)^n$, we have $C^{\star} = C^{-1} = (-1)^nC$.
Suppose $n$ is even.
There exists a unique form $T: V \times V \to E$ such that $T(y,x) = T(x,y)^{\star}$,
$T(ax,by) = ab^{\star}T(x,y)$, and, $\psi(x,y) = \Tr_{E/\mathbb Q} T(x,y)$ for $a,b \in E$, $x,y \in V$.
Identify $V$ with $E$, and let $\beta = T(1,1)$.
Then $\psi(x,Cx) = \Tr_{E \otimes_{\mathbb Q} \mathbb R/\mathbb R} T(x,Cx) = \Tr_{E \otimes_{\mathbb Q} \mathbb R/\mathbb R} C\beta xx^{\star} > 0$ for $x \neq 0$.
Note that $f := C\beta \in F \otimes_{\mathbb Q} \mathbb R$.
Suppose next that $n$ is odd.
Let $\alpha \in F$ be such that $E = F(\sqrt{\alpha})$.
Then $\sqrt{\alpha}^{\,\star} = -\sqrt{\alpha}$.
There exists a unique form $T: V \times V \to E$ such that $T(y,x) = -T(x,y)^{\star}$,
$T(ax,by) = ab^{\star}T(x,y)$, and, $\psi(x,y) = \Tr_{E/\mathbb Q} \sqrt{\alpha} T(x,y)$ for $a,b \in E$, $x,y \in V$.
Identify $V$ with $E$, and let $\beta = T(1,1)$.
Then
$$\psi(x,Cx) = \Tr_{E \otimes_{\mathbb Q} \mathbb R/\mathbb R} \sqrt{\alpha} T(x,Cx) = \Tr_{E \otimes_{\mathbb Q} \mathbb R/\mathbb R} C^{\star} \sqrt{\alpha} \beta xx^{\star} > 0$$ for $x \neq 0$.
Note that $f := C^{\star} \sqrt{\alpha} \beta \in F \otimes_{\mathbb Q} \mathbb R$.
Thus in all cases we have $\Tr_{E\otimes_{\mathbb Q} \mathbb R/\mathbb R} fxx^{\star} > 0$ for all $x \in (E \otimes_{\mathbb Q} \mathbb R)^{\times}$, and a fixed $f \in F \otimes_{\mathbb Q} \mathbb R$.
In particular, for all $x \in (F \otimes_{\mathbb Q} \mathbb R)^{\times}$ we have $\Tr_{F\otimes_{\mathbb Q} \mathbb R/\mathbb R} fx^2 > 0$.
Write $f = (f_{\sigma}) \in \oplus_{\sigma \in S} E^{\sigma}$.
Suppose $\tau: F \to \mathbb C$ were a nonreal embedding.
Write $$f_{\sigma} = r_{\sigma} e^{i\theta_{\sigma}} \otimes s_{\sigma} \in \mathbb C \otimes_{\mathbb Q} \mathbb R,$$
with all $r_{\sigma}, s_{\sigma} > 0$.
By the Artin-Whaples approximation theorem there exists $x \in F$ such that $\sigma(x)$ is close to $0$ for all embeddings of $F$ except $\tau$ and $\overline{\tau}$, while $\tau(x)$ has large absolute value, and argument close to $\frac{1}{2} (\pi - \theta_{\sigma})$.
Then $\Tr_{F\otimes_{\mathbb Q} \mathbb R/\mathbb R} fx^2 < 0$.
This contradiction shows that $F$ is totally real.
\end{proof}
\begin{lemma}
\label{typelemma}
Let $E$ be a CM-field, and $S$ the set of embeddings of $E$ into $\mathbb C$.
Let $V$ be a $1$-dimensional vector space over $E$.
There is a bijection $V_{\phi} \leftrightarrow \phi$ between Hodge structures on $V$ of CM-type of weight $n$ with endomorphisms by $E$, and, functions $\phi: S \to \mathbb Z$ satisfying
\begin{equation}
\label{type}
\phi(\sigma) + \phi(\bar{\sigma}) = n.
\end{equation}
\end{lemma}
\begin{proof}
Let $V$ be a Hodge structure which is $1$-dimensional as a vector space over $E$.
Let $G$ be the Hodge group of $V$.
Then,
$$V \otimes_{\mathbb Q} \mathbb C = E \otimes_{\mathbb Q} \mathbb C = \bigoplus_{\sigma \in S} V^{\sigma},$$
where, $V^{\sigma} := E \otimes_{E,\sigma} \mathbb C = E^{\sigma}$.
Since $G \subset T_E$, each $V^{\sigma}$ is a $G_{\mathbb C}$-submodule of $V_\mathbb C$ of complex dimension $1$.
Hence $V^{\sigma} \subset V^{p,q}$ for some $p,q$, with $p+q=n$.
Write $\phi(\sigma) = p$.
Since $\overline{V^{\sigma}} = V^{\bar{\sigma}}$, $\phi$ satisfies \eqref{type}.
Conversely, given a function $\phi: S \to \mathbb Z$ satisfying \eqref{type}, we can define a unique Hodge structure $V_{\phi}$ on $V$ such that $V^{\sigma}$ has Hodge type $(\phi(\sigma), \phi(\bar{\sigma}))$.
Let $H$ be the Hodge group of $V_{\phi}$.
Since each $V^{\sigma}$ is an $E$-submodule of $V_{\mathbb C}$, we see that $\End_H V_{\phi} \supset E$.
Then $E$ equals its own centralizer in $\End_{\mathbb Q} V$, so $H \subset T_E$, and $V_{\phi}$ is of CM-type.
\end{proof}
\section{Abelian varieties}
Let $X$ be a smooth projective variety over $\mathbb C$, and $\mathcal Y$ a family of smooth projective varieties over $\mathbb C$.
We say that $X$ is \emph{dominated} by $\mathcal Y$ if, for each irreducible Hodge substructure $V$ of the cohomology of $X$, there exists $Y \in \mathcal Y$, a nonnegative integer $n$, and a Hodge substructure $W \subset H^n(Y, \mathbb Q)$ such that $W$ is isomorphic to a Tate twist of $V$, and, $W_{\mathbb C}$ contains an element of Hodge type $(n,0)$.
If $X$ is dominated by $\mathcal Y$, then, the usual Hodge conjecture for all $X \times Y$ for all $Y \in \mathcal Y$ implies the general Hodge conjecture for $X$.
This observation is due to Grothendieck; see \cite{Abdulali1997}*{Prop.~2.1, p.~243} for a proof.
\begin{theorem}
\label{main}
Let $V$ be an effective Hodge structure with complex multiplication by $E$.
Then $V$ is isomorphic to a Hodge substructure of the cohomology of a product of abelian varieties with CM by $E$.
\end{theorem}
\begin{proof}
We shall prove this by induction on the weight $n$ of the Hodge structure $V$, the case $n=1$ being true by the classical theory of complex multiplication.
Let $\phi: S \to \mathbb Z$ be the function associated to $V$ by Lemma \ref{typelemma}, so that $V = V_{\phi}$.
Choose a CM-type $T$ of $E$ such that $\phi(\sigma) \geq \phi(\bar{\sigma})$ for all $\sigma \in T$.
Let $\chi$ be the characteristic function of $T$.
Then $V_{\chi} = H^1(A, \mathbb Q)$ for an abelian variety $A$ with CM by $E$.
Let $\psi = \phi - \chi$.
Then $V_{\psi}$ is an effective Hodge structure of weight $n-1$, so by induction, it is contained in $H^r(B, \mathbb Q)$, with $B$ a product of abelian varieties with CM by $E$.
Inside the K\"unneth component $H^1(A,\mathbb C) \otimes H^r(B, \mathbb C)$ of $H^{r+1}(A \times B, \mathbb C)$, we have the space
$$\bigoplus_{\sigma \in S} V_{\chi}^{\sigma} \otimes V_{\psi}^{\sigma} = \bigoplus_{\sigma \in S} V_{\chi + \psi}^{\sigma} = \bigoplus_{\sigma \in S} V^{\sigma} = V_{\mathbb C},$$
so, $H^{r+1}(A \times B, \mathbb Q)$ contains a Hodge substructure isomorphic to $V$.
\end{proof}
\begin{theorem}
Any abelian variety of CM-type is dominated by the class of all CM abelian varieties.
Let $E$ be a CM-field, and $A$ an abelian variety with CM by $E$.
Then any power of $A$ is dominated by the set of products of all CM abelian varieties with CM-field $E$.
\end{theorem}
\begin{proof}
Let $A$ be a CM abelian variety, and, $V$ an irreducible Hodge structure in $H^n(A, \mathbb Q)$.
Then $V$ is itself of CM-type since its Hodge group is a quotient of the Hodge group of $A$ (\cite{Abdulali2002}*{Lemma~2.1.1, p.~917}).
Let $r$ be the largest integer such that $V(r)$ is effective.
By the previous theorem, $V(r)$ occurs in the cohomology of a product of CM abelian varieties.
This shows that $A$ is dominated by the class of all CM abelian varieties.
Suppose now that $A$ has CM by a CM-field $E$.
We assume without loss of generality that $A$ is simple, and identify $U := H^1(A,\mathbb Q)$ with $E$.
Let $G$ be the Hodge group of $A$.
Let $V$ be an irreducible Hodge structure in the cohomology of a power of $A$.
Then $V$ is isomorphic to a Hodge substructure of $U^{\otimes n}$ for a positive integer $n$.
We note that the Hodge group of $U^{\otimes n}$ is also $G$.
The identification of $U$ with $E$ extends to an identification of $U^{\otimes n}$ with $E^{\otimes n}$.
Since $G \subset E^{\otimes n}$, any ideal of $E^{\otimes n}$ is also a $G$-submodule, and hence a Hodge structure.
Hence $V$ is contained in a simple ideal $W$ of $E^{\otimes n}$.
But $E^{\otimes n}$ is isomorphic to a direct sum of copies of $E$.
Hence $W$ is a Hodge structure of type $E$.
It follows that $F$, the CM-field of $V$, is contained in $E$.
Let $r$ be the largest integer such that $V(r)$ is effective.
By the previous theorem, $V(r)$ occurs in the cohomology of a product of abelian varieties with CM by $F$.
Since any CM-type for $F$ can be lifted to a CM-type for $E$, $V(r)$ also occurs in the cohomology of a product of abelian varieties with CM by $E$, and $A$ is dominated by such products.
\end{proof}
\begin{proposition}
\label{product}
Let $E_1, \dots, E_m$ be CM-fields whose Galois closures are linearly disjoint over $\mathbb Q$.
For each $i=1,\dots,m$, let $A_i$ be an abelian variety with complex multiplication by $E_i$, and let $\mathcal A_i$ be a class of varieties which dominates $A_i$.
Then $A = \prod_{i=1}^m A_i$ is dominated by $\prod_{i=1}^m \mathcal A_i$.
\end{proposition}
\begin{proof}
Let $E$ be the compositum of the $E_i$.
Let $\mathcal G = \Gal(\overline{E}/\mathbb Q)$, and, $\mathcal G_i = \Gal(\overline{E_i}/\mathbb Q)$, where bars denote Galois closure.
Then, the linear disjointness of the $\overline{E_i}$ implies that $\mathcal G = \prod_i \mathcal G_i$.
From $\mathcal G = \prod_i \mathcal G_i$, and, the explicit description of the cocharacter groups of the Mumford-Tate groups of abelian varieties of CM-type given by Deligne \cite{900}*{p.~47}, we immediately infer that the special Mumford-Tate group of $A$ is the product of the special Mumford-Tate groups of $A_i$.
Thus $G = \prod_i G_i$, where $G$ is the Hodge group of $A$, and $G_i$ the Hodge group of $A_i$.
Let $V$ be an irreducible Hodge structure in the cohomology of $A$.
Let $W$ be an irreducible $G_{\mathbb C}$-submodule of $V_{\mathbb C}$.
Write $W = \bigotimes_i W_i$, with $W_i$ an irreducible $G_{i,\mathbb C}$-module.
Since $\mathcal G = \prod_i \mathcal G_i$,
the set of Galois conjugates of $W$ consists of representations of the form
$W_1^{\sigma_1} \otimes \dots \otimes W_m^{\sigma_m}$,
with $\sigma_i \in \mathcal G_i$.
Let $V_i$ be the direct sum of all $\mathcal G_i$ conjugates of $W_i$.
Then, $V_{\mathbb C}$ contains $V' = V_1 \otimes V_2 \otimes \dots \otimes V_m$.
Since $V'$ is $\mathcal G$-invariant, it follows that $V'= V_0 \otimes E$ for some $G$-module $V_0$.
The irreducibility of $V$ then implies that $V = V_0$.
Now, each $W_i$ appears in $H^{a_i}(A_i,\mathbb C)$ for some $a_i$.
Hence $W_i$ occurs in a Hodge structure $M_i \subset H^{b_i} (B_i, \mathbb Q)$ for some $B_i \in \mathcal A_i$, and such that $M_i$ contains $(b_i,0)$-forms.
Let $M = \bigotimes_i M_i \subset H^n(B,\mathbb Q)$, where $n=\sum_i b_i$, and, $B=\prod_i B_i$.
Then, $M$ is equivalent to $V$ as a $G$-module, and contains $(n,0)$-forms.
\end{proof}
\section{Examples}
We give here all examples known to us of the general Hodge conjecture for CM abelian varieties.
This supersedes the listing in \cite{Abdulali2001}*{\S 4}.
We recall that a CM abelian variety is called \emph{nondegenerate} if its dimension equals the dimension of its Hodge group.
It is known (Hazama \cite{Hazama1983} and Murty \cite{Murty}) that if $A$ is a nondegenerate CM abelian variety, then, the Hodge ring of each power of $A$ is generated by divisors, so that the usual Hodge conjecture holds for all powers of $A$.
\begin{example}
\label{Tankeev}
Let $F$ be any totally real number field of degree $d$ over $\mathbb Q$.
Then there exists a totally imaginary quadratic extension $E$ of $F$ such that $[\overline{E} : \overline{F}] = 2^d$, where bars denote Galois closure (Shimura \cite{ShimuraCanonical}*{1.10.1, p.~155}).
$\Gal(\overline{E}/\mathbb Q)$ acts transitively on the set of CM-types of $E$ (Tankeev \cite{Tankeev2}*{Lemma~2.5, p.~187}).
Therefore, up to isogeny, there is only one abelian variety $A$ with complex multiplication by $E$.
Any power of $A$ is then dominated by the set of powers of $A$.
Since $A$ is nondegenerate (Ribet \cite{Ribet}*{Corollary 3.6, p.~87}), the general Hodge conjecture is true for all powers of $A$.
The general Hodge conjecture for $A$ was first proved by Tankeev \cite{Tankeev2}*{Theorem~2, p.~180}.
\end{example}
\begin{example}
If $F=\mathbb Q$, and, $K$ is an imaginary quadratic number field then the hypotheses of the previous example are satisfied.
Therefore, the general Hodge conjecture is true for any power of an elliptic curve with complex multiplication.
This result is due to Shioda \cite{Shioda}.
Let $K_1$, $K_2$, $K_3$ be distinct quadratic imaginary number fields.
Then it is easy to see that they are linearly disjoint.
Let $E_i$ be an elliptic curve with complex multiplication by $K_i$ for $i=1,2,3$.
Then Prop.~\ref{product} implies the general Hodge conjecture for all $E_1^j \times E_2^k \times E_3^{\ell}$.
\end{example}
\begin{example}
We will now see that the general Hodge conjecture holds for any power of an abelian surface of CM-type.
Let $F$ be a totally real quadratic number field, and, $E$ a totally imaginary quadratic extension of $F$.
Let $A$ be an abelian variety with CM by $E$.
If $A$ is not simple, then it is a product of two elliptic curves, and as in the previous example, we have the general Hodge conjecture for all powers of $A$.
Suppose now that $A$ is simple.
If $E$ is Galois over $\mathbb Q$, then $\mathcal G := \Gal(E/\mathbb Q)$ is cyclic of order $4$, acting transitively on the set of $4$ CM-types associated with $E$ (see \cite{ShimuraTaniyama}*{pp.~64--65}).
Thus, up to isogeny, $A$ is the only abelian variety with CM by $E$.
Therefore, $A$ is dominated by the set of powers of itself, and the general Hodge conjecture holds for all powers of $A$.
If $E$ is not Galois over $\mathbb Q$, then, up to isogeny, there are two abelian surfaces with CM by $E$ (Moonen and Zarhin \cite{MoonenZarhin1999}*{p.~725}).
Call them $A$ and $B$.
Any power of $A$ is dominated by the set of all abelian varieties of the form $A^i \times B^j$.
But the Hodge group of $A \times B$ equals the product of the Hodge groups of $A$ and $B$ \cite{MoonenZarhin1999}*{Prop.~4.2, p.~725}.
Hence the usual Hodge conjecture holds for all $A^i \times B^j$.
Thus, the general Hodge conjecture is true for any power of an abelian surface of CM-type.
\end{example}
\begin{example}
Let $A$ be a simple abelian variety of CM-type $(E,\Phi)$ such that $d := \dim A$ is odd.
Let $F$ be the maximal real subfield of $E$, and $\overline{F}$ its Galois closure.
If $\Gal(\overline{F} / \mathbb Q)$ is isomorphic to either the symmetric group or the alternating group on $d$ letters, then $A$ is nondegenerate (Dodson \cite{Dodson2}*{Prop.~2.1, p.~58}).
Suppose that $E$ does not contain any imaginary quadratic field.
Then, $[\overline{E}/\overline{F}] = 2^d$ (Dodson \cite{Dodson1}*{Prop.~2.2.2, p.~82}), so $A$ satisfies the hypotheses of Example~\ref{Tankeev}, and the general Hodge conjecture holds for all powers of~$A$.
\end{example}
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{"url":"https:\/\/johnhcochrane.blogspot.com\/2014\/03\/","text":"## Monday, March 31, 2014\n\n### EconTalk MOOC Podcast\n\n Russ Roberts\npodcast interview with Russ Roberts on EconTalk about my experience teaching a MOOC and thoughts on the economics of MOOCs. (The interview was based a bit on my last post here.)\n\nRuss is a very good interviewer, and the EconTalk series quite interesting.\n\n## Wednesday, March 26, 2014\n\n### The sign of monetary policy, part II\n\n(This blog post uses mathjax to show equations. You should see pretty equations, not ugly LaTex code.)\n\nThe ECB is in the news today. They want some inflation, yet the overnight rate is already zero. They're talking about negative interest rates, which leads to a great lunchroom discussion about bags of euros wandering around Europe. \u00a0All very interesting.\n\nYet it brings to mind a heretical thought I explored in an earlier blog post: What if we have the sign wrong on the effect of monetary policy? Could it be that to get more inflation, our central banks should raise rates not lower them? (Leave aside whether you think more inflation is good, which I don't. But suppose you want it, how do you get it?)\n\nIt's not as crazy as it sounds.\n\n### Interviews\n\nI did two interviews that blog readers might enjoy.\n\nThis is an interview with Jeff Garten at Yale, covering financial crises and reform\/regulation efforts rather broadly. Source here. It's part of a very interesting series of interviews on the \"future of global finance\" with lots of superstars. I give Niall Ferguson the prize for most creative \u00a0author photo.\n\nThis one\u00a0is a podcast interview on the ACA and how free-market health care can work,\u00a0with Don Watkins at the Ayn Rand institute's \"debt dialogues\" series. If you follow the link you get several formats.\n\n## Monday, March 24, 2014\n\n### Goodman Vs. Emanuel\n\nOn the fourth anniversary of the ACA, Saturday's Wall Street Journal had an excellent pair of pro and con OpEds from John Goodman \"A costly failed experiment\" and Ezekiel Emanuel \"Progress, with caveats.\"\n\n### Stein on Financial Stability in Monetary Policy\n\nFed governor and Harvard Professor Jeremy Stein gave an important speech on March 21,\u00a0Incorporating Financial Stability Considerations into a Monetary Policy Framework.\u00a0I have a few minor criticims, specifically on standard errors, causal mechanism, and Lucas critique. But it's great for Jeremy to think out loud this way, and give me occasion to do the same. You should read the whole thing.\n\nStein's bottom line:\n...all else being equal, monetary policy should be less accommodative--by which I mean that it should be willing to tolerate a larger forecast shortfall of the path of the unemployment rate from its full-employment level--when estimates of risk premiums in the bond market are abnormally low.\nThis view has put\u00a0Stein\u00a0a bit in the camps of the hawks, meaning simply those who for one reason or another\u00a0think the time to raise rates is sooner rather than later.\n\n## Friday, March 21, 2014\n\n### A World Without Banks?\n\nA graphic short story in this month's \"capital ideas.\"\u00a0 Click on the link or the image to read the whole thing (4 panels). If you can find the print magazine, the visual quality is much better. I think it does a great job of making economic ideas visual without too many talking heads and big balloons full of text. More of these to come in future \"Capital Ideas.\" More work from this unusually talented graphic novelist here. (My side of this \"debate\" is a bit captured here.)\n\n## Thursday, March 20, 2014\n\n### Hello Discretion\n\nToday, the much-anticipated first Fed policy statement of the Yellen era came out. FOMC statement, here.\n\nSome interesting tidbits:\nThe Committee will closely monitor incoming information on economic and financial developments in coming months and will continue its purchases of Treasury and agency mortgage-backed securities, and employ its other policy tools as appropriate, until the outlook for the labor market has improved substantially in a context of price stability. ... asset purchases are not on a preset course, and the Committee's decisions about their pace will remain contingent on the Committee's outlook for the labor market and inflation as well as its assessment of the likely efficacy and costs of such purchases.\nIn determining how long to maintain the current 0 to 1\/4 percent target range for the federal funds rate, the Committee will assess progress--both realized and expected--toward its objectives of maximum employment and 2 percent inflation. This assessment will take into account a wide range of information, including measures of labor market conditions, indicators of inflation pressures and inflation expectations, and readings on financial developments.\nWith the unemployment rate nearing 6-1\/2 percent, the Committee has updated its forward guidance.\nIn other words, the committee will do whatever it feels like doing, whenever it feels like doing it, based on whatever information it decides is relevant. The Committee updated its forward guidance by throwing it under a bus, or at least by clarifying that it is of the form \"here is what we think now we will want to do in the future, but we can change our minds at any time.\"\n\nThe larger context is the debate between commitment or rules and discretion. Discretion wins.\n\nYou might expect me to be fulminating. I'm not. (Though I'm waiting for a rules vs. discretion blast from\u00a0John Taylor! (Update: here it is.)\u00a0\u00a0I regard this as simply stating reality.\n\n### University Debt\n\nBloomberg has a story on the University of Chicago's big debt expansion. Obviously, it's a topic around faculty lounges too.\n\nA few thoughts. Why does a university simultaneously borrow $3.6 billion but have$6.7 billion Invested? If borrowing is such a big deal, why not just spend the endowment on new buildings?\n\nAnswer: universities can borrow at municipal rates, free of federal tax to the lender, if they are building something. Borrowing at tax-free rates makes financial sense, even you just stuff the marginal dollar into endowment. Of course the endowment is not invested in Treasuries -- universities don't do simple tax arbitrage. So the model is more that of a leveraged hedge fund -- borrow at low tax-free rates, up to the limit imposed by tax law, and invest in high risk, (hopefully) high-return projects like hedge funds, private equity, real estate etc. The fact that investment returns are also not taxed makes this a doubly advantageous strategy. Donors: if you give now, your gift grows tax-free, while if you earn the rate of return and then give the money to the university, you pay taxes on the intervening returns.\n\n## Monday, March 17, 2014\n\n### House of Debt\n\nAtif Mian and Amir Sufi have started a blog related to their new book, \"House of Debt.\" Amir and Atif are admirably data-oriented, which ought to make for good reading.\n\nToday's post \"Fed Meetings and Asset Prices\" is a good example. They put together one-day returns on the June 19 \"taper tantrum\" when the Fed announced it might (heavens) start tapering bond purchases. There is, of course, a large literature studying announcement effects. Atif and Amir \u00a0put together an unusually wide spectrum of asset classes.\n\n## Monday, March 10, 2014\n\n### Goodman Plan\n\nJohn Goodman has an excellent health-care piece at National Review Online. You don't have to subscribe to every element of his \"plan\" to appreciate many of his trenchant observations of coming Obamacare disasters. (Any \"plan\" that advertises it is crafted to meet perceived political constraints is bound to be less than perfect as a matter of economics.)\n\nThe slight weak point: he keeps community rating and guaranteed issue, but talks about how people need to sign up immediately or lose that benefit as they do in Medicare. I'm not sure just how he wants to do that or if that's realistic. But the big picture is right on: deregulated, individual, portable insurance.\n\nTransferability between plans is a nice point: \u00a0\"if an expensive-to-treat patient moves from Plan A to Plan B, the former has to compensate the latter for any above-average expected costs \u2014 just the way Medicare compensates private plans.\"\n\nBut read it for the mess we're in now. Lots of looming problems have not made headlines. Yet.\n\n### Asness and Liew on Efficiency\n\n Source: Institutional Investor\nCliff Asness and John Liew -- Chicago PhD's and now founding principals of AQR -- have a nice piece in Institutional Investor on Fama, Shiller, Nobel Prizes and efficiency.\n\nThey do a good job on the joint hypothesis theorem -- maybe a more important part of Fama's 1970 paper than efficiency itself -- and value and momentum strategies.\n\nThey point out one big difficulty for the inefficiency view (p.5). If value stocks are just overlooked and growth stocks irrationally overpriced, why do value stocks all subsequently rise or fall together, and growth stocks go the other way? \"Cheap stocks would get cheaper across the board at the same time. It didn't matter if the stock was an automaker or an insurance company. When value was losing it was losing everywhere.\"\n\nA second very important theorem: the average investor must hold the market portfolio, so alpha is a zero sum game. If you're going to profit, it helps a lot to identify just who the morons are whose money you are taking and why they're willing to give it to you. Everyone thinks the other guy is \"behavioral.\" Are you sure it's not you?\n\n## Saturday, March 8, 2014\n\n### Employment-Population Ratio: war of the graphs\n\nThe comments on my last post were particularly good, and pointed to some alternative graphs. And, I think, to the important conclusion, that there is no substitute really for sitting down and doing some economics.\n\n## Thursday, March 6, 2014\n\n### Employment-Population ratio\n\nTorsten Slok keeps making interesting graphs, which make a blogger's job easy.","date":"2022-01-29 10:53:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.183720201253891, \"perplexity\": 4557.780957851491}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320304883.8\/warc\/CC-MAIN-20220129092458-20220129122458-00428.warc.gz\"}"}
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Platformizing women's labour: Towards algorithms of empowerment
[By Pallavi Bansal]
As the fifth-born daughter to a poverty-stricken couple in a small village of Karnataka, Rinky would consider herself fortunate on days she wouldn't have to sleep on an empty stomach. Her parents pressurised her to take care of her younger brother while they struggled to make ends meet. As the siblings grew up, the brother started going to a nearby school, while Rinky managed the household chores along with her sisters. A curious teenager, Rinky coerced her brother to teach her every now and then, including how to operate a smartphone they had recently acquired. When she turned 19, she mustered the courage to move to Bengaluru in search of a better life. She survived by doing menial jobs in the beginning such as cleaning houses, cooking and washing dishes. She earned a mere amount of Rs 15,000 (about 200 USD) every month, enough just to get by. She always felt disrespected having to deal with constant humiliation until someone in the neighbourhood advised her to learn driving and partner with the ride-hailing platform Ola.
Image credit: Renate Köppel, Pixabay
While there were initial hiccups in procuring the vehicle and learning how the app works, this move dramatically changed her life as it turned her into a micro-entrepreneur with a lucrative income of Rs 60,000 to take home every month. Besides allowing flexible work hours, the job provided her a sense of independence which was missing when she worked for others. She wasn't really bothered about how the rides were assigned to her, though she always worried about her safety while boarding male passengers. At the same time, she was unable to comprehend how some of her colleagues earned more than her despite driving for similar number of hours.
"Rinky" is a composite character but represents stories of many such women for whom the platform-based economy has opened up a plethora of employment opportunities. The interesting aspect is that women workers are no longer confined to stereotypical jobs of salon or care workers; they are venturing into hitherto male domains such as cab driving and delivery services as well. The Babajob platform in fiscal 2016 recorded an increase of 153 per cent in women's applications for driver jobs. According to the Road Transport Yearbook for 2015-16 (the latest such report available), 17.73 % of the 15 million licensed female drivers ride professionally. Though there are no distinct figures available for how many women are registered with Ola and Uber as drivers, ride-hailing app Ola confirmed a rise of 40 % every quarter in the number of female drivers with them. Moreover, cab aggregative service Uber announced to tie up with a Singapore-based company to train 50,000 women taxi drivers in India by this year.
Clearly, the ride-sharing economy is helping Indian women to break the shackles of patriarchy and improve their livelihood. However, the potential of these platforms cannot be fully utilized unless researchers turn an eye on the algorithms that govern them. These algorithms not only act as digital matchmakers assigning passengers to the drivers but regulate almost all aspects of the job – from monitoring workers' behaviour to evaluating their performance. These machines often fail to treat workers as humans – as people who can fall sick, need a leisure break, socialise with others to stay motivated, de-route to pick up their kids from school, attend to an emergency at home, lose their temper occasionally, and moreover, coming to the work after facing physical abuse at home. In a normal work environment, employers tend to understand their team-members and often deal with compassion during tough times.
Image credit: Satvik Shahapur
Research shows how these data-driven management systems especially in context of ride-sharing apps impact human workers negatively as they lack human-centred design. They discovered that sometimes female drivers did not accept male passengers without pictures at night only to be penalized by these algorithms later. Moreover, drivers complain of rude passengers, which is seldom taken into consideration by platform companies and it only lowers the driver's acceptance rate and ratings.
Technology creators need to ask themselves how to ensure that algorithms are designed to enable workers and not just be optimized for customer satisfaction. Alternatively, they need to see the extension of worker's satisfaction as that of customer gratification given these two realms reinforce one another. By sensitizing to the needs of women like Rinky who are perhaps stepping out in this male-dominated world for the first time, programmers could create a more empowering pathway for such women workers. With the entrenched gender norms burdening women with familial duties and limiting their access to education and skills training, the intervention by platform designers can promise genuine change. While cultural change often takes a long course, by placing women at the centre, designers can accelerate this shift.
More concretely, what if platform companies did the following:
They create a feedback/resolution system which accounts for rejections and safeguards ratings when women drivers reject certain passengers if they consider them as potential threat.
They can institute flexibility in terms of wanting to go home early and this shouldn't be translated into 'lower incentives', after all this is the premise of gig economy.
AI should aim at promoting workers' well-being, which means following a demanding or intensive piece of work (a long ride in this case), AI could recommend a relatively easier task for drivers.
Another aspect is to ensure transparency in terms of how the wages are allocated to different people and an understanding of how the autonomous systems impact ratings with also a system of redressal, i.e. one that allows for corrections etc.
Algorithms should encourage a community-building culture rather than individualism-oriented – social incentives could be given to those drivers who pick up rides when an assigned driver is unable to reach the destination instead of penalizing him or her.
While the in-built GPS system in apps can help drivers track public toilets and other places that could be used for restroom breaks, algorithms could be trained to adjust routes according to drivers' needs and availability of amenities.
Moreover, popular ride-sharing platforms like Ola and Uber can consider assigning women passengers to women riders especially during the night-time. This move can make both the parties feel secure considering women dread boarding a taxi in an 'unsafe' country like India.
Across disciplines, if we brainstorm on reimagining these platforms as cooperative instead of competitive spaces, of human-centered versus optimization-centered, and as feminist-oriented and not just male-oriented, there may be more promise for our digital wellbeing.
Tags Gig economy, Pallavi Bansal, Platforms
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The statistic shows a comparison of display advertising market value and ad blocking losses worldwide in 2016 and 2017. According to the estimates, display ad market in 2017 was worth approximately 100 billion U.S. dollars. At the same time, 42 billion U.S. dollars out of this amount was believed to be lost due to ad blocking.
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Game Royale 2 - The Secret of Jannis Island Review
Game Royale 2 - The Secret of Jannis Island
Game Royale 2 - The Secret of Jannis Island is the highly anticipated sequel to our successful first adventure game Game Royale - Riders of the lost bald.
The game is a point-&-click adventure and a tribute to the great classics of this genre.
The second part of Game Royale continues the story right after the dramatic cliffhanger seen at the end of the first part. The player guides the main character Jan Bhmermann through a short and thrilling adventure full of puzzles and humorous dialogues. The story begins with Jan, as he wakes up on a remote island without any memories of previous events. He has to fight for survival until he is finally able to call for help. While the main storyline takes place on said island, we will also revisit the shows venue, the famous Studio Knig in Kln-Ehrenfeld. In a parallel storyline, the player takes on the shape of popular sidekick William Cohn, who makes it his own personal mission to find his missing friend. Apart from these two, there will be a reunion with other popular figures of the Neo Magazin Royale world. All of them are dubbed by the real people on the show.
***THIS GAME IS AVAILABLE IN GERMAN LANGUAGE ONLY***
Game Royale 2 - The Secret of Jannis Island FAQ
There's not a lot of questions about Game Royale 2 - The Secret of Jannis Island. Ask insistently in the comments so you can make it frequent! Yup,take me to comments
Miasmata
Temple of the Apsara
Evertree Inn
© 2016 btf GmbH © 2016 ZDF, licensed by ZDF Enterprises GmbH. Distributed by Headup Games GmbH & Co. KG
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{"url":"https:\/\/ask-public.com\/1370911\/","text":"# In a uniform magnetic field of induction B, a wire, in the form of semicircular of radius \u2018R\u2019, rotates about the diameter of the circle with an angular speed '\u03c9' . The axis of rotation is perpendicular to the field. If the total resistance of the circuit is R\u2019, the near power generated per period of rotation is (1) $$\\frac{B\\pi R^2 \u03c9}{2R'}$$ (2) $$\\frac{(B\\pi R^2 \u03c9)^2}{8R'}$$ (3) $$\\frac{(B\\pi R\u03c9)^2}{8R'}$$ (4) $$\\frac{B\\pi R^2\u03c9^2}{8R'}$$\n\n0 views\nIn a uniform magnetic field of induction B, a wire, in the form of semicircular of radius \u2018R\u2019, rotates about the diameter of the circle with an angular speed '\u03c9' . The axis of rotation is perpendicular to the field. If the total resistance of the circuit is R\u2019, the near power generated per period of rotation is (1) $$\\frac{B\\pi R^2 \u03c9}{2R'}$$ (2) $$\\frac{(B\\pi R^2 \u03c9)^2}{8R'}$$ (3) $$\\frac{(B\\pi R\u03c9)^2}{8R'}$$ (4) $$\\frac{B\\pi R^2\u03c9^2}{8R'}$$\n\n## Related questions\n\n0 views\n\nDescription : A circular coil, of radius a, (having N turns) is made to rotate about its vertical diameter with an angular speed \u03c9. The coil is present in a region where a uniform horizontal magnetic field B is present. If the coil has a resistance ... a^2 \u03c9B}{\\sqrt{2}R}\\) and $$\\frac{(N\\pi a^2 \u03c9B)}{\\sqrt{2}R}$$\n\n0 views\n\nDescription : A straight rod PQ, of length L, is rotating about an axis passing through O' and perpendicular to its plane. The rod rotates in a uniform (and normal) magnetic field B, with an angular speed \u03c9 (The point O' is at a perpendicular ... {B\u03c9L^2}{4}\\) (3) $$\\frac{B\u03c9L^2}{3}$$ (4) $$\\frac{1}{2}B\u03c9L^2$$\n\n0 views\n\nDescription : A planar circular coil, of radius a, has N turns and it can be made to rotate about its diameter as axis. The coil has a resistance R and is present in a region where a uniform magnetic field, B ... spin' at start, it would then keep on rotating at its initial rate, due to its rotational inertia'.\n\n0 views\n\nDescription : An A.C. voltage source (v = v0 sin \u03c9t) , is connected across a series LCR Circuit. If the values of L and C, in the circuit, equal $$(\\frac{2\\sqrt{3}R}{\u03c9})$$ and $$(\\frac{1}{\\sqrt{3}(R\u03c9)})$$ respectively; the phase ... ((\\frac{1}{\\sqrt{2}})\\) (4) $$(\\frac{\\pi}{3})$$ and $$(\\frac{\\sqrt{3}}{6})$$\n\n0 views\n\nDescription : A uniform magnetic field B exists in a cylindrical region of radius 0.1 m as shown in the figure. A uniform wire of length 0.80 m and resistance 4.0\u03a9 is bent into a square frame and is placed with one side along a diameter ... frame. [Hint: |\u03b5|= $$\\frac{d\u03d5_B}{dt}=\\frac{1}{2}\\pi^2\\frac{dB}{dt}$$ ]\n\n0 views\n\nDescription : A rectangular coil, of 300 turns, has an average area of 25 cm x 10 cm. The coil rotates, with a speed of 50 cycles per second, in a uniform magnetic field of 4 x 10-2T, about an axis perpendicular to the field. The peak value of induced emf (in volt), is (1) 3 \u03c0 (2) 30 \u03c0 (3) 300 \u03c0 (4) 3000 \u03c0\n\n0 views\n\nDescription : A thin semicircular conducting ring, of radius R, in falling with its plane vertical in a uniform horizontal magnetic field of strength B. At the position MNQ the speed of the ring is V. The potential ... higher potential (3) 2RBV, with Q at higher potential (4) 2R2BV, with M at higher potential\n\n0 views\n\nDescription : A conducting circular loop is placed in a uniform magnetic filed (of induction B'tesla), with its plane normal to the field. If the radius of the loop were to start shrinking at a constant rate dr\/dt. The induced emf, at the instant ... {dr}{dt})B\\) (4) $$-\\frac{1}{2}\\pi r^2 B(\\frac{dr}{dt})B$$\n\n0 views\n\nDescription : A mini generator has a coil of 1000 turns, each of area 10-2m2 . The coil is placed with its plane perpendicular to a uniform magnetic field of intensity 25 mT and is rotated at a uniform rate of 100 rotations per ... 3) 100 volt, 0 volt, 100 volt, 0 volt (4) 100 volt, 100 volt, 100 volt, 100 volt\n\n0 views\n\nDescription : A conducting circular loop is placed in a uniform magnetic field B = 0.25 T, with its plane perpendicular to the field. The radius of the loop is made to shrink at a constant rate of 1 mm\/s. The induced emf, when the radius is 2 cm, ... })\u03bcV\\) (2) $$(\\pi)\u03bcV$$ (3) $$(2\\pi)\u03bcV$$ (4) $$(2.5)\\pi \u03bcV$$\n\n0 views\n\nDescription : A uniform disc of mass M, radius R is rotating in a horizontal plane about its own axis with a constant angular speed \u03c90 . A ring of mass M\/2 ; radius R\/2 is initially at rest. The ring is gently placed coincentrically on rotating ... \\) (3) $$\\frac{MR^2\u03c9^2_ 0}{5}$$ (4) $$\\frac{MR^2\u03c9^2_ 0}{4}$$\n\n0 views\n\nDescription : The set-up, shown in the figure, is present in a uniform magnetic field, B, directed perpendicular to the plane of this set-up'. The rod, PQ, of length L, is allowed to slide down vertically, under its own weight, on the (shaded) ... })\\) (3) $$(\\frac{B^2l^2}{mgR})$$ (4) $$(\\frac{mgR}{B^2l^2})$$\n\n0 views\n\nDescription : Two circular loops, of radii R and r (R >> r), respectively, are positioned, parallel to each other, in the yz plane. The centres and axis, of both the loops, lie on the x-axis, a distance x apart. The larger loop has N turns, ... {\u03bc_0\\pi}{2x^2}(nN)(Rr^2)\\) (4) $$\\frac{\u03bc_0\\pi}{2x^3}(nr^2)(NR^2)$$\n\n0 views\n\nDescription : The magnetic flux \u03d5 , through a stationary loop of wire, having a resistance R, varies with time as \u03d5 = (at2 + bt) where a and b are positive constants. The average emf, and the total charge flowing in the loop in the time interval t ... b \\tau}{R})\\) (4) $$2(a\\tau +b),(\\frac{a\\tau ^2+b \\tau}{2R})$$\n\n0 views\n\nDescription : A particle moves in x-y plane in a circular path of radius R with a constant angular velocity \u03c9 . At t = t, the radius vector joining particle with centre of circle makes an angle \u03b8 with x-axis. The instantaneous linear velocity v ... )$$\\hat{i}$$ - (R\u03c9 sin\u03b8) (4) (-R\u03c9 sin\u03b8) - (R\u03c9 sin\u03b8)$$\\hat{j}$$\n\n0 views\n\nDescription : A solid conducting sphere, of radius R and total charge q, rotates about its diametric axis with a constant angular speed '\u03c9' . The equivalent magnetic moment of the sphere is (1) 1\/3 qR2 \u03c9 (2) 2\/3 qR2 \u03c9 (3) 1\/5 qR2 \u03c9 (4) 2\/5 qR2 \u03c9\n\n0 views\n\nDescription : A very small circular loop of area 5 x 10-4m2 , resistance 2 ohm and negligible inductance, is initially coplanar and cocentric, with a much larger fixed circular loop of radius 0.1m. A constant ... diameter. Calculate the induced emf and induced current, as a function of time, in the smaller loop.\n\n0 views\n\nDescription : A rigid wire consists of semicircular portion of radius R' and two straight sections as shown in figure. It is present in a uniform perpendicular magnetic field B'. The magnetic force on the wire, when it carries a ... (1) (2i RB) upward (2) (2iRB) downward (3) (i\u03c0RB) upward (4) (i\u03c0RB) downward\n\n0 views\n\nDescription : A long cylindrical conductor, of radius a', has two cylindrical cavities of diameter a', through its entire length, as shown in cross section in figure. A current I is directed out of the page and is uniform throughout the cross ... $$\\frac{\u03bc_0I}{2\\pi r}[\\frac{2r^2 +a^2}{4r^2 + a^2}]$$ downwards\n\n0 views\n\nDescription : A uniform disc of mass 2m, radius R is at rest in a vertical position. It is free to rotate about a horizontal axis through center O. A particle of m moving with speed v is aimed towards the edge of the disc as shown in ... gets embedded is (neglect gravity) (1) v\/R (2) v\/2R (3) 2v\/R (4) 3\/2 v\/R\n\n1 view\n\nDescription : A (infinite) long straight wire, carrying a current I, is placed parallel to the x-axis. If a straight conductor, of length l , positioned along the y-axis, as shown, starts moving with a uniform velocity v = v$$\\hat{i}$$ , the ... +\\frac{l}{a})\\) (4) $$\\frac{\u03bc_0Iv}{2\\pi}\\frac{l}{(a+\\frac{l}{2})}$$\n\n0 views\n\nDescription : A conducting loop of area 5.0 cm2 is placed in a magnetic field which varies sinosoudally with time as B = B0 sin \u03c9t where B0 = 0.20 T and \u03c9 300\/s, such that the normal to the coil makes an angle of 60\u00b0 with ... ) the maximum emf induced in the coil (b) the emf induced at $$t =(\\frac{\\pi}{600})s.$$\n\n0 views\n\nDescription : A conducting wire xy, of mass m, (and negligible resistance) slides smoothly on two parallel conducting wires as shown in the diagram. The closed circuit has resistance R' due to side AC. Sides AB and CD are perfect conductors. If ... frac{l^2B^2}{mR}\\frac{dx}{dt}+\\frac{l^2B}{mR}\\frac{dB}{dt}x(t)\\)\n\n0 views\n\nDescription : Instantaneous voltage of the amplitude modulated wave is given by $$\\nu_{AM}=V_c(1+m\\,sin\\,\u03c9t)sin\\,\u03c9_ct$$ Assuming that the effective resistance of the modulator circuit is R, the total power of the amplitude modulated wave will be equal ... {m^2V^2_C}{4R}\\) (4) $$\\frac{V^2_c}{2R}[\\frac{1+m^2}{2}]$$\n\n0 views\n\nDescription : A uniformly charged disc, whose total charge has a magnitude \u2018q\u2019, and whose radius is \u2018r\u2019, rotates with a constant angular velocity of magnitude '\u03c9'. The magnetic dipole moment of the ring is (1) $$\\frac{q\u03c9r^2}{4}$$ (2) $$\\frac{q\u03c9r^2}{2}$$ (3) $$\\frac{q\u03c9r^2}{8}$$ (4) $$q\u03c9r^2$$\n\n0 views\n\nDescription : A rod of mass M; length L is made of material of Young's modulus Y. The rod rotates in a horizontal plane about an axis through its one end and perpendicular to length of rod with a constant angular speed \u03c9. The increase in length of ... (\\frac{p\u03c9^2}{Y})L^3\\) (d) $$\\frac{2}{3}(\\frac{p\u03c9^2}{Y})L^3$$\n\n0 views\n\nDescription : A rod, PQ, of length l , slides with a constant velocity on two conducting rails. The system of the rod and conducting rails, is present in a region where a uniform (normal) magnetic field, of strength B, is present. The two ... {16Ri_0}{Bl}\\) (3) $$\\frac{Bl}{(16Ri_0)}$$ (4) $$\\frac{3Bl}{(16Ri_0)}$$\n\n0 views\n\nDescription : A sinusoidal A.C. voltage source, of adjustable frequency, is connected across a series LCR' combination. The inductance (L) and the capacitance (C), present in the circuit, can be expresssed in terms of the quality factor (Q) the ... $$L=\\frac{1}{\u03c9_0Q^2R^2}$$ and $$C = (\\frac{Q^2R^2}{\u03c9_0})$$\n\n0 views\n\nDescription : A uniform disc of mass 2m radius R is pivoted at its center O on a smooth horizontal surface. The disc in free to rotate about a vertical axis ZOZ\u00b4 through its center O. The disc is initially at rest. A particle of mass m is moving ... $$\\frac{2v}{R};\\frac{v}{R}$$ (4) $$\\frac{2v}{R};\\frac{v}{2R}$$\n\n0 views\n\nDescription : A uniform cylinder of mass M, radius R is rotating about its own axis with a speed of n r.p.s. It is gently placed against a corner as shown in Fig.. Coefficient of freection between walls and cylinder is \u03bc. The number of revolutions ... {48\\pi^2\u03bcg(\u03bc+1)}\\) (4) $$\\frac{n^2R(\u03bc^2+1)}{32\\pi^2\u03bcg(\u03bc-1)}$$\n\n0 views\n\nDescription : A voltage source, v = v0 sin \u03c9t, is connected across a series LCR circuit. The values of L and C, in the circuit, equal $$\\frac{9R}{\u03c9}$$ and $$\\frac{1}{10R\u03c9}$$ , The phase angle (between the current and voltage in ... ) and $$3\\sqrt{10}$$ (4) $$\\frac{\\pi}{4}$$ (Current lagging) and $$2\\sqrt{10}$$\n\n0 views\n\nDescription : A charge \u2018q\u2019 is uniformly distributed on a non-conducting disc, of radius R, it is rotated with angular speed '\u03c9' about an axis passing through the centre of mass of the disc and perpendicular to its plane. The magnetic moment of the disc will be (1) 1\/4 \u03c9qR2 (2) 1\/2 \u03c9qR2 (3) \u03c9qR2 (4) 1\/8 \u03c9qR2\n\n0 views\n\nDescription : A 10 meter long wire is kept in east west direction. It is falling down with a speed of 5 m\/s, perpendicular to the horizontal component of earth's magnetic field of 0.30 x 10-4 Wb\/m2 . (i) what ... .d induced between the ends of the wire? (ii) which end of the wire will be at a higher potential?\n\n0 views\n\nDescription : A uniform Catherine Wheel consists of many thin circular turns of a combustible material. The wheel is free to rotate about a vertical axis through its center in a horizontal plane. The combustile material burns at a constant ... (\\frac{4\\pi F\u03c3R}{M_1\\alpha}\\) (4) $$\\frac{2\\pi F\u03c3R}{M_1\\alpha}$$\n\n0 views\n\nDescription : A conducting ring, of radius r, is placed in a varying magnetic field perpendicular to the plane of the ring. If the rate at which the magnetic field, varies, is x, the (average) electric field intensity, at any point of the ring, would be (1) r x (2) rx\/2 (3) 2rx (4) 4r\/x\n\n0 views\n\nDescription : A uniform (normal) magnetic field, B, bends the path of the most energatic photoelectrons, (emitted from a given photosensitive surface due to the action of monochromatic radiations of wave length \u03bb ) into a circle of radius r. The work ... 2r^2}{2m}]\\) (4) $$[\\frac{hc}{\u03bb}+\\frac{e^2B^2r^2}{2m}]$$\n\n0 views\n\nDescription : A small coil, having n turns each of (everage) radius r, is held normal to the magnetic field liens in the region between the (flat) pole pieces of a horse shoe magnet. The terminals ofthe coil are connected to a ... the magnetic field), is the graph labelled as graph. (1) K (2) L (3) M (4) N\n\n0 views\n\nDescription : A wire of mass m is bent into an equilateral triangle of side l . Two beads (identical) each of mass m0 can slide freely along sides PQ and QR of triangle. The triangle is set into ... i.e. sum of kinetic and potential energy) and total angular momentum (4) Kinetic energy and angular momentum\n\n0 views\n\nDescription : The magnetic flux, through a coil, present in a magnetic field, directed perpendicular to its plane, varies with time (in second) according to the relation. \u03d5B = 6t2 + 7t + 1 (\u03d5B in miliweber) Find the ... at t = 2s. Also find the current, and its direction, in the resistance R, if R = 10\u03a9\n\n0 views\n\nDescription : In A.C. voltage source, v = v0 sin t; having a fixed value for v0 , but of variable frequency, is connected across a given series LCR circuit. The variation of the current I, flowing in the circuit, ... \u03c90 and the difference, (\u03c92- \u03c91) , increases with an increase in the Q-factor' of the circuit\n\n0 views\n\nDescription : In A.C. voltage source, v = v0 sin t; having a fixed value for v0 , but of variable frequency, is connected across a given series LCR circuit. The variation of the current I, flowing in the circuit, ... \u03c90 and the difference, (\u03c92- \u03c91) , increases with an increase in the Q-factor' of the circuit\n\n0 views\n\nDescription : A regular polygon of n-sides is formed by bending a wire of total length 2\u03c0r. If this wire now carries a current i, the magnetic field B, at the centre of this polygon, would be (1) $$\\frac{\u03bc_0in^2 sin^2 (\\pi \/n)}{2r^2 \\pi ... (3) \\(\\frac{\u03bc_0in^2 sin^2 (\\pi \/n)}{2\\pi^2 r \\,cos (\\pi \/ n)}$$ (4) zero\n\n0 views\n\nDescription : A thin uniform rod AB of mass 2m has length 2L. The rod is pivoted at midpoint O on a smooth horizontal surface. A particle P of mass m moving with speed v0 as shown in Fig ; hits rod at point C and sticks to it. The rod-mass ... ) (3) $$v_0=\\frac{7}{12}\\sqrt{\u03c9}L$$ (4) $$v_0=\\frac{5}{12}\\sqrt{\u03c9}L$$\n\n0 views\n\nDescription : A given AC source, v = v0 sin \u03c9t, is connected across a given series LCR circuit. The graph, showing simultaneously, the variation of the voltage and current in the circuit, with time, has the form shown. The current, ... ) (4) $$(\\frac{\u03c9^2 C}{1-\u03c9^2LC})$$ in series with the given inductor (L)\n\n0 views\n\nDescription : An electric current i' enters, and leaves, a uniform circular wire of radius a' through diametrically opposite points, A charged particle q, moving along the axis of circular wire, passes through its centre at speed v'. The ... $$qv\\,\\frac{\u03bc_0i}{4a}$$ (3) $$qv\\frac{\u03bc_0i}{2\\pi a}$$ (4) zero\n\n0 views\n\nDescription : A series LCR circuit is connected to an a.c. voltage source (v = v0 sin\u03c9t) and the value of C is such that 1\/\u03c9C \u2243 0. The phase angle (\u03b8), between the current flowing and the voltage applied, then equals. (1 ... tan-1 (1\/\u03c9R); current lagging; \u03b8 can be made zero by taking a value of C equal to (1\/\u03c92L)\n\n0 views\n\nDescription : A wire of uniform resistance z r m-1 is sent into a circle of radius r. The same wire is connected between point A and B as shown. The equivalent resistance between A and B is (1) $$\\frac{Zr}{(3\\pi +16)}$$ (2) $$\\frac{6\\pi (z\/ ... (3) \\(\\frac{6\\pi z r}{(3\\pi + 16)}$$ (4) $$(\\frac{z}{r})(3\\pi + 16)$$\n\n0 views\n\nDescription : A thin wire of uniform linear density \u03bb is bent into a semi-circle of radius R. What is the intensity of gravitational field at center O? (1) $$\\frac{2GM}{R^2}j$$ (2) $$\\frac{2GM}{\\pi R^2}j$$ (3) $$\\frac{\\sqrt{2}GM}{R^2}j$$ (4) $$\\frac{3GM}{R^2}j$$\n\nDescription : Consider two coplanar, co-centric circular coils having radius r1 and r2 (r2 > r1) as shown in the figure. The mutual inductance between these two coils will be (1) $$\\frac{\u03bc_0\\pi r^2_1}{2r_2}$$ (2) $$\\frac{\u03bc_0\\pi r_1}{2r_2}$$ (3) $$\\frac{\u03bc_0\\pi r^2_2}{2r_2}$$ (4) $$\\frac{\u03bc_0\\pi r^2_1}{2r_1}$$","date":"2023-01-31 02:51:48","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\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8094252347946167, \"perplexity\": 817.889954120462}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499842.81\/warc\/CC-MAIN-20230131023947-20230131053947-00864.warc.gz\"}"}
| null | null |
Pronous tetralobus är en spindelart som beskrevs av Simon 1895. Pronous tetralobus ingår i släktet Pronous och familjen hjulspindlar.
Artens utbredningsområde är Madagaskar. Inga underarter finns listade i Catalogue of Life.
Källor
Hjulspindlar
tetralobus
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 5,963
|
{"url":"https:\/\/homework.cpm.org\/category\/CCI_CT\/textbook\/pc\/chapter\/4\/lesson\/4.2.4\/problem\/4-118","text":"Home > PC > Chapter 4 > Lesson 4.2.4 > Problem4-118\n\n4-118.\n\nAn exponential function passes through the points $(1, 70)$ and $(3, 145)$. The function has a horizontal asymptote at $y = 10$. Find an equation for the function.\n\nWrite the general equation for this problem.\n\nSubstitute the points in for $(x, y)$, creating two equations and two unknowns.\n\nSubtract $10$ on both sides.\nThen divide one equation by the other.\n\nSubstitute the $b$-value back into one of the original equations and solve for $a$. Then substitute $a$ and $b$ into the general equation.\n\n$y = a(b^x) + 10$\n\n$145 = a(b^3) + 10$\n$70 = a(b^1) + 10$\n\n$135 = a(b^3)$\n$60 = a(b^1)$\n$2.25 = b^2$\n$b = 1.5$","date":"2022-05-16 04:41:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 16, \"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\": 0.808506190776825, \"perplexity\": 259.72350522394424}, \"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-2022-21\/segments\/1652662509990.19\/warc\/CC-MAIN-20220516041337-20220516071337-00357.warc.gz\"}"}
| null | null |
package Player;
public interface _MonitorOperationsNC
{
void report(String action, Song s);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,379
|
Tag Archives: Gerald Stubbs
Searching the Callaghan
1956 was the year that a T33 military jet mysteriously disappeared over the Callaghan Valley area. The two pilots inside were never found and 60 years later only a few pieces have been found that would give us any clue as to what happened to the two men inside the plane when it went down.
The two men were First Officers James Miller and Gerald Stubbs of the 409 Squadron of the Royal Canadian Air Force. Their flight was only scheduled to take an hour and a half and be within a hundred mile radius of the Comox base on Vancouver Island. Yet just seventeen minutes into their flight they were documented by radar as entering bad weather and were never seen again.
That is until eighteen years later in November of 1974 when the canopy of their plane was found in the Callaghan Valley, nowhere near where the search teams had been looking for the men. Forty-two years after the plane disappeared its fuselage was found not far from the Callaghan Country lodge, and then twelve years later in October 2010, remnants of one of the pilot's helmets was found and identified by its colours.
Google Earth image of the location of the T33 crash debris. GPS data courtesy Whistler Search & Rescue.
The Whistler Museum now preserves those fragments of helmet in our archive room. It is likely they will have to be sent off to be cleaned at the Royal BC Museum though as our small museum does not have the resources to properly clean them. Archival-level preservation becomes especially challenging when you have multiple types of materials in a single artifact, like, for example, the plastic, foam, metal, and leather in a pilot's helmet. Fifty-four years in the elements has not been kind to the pieces of the flight helmet and it will take a lot of care for them to be able to be displayed in the future.
The remnants of the flight helmet.
The Whistler Museum also has what we can only assume is a piece from the windshield of the plane as well; a large jagged piece of curved plexiglass as well as a chunk of metal tubing. These pieces along with the helmet fragments were donated to the Museum from the RCMP after they were found in 2010.
Callaghan Valley in the 1960s.
A television show called "Callout: Search and Rescue" even did an episode on this mysterious crash for the first episode of their third season. The episode covers the Search and Rescue team scouring the Callaghan Valley looking for any missed clues as to what may have happened to the pilots.
In October the Search and Rescue team does an annual search of the Valley and they continuously look for things like ejection seats, helmets or boots. Things that will withstand the elements and will also stand out in the forest. As of the last search in October 2015 nothing else has been found but the search still continues.
For more information check out this feature article written by Pique Newsmagazine in 2015.
Posted in Environment & Biodiversity, From the Archives, Mountain Culture
Tagged artefact, Callaghan Valley, Gerald Stubbs, James Miller, plane crash, Royal Canadian Air Force, Whistler history, Whistler Museum, Whistler Search and Rescue
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 3,689
|
The Crewe by-election was a Parliamentary by-election held on 26 July 1912. The constituency returned one Member of Parliament (MP) to the House of Commons of the United Kingdom, elected by the first past the post voting system.
Vacancy
Walter McLaren had been Liberal MP for the seat of Crewe since the April 1910 By-Election. In 1912, he died causing the vacancy.
History
The Liberal party had won every election in Crewe, since the seat was created in 1885 apart from the 1895 election, when a Conservative won.
Candidates
The Liberal candidate was 30-year-old Harold Lawson Murphy, a lecturer in Political Economy in Trinity College Dublin. He had trained as a solicitor and was secretary to the Liberal Cabinet Minister, Sir John Simon.
The Unionist candidate was Ernest Craig, who had been the unsuccessful Liberal Unionist candidate here in December 1910.
The Labour party, who had not fielded a candidate in December 1910 having fielded a candidate in January 1910, decided to re-enter the contest. Their candidate was James Holmes who was a member of the Amalgamated Society of Railway Servants.
Given the intervention of the Labour party, the result of the previous three-way contest is relevant;
Result
The intervention of the Labour candidate took enough votes from the Liberal candidate to help the Unionist candidate win.
Aftermath
A General Election was due to take place by the end of 1915. By the autumn of 1914, the following candidates had been adopted to contest that election;
Unionist: Ernest Craig
Liberal: Joseph Davies
Following the by-election the Labour party decided to re-adopt Holmes as their prospective parliamentary candidate. The National Union of Railwaymen agreed to be his sponsor. However Holmes was concerned that another election where the progressive vote was split would result in another Unionist victory. He called for the Labour and Liberal parties to come to some sort of electoral arrangement as had been the practice in the past. In response, the Labour party decided to drop him as prospective candidate.
Due to the outbreak of war, the election never took place.
Joseph Davies, who had been adopted as Liberal candidate back in 1913 was a supporter of David Lloyd George and in 1918 was granted the 'Coalition Coupon'. As a result, Ernest Craig withdrew and did not defend the seat he had won 6 years earlier.
References
Craig, F. W. S. (1974). British parliamentary election results 1885-1918 (1 ed.). London: Macmillan.
The Social and Economic Development of Crewe, 1780-1923 by William Henry Chaloner [1950]
The Socialist review - Volume 10 - [1913]
Wikipedia: en.wikipedia.org
Who's Who: www.ukwhoswho.com
Debrett's House of Commons 1916
1912 elections in the United Kingdom
1912 in England
20th century in Cheshire
By-elections to the Parliament of the United Kingdom in Cheshire constituencies
Crewe
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,635
|
{"url":"https:\/\/devblogs.nvidia.com\/parallelforall\/tag\/cuda-c\/","text":"# High-Performance Geometric Multi-Grid with GPU Acceleration\n\nLinear solvers are probably the most common tool\u00a0in scientific computing applications. There are two basic classes of methods that can be used to solve an $Ax=b$ equation: direct and iterative. Direct methods are usually robust, but have additional computational complexity and memory capacity requirements. Unlike direct solvers, iterative solvers require minimal memory overhead and feature\u00a0better computational complexity. However, these solvers are still super-linear in the number of variables and often have a slow rate of convergence of low-frequency errors. Finally, there are multi-grid iterative methods\u00a0that can deliver linear complexity by solving a problem at different resolutions and smoothing low frequency errors using\u00a0coarser grids.\n\nBroadly speaking, multi-grid methods can be differentiated into more general algebraic multi-grid (AMG) and the specialized geometric multi-grid (GMG). AMG is a perfect \u201cblack-box\u201d solver\u00a0for problems with unstructured meshes, where elements or volumes can have different numbers of neighbors, and it is difficult to identify a subproblem. There is\u00a0an interesting blog post\u00a0demonstrating that\u00a0GPU accelerators show good performance\u00a0in AMG using the\u00a0NVIDIA AmgX\u00a0library. GMG methods are more efficient than AMG on structured problems, since they can take advantage of the additional information from the geometric representation of the problem. GMG solvers have significantly lower memory requirements, deliver higher computational throughput and also show good scalability. Moreover, these methods require less tuning in general and have a simpler setup\u00a0than AMG. Let\u2019s take a closer look at GMG and see how well it maps to GPU accelerators. Continue reading\n\n# Cutting Edge Parallel Algorithms Research with CUDA\n\nLeyuan Wang, a Ph.D. student in the UC Davis Department of Computer Science, presented one of only two \u201cDistinguished Papers\u201d of the 51 accepted at Euro-Par 2015.\u00a0\u00a0Euro-Par is a European conference devoted to all aspects of parallel and distributed processing held\u00a0August 24-28 at Austria\u2019s Vienna University of Technology.\n\nLeyuan\u2019s\u00a0paper\u00a0Fast Parallel Suffix Array on the GPU, co-authored by her advisor John Owens and Sean Baxter, a research scientist at New York\u2019s DE Shaw Research, details their\u00a0efforts to implement a linear-time suffix array construction algorithm on NVIDIA GPUs, resulting in algorithmic improvements and significant speedups over the existing state of the art.\n\nWang completed her master\u2019s degree in electrical and computer engineering at UC Davis in October 2014, after having earned her undergraduate degree in electronics science and technology at China\u2019s Zhejiang University.\n\nLeyuan Wang: I work on high-performance string processing and graph processing algorithms, mostly in string and graph queries. My current research focus is on GPGPU (general-purpose computing on graphics processing units) and the benchmark I care about most is speed. I\u2019ve been working on designing and improving parallel suffix array construction algorithms (SACAs) and incorporating the implementations in a Burrows-Wheeler transform-based lossless data compression (bzip2) and a parallel FM index for pattern searching. The suffix array (SA) of a string is the sorted set of all suffixes of the string. The inverse suffix array (ISA) is also the lexicographic ranks of suffixes.\n\nThe Burrows-Wheeler transform (BWT) of a string is generated by lexicographically sorting the cyclic shift of the string to form a string matrix and taking the last column of the matrix. The BWT groups repeated characters together by permuting the string; it is also reversible, which means the original string can be recovered. These two characteristics make BWT a popular choice for a compression pipeline stage (for instance, bzip2). It is directly related to the suffix array: the sorted rows in the matrix are essentially the sorted suffixes of the string and the first column of the matrix reflects a suffix array. Table 1 shows an example of the SA, ISA and BWT of\u00a0the input string \u201cbanana\\$\u201d\n\nThe suffix array data structure is a building block in a spectrum of applications, including data compression, bioinformatics, text indexing, etc. I\u2019ve studied the taxonomy of all classes of SACAs and\u00a0compared\u00a0them in order to find the best candidate for the GPU.\u00a0I revisited the previous conclusion that skew SACAs are best suited on the GPU by demonstrating that prefix-doubling SACAs are actually better both in theoretical analysis and experimental benchmarks. Our hybrid skew\/prefix-doubling suffix array implementation (with our amazing research collaborator Sean Baxter, formerly of NVIDIA Research) using a Tesla K20 achieves a 7.9x speedup against the previous state-of-the-art skew implementation. Our optimized skew SACA implementation has been added as a primitive to CUDPP 2.2 (CUDA Data Parallel Primitives Library) and incorporated into the BWT and bzip2 data compression application, resulting in great speedups compared with bzip2 in CUDPP 2.1. Figure 1 shows pseudocode for our two approaches. Continue reading\n\n# Accelerating Materials Discovery with CUDA\n\nIn this post, we discuss how CUDA has facilitated materials research in the Department of Chemical and Biomolecular Engineering at UC Berkeley and Lawrence Berkeley National Laboratory. This post is a collaboration between Cory Simon,\u00a0Jihan Kim, Richard L. Martin, Maciej Haranczyk, and Berend Smit.\n\n## Engineering Applications of Nanoporous Materials\n\nNanoporous materials have nano-sized pores such that only a few molecules can fit inside. Figure 1 shows the chemical structure of metal-organic framework IRMOF-1, just one of the many thousands of nanoporous materials that have been synthesized.\n\nNanoporous materials have many potential engineering applications based on gas adsorption: the process by which gas molecules adhere to a surface. In this case, the walls of the material\u2019s pores form the surface to which gas molecules stick. Figure 2 shows the unit cell of the IRMOF-1 crystal structure and the corresponding depiction of IRMOF-1 as a raveled-up surface.\n\nIf we could unravel and flatten out the surface of IRMOF-1 in Figure 2, the surface area contained in a single gram of it could cover more than a soccer field! This provides a lot of surface area on which gas molecules can adsorb. These high surface areas are part of the reason that nanoporous materials are so promising for many engineering applications.\n\n# Voting and Shuffling to Optimize Atomic Operations\n\n2iSome years ago I started work on my first CUDA implementation of the Multiparticle Collision Dynamics (MPC) algorithm, a particle-in-cell code used to simulate hydrodynamic interactions between solvents and solutes. As part of this algorithm, a number of particle parameters are summed to calculate certain cell parameters. This was in the days of the Tesla GPU architecture (such as GT200 GPUs, Compute Capability 1.x), which had poor atomic operation performance. A linked list approach I developed worked well on Tesla and Fermi as an alternative to atomic adds but performed poorly on Kepler GPUs. However, atomic operations are much faster on the Kepler and Maxwell architectures, so it makes sense to use atomic adds.\n\nThese types of summations are not limited to MPC or particle-in-cell codes, but, to some extent, occur whenever data elements are aggregated by key. For data elements sorted and combined by key with a large number of possible values, pre-combining elements with the same key at warp level can lead to a significant speed-up.\u00a0In this post, I will describe algorithms for speeding up your summations (or similar aggregations) for problems with a large number of keys where there is a reasonable correlation between the thread index and the key. This is usually the case for elements that are at least partially sorted. Unfortunately, this argument works in both directions: these algorithms are not for you if your number of keys is small or your distribution of keys is random. \u00a0To clarify: by a \u201clarge\u201d number of keys I mean more than could be handled if all bins were put into shared memory.\n\nNote that this technique is related to a previously posted technique called warp-aggregated atomics by Andrey Adinetz, and also to the post Fast Histograms Using Shared Atomics on Maxwell by\u00a0Nikolay Sakharnykh. The main difference here is that we are aggregating many groups, each designated by a key (to compute a histogram, for example). So you could consider this technique \u201cwarp-aggregated atomic reduction by key\u201d. Continue reading\n\n# New Features in CUDA 7.5\n\nToday I\u2019m happy to announce that the CUDA Toolkit 7.5 Release Candidate is now available. The CUDA Toolkit 7.5 adds support for FP16 storage for up to 2x larger data sets and reduced memory bandwidth, cuSPARSE GEMVI routines, instruction-level profiling and more. Read on for full details.\n\n## 16-bit Floating Point (FP16) Data\n\nCUDA 7.5 expands support for 16-bit floating point (FP16) data storage and arithmetic, adding new `half` and `half2` datatypes and intrinsic functions for operating on them. 16-bit \u201chalf-precision\u201d floating point types are useful in applications that can process larger datasets or gain performance by choosing to store and operate on lower-precision data. Some large neural network models, for example, may be constrained by available GPU memory; and some signal processing kernels (such as FFTs) are bound by memory bandwidth.\n\nMany applications can benefit by storing data in half precision, and processing it in 32-bit (single) precision. At GTC 2015 in March, NVIDIA CEO Jen-Hsun Huang announced that future Pascal architecture GPUs will include full support for such \u201cmixed precision\u201d computation, with FP16 (half) computation at higher throughput than FP32 (single) or FP64 (double) .\n\nWith CUDA 7.5, applications can benefit by storing up to 2x larger models in GPU memory. Applications that are bottlenecked by memory bandwidth may get up to 2x speedup. And applications on Tegra X1 GPUs bottlenecked by FP32 computation may benefit from 2x faster computation on `half2` data.\n\nCUDA 7.5 provides 3 main FP16 features: Continue reading\n\n# BIDMach: Machine Learning at the Limit with GPUs\n\nDeep learning has made enormous leaps forward thanks to GPU hardware. But much Big Data analysis is still done with classical methods on sparse data. Tasks like click prediction, personalization, recommendation, search ranking, etc. still account for most of the revenue from commercial data analysis. The role of GPUs in that realm has been less clear. In the BIDMach project (part of the BID Data Project at UC Berkeley), we have been exploring general machine learning with GPUs. The results are remarkable: not only do we see order-of-magnitude speedups for most problems, but our system also outperforms today\u2019s cluster computing systems running up to several hundred nodes on typical workloads. As well as the incentives to adopt GPU technology for deep learning tasks, there is now a strong incentive for organizations to migrate to GPUs for the remainder of their analytics workload.\n\n## Roofline Design\n\nTo build the fastest system, we borrowed the approach of roofline design from computer architecture. Roofline design involves designing to fundamental limits (e.g. ALU throughput, memory speed, network speed, I\/O speed etc). A rooflined system is fast, and no other system can be much faster, since both have to respect the same hardware limits. A roofline diagram for matrix multiply is shown below:\n\nThe y-axis shows the potential throughput in arithmetic operations\/second. The x-axis is \u201coperational intensity\u201d which is the number of operations applied to each data value (in units of operations per byte). The intensity is much lower for sparse operations \u2013 e.g. sparse matrix multiply typically involves only a multiply and add for each input datum, while dense matrix multiply uses each datum many times. The horizontal lines reflect the maximum ALU throughput for each type of processor (the graph is drawn for Intel i7 and NVIDIA GeForce GTX 680 processors). GPUs have much higher ALU throughput since the GPU chip area is almost entirely ALU. For dense matrix multiply, GPUs are10x faster thanks to this higher computing area.\n\nThe diagonal lines reflect memory bandwidth. Since bandwidth is flow in bytes\/second it defines a linear relationship between the x-axis (flops\/byte) and the y-axis (in flops\/sec). A less well-known feature of GPUs is their higher main-memory bandwidth. This leads to a (potential) 10x gap in sparse matrix operations, which are the most important for many machine learning tasks. CuSparse achieves this ceiling for typical scientific data, but we found it was less well fit to very sparse data (text, web logs etc.). We wrote our own sparse kernels and were able to get them close to the roofline limits over a full range of sparseness. These kernels form the basis for the high throughput in most of BIDMach\u2019s algorithms. Continue reading\n\n# CUDA 7 Release Candidate Feature Overview: C++11, New Libraries, and More\n\nIt\u2019s almost time for the next major release of the CUDA Toolkit, so I\u2019m excited to tell you about the CUDA 7 Release Candidate, now available to all CUDA Registered Developers. The CUDA Toolkit version 7 expands the capabilities and improves the performance of the Tesla Accelerated Computing Platform and of accelerated computing on NVIDIA GPUs.\n\nRecently NVIDIA released the CUDA Toolkit version 5.5 with support for the IBM POWER architecture. Starting with CUDA 7, all future CUDA Toolkit releases will support POWER CPUs.\n\nCUDA 7 is a huge update to the CUDA platform; there are too many new features and improvements to describe in one blog post, so I\u2019ll touch on some of the most significant ones today. Please refer to the CUDA 7 release notes and documentation for more information. We\u2019ll be covering many of these features in greater detail in future Parallel Forall posts, so check back often!\n\n## Support for Powerful C++11 Features\n\nC++11 is a major update to the popular C++ language standard. C++11 includes a long list of new features for simpler, more expressive C++ programming with fewer errors and higher performance. I think Bjarne Stroustrup, the creator of C++, put it best:\n\nC++11 feels like a new language: The pieces just fit together better than they used to and I find a higher-level style of programming more natural than before and as efficient as ever.\nOpenACC is a high-level programming model for accelerating applications with GPUs and other devices using compiler directives compiler directives to specify loops and regions of code in standard C, C++ and Fortran to offload from a host CPU to an attached accelerator.\u00a0OpenACC simplifies accelerating applications with GPUs. An often-overlooked feature of OpenACC is its ability to interoperate with the broader parallel programming ecosystem. In this post I\u2019ll teach you 3 powerful interoperability techniques for combining OpenACC and CUDA: the `host_data` construct, the `deviceptr` clause, and the `acc_map_data()` API function.","date":"2016-05-26 00:38:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 1, \"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\": 0.37290558218955994, \"perplexity\": 2080.5109770474874}, \"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-2016-22\/segments\/1464049275429.29\/warc\/CC-MAIN-20160524002115-00043-ip-10-185-217-139.ec2.internal.warc.gz\"}"}
| null | null |
#include "BUpdaterLSF.h"
int main(int argc, char *argv[]){
FILE *fd;
job_registry_entry *en;
time_t now;
time_t purge_time=0;
time_t last_consistency_check=0;
char *pidfile=NULL;
char *first_duplicate=NULL;
struct pollfd *remupd_pollset = NULL;
int remupd_nfds;
int version=0;
int tmptim;
time_t finalquery_start_date;
int loop_interval=DEFAULT_LOOP_INTERVAL;
int rc;
int c;
pthread_t RecUpdNetThd;
int confirm_time=0;
static int help;
static int short_help;
struct stat sbuf;
char *s;
bact.njobs = 0;
bact.jobs = NULL;
while (1) {
static struct option long_options[] =
{
{"help", no_argument, &help, 1},
{"usage", no_argument, &short_help, 1},
{"nodaemon", no_argument, 0, 'o'},
{"version", no_argument, 0, 'v'},
{"prefix", required_argument, 0, 'p'},
{0, 0, 0, 0}
};
int option_index = 0;
c = getopt_long (argc, argv, "vop:",long_options, &option_index);
if (c == -1){
break;
}
switch (c)
{
case 0:
if (long_options[option_index].flag != 0){
break;
}
case 'v':
version=1;
break;
case 'o':
nodmn=1;
break;
case 'p':
break;
case '?':
break;
default:
abort ();
}
}
if(help){
usage();
}
if(short_help){
short_usage();
}
argv0 = argv[0];
signal(SIGHUP,sighup);
if(version) {
printf("%s Version: %s\n",progname,VERSION);
exit(EXIT_SUCCESS);
}
/* Checking configuration */
check_config_file("UPDATER");
cha = config_read(NULL);
if (cha == NULL)
{
fprintf(stderr,"Error reading config: ");
perror("");
return -1;
}
ret = config_get("bupdater_child_poll_timeout",cha);
if (ret != NULL){
tmptim=atoi(ret->value);
if (tmptim > 0) bfunctions_poll_timeout = tmptim*1000;
}
ret = config_get("bupdater_debug_level",cha);
if (ret != NULL){
debug=atoi(ret->value);
}
ret = config_get("bupdater_debug_logfile",cha);
if (ret != NULL){
debuglogname=strdup(ret->value);
if(debuglogname == NULL){
sysfatal("strdup failed for debuglogname in main: %r");
}
}
if(debug <=0){
debug=0;
}
if(debuglogname){
if((debuglogfile = fopen(debuglogname, "a+"))==0){
debug = 0;
}
} else {
debug = 0;
}
ret = config_get("lsf_binpath",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key lsf_binpath not found\n",argv0);
} else {
lsf_binpath=strdup(ret->value);
if(lsf_binpath == NULL){
sysfatal("strdup failed for lsf_binpath in main: %r");
}
}
ret = config_get("job_registry",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key job_registry not found\n",argv0);
sysfatal("job_registry not defined. Exiting");
} else {
registry_file=strdup(ret->value);
if(registry_file == NULL){
sysfatal("strdup failed for registry_file in main: %r");
}
}
ret = config_get("purge_interval",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key purge_interval not found using the default:%d\n",argv0,purge_interval);
} else {
purge_interval=atoi(ret->value);
}
ret = config_get("bhist_finalstate_interval",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key bhist_finalstate_interval not found using the default:%d\n",argv0,bhist_finalstate_interval);
} else {
bhist_finalstate_interval=atoi(ret->value);
}
ret = config_get("finalstate_query_interval",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key finalstate_query_interval not found using the default:%d\n",argv0,finalstate_query_interval);
} else {
finalstate_query_interval=atoi(ret->value);
}
ret = config_get("alldone_interval",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key alldone_interval not found using the default:%d\n",argv0,alldone_interval);
} else {
alldone_interval=atoi(ret->value);
}
ret = config_get("bupdater_consistency_check_interval",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key bupdater_consistency_check_interval not found using the default:%d\n",argv0,bupdater_consistency_check_interval);
} else {
bupdater_consistency_check_interval=atoi(ret->value);
}
ret = config_get("bhist_logs_to_read",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key bhist_logs_to_read not found using the default:%d\n",argv0,bhist_logs_to_read);
} else {
bhist_logs_to_read=atoi(ret->value);
}
ret = config_get("bupdater_pidfile",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key bupdater_pidfile not found\n",argv0);
} else {
pidfile=strdup(ret->value);
if(pidfile == NULL){
sysfatal("strdup failed for pidfile in main: %r");
}
}
ret = config_get("bupdater_loop_interval",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key bupdater_loop_interval not found - using the default:%d\n",argv0,loop_interval);
} else {
loop_interval=atoi(ret->value);
}
ret = config_get("bupdater_bjobs_long_format",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key bupdater_bjobs_long_format not found - using the default:%s\n",argv0,bjobs_long_format);
} else {
bjobs_long_format=strdup(ret->value);
if(bjobs_long_format == NULL){
sysfatal("strdup failed for bjobs_long_format in main: %r");
}
}
ret = config_get("bupdater_use_bhist_for_susp",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key bupdater_use_bhist_for_susp not found - using the default:%s\n",argv0,use_bhist_for_susp);
} else {
use_bhist_for_susp=strdup(ret->value);
if(use_bhist_for_susp == NULL){
sysfatal("strdup failed for use_bhist_for_susp in main: %r");
}
}
ret = config_get("bupdater_use_bhist_time_constraint",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key bupdater_use_bhist_time_constraint not found - using the default:%s\n",argv0,use_bhist_time_constraint);
} else {
use_bhist_time_constraint=strdup(ret->value);
if(use_bhist_time_constraint == NULL){
sysfatal("strdup failed for use_bhist_time_constraint in main: %r");
}
}
ret = config_get("bupdater_use_btools",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key bupdater_use_btools not found - using the default:%s\n",argv0,use_btools);
} else {
use_btools=strdup(ret->value);
if(use_btools == NULL){
sysfatal("strdup failed for use_btools in main: %r");
}
}
ret = config_get("bupdater_btools_path",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key bupdater_btools_path not found - using the default:%s\n",argv0,btools_path);
} else {
btools_path=strdup(ret->value);
if(btools_path == NULL){
sysfatal("strdup failed for btools_path in main: %r");
}
}
if(use_btools && strcmp(use_btools,"yes")==0){
s=make_message("%s/bjobsinfo",btools_path);
rc=stat(s,&sbuf);
if(rc) {
do_log(debuglogfile, debug, 1, "%s not found. Fall back to normal bjobs mode.\n",s);
use_btools="no";
if( ! (sbuf.st_mode & (S_IXUSR|S_IXGRP|S_IXOTH)) ) {
do_log(debuglogfile, debug, 1, "%s is not executable, but mode %05o. Fall back to normal bjobs mode.\n",s,(int)sbuf.st_mode);
use_btools="no";
}
}
free(s);
}
ret = config_get("bupdater_use_bhist_for_killed",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key bupdater_use_bhist_for_killed not found - using the default:%s\n",argv0,use_bhist_for_killed);
} else {
use_bhist_for_killed=strdup(ret->value);
if(use_bhist_for_killed == NULL){
sysfatal("strdup failed for use_bhist_for_killed in main: %r");
}
}
ret = config_get("bupdater_use_bhist_for_idle",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key bupdater_use_bhist_for_idle not found - using the default:%s\n",argv0,use_bhist_for_idle);
} else {
use_bhist_for_idle=strdup(ret->value);
if(use_bhist_for_idle == NULL){
sysfatal("strdup failed for use_bhist_for_idle in main: %r");
}
}
ret = config_get("lsf_batch_caching_enabled",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key lsf_batch_caching_enabled not found using default\n",argv0,lsf_batch_caching_enabled);
} else {
lsf_batch_caching_enabled=strdup(ret->value);
if(lsf_batch_caching_enabled == NULL){
sysfatal("strdup failed for lsf_batch_caching_enabled in main: %r");
}
}
ret = config_get("batch_command_caching_filter",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key batch_command_caching_filter not found using default\n",argv0,batch_command_caching_filter);
} else {
batch_command_caching_filter=strdup(ret->value);
if(batch_command_caching_filter == NULL){
sysfatal("strdup failed for batch_command_caching_filter in main: %r");
}
}
batch_command=(strcmp(lsf_batch_caching_enabled,"yes")==0?make_message("%s ",batch_command_caching_filter):make_message(""));
ret = config_get("job_registry_use_mmap",cha);
if (ret == NULL){
do_log(debuglogfile, debug, 1, "%s: key job_registry_use_mmap not found. Default is NO\n",argv0);
} else {
do_log(debuglogfile, debug, 1, "%s: key job_registry_use_mmap is set to %s\n",argv0,ret->value);
}
remupd_conf = config_get("job_registry_add_remote",cha);
if (remupd_conf == NULL){
do_log(debuglogfile, debug, 1, "%s: key job_registry_add_remote not found\n",argv0);
}else{
if (job_registry_updater_setup_receiver(remupd_conf->values,remupd_conf->n_values,&remupd_head) < 0){
do_log(debuglogfile, debug, 1, "%s: Cannot set network receiver(s) up for remote update\n",argv0);
fprintf(stderr,"%s: Cannot set network receiver(s) up for remote update \n",argv0);
}
if (remupd_head == NULL){
do_log(debuglogfile, debug, 1, "%s: Cannot find values for network endpoints in configuration file (attribute 'job_registry_add_remote').\n",argv0);
fprintf(stderr,"%s: Cannot find values for network endpoints in configuration file (attribute 'job_registry_add_remote').\n", argv0);
}
if ((remupd_nfds = job_registry_updater_get_pollfd(remupd_head, &remupd_pollset)) < 0){
do_log(debuglogfile, debug, 1, "%s: Cannot setup poll set for receiving data.\n",argv0);
fprintf(stderr,"%s: Cannot setup poll set for receiving data.\n", argv0);
}
if (remupd_pollset == NULL || remupd_nfds == 0){
do_log(debuglogfile, debug, 1, "%s: No poll set available for receiving data.\n",argv0);
fprintf(stderr,"%s: No poll set available for receiving data.\n",argv0);
}
}
if( !nodmn ) daemonize();
if( pidfile ){
writepid(pidfile);
free(pidfile);
}
rha=job_registry_init(registry_file, BY_BATCH_ID);
if (rha == NULL){
do_log(debuglogfile, debug, 1, "%s: Error initialising job registry %s\n",argv0,registry_file);
fprintf(stderr,"%s: Error initialising job registry %s :",argv0,registry_file);
perror("");
}
if (remupd_conf != NULL){
pthread_create(&RecUpdNetThd, NULL, (void *(*)(void *))ReceiveUpdateFromNetwork, (void *)NULL);
if (job_registry_updater_setup_sender(remupd_conf->values,remupd_conf->n_values,0,&remupd_head_send) < 0){
do_log(debuglogfile, debug, 1, "%s: Cannot set network sender(s) up for remote update\n",argv0);
fprintf(stderr,"%s: Cannot set network sender(s) up for remote update \n",argv0);
}
if (remupd_head_send == NULL){
do_log(debuglogfile, debug, 1, "%s: Cannot find values for network endpoints in configuration file (attribute 'job_registry_add_remote').\n",argv0);
fprintf(stderr,"%s: Cannot find values for network endpoints in configuration file (attribute 'job_registry_add_remote').\n", argv0);
}
}
config_free(cha);
for(;;){
/* Purge old entries from registry */
now=time(0);
if(now - purge_time > 86400){
if((rc=job_registry_purge(registry_file, now-purge_interval,0))<0){
do_log(debuglogfile, debug, 1, "%s: Error purging job registry %s:%d\n",argv0,registry_file,rc);
fprintf(stderr,"%s: Error purging job registry %s :",argv0,registry_file);
perror("");
}else{
purge_time=time(0);
}
}
now=time(0);
if(now - last_consistency_check > bupdater_consistency_check_interval){
if(job_registry_check_index_key_uniqueness(rha,&first_duplicate)==JOB_REGISTRY_FAIL){
do_log(debuglogfile, debug, 1, "%s: Found job registry duplicate entry. The first one is:%s\n",argv0,first_duplicate);
fprintf(stderr,"%s: Found job registry duplicate entry. The first one is:%s",argv0,first_duplicate);
}else{
last_consistency_check=time(0);
}
}
if(use_btools && strcmp(use_btools,"yes")==0){
IntStateQueryCustom();
}else if(bjobs_long_format && strcmp(bjobs_long_format,"yes")==0){
IntStateQuery();
}else{
IntStateQueryShort();
}
fd = job_registry_open(rha, "r");
if (fd == NULL){
do_log(debuglogfile, debug, 1, "%s: Error opening job registry %s\n",argv0,registry_file);
fprintf(stderr,"%s: Error opening job registry %s :",argv0,registry_file);
perror("");
sleep(loop_interval);
continue;
}
if (job_registry_rdlock(rha, fd) < 0){
do_log(debuglogfile, debug, 1, "%s: Error read locking job registry %s\n",argv0,registry_file);
fprintf(stderr,"%s: Error read locking job registry %s :",argv0,registry_file);
perror("");
sleep(loop_interval);
continue;
}
job_registry_firstrec(rha,fd);
fseek(fd,0L,SEEK_SET);
finalquery_start_date = time(0);
while ((en = job_registry_get_next(rha, fd)) != NULL){
if((bupdater_lookup_active_jobs(&bact,en->batch_id) != BUPDATER_ACTIVE_JOBS_SUCCESS) && en->status!=REMOVED && en->status!=COMPLETED){
confirm_time=atoi(en->updater_info);
if(confirm_time==0){
confirm_time=en->mdate;
}
/* Assign Status=4 and ExitStatus=999 to all entries that after alldone_interval are still not in a final state(3 or 4)*/
if(now-confirm_time>alldone_interval){
AssignFinalState(en->batch_id);
free(en);
continue;
}
/* Try to run FinalStateQuery reading older log files*/
if(now-confirm_time>bhist_finalstate_interval && use_bhist_for_idle && strcmp(use_bhist_for_idle,"yes")==0){
do_log(debuglogfile, debug, 2, "%s: FinalStateQuery needed for jobid=%s with status=%d from old logs\n",argv0,en->batch_id,en->status);
runfinal_oldlogs=TRUE;
free(en);
continue;
}
if(en->status==IDLE && strlen(en->updater_info)>0 && use_bhist_for_idle && strcmp(use_bhist_for_idle,"yes")==0){
if (en->mdate < finalquery_start_date){
finalquery_start_date=en->mdate;
}
do_log(debuglogfile, debug, 2, "%s: FinalStateQuery needed for jobid=%s with status=%d v1\n",argv0,en->batch_id,en->status);
runfinal=TRUE;
}else if((now-confirm_time>finalstate_query_interval) && (now > next_finalstatequery) && use_bhist_for_idle && strcmp(use_bhist_for_idle,"yes")==0){
if (en->mdate < finalquery_start_date){
finalquery_start_date=en->mdate;
}
do_log(debuglogfile, debug, 2, "%s: FinalStateQuery needed for jobid=%s with status=%d v2\n",argv0,en->batch_id,en->status);
runfinal=TRUE;
}
}
free(en);
}
if(runfinal_oldlogs){
FinalStateQuery(0,bhist_logs_to_read);
runfinal_oldlogs=FALSE;
runfinal=FALSE;
}else if(runfinal){
FinalStateQuery(finalquery_start_date,1);
runfinal=FALSE;
}
fclose(fd);
sleep(loop_interval);
}
job_registry_destroy(rha);
return 0;
}
int
ReceiveUpdateFromNetwork()
{
char *proxy_path, *proxy_subject;
int timeout_ms = 0;
int ret, prret, rhret;
job_registry_entry *nen;
job_registry_entry *ren;
proxy_path = NULL;
proxy_subject = NULL;
while ((nen = job_registry_receive_update(remupd_pollset, remupd_nfds,timeout_ms, &proxy_subject, &proxy_path))){
JOB_REGISTRY_ASSIGN_ENTRY(nen->subject_hash,"\0");
JOB_REGISTRY_ASSIGN_ENTRY(nen->proxy_link,"\0");
if ((ren=job_registry_get(rha, nen->batch_id)) == NULL){
if ((ret=job_registry_append(rha, nen)) < 0){
fprintf(stderr,"%s: Warning: job_registry_append returns %d: ",argv0,ret);
perror("");
}
}else{
if(ren->subject_hash!=NULL && strlen(ren->subject_hash) && ren->proxy_link!=NULL && strlen(ren->proxy_link)){
JOB_REGISTRY_ASSIGN_ENTRY(nen->subject_hash,ren->subject_hash);
JOB_REGISTRY_ASSIGN_ENTRY(nen->proxy_link,ren->proxy_link);
}else{
if (proxy_path != NULL && strlen(proxy_path) > 0){
prret = job_registry_set_proxy(rha, nen, proxy_path);
if (prret < 0){
do_log(debuglogfile, debug, 1, "%s: warning: setting proxy to %s\n",argv0,proxy_path);
fprintf(stderr,"%s: warning: setting proxy to %s: ",argv0,proxy_path);
perror("");
/* Make sure we don't renew non-existing proxies */
nen->renew_proxy = 0;
}
free(proxy_path);
nen->subject_hash[0] = '\000';
if (proxy_subject != NULL && strlen(proxy_subject) > 0){
job_registry_compute_subject_hash(nen, proxy_subject);
rhret = job_registry_record_subject_hash(rha, nen->subject_hash, proxy_subject, TRUE);
if (rhret < 0){
do_log(debuglogfile, debug, 1, "%s: warning: recording proxy subject %s (hash %s)\n",argv0, proxy_subject, nen->subject_hash);
fprintf(stderr,"%s: warning: recording proxy subject %s (hash %s): ",argv0, proxy_subject, nen->subject_hash);
perror("");
}
}
free(proxy_subject);
}
}
if(job_registry_need_update(ren,nen,JOB_REGISTRY_UPDATE_ALL)){
if ((ret=job_registry_update(rha, nen)) < 0){
fprintf(stderr,"%s: Warning: job_registry_update returns %d: ",argv0,ret);
perror("");
}
}
}
free(nen);
}
return 0;
}
int
IntStateQueryCustom()
{
/*
IntStateQuery to use btools command bjobsinfo
*/
FILE *fp;
char *line=NULL;
char **token;
int maxtok_l=0;
job_registry_entry en;
int ret;
char *tmp=NULL;
char *cp=NULL;
char *command_string=NULL;
job_registry_entry *ren=NULL;
int first=TRUE;
time_t now;
char *string_now=NULL;
int wexitcode=0;
int wexitinfo=0;
command_string=make_message("%s%s/bjobsinfo -a",batch_command,btools_path);
fp = popen(command_string,"r");
do_log(debuglogfile, debug, 3, "%s: command in IntStateQueryCustom is:%s\n",argv0,command_string);
en.status=UNDEFINED;
JOB_REGISTRY_ASSIGN_ENTRY(en.wn_addr,"\0");
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,"\0");
en.exitcode=-1;
bupdater_free_active_jobs(&bact);
if(fp!=NULL){
while(!feof(fp) && (line=get_line(fp))){
if(line && strlen(line)==0){
free(line);
continue;
}
if ((cp = strrchr (line, '\n')) != NULL){
*cp = '\0';
}
do_log(debuglogfile, debug, 3, "%s: line in IntStateQueryCustom is:%s\n",argv0,line);
tmp=strdup(line);
if(tmp == NULL){
sysfatal("strdup failed for tmp in IntStateQueryCustom: %r");
}
if ((cp = strchr (tmp, ' ')) != NULL){
*cp = '\0';
}
now=time(0);
string_now=make_message("%d",now);
if(!first && en.status!=UNDEFINED && ren && ren->status!=REMOVED && ren->status!=COMPLETED){
if ((ret=job_registry_update_recn_select(rha, &en, ren->recnum,
JOB_REGISTRY_UPDATE_WN_ADDR|
JOB_REGISTRY_UPDATE_STATUS|
JOB_REGISTRY_UPDATE_UDATE|
JOB_REGISTRY_UPDATE_UPDATER_INFO|
JOB_REGISTRY_UPDATE_EXITCODE|
JOB_REGISTRY_UPDATE_EXITREASON)) < 0){
if(ret != JOB_REGISTRY_NOT_FOUND){
fprintf(stderr,"Update of record returns %d: ",ret);
perror("");
}
} else {
if(ret==JOB_REGISTRY_SUCCESS){
if (en.status == REMOVED || en.status == COMPLETED) {
do_log(debuglogfile, debug, 2, "%s: registry update in IntStateQueryCustom for: jobid=%s creamjobid=%s wn=%s status=%d exitcode=%d\n",argv0,en.batch_id,en.user_prefix,en.wn_addr,en.status,en.exitcode);
job_registry_unlink_proxy(rha, &en);
}else{
do_log(debuglogfile, debug, 2, "%s: registry update in IntStateQueryCustom for: jobid=%s creamjobid=%s wn=%s status=%d\n",argv0,en.batch_id,en.user_prefix,en.wn_addr,en.status);
}
if (remupd_conf != NULL){
if (job_registry_send_update(remupd_head_send,&en,NULL,NULL)<=0){
do_log(debuglogfile, debug, 2, "%s: Error creating endpoint in IntStateQueryCustom\n",argv0);
}
}
}
}
en.status = UNDEFINED;
JOB_REGISTRY_ASSIGN_ENTRY(en.wn_addr,"\0");
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,"\0");
en.exitcode=-1;
}
en.status = UNDEFINED;
JOB_REGISTRY_ASSIGN_ENTRY(en.batch_id,tmp);
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,string_now);
en.exitcode=-1;
bupdater_push_active_job(&bact, en.batch_id);
free(tmp);
maxtok_l = strtoken(line, ' ', &token);
JOB_REGISTRY_ASSIGN_ENTRY(en.wn_addr,token[14]);
if(!first) free(ren);
if ((ren=job_registry_get(rha, en.batch_id)) == NULL){
fprintf(stderr,"Get of record returns error for %s ",en.batch_id);
perror("");
}
first=FALSE;
if(token[10] && strcmp(token[10],"PEND")==0){
en.status=IDLE;
en.exitcode=-1;
en.udate=strtoul(token[4],NULL,10);
}else if(token[10] && ((strcmp(token[10],"USUSP")==0) || (strcmp(token[10],"PSUSP")==0) ||(strcmp(token[10],"SSUSP")==0))){
en.status=HELD;
en.exitcode=-1;
}else if(token[10] && strcmp(token[10],"RUN")==0){
en.status=RUNNING;
en.exitcode=-1;
en.udate=strtoul(token[5],NULL,10);
}else if(token[10] && strcmp(token[10],"DONE")==0){
en.status=COMPLETED;
en.exitcode=0;
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
en.udate=strtoul(token[6],NULL,10);
}else if(token[10] && strcmp(token[10],"EXIT")==0){
if(use_bhist_for_killed && strcmp(use_bhist_for_killed,"yes")==0){
if(en.status == UNDEFINED){
en.status=IDLE;
en.exitcode=-1;
}
bupdater_remove_active_job(&bact, en.batch_id);
}else{
wexitinfo=atoi(token[13]);
/*13 because (see lsbatch.h) all the signal greater than 13 are some kind of TERM signals*/
if(wexitinfo>13){
en.status=REMOVED;
en.exitcode=-999;
}else{
wexitcode=WEXITSTATUS(atoi(token[12]));
if(wexitcode==255 || wexitcode==130){
en.status=REMOVED;
en.exitcode=-999;
}else{
en.status=COMPLETED;
en.exitcode=wexitcode;
}
}
}
en.udate=strtoul(token[6],NULL,10);
}
freetoken(&token,maxtok_l);
free(line);
free(string_now);
}
pclose(fp);
}
if(en.status!=UNDEFINED && ren && ren->status!=REMOVED && ren->status!=COMPLETED){
if ((ret=job_registry_update_recn_select(rha, &en, ren->recnum,
JOB_REGISTRY_UPDATE_WN_ADDR|
JOB_REGISTRY_UPDATE_STATUS|
JOB_REGISTRY_UPDATE_UDATE|
JOB_REGISTRY_UPDATE_UPDATER_INFO|
JOB_REGISTRY_UPDATE_EXITCODE|
JOB_REGISTRY_UPDATE_EXITREASON)) < 0){
if(ret != JOB_REGISTRY_NOT_FOUND){
fprintf(stderr,"Update of record returns %d: ",ret);
perror("");
}
} else {
if(ret==JOB_REGISTRY_SUCCESS){
if (en.status == REMOVED || en.status == COMPLETED) {
do_log(debuglogfile, debug, 2, "%s: registry update in IntStateQueryCustom for: jobid=%s creamjobid=%s wn=%s status=%d exitcode=%d\n",argv0,en.batch_id,en.user_prefix,en.wn_addr,en.status,en.exitcode);
job_registry_unlink_proxy(rha, &en);
}else{
do_log(debuglogfile, debug, 2, "%s: registry update in IntStateQueryCustom for: jobid=%s creamjobid=%s wn=%s status=%d\n",argv0,en.batch_id,en.user_prefix,en.wn_addr,en.status);
}
if (remupd_conf != NULL){
if (job_registry_send_update(remupd_head_send,&en,NULL,NULL)<=0){
do_log(debuglogfile, debug, 2, "%s: Error creating endpoint in IntStateQueryCustom\n",argv0);
}
}
}
}
}
free(ren);
free(command_string);
return 0;
}
int
IntStateQueryShort()
{
/*
Filled entries:
batch_id
wn_addr
status
udate
Filled by submit script:
blah_id
Unfilled entries:
exitreason
exitcode
*/
FILE *fp;
char *line=NULL;
char **token;
int maxtok_l=0;
job_registry_entry en;
int ret;
char *timestamp;
time_t tmstampepoch;
char *tmp=NULL;
char *cp=NULL;
char *command_string=NULL;
job_registry_entry *ren=NULL;
int first=TRUE;
time_t now;
char *string_now=NULL;
command_string=make_message("%s%s/bjobs -u all -w -a",batch_command,lsf_binpath);
fp = popen(command_string,"r");
en.status=UNDEFINED;
JOB_REGISTRY_ASSIGN_ENTRY(en.wn_addr,"\0");
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,"\0");
en.exitcode=-1;
bupdater_free_active_jobs(&bact);
if(fp!=NULL){
while(!feof(fp) && (line=get_line(fp))){
if(line && (strlen(line)==0 || strncmp(line,"JOBID",5)==0)){
free(line);
continue;
}
if ((cp = strrchr (line, '\n')) != NULL){
*cp = '\0';
}
tmp=strdup(line);
if(tmp == NULL){
sysfatal("strdup failed for tmp in IntStateQueryShort: %r");
}
if ((cp = strchr (tmp, ' ')) != NULL){
*cp = '\0';
}
now=time(0);
string_now=make_message("%d",now);
if(!first && en.status!=UNDEFINED && ren && ren->status!=REMOVED && ren->status!=COMPLETED){
if ((ret=job_registry_update_recn_select(rha, &en, ren->recnum,
JOB_REGISTRY_UPDATE_WN_ADDR|
JOB_REGISTRY_UPDATE_STATUS|
JOB_REGISTRY_UPDATE_UDATE|
JOB_REGISTRY_UPDATE_UPDATER_INFO|
JOB_REGISTRY_UPDATE_EXITCODE|
JOB_REGISTRY_UPDATE_EXITREASON)) < 0){
if(ret != JOB_REGISTRY_NOT_FOUND){
fprintf(stderr,"Update of record returns %d: ",ret);
perror("");
}
} else {
if(ret==JOB_REGISTRY_SUCCESS){
if (en.status == REMOVED || en.status == COMPLETED) {
do_log(debuglogfile, debug, 2, "%s: registry update in IntStateQueryShort for: jobid=%s creamjobid=%s wn=%s status=%d exitcode=%d\n",argv0,en.batch_id,en.user_prefix,en.wn_addr,en.status,en.exitcode);
job_registry_unlink_proxy(rha, &en);
}else{
do_log(debuglogfile, debug, 2, "%s: registry update in IntStateQueryShort for: jobid=%s creamjobid=%s wn=%s status=%d\n",argv0,en.batch_id,en.user_prefix,en.wn_addr,en.status);
}
if (remupd_conf != NULL){
if (job_registry_send_update(remupd_head_send,&en,NULL,NULL)<=0){
do_log(debuglogfile, debug, 2, "%s: Error creating endpoint in IntStateQueryShort\n",argv0);
}
}
}
}
en.status = UNDEFINED;
JOB_REGISTRY_ASSIGN_ENTRY(en.wn_addr,"\0");
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,"\0");
en.exitcode=-1;
}
en.status = UNDEFINED;
JOB_REGISTRY_ASSIGN_ENTRY(en.batch_id,tmp);
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,string_now);
en.exitcode=-1;
bupdater_push_active_job(&bact, en.batch_id);
free(tmp);
maxtok_l = strtoken(line, ' ', &token);
JOB_REGISTRY_ASSIGN_ENTRY(en.wn_addr,token[5]);
if(!first) free(ren);
if ((ren=job_registry_get(rha, en.batch_id)) == NULL){
fprintf(stderr,"Get of record returns error for %s ",en.batch_id);
perror("");
}
first=FALSE;
if(token[2] && strcmp(token[2],"PEND")==0){
en.status=IDLE;
en.exitcode=-1;
}else if(token[2] && ((strcmp(token[2],"USUSP")==0) || (strcmp(token[2],"PSUSP")==0) ||(strcmp(token[2],"SSUSP")==0))){
en.status=HELD;
en.exitcode=-1;
}else if(token[2] && strcmp(token[2],"RUN")==0){
en.status=RUNNING;
en.exitcode=-1;
}else if(token[2] && strcmp(token[2],"DONE")==0){
en.status=COMPLETED;
en.exitcode=0;
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
}else if(token[2] && strcmp(token[2],"EXIT")==0){
if(en.status == UNDEFINED){
en.status=IDLE;
en.exitcode=-1;
}
bupdater_remove_active_job(&bact, en.batch_id);
}
timestamp=make_message("%s %s %s",token[7],token[8],token[9]);
tmstampepoch=str2epoch(timestamp,"V");
free(timestamp);
en.udate=tmstampepoch;
freetoken(&token,maxtok_l);
free(line);
free(string_now);
}
pclose(fp);
}
if(en.status!=UNDEFINED && ren && ren->status!=REMOVED && ren->status!=COMPLETED){
if ((ret=job_registry_update_recn_select(rha, &en, ren->recnum,
JOB_REGISTRY_UPDATE_WN_ADDR|
JOB_REGISTRY_UPDATE_STATUS|
JOB_REGISTRY_UPDATE_UDATE|
JOB_REGISTRY_UPDATE_UPDATER_INFO|
JOB_REGISTRY_UPDATE_EXITCODE|
JOB_REGISTRY_UPDATE_EXITREASON)) < 0){
if(ret != JOB_REGISTRY_NOT_FOUND){
fprintf(stderr,"Update of record returns %d: ",ret);
perror("");
}
} else {
if(ret==JOB_REGISTRY_SUCCESS){
if (en.status == REMOVED || en.status == COMPLETED) {
do_log(debuglogfile, debug, 2, "%s: registry update in IntStateQueryShort for: jobid=%s creamjobid=%s wn=%s status=%d exitcode=%d\n",argv0,en.batch_id,en.user_prefix,en.wn_addr,en.status,en.exitcode);
job_registry_unlink_proxy(rha, &en);
}else{
do_log(debuglogfile, debug, 2, "%s: registry update in IntStateQueryShort for: jobid=%s creamjobid=%s wn=%s status=%d\n",argv0,en.batch_id,en.user_prefix,en.wn_addr,en.status);
}
if (remupd_conf != NULL){
if (job_registry_send_update(remupd_head_send,&en,NULL,NULL)<=0){
do_log(debuglogfile, debug, 2, "%s: Error creating endpoint in IntStateQueryShort\n",argv0);
}
}
}
}
}
free(ren);
free(command_string);
return 0;
}
int
IntStateQuery()
{
/*
Filled entries:
batch_id
wn_addr
status
udate
Filled by submit script:
blah_id
Unfilled entries:
exitcode
exitreason
*/
FILE *fp;
char *line=NULL;
char **token;
int maxtok_t=0;
job_registry_entry en;
int ret;
char *timestamp;
time_t tmstampepoch;
char *cp=NULL;
char *wn_str=NULL;
char *ex_str=NULL;
char *batch_str=NULL;
char *command_string=NULL;
job_registry_entry *ren=NULL;
int isresumed=FALSE;
int first=TRUE;
time_t now;
char *string_now=NULL;
int wexitcode=0;
command_string=make_message("%s%s/bjobs -u all -l -a",batch_command,lsf_binpath);
fp = popen(command_string,"r");
en.status=UNDEFINED;
JOB_REGISTRY_ASSIGN_ENTRY(en.wn_addr,"\0");
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,"\0");
en.exitcode=-1;
bupdater_free_active_jobs(&bact);
if(fp!=NULL){
while(!feof(fp) && (line=get_line(fp))){
if(line && strlen(line)==0){
free(line);
continue;
}
if ((cp = strrchr (line, '\n')) != NULL){
*cp = '\0';
}
do_log(debuglogfile, debug, 3, "%s: line in IntStateQuery is:%s\n",argv0,line);
now=time(0);
string_now=make_message("%d",now);
if(line && strstr(line,"Job <")){
isresumed=FALSE;
if(!first && en.status!=UNDEFINED && ren && ren->status!=REMOVED && ren->status!=COMPLETED){
if ((ret=job_registry_update_recn_select(rha, &en, ren->recnum,
JOB_REGISTRY_UPDATE_WN_ADDR|
JOB_REGISTRY_UPDATE_STATUS|
JOB_REGISTRY_UPDATE_UDATE|
JOB_REGISTRY_UPDATE_UPDATER_INFO|
JOB_REGISTRY_UPDATE_EXITCODE|
JOB_REGISTRY_UPDATE_EXITREASON)) < 0){
if(ret != JOB_REGISTRY_NOT_FOUND){
fprintf(stderr,"Update of record returns %d: ",ret);
perror("");
}
} else {
if(ret==JOB_REGISTRY_SUCCESS){
if (en.status == REMOVED || en.status == COMPLETED) {
do_log(debuglogfile, debug, 2, "%s: registry update in IntStateQuery for: jobid=%s creamjobid=%s wn=%s status=%d exitcode=%d\n",argv0,en.batch_id,en.user_prefix,en.wn_addr,en.status,en.exitcode);
job_registry_unlink_proxy(rha, &en);
}else{
do_log(debuglogfile, debug, 2, "%s: registry update in IntStateQuery for: jobid=%s creamjobid=%s wn=%s status=%d\n",argv0,en.batch_id,en.user_prefix,en.wn_addr,en.status);
}
if (remupd_conf != NULL){
if (job_registry_send_update(remupd_head_send,&en,NULL,NULL)<=0){
do_log(debuglogfile, debug, 2, "%s: Error creating endpoint in IntStateQuery\n",argv0);
}
}
}
}
en.status = UNDEFINED;
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,"\0");
JOB_REGISTRY_ASSIGN_ENTRY(en.wn_addr,"\0");
en.exitcode=-1;
}
en.status = UNDEFINED;
maxtok_t = strtoken(line, ',', &token);
batch_str=strdel(token[0],"Job <>");
JOB_REGISTRY_ASSIGN_ENTRY(en.batch_id,batch_str);
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,string_now);
en.exitcode=-1;
bupdater_push_active_job(&bact, en.batch_id);
free(batch_str);
freetoken(&token,maxtok_t);
if(!first) free(ren);
if ((ren=job_registry_get(rha, en.batch_id)) == NULL){
fprintf(stderr,"Get of record returns error ");
perror("");
}
if(ren){
if(strlen(ren->updater_info)>0){
en.udate=ren->udate;
}else{
en.udate=time(0);
}
}
first=FALSE;
}else if(line && strstr(line," Submitted from host ") && (en.status == IDLE)){
maxtok_t = strtoken(line, ' ', &token);
timestamp=make_message("%s %s %s %s",token[0],token[1],token[2],token[3]);
timestamp[strlen(timestamp)-1]='\0';
tmstampepoch=str2epoch(timestamp,"W");
en.udate=tmstampepoch;
free(timestamp);
freetoken(&token,maxtok_t);
}else if(line && strstr(line," <PEND>, ")){
en.status=IDLE;
en.exitcode=-1;
if(use_bhist_for_susp && strcmp(use_bhist_for_susp,"yes")==0){
/*If status was HELD we have to check timestamp of resume to pend with bhist (the info is not there with bjobs)*/
if(ren && ren->status==HELD){
tmstampepoch=get_pend_timestamp(en.batch_id);
en.udate=tmstampepoch;
}
}
if(ren && (ren->status==IDLE || ren->status==HELD)){
JOB_REGISTRY_ASSIGN_ENTRY(en.wn_addr,"\0");
}
}else if(line && strstr(line," <RUN>, ")){
en.status=RUNNING;
en.exitcode=-1;
if(use_bhist_for_susp && strcmp(use_bhist_for_susp,"yes")==0){
/*If status was HELD we have to check timestamp of resume with bhist (the info is not there with bjobs)*/
if(ren && ren->status==HELD){
tmstampepoch=get_resume_timestamp(en.batch_id);
en.udate=tmstampepoch;
isresumed=TRUE;
}
}
}else if(line && (strstr(line," <USUSP>,") || strstr(line," <PSUSP>,") || strstr(line," <SSUSP>,"))){
en.status=HELD;
en.exitcode=-1;
if(ren && ren->status==IDLE){
JOB_REGISTRY_ASSIGN_ENTRY(en.wn_addr,"\0");
}
/*If status is HELD we check timestamp of suspension with bhist (the info is not there with bjobs)*/
if(use_bhist_for_susp && strcmp(use_bhist_for_susp,"yes")==0){
tmstampepoch=get_susp_timestamp(en.batch_id);
en.udate=tmstampepoch;
}
}else if(line && strstr(line,"Started on ") && (en.status == RUNNING) && (!isresumed)){
maxtok_t = strtoken(line, ' ', &token);
timestamp=make_message("%s %s %s %s",token[0],token[1],token[2],token[3]);
timestamp[strlen(timestamp)-1]='\0';
tmstampepoch=str2epoch(timestamp,"W");
en.udate=tmstampepoch;
en.status=RUNNING;
en.exitcode=-1;
free(timestamp);
wn_str=strdel(token[6],"<>,;");
JOB_REGISTRY_ASSIGN_ENTRY(en.wn_addr,wn_str);
free(wn_str);
freetoken(&token,maxtok_t);
}else if(line && strstr(line," Exited with exit code") && en.status != REMOVED){
if(use_bhist_for_killed && strcmp(use_bhist_for_killed,"yes")==0){
if(en.status == UNDEFINED){
en.status=IDLE;
en.exitcode=-1;
}
bupdater_remove_active_job(&bact, en.batch_id);
}else{
maxtok_t = strtoken(line, ' ', &token);
timestamp=make_message("%s %s %s %s",token[0],token[1],token[2],token[3]);
timestamp[strlen(timestamp)-1]='\0';
tmstampepoch=str2epoch(timestamp,"W");
en.udate=tmstampepoch;
free(timestamp);
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,string_now);
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
ex_str=strdel(token[8],".");
wexitcode=atoi(ex_str);
free(ex_str);
freetoken(&token,maxtok_t);
if(wexitcode==255 || wexitcode==130){
en.status=REMOVED;
en.exitcode=-999;
}else{
en.status=COMPLETED;
en.exitcode=wexitcode;
}
}
}else if(line && strstr(line," Exited by signal") && en.status != REMOVED){
maxtok_t = strtoken(line, ' ', &token);
timestamp=make_message("%s %s %s %s",token[0],token[1],token[2],token[3]);
timestamp[strlen(timestamp)-1]='\0';
tmstampepoch=str2epoch(timestamp,"W");
en.udate=tmstampepoch;
en.status=COMPLETED;
free(timestamp);
ex_str=strdel(token[7],".");
en.exitcode=atoi(ex_str);
free(ex_str);
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,string_now);
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
freetoken(&token,maxtok_t);
}else if(line && strstr(line," Done successfully")){
maxtok_t = strtoken(line, ' ', &token);
timestamp=make_message("%s %s %s %s",token[0],token[1],token[2],token[3]);
timestamp[strlen(timestamp)-1]='\0';
tmstampepoch=str2epoch(timestamp,"W");
en.udate=tmstampepoch;
en.status=COMPLETED;
free(timestamp);
en.exitcode=0;
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
freetoken(&token,maxtok_t);
}else if(line && strstr(line," Post job process failed") && en.status == COMPLETED){
maxtok_t = strtoken(line, ' ', &token);
timestamp=make_message("%s %s %s %s",token[0],token[1],token[2],token[3]);
timestamp[strlen(timestamp)-1]='\0';
tmstampepoch=str2epoch(timestamp,"W");
en.udate=tmstampepoch;
en.status=COMPLETED;
free(timestamp);
en.exitcode=-998;
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,string_now);
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"LSF Postjob failed");
freetoken(&token,maxtok_t);
}
free(line);
free(string_now);
}
pclose(fp);
}
if(en.status!=UNDEFINED && ren && ren->status!=REMOVED && ren->status!=COMPLETED){
if ((ret=job_registry_update_recn_select(rha, &en, ren->recnum,
JOB_REGISTRY_UPDATE_WN_ADDR|
JOB_REGISTRY_UPDATE_STATUS|
JOB_REGISTRY_UPDATE_UDATE|
JOB_REGISTRY_UPDATE_UPDATER_INFO|
JOB_REGISTRY_UPDATE_EXITCODE|
JOB_REGISTRY_UPDATE_EXITREASON)) < 0){
if(ret != JOB_REGISTRY_NOT_FOUND){
fprintf(stderr,"Update of record returns %d: ",ret);
perror("");
}
} else {
if(ret==JOB_REGISTRY_SUCCESS){
if (en.status == REMOVED || en.status == COMPLETED) {
do_log(debuglogfile, debug, 2, "%s: registry update in IntStateQuery for: jobid=%s creamjobid=%s wn=%s status=%d exitcode=%d\n",argv0,en.batch_id,en.user_prefix,en.wn_addr,en.status,en.exitcode);
job_registry_unlink_proxy(rha, &en);
}else{
do_log(debuglogfile, debug, 2, "%s: registry update in IntStateQuery for: jobid=%s creamjobid=%s wn=%s status=%d\n",argv0,en.batch_id,en.user_prefix,en.wn_addr,en.status);
}
if (remupd_conf != NULL){
if (job_registry_send_update(remupd_head_send,&en,NULL,NULL)<=0){
do_log(debuglogfile, debug, 2, "%s: Error creating endpoint in IntStateQuery\n",argv0);
}
}
}
}
}
free(ren);
free(command_string);
return 0;
}
int
FinalStateQuery(time_t start_date, int logs_to_read)
{
/*
bhist -u all -a -l
In line:
Tue Mar 18 13:47:32: Exited with exit code 2. The CPU time used is 1.8 seconds;
or
Tue Mar 18 12:48:24: Done successfully. The CPU time used is 2.1 seconds;
there are:
udate for the final state (Tue Mar 18 13:47:32):
exitcode (=0 if Done successfully) or (from Exited with exit code 2)
*/
/*
Filled entries:
batch_id
status
exitcode
wn_addr
udate
Filled by submit script:
blah_id
Unfilled entries:
exitreason
*/
FILE *fp;
char *line=NULL;
char **token;
int maxtok_t=0;
job_registry_entry en;
int ret;
char *timestamp;
time_t tmstampepoch;
char *batch_str=NULL;
char *ex_str=NULL;
char *cp=NULL;
struct tm start_date_tm;
char start_date_str[80];
char *start_date_flagged=NULL;
char *command_string=NULL;
int failed_count=0;
int time_to_add=0;
time_t now;
char *string_now=NULL;
int first=TRUE;
job_registry_entry *ren=NULL;
if(strcmp(use_bhist_time_constraint,"yes")==0){
if(start_date != 0){
localtime_r(&start_date, &start_date_tm);
strftime(start_date_str, sizeof(start_date_str), "%Y/%m/%d/%H:%M,", &start_date_tm);
start_date_flagged=make_message("-C %s",start_date_str);
}else{
start_date_flagged=make_message("");
}
}else{
start_date_flagged=make_message("");
}
command_string=make_message("%s%s/bhist -u all -d -l -n %d %s",batch_command,lsf_binpath,logs_to_read,start_date_flagged);
free(start_date_flagged);
fp = popen(command_string,"r");
do_log(debuglogfile, debug, 3, "%s: command_string in FinalStateQuery is:%s\n",argv0,command_string);
en.status=UNDEFINED;
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
if(fp!=NULL){
while(!feof(fp) && (line=get_line(fp))){
if(line && strlen(line)==0){
free(line);
continue;
}
if ((cp = strrchr (line, '\n')) != NULL){
*cp = '\0';
}
do_log(debuglogfile, debug, 3, "%s: line in FinalStateQuery is:%s\n",argv0,line);
now=time(0);
string_now=make_message("%d",now);
if(line && strstr(line,"Job <")){
if(!first && en.status!=UNDEFINED && en.status!=IDLE && ren && ren->status!=REMOVED && ren->status!=COMPLETED){
if ((ret=job_registry_update_select(rha, &en,
JOB_REGISTRY_UPDATE_UDATE |
JOB_REGISTRY_UPDATE_STATUS |
JOB_REGISTRY_UPDATE_UPDATER_INFO |
JOB_REGISTRY_UPDATE_EXITCODE |
JOB_REGISTRY_UPDATE_EXITREASON )) < 0){
if(ret != JOB_REGISTRY_NOT_FOUND){
fprintf(stderr,"Update of record returns %d: ",ret);
perror("");
}
} else {
do_log(debuglogfile, debug, 2, "%s: registry update in FinalStateQuery for: jobid=%s creamjobid=%s status=%d\n",argv0,en.batch_id,en.user_prefix,en.status);
if (en.status == REMOVED || en.status == COMPLETED){
job_registry_unlink_proxy(rha, &en);
}
}
if (remupd_conf != NULL){
if (job_registry_send_update(remupd_head_send,&en,NULL,NULL)<=0){
do_log(debuglogfile, debug, 2, "%s: Error creating endpoint in FinalStateQuery\n",argv0);
}
}
}
en.status = UNDEFINED;
maxtok_t = strtoken(line, ',', &token);
batch_str=strdel(token[0],"Job <>");
JOB_REGISTRY_ASSIGN_ENTRY(en.batch_id,batch_str);
if(!first) free(ren);
if ((ren=job_registry_get(rha, en.batch_id)) == NULL){
fprintf(stderr,"Get of record returns error ");
perror("");
}
first=FALSE;
free(batch_str);
freetoken(&token,maxtok_t);
}else if(line && strstr(line," Signal <KILL>")){
maxtok_t = strtoken(line, ' ', &token);
timestamp=make_message("%s %s %s %s",token[0],token[1],token[2],token[3]);
timestamp[strlen(timestamp)-1]='\0';
tmstampepoch=str2epoch(timestamp,"W");
en.udate=tmstampepoch;
en.status=REMOVED;
free(timestamp);
en.exitcode=-999;
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,string_now);
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
freetoken(&token,maxtok_t);
}else if(line && strstr(line," Exited with exit code") && en.status != REMOVED){
maxtok_t = strtoken(line, ' ', &token);
timestamp=make_message("%s %s %s %s",token[0],token[1],token[2],token[3]);
timestamp[strlen(timestamp)-1]='\0';
tmstampepoch=str2epoch(timestamp,"W");
en.udate=tmstampepoch;
en.status=COMPLETED;
free(timestamp);
ex_str=strdel(token[8],".");
en.exitcode=atoi(ex_str);
free(ex_str);
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,string_now);
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
freetoken(&token,maxtok_t);
}else if(line && strstr(line," Exited by signal") && en.status != REMOVED){
maxtok_t = strtoken(line, ' ', &token);
timestamp=make_message("%s %s %s %s",token[0],token[1],token[2],token[3]);
timestamp[strlen(timestamp)-1]='\0';
tmstampepoch=str2epoch(timestamp,"W");
en.udate=tmstampepoch;
en.status=COMPLETED;
free(timestamp);
ex_str=strdel(token[7],".");
en.exitcode=atoi(ex_str);
free(ex_str);
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,string_now);
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
freetoken(&token,maxtok_t);
}else if(line && strstr(line," Done successfully") && en.status != REMOVED){
maxtok_t = strtoken(line, ' ', &token);
timestamp=make_message("%s %s %s %s",token[0],token[1],token[2],token[3]);
timestamp[strlen(timestamp)-1]='\0';
tmstampepoch=str2epoch(timestamp,"W");
en.udate=tmstampepoch;
en.status=COMPLETED;
free(timestamp);
en.exitcode=0;
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,string_now);
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
freetoken(&token,maxtok_t);
}else if(line && strstr(line," Post job process failed") && en.status == COMPLETED){
maxtok_t = strtoken(line, ' ', &token);
timestamp=make_message("%s %s %s %s",token[0],token[1],token[2],token[3]);
timestamp[strlen(timestamp)-1]='\0';
tmstampepoch=str2epoch(timestamp,"W");
en.udate=tmstampepoch;
en.status=COMPLETED;
free(timestamp);
en.exitcode=-998;
JOB_REGISTRY_ASSIGN_ENTRY(en.updater_info,string_now);
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"LSF Postjob failed");
freetoken(&token,maxtok_t);
}
free(string_now);
free(line);
}
pclose(fp);
}
if(en.status!=UNDEFINED && en.status!=IDLE && ren && ren->status!=REMOVED && ren->status!=COMPLETED){
if ((ret=job_registry_update_select(rha, &en,
JOB_REGISTRY_UPDATE_UDATE |
JOB_REGISTRY_UPDATE_STATUS |
JOB_REGISTRY_UPDATE_UPDATER_INFO |
JOB_REGISTRY_UPDATE_EXITCODE |
JOB_REGISTRY_UPDATE_EXITREASON )) < 0){
if(ret != JOB_REGISTRY_NOT_FOUND){
fprintf(stderr,"Update of record returns %d: ",ret);
perror("");
}
} else {
do_log(debuglogfile, debug, 2, "%s: f registry update in FinalStateQuery for: jobid=%s creamjobid=%s status=%d\n",argv0,en.batch_id,en.user_prefix,en.status);
if (en.status == REMOVED || en.status == COMPLETED){
job_registry_unlink_proxy(rha, &en);
}
if (remupd_conf != NULL){
if (job_registry_send_update(remupd_head_send,&en,NULL,NULL)<=0){
do_log(debuglogfile, debug, 2, "%s: Error creating endpoint in FinalStateQuery\n",argv0);
}
}
}
}else{
failed_count++;
}
now=time(0);
time_to_add=pow(failed_count,1.5);
next_finalstatequery=now+time_to_add;
do_log(debuglogfile, debug, 3, "%s: next FinalStatequery will be in %d seconds\n",argv0,time_to_add);
free(ren);
free(command_string);
return 0;
}
time_t
get_susp_timestamp(char *jobid)
{
FILE *fp;
char *line=NULL;
char **token;
int maxtok_t=0;
char *timestamp;
time_t tmstampepoch = 0;
char *cp=NULL;
char *command_string=NULL;
command_string=make_message("%s/bhist -u all -l %s",lsf_binpath,jobid);
fp = popen(command_string,"r");
if(fp!=NULL){
while(!feof(fp) && (line=get_line(fp))){
if(line && strlen(line)==0){
free(line);
continue;
}
if ((cp = strrchr (line, '\n')) != NULL){
*cp = '\0';
}
if(line && strstr(line," Suspended by")){
maxtok_t = strtoken(line, ' ', &token);
timestamp=make_message("%s %s %s %s",token[0],token[1],token[2],token[3]);
timestamp[strlen(timestamp)-1]='\0';
tmstampepoch=str2epoch(timestamp,"W");
free(timestamp);
freetoken(&token,maxtok_t);
}
free(line);
}
pclose(fp);
}
free(command_string);
return tmstampepoch;
}
time_t
get_resume_timestamp(char *jobid)
{
FILE *fp;
char *line=NULL;
char **token;
int maxtok_t=0;
char *timestamp;
time_t tmstampepoch = 0;
char *cp=NULL;
char *command_string=NULL;
command_string=make_message("%s/bhist -u all -l %s",lsf_binpath,jobid);
fp = popen(command_string,"r");
if(fp!=NULL){
while(!feof(fp) && (line=get_line(fp))){
if(line && strlen(line)==0){
free(line);
continue;
}
if ((cp = strrchr (line, '\n')) != NULL){
*cp = '\0';
}
if(line && strstr(line," Running;")){
maxtok_t = strtoken(line, ' ', &token);
timestamp=make_message("%s %s %s %s",token[0],token[1],token[2],token[3]);
timestamp[strlen(timestamp)-1]='\0';
tmstampepoch=str2epoch(timestamp,"W");
free(timestamp);
freetoken(&token,maxtok_t);
}
free(line);
}
pclose(fp);
}
free(command_string);
return tmstampepoch;
}
time_t
get_pend_timestamp(char *jobid)
{
FILE *fp;
char *line=NULL;
char **token;
int maxtok_t=0;
char *timestamp;
time_t tmstampepoch = 0;
char *cp=NULL;
char *command_string=NULL;
command_string=make_message("%s/bhist -u all -l %s",lsf_binpath,jobid);
fp = popen(command_string,"r");
if(fp!=NULL){
while(!feof(fp) && (line=get_line(fp))){
if(line && strlen(line)==0){
free(line);
continue;
}
if ((cp = strrchr (line, '\n')) != NULL){
*cp = '\0';
}
if(line && strstr(line," Pending: Waiting for scheduling after resumed")){
maxtok_t = strtoken(line, ' ', &token);
timestamp=make_message("%s %s %s %s",token[0],token[1],token[2],token[3]);
timestamp[strlen(timestamp)-1]='\0';
tmstampepoch=str2epoch(timestamp,"W");
free(timestamp);
freetoken(&token,maxtok_t);
}
free(line);
}
pclose(fp);
}
free(command_string);
return tmstampepoch;
}
int AssignFinalState(char *batchid){
job_registry_entry en;
int ret,i;
time_t now;
now=time(0);
JOB_REGISTRY_ASSIGN_ENTRY(en.batch_id,batchid);
en.status=COMPLETED;
en.exitcode=999;
en.udate=now;
JOB_REGISTRY_ASSIGN_ENTRY(en.wn_addr,"\0");
JOB_REGISTRY_ASSIGN_ENTRY(en.exitreason,"\0");
if ((ret=job_registry_update(rha, &en)) < 0){
if(ret != JOB_REGISTRY_NOT_FOUND){
fprintf(stderr,"Update of record %d returns %d: ",i,ret);
perror("");
}
} else {
do_log(debuglogfile, debug, 2, "%s: registry update in AssignStateQuery for: jobid=%s creamjobid=%s status=%d\n",argv0,en.batch_id,en.user_prefix,en.status);
job_registry_unlink_proxy(rha, &en);
if (remupd_conf != NULL){
if (job_registry_send_update(remupd_head_send,&en,NULL,NULL)<=0){
do_log(debuglogfile, debug, 2, "%s: Error creating endpoint in AssignFinalState\n",argv0);
}
}
}
return 0;
}
void sighup()
{
if(debug){
fclose(debuglogfile);
if((debuglogfile = fopen(debuglogname, "a+"))==0){
debug = 0;
}
}
}
int
usage()
{
printf("Usage: BUpdaterLSF [OPTION...]\n");
printf(" -o, --nodaemon do not run as daemon\n");
printf(" -v, --version print version and exit\n");
printf("\n");
printf("Help options:\n");
printf(" -?, --help Show this help message\n");
printf(" --usage Display brief usage message\n");
exit(EXIT_SUCCESS);
}
int
short_usage()
{
printf("Usage: BUpdaterLSF [-ov?] [-o|--nodaemon] [-v|--version] [-?|--help] [--usage]\n");
exit(EXIT_SUCCESS);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,402
|
{"url":"http:\/\/mathoverflow.net\/revisions\/71890\/list","text":"## Return to Answer\n\n2 deleted 509 characters in body\n\nI assume you want $q$ and $r$ to be odd primes. Also, note that I will be using the notation that $\\zeta_m$ means an arbitrary primitive $m$-th root of unity (but the same one every time it appears in an equation), and will be proving the statement in that generality.\n\nLemma: For any odd $m>1$ and any $\\zeta_m$, the number $\\zeta_m+1$ is a unit.\n\nProof: Let $r$ be such that $m | 2^r-1$. We'll abbreviate $\\zeta_m$ to $\\zeta$.\n\nThen $\\zeta^{2^r} = \\zeta$ so $$1 = \\left( \\frac{\\zeta^{2}-1}{\\zeta -1} \\right) \\left( \\frac{\\zeta^{4}-1}{\\zeta^{2} -1} \\right) \\cdots \\left( \\frac{\\zeta^{2^r}-1}{\\zeta^{2^{r-1}} -1} \\right)=$$ $$\\left( \\zeta+1 \\right) \\left( \\zeta^{2} + 1 \\right) \\cdots \\left( \\zeta^{2^{r-1}} +1 \\right),$$ exhibiting an explicit inverse for $\\zeta+1$.\n\nNote that, for $m$ odd, the negation of a primitive\n\nLet $2m$-th root of unity is \\eta$be a primitive$m$-th 2qr$ root of unity. So the negation of Then your proposed unit is $\\zeta_q+\\zeta_{-q} + \\zeta_{r} + \\zeta_{-r}$. There is a primitive $qr$-th root of unity, $\\eta$, such that $\\zeta_q = \\eta^r$ and $\\zeta_r = \\eta^q$. So the claim is that $$\\eta^r + \\eta^{-r} \\eta^{r}+\\eta^{-r} + \\eta^q + \\eta^{-q}$$ is a unit. This eta^{-q}$and factors as $$\\eta^r \\left(1 + \\eta^{q-r} +\\eta^{-q-r} + \\eta^{-2r} \\right) = \\eta^r \\left( 1+ \\eta^{q-r} \\right) \\left( 1+\\eta^{-q-r} \\right).$$ Using that (1+\\eta^{q-r})(1+\\eta^{-q-r}).$$Since q and r are primes, we have GCD(q-r, qr) = GCD(q+r,qr)=1odd and relatively prime, so \\eta^{q-r} and \\eta^{q+r} are primitive qr-th roots of unity , and we are done by the lemma. Note: I don't think I used that q and r were odd primes, only that they were odd, relatively prime, and >1. 1 I assume you want q and r to be odd primes. Also, note that I will be using the notation that \\zeta_m means an arbitrary primitive m-th root of unity (but the same one every time it appears in an equation), and will be proving the statement in that generality. Lemma: For any odd m>1 and any \\zeta_m, the number \\zeta_m+1 is a unit. Proof: Let r be such that m | 2^r-1. We'll abbreviate \\zeta_m to \\zeta. Then \\zeta^{2^r} = \\zeta so $$1 = \\left( \\frac{\\zeta^{2}-1}{\\zeta -1} \\right) \\left( \\frac{\\zeta^{4}-1}{\\zeta^{2} -1} \\right) \\cdots \\left( \\frac{\\zeta^{2^r}-1}{\\zeta^{2^{r-1}} -1} \\right)=$$ $$\\left( \\zeta+1 \\right) \\left( \\zeta^{2} + 1 \\right) \\cdots \\left( \\zeta^{2^{r-1}} +1 \\right),$$ exhibiting an explicit inverse for \\zeta+1. Note that, for m odd, the negation of a primitive 2m-th root of unity is a primitive m-th root of unity. So the negation of your proposed unit is \\zeta_q+\\zeta_{-q} + \\zeta_{r} + \\zeta_{-r}. There is a primitive qr-th root of unity, \\eta, such that \\zeta_q = \\eta^r and \\zeta_r = \\eta^q. So the claim is that$$\\eta^r + \\eta^{-r} + \\eta^q + \\eta^{-q}$$is a unit. This factors as $$\\eta^r \\left(1 + \\eta^{q-r} +\\eta^{-q-r} + \\eta^{-2r} \\right) = \\eta^r \\left( 1+ \\eta^{q-r} \\right) \\left( 1+\\eta^{-q-r} \\right).$$ Using that$q$and$r$are primes, we have$GCD(q-r, qr) = GCD(q+r,qr)=1$, so$\\eta^{q-r}$and$\\eta^{q+r}$are primitive$qr$-th roots of unity, and we are done by the lemma. Note: I don't think I used that$q$and$r$were odd primes, only that they were odd, relatively prime, and$>1\\$.","date":"2013-05-21 11:02:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.976094663143158, \"perplexity\": 493.3163478215211}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368699899882\/warc\/CC-MAIN-20130516102459-00072-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
| null | null |
Q: Class method return iterator I implemented an iterator class as following:
import numpy as np
import time
class Data:
def __init__(self, filepath):
# Computationaly expensive
print("Computationally expensive")
time.sleep(10)
print("Done!")
def __iter__(self):
return self
def __next__(self):
return np.zeros((2,2)), np.zeros((2,2))
count = 0
for batch_x, batch_y in Data("hello.csv"):
print(batch_x, batch_y)
count = count + 1
if count > 5:
break
count = 0
for batch_x, batch_y in Data("hello.csv"):
print(batch_x, batch_y)
count = count + 1
if count > 5:
break
However the constructor is computationally expensive, and the for loop might be called multiple times. For example, in above code the constructor is called twice (each for loop create a new Data object).
How do I separate constructor and iterator? I am hoping to have the following code, where constructor is called once only:
data = Data(filepath)
for batch_x, batch_y in data.get_iterator():
print(batch_x, batch_y)
for batch_x, batch_y in data.get_iterator():
print(batch_x, batch_y)
A: You can just iterate over an iterable object directly, for..in doesn't require anything else:
data = Data(filepath)
for batch_x, batch_y in data:
print(batch_x, batch_y)
for batch_x, batch_y in data:
print(batch_x, batch_y)
That said, depending on how you implement __iter__(), this could be buggy.
E.g.:
Bad
class Data:
def __init__(self, filepath):
self._items = load_items(filepath)
self._i = 0
def __iter__(self): return self
def __next__(self):
if self._i >= len(self._items): # Or however you check if data is available
raise StopIteration
result = self._items[self._i]
self._i += 1
return result
Because then you couldn't iterate over the same object twice, as self._i would still point at the end of the loop.
Good-ish
class Data:
def __init__(self, filepath):
self._items = load_items(filepath)
def __iter__(self):
self._i = 0
return self
def __next__(self):
if self._i >= len(self._items):
raise StopIteration
result = self._items[self._i]
self._i += 1
return result
This resets the index every time you're about to iterate, fixing the above. This won't work if you're nesting iteration over the same object.
Better
To fix that, keep the iteration state in a separate iterator object:
class Data:
class Iter:
def __init__(self, data):
self._data = data
self._i = 0
def __next__(self):
if self._i >= len(self._data._items): # check for available data
raise StopIteration
result = self._data._items[self._i]
self._i = self._i + 1
def __init__(self, filepath):
self._items = load_items(filepath)
def __iter__(self):
return self.Iter(self)
This is the most flexible approach, but it's unnecessarily verbose if you can use either of the below ones.
Simple, using yield
If you use Python's generators, the language will take care of keeping track of iteration state for you, and it should do so correctly even when nesting loops:
class Data:
def __init__(self, filepath):
self._items= load_items(filepath)
def __iter__(self):
for it in self._items: # Or whatever is appropriate
yield return it
Simple, pass-through to underlying iterable
If the "computationally expensive" part is loading all the data into memory, you can just use the cached data directly.
class Data:
def __init__(self, filepath):
self._items = load_items(filepath)
def __iter__(self):
return iter(self._items)
A: Instead of creating a new instance of Data, create a second class IterData that contains an __init__ method that runs a process which is not as computationally expensive as instantiating Data. Then, create a classmethod in Data as an alternative constructor for IterData:
class IterData:
def __init__(self, filepath):
#only pass the necessary data
def __iter__(self):
#implement iter here
class Data:
def __init__(self, filepath):
# Computationaly expensive
@classmethod
def new_iter(cls, filepath):
return IterData(filepath)
results = Data.new_iter('path')
for batch_x, batch_y in results:
pass
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 655
|
{"url":"https:\/\/www.physicsforums.com\/threads\/calculate-current-rms.842970\/","text":"# Calculate Current (RMS)\n\nTags:\n1. Nov 13, 2015\n\n### elctronoob\n\n1. The problem statement, all variables and given\/known data\nSorry for what will likely be considered a noob question but I've been stuck on it for a while and its only at the start of my assignment :(\nSimple circuit\nVac = 50Sin200\u03c0t, R1 = 40\u03a9\nCalculate the value of the RMS current\n\n2. Relevant equations\nVt = Vmax sin2\u03c0ft = Vmax sin\u03c9t\n\n3. The attempt at a solution\nI can't figure out how to do this when we are given no value for time at all. I think i have it worked out at say 1ms\nVt = 50sin((200\u03c0)(0.001))\n= 50sin(.628318)\n= 54.83V\n\nVrms = 54.83 * .7071 = 38.77\n\nIrms = Vrms \/ R = 38.77 \/ 40 = 0.969 amperes\n\nWe have been told that time is intended to be an unknown variable. It's been a long time since I've look at any of these types of problems, can someone help me out on what I'm missing here please\n\nThanks\n\n2. Nov 13, 2015\n\n### PWiz\n\nThe r.m.s. can be calculated just from the peak value if the current is sinusoidal, as is the case here. You can calculate the peak current from the peak voltage. Also remember that the average of the sin^2 function is half. Can you take it from here?\n\n3. Nov 13, 2015\n\n### elctronoob\n\nThanks for the reply PWiz...any formula I can find for these calculations seem to require a time.\nThe only one I can find without t is Vpeak = Vrms\u221a2...=> Vpeak\/\u221a2 = Vrms\nVrms \/ R = Irms\n\nI was thinkin maybe I could half the Vac given in the question, but that still involves a t variable...so frustrating :(\n\n4. Nov 13, 2015\n\n### PWiz\n\nThat's pretty much what I was talking about.\n\n5. Nov 13, 2015\n\n### elctronoob\n\nStill a bit confused, so can I say\nVpeak = 50Sin200\u03c0t \/ 2 = 25Sin200\u03c0t\nVrms = 25Sin200\u03c0t \/ \u221a2\nThis still requires a value for t, no?\n\n6. Nov 13, 2015\n\n### elctronoob\n\n7. Nov 16, 2015\n\n### elctronoob\n\nCould someone confirm that my thinking on this question is correct for me please? I'm basically just disregarding the Sin200\u03c0t and calculating the Vrms and Irms derived from from the 50 value at the start of the expression, i.e 50V. This seems incorrect to me, could someone just check my logic here please\n\n8. Nov 16, 2015\n\n### Staff: Mentor\n\nelectronoob,\n\nFor a sinusoid (sine or cosine) the rms value is $1 \/ \\sqrt{2}$ its peak value. This is entirely independent of the frequency of the sinusoid. In this problem you're given a sinusoidal voltage signal with a peak value of $50 ~V$, so the rms voltage is $50 \/ \\sqrt{2}~~V$.\n\nIt really is that simple for sinusoids. There are other relationships between peak and rms values for other signal shapes.\n\n9. Nov 16, 2015\n\n### elctronoob\n\nThanks for that gneill, much appreciated","date":"2018-02-21 06:24:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.5758611559867859, \"perplexity\": 1068.9538657159553}, \"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-2018-09\/segments\/1518891813431.5\/warc\/CC-MAIN-20180221044156-20180221064156-00690.warc.gz\"}"}
| null | null |
{"url":"https:\/\/tex.stackexchange.com\/questions\/174125\/custom-page-numbering-in-lyx-for-thesis-requirement","text":"# Custom Page Numbering in LyX (for thesis requirement)\n\nI would like to place the page number in the top right corner of my page (inside the margin). My margins are 1 inch from the top and 1 inch from the right. I can put page numbers in the upper right corner by using a horizontal fill, but I would like the page number to be as close to my margin as possible. When I print out the document it is over 1.5 inches from the top of the page. How can I make the number appear higher in the page?\n\nMWE My source code for the following LyX screen and pdf is\n\n\\begin{center}\n\\newpage{}\\hfill{}\\hfill{}1\\thispagestyle{empty}\\vspace{1in}\n\\par\\end{center}\n\\section{\\noindent {\\normalsize{INTRODUCTION}}}\n\n\nMy LyX screen is\n\nAnd my PDF is\n\n\u2022 You might want to have a look at \\frontmatter, \\mainmatter, and friends. \u2013\u00a0Sean Allred Apr 29 '14 at 3:55\n\u2022 Pages that should have their numbers not typed (title, copyright, dedication?) would generally use \\thispagestyle{empty} to remove the numbers. They would still increment the count, however, which seems to be as desired. \u2013\u00a0cslstr Apr 29 '14 at 4:34\n\u2022 @cslstr if I insert that in my Latex preamble, how will it know which page numbers to hide? \u2013\u00a0The Substitute Apr 29 '14 at 4:47\n\u2022 @TheSubstitute: Insert the command at the places of the corresponding pages, not in the preamble. You have to do this multiple times, at all positions\/pages you don't want to have visually numbered. \u2013\u00a0user31729 Apr 29 '14 at 4:55\n\u2022 thank you, this takes care of the pages that I don't want to have visible numbers. \u2013\u00a0The Substitute Apr 29 '14 at 5:05\n\nIn Page Margins set margins to 1 in. (you may need to adjust to get it exactly right). In Page Layout set headings style to default. In Preamble\n\n\\pagestyle{myheadings}\n\\markboth{}{}\n\n\nAdd a thin space and a vertical space if you want the introduction to be a little down.\n\nPage numbers should appear at right edge, starting at 1 on first page.\n\nAfter that, I believe you can put a\n\n\\pagestyle{empty}\n\n\nor otherwise, to remove the page numbers...\n\n% Preview source code\n\n%% Do not edit unless you really know what you are doing.\n\\documentclass[english]{article}\n\\usepackage[latin9]{inputenc}\n\\usepackage{geometry}\n\\geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}\n\n\\makeatletter\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.\n\\markboth{}{}\n\\usepackage{lipsum}\n\n\\makeatother\n\n\\usepackage{babel}\n\\begin{document}\n~ \\vspace{3in}\n\n\\section{Introduction}\n\n\\lipsum\n\n\\lipsum\n\\pagestyle{empty}\n\\lipsum\n\\end{document}","date":"2019-07-23 03:52:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7231823801994324, \"perplexity\": 1337.183618822177}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195528687.63\/warc\/CC-MAIN-20190723022935-20190723044935-00251.warc.gz\"}"}
| null | null |
Originally built by the Archduke Luis Salvador around 140 years ago, this exceptional Mallorquin finca is positioned in an unrivalled coastal location of the wonderful village of Valldemossa, boasting outstanding panoramic sea views.
The spacious interior of the residence is gracefully refurbished in an inviting traditional style and offers grand living areas including six superb bedrooms suites with private bathrooms. The residence is surrounded by an impeccably kept garden offering many wonderful lounging areas, some overlooking the coastline, others set amidst the peaceful garden. The splendid pool area with a covered terrace overlooks the sea. Several outbuildings including a tower complete this outstanding property.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 590
|
A referendum is a blunt political instrument to deal with complex political questions, but now that it is over Parliament will have to deal with the consequences after what has been a difficult campaign. Below are some of our early thoughts.
The Fixed Term Parliament Act 2011 sets the date of the general election in 2020. This legislative straitjacket makes it difficult but not impossible to hold an early election should the new Prime Minister and government decide they need a fresh mandate. There is no discretion for the Prime Minister or the Monarch on this issue. There are legally only two routes to secure it.
The first is that two thirds of all MPs – 433 – vote for a motion calling for one. No single party can command that level of support so it would require a significant proportion of both Conservative and opposition MPs to vote for it. But to do so would need the parties to mutually conclude that an election would be in their interests at this point in time.
The alternative route is for a simple majority to be secured on a motion "That this House has no confidence in Her Majesty's government". There would then be a 14-day period during which consideration could be given to possible alternative administrations and who can command the confidence of the House of Commons. To enjoy the confidence of the House of Commons does not require that a government command the positive support of a majority of the House, merely that no combination of parties can form a majority against it. If no alternative emerges, and therefore a motion of confidence cannot be secured within the 14 days, a general election would be called.
The Electoral Registration and Administration Act 2013 made provision for lengthening the general election timetable from the previous 17 days to 25 days. This came into force with The Electoral Registration and Administration Act 2013 (Commencement No 5 and Transitory Provisions) Order 2014. The 25-day timetable does not include weekends and bank holidays (and any day appointed for public thanksgiving or mourning). In practice it therefore means a five-week election timetable.
Throughout this period Parliament would be dissolved and, unlike during recess periods, it could not be recalled. There is no legal mechanism to restore MPs in the old Parliament on a temporary basis. The proclamation dissolving Parliament for the election will set out the date of summons for the new Parliament. In the interim, no Parliament exists.
So, if politically, a general election could be secured under the terms of the Fixed Term Parliament Act, there would follow a minimum five-week period between dissolution and assembly of the new Parliament when there would be no parliamentary oversight of the executive. Given the volatile economic and constitutional environment this may be deemed unsatisfactory.
If it was deemed politically desirable, and there was sufficient support for it in both Houses of Parliament, the Fixed Term Parliaments Act and the Electoral Registration and Administration Act 2013 (Commencement No 5 and Transitory Provisions) Order 2014 could both be amended or replaced to facilitate a general election in the new circumstances. The Fixed Term Parliaments Act cannot simply be repealed as Parliament would exist in perpetuity for there would be no legal provision to bring it to an end.
The decision to leave the EU may herald significant changes in the way Parliament, and particularly the House of Commons, operates.
MPs will want some oversight of negotiations over the two-year departure period once Article 50 of the Treaty of Lisbon is triggered and they will ultimately have a vote on the final exit Treaty(s).
The scrutiny work that lies ahead will be detailed, complex and technical. MPs already struggle to effectively scrutinise financial and delegated legislation and this will add to the burden. Serious consideration therefore needs to be given to a bi-cameral solution to the scrutiny process given that Peers tend to have greater appetite for and experience of such scrutiny. It would be in both Houses' interest not to duplicate work.
In the Commons a regular ministerial question time may well be demanded, debates about the way ahead will be a regular feature and the Urgent Question is likely to be a key tool as backbenchers try to hold ministers to account. Similar provisions for question time and debates are likely to be requested in the House of Lords.
A stand-alone select committee or a joint committee of both Houses may be required to monitor the negotiations and decision-making, although existing departmental select committees will want to oversee their own particular departmental policy area. As a consequence we will likely see greater efforts at joint working across committees. Consideration will need to be given to how to knit together the new scrutiny work with that of existing committees that are tasked with scrutinising EU legislation, specifically the European Scrutiny Committee in the Commons and the EU Committee in the Lords.
The Liaison Committee of select committee chairs currently questions the Prime Minister three times a year – they may want an increase in these sessions.
Parliament may require additional staff and support and there will, from time to time, be a need to bring in specialist expertise, which will bring its own costs. When the bicameral Banking Commission inquiry, modelled similarly to a select committee was set up a couple of years ago it consumed resources. The requirement for scrutinising our departure from the EU will be on a much greater, and longer-term scale and the resource need will therefore be higher. Effective scrutiny will not be cheap but should not be stinted upon.
Ironically, although advocates of leave campaigned ostensibly to restore the influence of Parliament, in reality the government not the legislature will be empowered in the short term by Brexit.
Reviewing the corpus of EU law will be an enormously challenging task if mistakes and anomalies that plunge people into grey areas of legal uncertainty are to be avoided. At the heart of the leave campaign's roadmap for navigating an exit path is their proposal for a European Union Law (Emergency Provisions) Bill which would amend but not repeal the European Communities Act 1972. But decisions will have to be taken about whether to keep, amend or repeal all the regulations made under Section 2.2 of the 1972 Act otherwise they will automatically lapse on its repeal.
Given the volume of legislation involved, in practice much of the heavy lifting will probably have to be done via delegation and through statutory instruments. This will empower the executive not Parliament and, given the complete inadequacies of Commons procedures for scrutinising delegated legislation, will frustrate MPs, not least because they do not currently have the power of amendment. The passage of this legislation may also be unduly complicated and time consuming given the need to determine the EVEL designation of each new SI for scrutiny purposes.
The process may also give rise, once again, to questions about the relationship between the Commons and Lords. As things stand, the Upper House retains its right to reject Statutory Instruments but does so very rarely. If the government does not enact the proposals in the Strathclyde Review then it is not inconceivable that the two Houses could clash on the detail of these SI's in future if Peers feel that errors are being made in the review and amendment process. Delegated legislation, unlike primary legislation is also subject to judicial review so the door will remain open to potential legal challenge if problems arise.
The Procedure Committee of the House of Commons initiated an inquiry into the scrutiny of delegated legislation before the last election but it made little headway. The Committee may wish to restart this inquiry and, once the government confirms the exit negotiating process, it may wish to initiate an inquiry into how the House of Commons scrutiny machinery can best be established to mirror the negotiating model.
If no changes to the scrutiny process are implemented, then under current arrangements the bulk of the scrutiny burden for SIs will fall on the Joint Committee on Statutory Instruments and the Secondary Legislation Scrutiny Committee. In the last session (2015-16), 741 SIs were laid by the government before Parliament for scrutiny. In the second session of a Parliament, the number tends to creep up to nearer 1,000. In their current form it is therefore likely to be beyond the capacity of these committees to consider the existing output of SIs alongside new SIs arising from decisions made in relation to the European Communities Act.
What happens to the current legislative programme?
Nine bills have been published and await the conclusion of the parliamentary scrutiny process, and a further 16 have been promised but have not yet emerged.
It is currently unclear what will happen to them, having been drafted with regard to the UK's obligations under EU law.
As the Prime Minister has announced his intention to resign we effectively have a caretaker government in all but name. It is likely that little further legislative progress will therefore be made pending the appointment of the new Prime Minister and Cabinet. Key decisions that were expected after the referendum such as a new runway at Heathrow and renewal of Trident may similarly be put on ice.
The Leader of the House of Commons would normally make a business statement to the House on Thursday morning. In the circumstances, however, an earlier statement may need to be made when Parliament reconvenes on Monday. Clarification may emerge in any statement the Prime Minister makes to the House on Monday or the Leader of the House may follow up with a separate statement setting out any changes to business in the coming weeks. A similar statement would be required in the House of Lords.
Once a new Prime Minister and Cabinet are chosen, if there is to be no general election, then they will have to set out their plans through statements to Parliament and, if deemed necessary, a new budget and accompanying Finance Bill.
A number of key constitutional challenges will also need to be addressed. For example, under the devolved model, key areas of policy where EU law applies are subject not to Westminster but the devolved legislatures.
The Scottish Parliament is required to legislate compatibly with the EU. The Scotland Act 1998 will therefore need to be amended which will require the consent of the Scottish Parliament. Given Scottish support for staying in the EU, it is unclear where such a constitutional stand off would lead if that consent was withheld. Similar issues will also arise with the National Assembly for Wales and the Northern Ireland Assembly.
Historically, communication between Westminster and the three devolved legislatures has been poor. Relations tend to be conducted on a government to government basis, and the devolved government then engages with its own legislature on the issues concerned. This has resulted in misunderstandings in legislative drafting and the devolved legislatures being left out of the loop about developments with one of their sister legislatures that might have longer-term implications for themselves. In addition to whatever processes are put in place at the Whitehall level for the inter-governmental negotiations, either through the Inter-governmental Council or a new body, Parliament may therefore wish to set up a four member inter-parliamentary committee arrangement to shadow the negotiating process.
So the constitutional and legislative challenges and the sheer volume of work will be enormous, and it is unclear how it will all fit alongside the current day-to-day workload of the government's existing legislative plans. And of course, all this will have to be done alongside preparations for the proposed multi-billion pound refurbishment of the Palace of Westminster. The 'restoration and renewal' programme is a one in 150 year opportunity to reform the leading institution of our democracy. It will now take place in a very different political environment to the one we were thinking about just weeks ago. But it also presents a real opportunity to reinvent the way we legislate and the culture and practice of parliamentary politics in a manner fit for our new future.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,599
|
Sometimes you are just in the right place at the right time. Like I was with this construction site just as the sun was going down. And it had all of that green detail, it was just meant to be. But let's talk about this outfit. It's one of my favorites this winter because a long lined mens wear inspired blazer paired with a chunky heel bootie gives me all of the feels right now. You most likely know about my obsession with blazers already and I can absolutely see myself wearing this one all winter long. It's got the kind of structured fit that make it great for every day work meetings but I think it would look incredible over a silk flowy shirt with a red lip and some pumps for a party or get together.
Houndstooth Long Line Blazer- Forever 21- Wearing a size small. This particular one is unfortunately no longer in stock, but this one is here is super similar and actually quite adorable with the frill details on the shoulders, it's just a bit shorter.
Skirt: Ribbed Green Pencil Skirt- Forever 21 from a few years ago- Size small- similar but in a different color here.
Necklace: A Silver Statement Necklace from Forever 21, not new. Very similar one here.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,642
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<?php
namespace FlorianKoerner\ChimpDrill\Tests;
use FlorianKoerner\ChimpDrill\ChimpDrill;
class EscapingTest extends \PHPUnit_Framework_TestCase
{
/**
* @return array
*/
public function dataProvider()
{
return array(
array(
'message' => 'Damn! Very cool PHP-Code: <?php echo "evil"; ?>',
'placeholder' => array(),
'expected' => 'Damn! Very cool PHP-Code: <?php echo "evil"; ?>'
),
array(
'message' => 'And take a look at these HTML-Tags: <strong>Unbelievable!</strong>',
'placeholder' => array(),
'expected' => 'And take a look at these HTML-Tags: <strong>Unbelievable!</strong>'
),
array(
'message' => 'Your input was *|INPUT|*. Thank you, *|NAME|*.',
'placeholder' => array(
'NAME' => 'John <hacks>',
'INPUT' => '<?php var_dump($_SERVER); ?> <strong>I\'m a very bad boy</strong>'
),
'expected' => 'Your input was <?php var_dump($_SERVER); ?> <strong>I\'m a very bad boy</strong>. Thank you, John <hacks>.'
),
array(
'message' => 'Your input was *|HTML:INPUT|*. Thank you, *|HTML:NAME|*.',
'placeholder' => array(
'NAME' => 'John <hacks>',
'INPUT' => '<?php var_dump($_SERVER); ?> <strong>I\'m a very bad boy</strong>'
),
'expected' => 'Your input was <?php var_dump($_SERVER); ?> <strong>I\'m a very bad boy</strong>. Thank you, John <hacks>.'
)
);
}
/**
* @param string $message
* @param array $placeholder
* @param string $expected
*
* @dataProvider dataProvider
*/
public function testEscaping($message, array $placeholder, $expected)
{
$this->assertEquals($expected, (string) new ChimpDrill($message, $placeholder));
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,749
|
Hylesia praeda är en fjärilsart som beskrevs av Paul Dognin 1901. Hylesia praeda ingår i släktet Hylesia och familjen påfågelsspinnare. Inga underarter finns listade i Catalogue of Life.
Källor
Externa länkar
Påfågelsspinnare
praeda
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,128
|
\section{Introduction}
\label{sec:introduction}
\begin{figure}[t]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=\textwidth]{images/process_v2.png}}
\label{fig:process_v1}
\end{minipage}
\caption{Continual DA paradigm. Initial training is performed with labeled data in the source domain and the trained model is deployed in the target domain. During deployment, unlabelled target domain data are received in streaming batches and the model is continuously adapted with each new batch of target data.}
\label{fig:process}
\end{figure}
Domain adaptation (DA) methods based on deep learning have received significant attention in recent years for mitigating
the domain shift from the training domain (source) to the inference domain (target) \cite{ganin2015unsupervised,adda,gvb,shot,hdmi,mdd}.
In unsupervised domain adaptation (UDA), where the same classes are present in the source and target domains (closed set), the gap between the annotated source domain data and unlabeled target domain data is the main cause of reduction in classification accuracy. Many of the recent popular deep learning based DA methods \cite{gvb, alda, kurmi2019attending, cdan} employ adversarial training with both the source and target data to learn domain agnostic features, as proposed in \cite{ganin2015unsupervised}, or to align the feature spaces of the source and target domains, as was done in \cite{adda}. Inspired by Hypothesis Transfer Learning (HTL) \cite{kuzborskij2013stability}, some recent methods \cite{shot, kundu2020universal, hdmi} transfer only the source trained model for target adaptation, thus greatly reducing the data storage footprint.
Current DA methods operate under the assumption that the entire target dataset is available during adaptation, which may not be feasible in practice, e.g., when an autonomous vehicle operates in a new environment.
In this paper, we consider the scenario depicted in Figure \ref{fig:process}, where the network model is initially trained using source domain data and is then deployed in a new domain where target data are collected incrementally in small batches and the model is updated continually.
In our continual DA framework, the shift between the source and target can be sudden and, depending on the datasets considered, the target distribution can be significantly different than the source distribution.
In a related approach, Hoffman et al. \cite{hoffman2014continuous} proposed a manifold-based method that deals with streaming target data from an evolving target domain that is changing slowly. However, this work did not consider deep learning methods, it was not applied on standard DA datasets, and assumed there is no sudden domain shift between the source and target domains or between two consecutive time instances within the target domain.
In contrast, we present a scenario where the target distribution is not directly related to the source distribution and the target data are received in a series of smaller batches over time, as shown in Figure \ref{fig:process}. Our approach is broader in scope, introduces a deep learning framework, and is applicable to standard DA datasets, making comparison with existing DA methods possible.
To illustrate the impact of continual DA on current methods, we performed experiments in a continual setting with two state-of-the-art (SOTA) methods: Source Hypothesis Transfer (SHOT) \cite{shot}, a source-free DA method, and Gradually Vanishing Bridge (GVB) \cite{gvb}, a source dependent adversarial DA method.
Using the Office-31 \cite{office} dataset, we considered Amazon as the source domain and adapted contiually on small batches of independent and identically distributed (i.i.d.) samples drawn from the target domain (DSLR or Webcam).
The results of Figure \ref{fig:contda_results} indicate that continual adaptation performance drops significantly for both methods, compared to standard DA using the entire target data. The drop in performance is more pronounced as the batch size gets smaller.
These results demonstrate the need for new continual DA approaches
that can maintain high performance while adapting to new target batches.
To solve this problem, we take cues from continual learning methods \cite{rebuffi2017icarl,wu2019large, hayes2020remind} and propose a Continual DA (ConDA) framework with a buffer to store processed samples and their predicted labels, and buffer management strategies to selectively store and replay previously seen target samples. Furthermore, our method incorporates better features with higher generalization capabilities that improve upon the performance of SOTA source-free DA methods.
The ConDA approach continually adapts the source model to the target domain as data arrive in batches, which greatly reduces the data storage requirements. Our method does not require any source data during adaptation, and additionally does not need to store the whole target domain at any time.
During adaptation, ConDA only requires the incoming batch of target data along with the data stored in the buffer.
We evaluate several buffer configurations, along with specific loss functions for continual adaptation, and propose a buffer management strategy and associated adaptation procedure that is well-suited for continual DA.
ConDA outperforms many non-continual DA methods that utilize the full target domain, yet it operates at a fraction of their data storage footprint.
The main contributions of our paper are outlined below.
\begin{figure}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=0.95\linewidth]{images/amazon_to_dslr_cont.png}
\label{fig:Ng1}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=0.95\linewidth]{images/amazon_to_webcam_cont.png}
\label{fig:Ng2}
\end{subfigure}
\caption[]{Evaluation of SOTA DA methods SHOT \cite{shot} and GVB \cite{gvb} in a continual DA framework from Amazon to DSLR (top) and Amazon to Webcam (bottom).
The horizontal dotted lines represent DA performance when the full target domain is available.
The solid lines are based on adaptation using incoming target data batches of different size and performance evaluated on the entire target domain after all of the batches are seen by the network.
}
\label{fig:contda_results}
\end{figure}
\begin{enumerate}
\vspace{-0.05in}
\item We propose a new paradigm for continual unsupervised domain adaptation performed on new batches of target samples.
\vspace{-0.01in}
\item We propose the ConDA framework for source-free continual DA that adapts on incoming batches of unlabeled target data and utilizes a buffer for selective replay of previous samples.
\vspace{-0.1in}
\item During ConDA adaptation, we utilize sample mixup and equal diversity loss along with our buffer management strategy for effective adaptation.
\vspace{-0.1in}
\item The performance of ConDA is superior to many SOTA DA methods, while utilizing a much smaller data storage footprint, even though other methods have access to the entire target and source domains.
\vspace{-0.1in}
\item We demonstrate that high resolution features are useful for generalization across domains and achieve significant performance gains for UDA on standard datasets.
\end{enumerate}
\section{Related Work}
\label{sec:relatedwork}
\subsection{Unsupervised Domain Adaptation}
A domain gap manifests due to the dataset bias when the data distributions in the source and target domains are significantly different \cite{torralba2011unbiased}.
Many UDA techniques have been proposed to mitigate this domain gap for computer vision tasks such as object detection and semantic segmentation \cite{shot, khodabandeh2019robust, chen2019domain}. Long et al. \cite{long2015learning} and Tzeng et al. \cite{tzeng2014deep} proposed minimizing the maximum mean discrepancy (MMD) for UDA.
Zellinger et al. \cite{zellinger} proposed minimizing central moment discrepancy (CMD) by matching higher order central moments of probability distributions in the source and target data. Ganin et al. \cite{ganin2016domain} aligned distributions of source and target domains via an adversarial domain discriminator.
Tzeng et al. \cite{tzeng2017adversarial} adversarially aligned features of source and target domain data while transferring the source domain classifier to the target domain. Likewise, generative models have also been employed to create source-like images at the pixel level for domain adaptation \cite{cyclegan}.
Adversarial methods require access to source data at the time of adaptation, but this is likely to create issues related to storage requirements or privacy when sharing of sensitive and private data. Domain adaptation research has been exploring such practical scenarios where adaptation is done without using source data. Source-free UDA methods consist of an initialization stage with access to source data for training and an adaptation stage with access only to the target data without any of the source data \cite{kundu2020towards}. Chidlovskii et al. \cite{chidlovskii2016domain} proposed a semi-supervised source-free DA framework where no source domain data are available during adaptation, but some representation of the source domain is available, such as class means or a few annotated target samples. Liang et al. \cite{liang2019distant} identified a subspace where target and source centroids are only modestly shifted and used class-wise distribution estimator of the source data to conduct distant supervision for target adaptation. An end-to-end, source-free DA method based on information maximization was proposed in \cite{shot}.
\subsection{Continual Learning}
\begin{figure*}[htbp]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=\textwidth]{images/network.png}}
\end{minipage}
\caption{Proposed ConDA framework adapting on target domain data that arrive in small batches. A subset of the samples that are already seen by the network are stored in a buffer for replay with the incoming batches. The buffer manager is responsible for selecting the samples that populate the buffer. The incoming target samples are mixed with the current buffer samples and sent to the network for adaptation.}
\label{fig:architecture}
\end{figure*}
Mammals, as opposed to artificial neural networks trained within the standard deep leaning framework, learn continuously so that their intelligence increases gradually over time. When neural networks are subjected to such continual learning, they run the risk of catastrophic forgetting, where they forget the knowledge gained in earlier training stages \cite{mccloskey1989catastrophic}. Continual or lifelong learning methods have proposed a few mechanisms to mitigate catastrophic forgetting in deep neural networks. Among them, the most prominent are (i) replay of previously seen data \cite{rebuffi2017icarl, wu2019large, hayes2020remind}, (ii) constraining network parameter updates according to a regularization scheme \cite{kirkpatrick2017overcoming, zenke2017continual}, and (iii) network expansion with increasing data \cite{progressive, dynamic, distillation}. Memory replay mimics the mechanism of the human brain, where during both the sleeping \cite{sleep} and awake \cite{awake} phases, past experiences are regenerated from encoded representations and the neocortex is trained on them \cite{stickgold2001sleep,play}. Rebuffi et al. first applied memory replay in iCaRL \cite{rebuffi2017icarl}, for class-incremental learning in the context of neural networks, where 20 raw samples from each class were stored for later replay. More recent replay methods extended iCaRL to make it end-to-end trainable \cite{endtoend}, introduced a loss function to correct for class bias \cite{wu2019large}, and stored mid-level features instead of raw images to reduce storage footprint \cite{hayes2020remind}.
Regularization based models learn new tasks incrementally while preserving knowledge from previous tasks by varying the plasticity of the network's convolutional filter weights, which are significant for retaining earlier knowledge.
Kirkpatric et al. \cite{kirkpatrick2017overcoming} proposed to selectively lower the learning rate from one task to the next.
In this work, we mainly draw from the concept of memory replay. We present a way to continually adapt a source trained model to a new target domain when the target data are received in batches and not all available at the same time. This is an area of domain adaptation that, to the best of our knowledge, has not yet explored. We showed in Figure \ref{fig:contda_results} that when incoming target data are received in batches, state-of-the-art DA methods suffer from performance degradation. We next present our ConDA method to overcome these limitations, and discuss strategies to configure the buffer and corresponding loss functions for continual DA. We benchmark our approach against standard DA methods and obtain SOTA results on some popular DA datasets.
\section{Method}
\label{sec:method}
We consider a source domain $\mathcal{D}_s$ with labelled source samples $\{x_s^i, y_s^i\}_{i=1}^{n_s}$ where $n_s$ is the total number of
source samples $x_s^i \in \mathcal{X}_s$ with corresponding labels $y_s^i \in \mathcal{Y}_s$. We are given an unlabelled target domain $\mathcal{D}_t$ with $n_t$ samples $\{x_t^i\}_{i=1}^{n_t}$ and $x_t \in \mathcal{X}_t$. In closed-set UDA, we assume that the number of classes $\mathcal{C}_s$ present in the source domain is same as the number of classes $\mathcal{C}_t$ present in the target domain, and the task is to predict the target labels $\{y_t^i\}_{i=1}^{n_t}$ where $y_t \in \mathcal{Y}_t$.
In the continual UDA setting, the target domain $\mathcal{D}_t$ is divided into $m$ batches, i.e., $\mathcal{X}_t = \{\mathcal{X}_t^1, \mathcal{X}_t^2, \mathcal{X}_t^3, .... , \mathcal{X}_t^m \}$
with samples $\{x_t^{j,i}\}_{j=1, i=1}^{m, n_t^j}$ where $n_t^j$ is the number of samples in the $j^{th}$ batch and $j \in \{1, 2, 3, ....., m\}$.
We consider that the source trained model $f_s: \mathcal{X}_s \rightarrow \mathcal{Y}_s$ is available with only a batch of target samples $\mathcal{X}_t^j$ at a time and our objective is to learn a model $f_t: \mathcal{X}_t^j \rightarrow \mathcal{Y}_t^j$ where $\mathcal{Y}_t^j$ is the predicted labels of $\mathcal{X}_t^j$.
The continual DA scenario runs the risk of the model overfitting to the current batch of target samples and failing to adapt to the marginal distribution of the entire target domain due to the continual nature of the incoming samples. Therefore, our task is to reduce the performance gap between the model that is adapted based on continuous batches of target data, i.e., $f_t: \mathcal{X}_t^m \rightarrow \mathcal{Y}_t^m$ and the model that is adapted given the entire target domain simultaneously (standard DA framework), i.e., $f_t: \mathcal{X}_t \rightarrow \mathcal{Y}_t$, both evaluated on the full target domain $\mathcal{X}_t$.
Our ConDA framework for continual adaptation is shown in Figure \ref{fig:architecture}. The source model $f_s(x) = h_s(g_s(x))$ consists of two parts: a feature generator model $g_s$ that includes a backbone and a fully-connected layer followed by a batch normalization layer, and a hypothesis model $h_s$ that includes a fully connected layer and a weight normalization layer.
Inspired by \cite{shot}, we train the source model $f_s$
in a supervised manner with label smoothing \cite{muller2019does}.
During target adaptation, we initialize the target hypothesis model with the source hypothesis, $h_t = h_s$, and the parameters of the hypothesis model remain unchanged over the adaptation procedure. We initialize the target feature generation model $g_t$ with the source feature generation model $g_s$ and adapt it with an incoming batch of target samples.
In Section \ref{sec:introduction}, Figure \ref{fig:contda_results}, we showed that SOTA methods do not reach their full performance during continual adaptation. In ConDA, we propose to use a buffer to store selected target samples and replay them with the incoming batch of samples so that the network can generalize effectively over the entire target domain.
\subsection{Buffer}
To conduct continual domain adaptation, we introduce a buffer $\mathcal{B}_t$ with states $\{\mathcal{B}_t^1, \mathcal{B}_t^2, ...., \mathcal{B}_t^m\}$ each corresponding to $m$ batches of target data. We maintain a class-balanced $\mathcal{B}_t$, i.e., an equal number of buffer slots are allocated for each class calculated from buffer length and the number of classes present in the target domain assuming that $\mathcal{C}_t = \mathcal{C}_s$. The buffer is populated after the network is trained on a batch of target samples. The buffer stores the samples and their corresponding class labels predicted by the network. Our model only requires access to the samples stored in the buffer for subsequent adaptation along with new target batches that arrive. The sample selection process to populate the buffer is handled by a buffer manager discussed in the following section.
\subsection{Buffer Manager}
Let's assume that the network is adapted on a batch $\mathcal{X}_t^j$ and outputs $\{\mathcal{Y}_t^j, \mathcal{U}_t^j\}$ where $\mathcal{U}_t$ is the softmax classification score. We compute the buffer sample labels $\mathcal{V}_t^{j-1}$ with the current state of the model $f_t: \mathcal{B}_t^{j-1} \rightarrow \mathcal{V}_t^{j-1}$.
The buffer manager takes in $\{\mathcal{X}_t^j$, $\mathcal{Y}_t^j$, $\mathcal{U}_t^j$, $\mathcal{B}_t^{j-1}$, and $\mathcal{V}_t^{j-1}\}$ and outputs $\mathcal{X}_t^\prime \subseteq \mathcal{X}_t^j \bigcup \mathcal{B}_t^{j-1}$ and corresponding labels to populate the buffer state $\mathcal{B}_t^j$.
At first, the incoming batch samples are grouped based on the output label $\mathcal{Y}_t^j$, and samples of each class are sorted based on the confidence $\mathcal{U}_t^j$. Then, the buffer manager only picks the high confidence samples if the number of samples for any class exceeds the allotted number of slots for that class in the buffer. Finally, if available, the remaining space for that class is filled with randomly drawn samples from $\mathcal{B}_t^{j-1}$ of that class.
We conducted multiple experiments with a few other buffer selection techniques, such as choosing the incoming samples randomly, or selecting the buffer samples based on the cosine distance to the nearest self-supervised cluster centers. We did not find any significant performance variation with various buffer sample selection techniques. We found a slight increase in performance with the sample selection mechanism based on the higher confidence scores.
In the $(j+1)^{th}$ batch, the current buffer samples $\mathcal{B}_t^j$ and the incoming batch samples $\mathcal{X}_t^{j+1}$ are appended and provided to the network.
We do not use any label information of the buffer samples when they are concatenated with the incoming batch samples.
During adaptation with the incoming batch and buffer samples, we performed clustering to compute pseudo labels. The clustering technique is described as follows.
\subsection{Clustering}
We adopted a self-supervised clustering method introduced in \cite{shot} as an extension of the Deep Cluster \cite{caron2018deep} method. The combination of the batch and the buffer samples is denoted as $\mathcal{X}_t^* = \mathcal{X}_t^j \bigcup \mathcal{B}_t^{j-1}$. The initial cluster center is obtained by utilizing the softmax output of the input target samples as follows.
\begin{equation}
c_k^{(0)} = \frac{\sum_{x_t \in \mathcal{X}_t^*}\hat{f}_t(x_t)\hat{g}_t(x_t)}{\sum_{x_t \in \mathcal{X}_t^*}\hat{f}_t(x_t)}
\end{equation}
After computing the initial estimate of the centroids, the initial estimate of the pseudo labels $\hat{y}_t^{(0)}$ is found using the cosine distance function.
\begin{equation}
\hat{y}_t^{(0)} = \argmin_k d(\hat{g}_t(x_t), c_k^0)
\end{equation}
where $d(\cdot,\cdot)$ is the cosine distance function. After computing the initial estimates of the pseudo labels, the cluster centers are recomputed as follows.
\begin{equation}
c_k^{(1)} = \frac{\sum_{x_t \in \mathcal{X}_t^*}\mathbbm{1}{(\hat{y}_t=k)}\hat{g}_t(x_t)}{\sum_{x_t \in \mathcal{X}_t^*}\mathbbm{1}{(\hat{y}_t = k)}}
\end{equation}
where $\mathbbm{1}(\cdot)$ is the indicator function. The final pseudo labels are computed using the updated cluster centers.
\begin{equation}
\hat{y}_t^{(1)} = \argmin_k d(\hat{g}_t(x_t), c_k^{(1)})
\end{equation}
where $\hat{y}_t^{(1)} \in \hat{\mathcal{Y}}_t^*$. However, computing pseudo-labels this way may lead to some noisy labels.
This effect can be more pronounced in continual DA, since each target batch contains only a partial representation of the overall target distribution because batches are composed of a small number of target samples per class. We deal with noisy pseudo-labels using sample mixup, as described next.
\subsection{Sample Mixup}
\begin{table*}[htbp]
\begin{center}
\begin{tabular}{lccccccc>{\columncolor[gray]{0.8}}c}
\toprule
Method & Target &A $\longrightarrow$ D & A $\longrightarrow$ W & D $\longrightarrow$ A & D $\longrightarrow$ W & W $\longrightarrow$ A & W $\longrightarrow$ D & Mean\\
\midrule
DANN \cite{ganin2015unsupervised} & Full & 79.7 & 82.0 & 68.2 & 96.9 & 67.4 & 99.1 & 82.2 \\
SAFN+ENT \cite{xu2019larger} & Full & 92.1 & 90.3 & 73.4 & 98.7 & 71.2 & 100.0 & 87.6 \\
ALDA \cite{alda} & Full & 94.0 & 95.6 & 72.2 & 97.7 & 72.5 & 100.0 & 88.7 \\
MDD+IA \cite{mdd} & Full & 92.1 & 90.3 & 75.3 & 98.7 & 74.9 & 99.8 & 88.8 \\
GVB-GD \cite{gvb} & Full & 95.0 & 94.8 & 73.4 & 98.7 & 73.7 & 100.0 & 89.4\\
CADA-P \cite{kurmi2019attending} & Full & 95.6 & 97.0 & 71.5 & 99.3 & 73.1 & 100.0 & 89.5 \\
HDMI \cite{hdmi} & Full & 94.4 & 94.0 & 73.7 & 98.9 & 75.9 & 99.8 & 89.5 \\
SPL \cite{wang2020unsupervised} & Full & 93.0 & 92.7 & 76.4 & 98.7 & 76.8 & 99.8 & 89.6 \\
CAN+A$^2$LP \cite{zhang2020label} & Full & 96.1 & 93.4 & 78.1 & 98.8 & 77.6 & 99.8 & 90.7 \\
SRDC \cite{tang2020unsupervised} & Full & 95.8 & 95.7 & 76.7 & 99.2 & 77.1 & 100.0 & 90.9 \\
SHOT \cite{shot} & Full & 94.0 & 90.1 & 74.7 & 98.4 & 74.3 & 99.9 & 88.6 \\
\midrule
HR-SHOT (Ours) & Full & 98.2 & 97.2 & 80.0 & 99.0 & 80.2 & 99.8 & 92.4 \\
\midrule
HR-SHOT (Ours) & Cont. & 95.8 & 90.6 & 73.8 & 96.9 & 76.7 & 99.8 & 88.9 \\
ConDA (Ours) & Cont. & 94.8 & 94.7 & 79.1 & 98.4 & 77.2 & 99.8 & 90.7 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Mean accuracy on the Office-31 dataset. The ConDA experiments are performed with a continual batch size of 62 and buffer size of 124 (4 samples per class).}
\label{res:office}
\end{table*}
In the context of information maximization, since we rely on pseudo-labels that are likely to be somewhat corrupted, we employ sample and label mixup \cite{zhang2017mixup} to alleviate prediction sensitivity and achieve better generalization.
Virtual target samples $ (\Tilde{x}_t, \Tilde{y}_t) $ are constructed via mixup as follows.
\begin{equation}
\begin{gathered}
\Tilde{x}_t = \lambda x_t^\alpha + (1-\lambda) x_t^\beta \\
\Tilde{y}_t = \lambda \hat{y}_t^\alpha + (1-\lambda) \hat{y}_t^\beta
\end{gathered}
\end{equation}
where $(x_t^\alpha, \hat{y}_t^\alpha)$ and $(x_t^\beta, \hat{y}_t^\beta)$ are drawn randomly from $\{\mathcal{X}_t^*, \hat{\mathcal{Y}}_t^*\}$ and $\Tilde{x}_t \in \Tilde{\mathcal{X}}_t^*$ and $\Tilde{y}_t \in \Tilde{\mathcal{Y}}_t^*$.
Also, $\lambda \in [0,1]$ is drawn from a $Beta(\rho,\rho)$ distribution, where $\rho \in (0,\infty)$.
\subsection{Adaptation Objective Function}
For our objective function, we consider the information maximization (IM) loss from \cite{gomes2010discriminative, shisha, hu, shot} to produce individually precise predictions while maintaining a global diversity of the network outputs. The IM loss is a combination of the entropy loss $\mathcal{L}_{ent}$ and equal diversity loss $\mathcal{L}_{eqdiv}$ functions shown below.
\begin{equation}
\begin{gathered}
\mathcal{L}_{ent}(f_t;\mathcal{X}_t) = -\mathbb{E}_{\Tilde{x}_t \in \Tilde{\mathcal{X}}_t^*} \sum_{k=1}^{C_s} \sigma_k(f_t(\Tilde{x}_t))log(\sigma_k(f_t(\Tilde{x}_t))) \\
\mathcal{L}_{eqdiv}(f_t;\mathcal{X}_t) = \sum_{k=1}^{C_s} q_k log \left ( \frac{q_k}{\hat{q}_k} \right )
\end{gathered}
\end{equation}
where $\sigma_k(a) = \frac{exp(a_k)}{\sum_i exp(a_i)}$ is the softmax function. Since we maintain a class-balanced buffer, we take $q_k$ as the ideally uniform mean response, such that $q_k$ is a $C_s$ dimensional vector with all values of ${1}/{C_s}$ and $\hat{q}_k = \mathbb{E}_{\Tilde{x}_t \in \Tilde{\mathcal{X}}_t^*} [ \sigma(f_t(\Tilde{x}_t)) ] $ is the mean of the softmax output for the incoming target batch and buffer samples. The equal diversity loss $L_{eqdiv}$ attempts to make network predictions equally diverse for all classes and is calculated as the KL divergence between the ideal uniform distribution and the softmax distribution from the network outputs.
Additionally, $f_t(\Tilde{x}_t) = h_t(g_t(\Tilde{x}_t))$ is a $C_s$-dim output for each virtual target sample generated by sample and label mixup.
We further minimize $\mathcal{L}_{mixup}$, the mixup cross-entropy loss for the generated virtual target samples, shown below.
\begin{multline}
\mathcal{L}_{mixup}(f_t;\mathcal{X}_t) = \\ - \lambda \mathbb{E}_{\Tilde{x}_t \in \Tilde{\mathcal{X}}_t^*, \hat{y}_t^\alpha \in \hat{\mathcal{Y}}_t^*} \sum_{k=1}^{C_s} \mathbf{1}_{[k=\hat{y}_t^\alpha]} log(\sigma_k(f_t(\Tilde{x}_t))) \\ - (1-\lambda) \mathbb{E}_{\Tilde{x}_t \in \Tilde{\mathcal{X}}_t^*, \hat{y}_t^\beta \in \hat{\mathcal{Y}}_t^*} \sum_{k=1}^{C_s} \bold{1}_{[k=\hat{y}_t^\beta]} log(\sigma_k(f_t(\Tilde{x}_t)))
\end{multline}
where $\hat{y}_t^\alpha$ and $\hat{y}_t^\beta$ are the respective clustering pseudolabels for samples $x_t^\alpha$ and $x_t^\beta$ such that $\Tilde{x}_t = \lambda x_t^\alpha + (1-\lambda)x_t^\beta$. Our final objective function therefore becomes,
\begin{equation}
\mathcal{L}(g_t) = \mathcal{L}_{ent} + \gamma_1 \mathcal{L}_{eqdiv} + \gamma_2 \mathcal{L}_{mixup}
\end{equation}
where $\gamma_1$ and $\gamma_2$ are hyper-parameters.
\begin{table*}[htbp]
\begin{center}
\setlength\tabcolsep{0.9pt}
\resizebox{\textwidth}{!}{\begin{tabular}{lccccccccccccc>{\columncolor[gray]{0.8}}c}
\toprule
Method & Target & Ar $\rightarrow$ Cl & Ar $\rightarrow$ Pr & Ar $\rightarrow$ Rw & Cl $\rightarrow$ Ar & Cl $\rightarrow$ Pr & Cl $\rightarrow$ Rw & Pr $\rightarrow$ Ar & Pr $\rightarrow$ Cl & Pr $\rightarrow$ Rw & Rw $\rightarrow$ Ar & Rw $\rightarrow$ Cl & Rw $\rightarrow$ Pr & Mean\\
\midrule
DANN \cite{ganin2016domain} & Full & 45.6 & 59.3 & 70.1 & 47.0 & 58.5 & 60.9 & 46.1 & 43.7 & 68.5 & 63.2 & 51.8 & 76.8 & 57.6 \\
ALDA \cite{alda} & Full & 53.7 & 70.1 & 76.4 & 60.2 & 72.6 & 71.5 & 56.8 & 51.9 & 77.1 & 70.2 & 56.3 & 82.1 & 66.6 \\
SAFN \cite{xu2019larger} & Full & 54.4 & 73.3 & 77.9 & 65.2 & 71.5 & 73.2 & 63.6 & 52.6 & 78.2 & 72.3 & 58.0 & 82.1 & 68.5 \\
MDD+IA \cite{mdd} & Full & 56.2 & 77.9 & 79.2 & 64.4 & 73.1 & 74.4 & 64.2 & 54.2 & 79.9 & 71.2 & 58.1 & 83.1 & 69.5 \\
CADA-P \cite{kurmi2019attending} & Full & 56.9 & 76.4 & 80.7 & 61.3 & 75.2 & 75.2 & 63.2 & 54.5 & 80.7 & 73.9 & 61.5 & 84.1 & 70.2 \\
GVB-GD \cite{gvb} & Full & 57.0 & 74.7 & 79.8 & 64.6 & 74.1 & 74.6 & 65.2 & 55.1 & 81.0 & 74.6 & 59.7 & 84.3 & 70.4 \\
HDAN \cite{hdan} & Full & 56.8 & 75.2 & 79.8 & 65.1 & 73.9 & 75.2 & 66.3 & 56.7 & 81.8 & 75.4 & 59.7 & 84.7 & 70.9 \\
SPL \cite{wang2020unsupervised} & Full & 54.5 & 77.8 & 81.9 & 65.1 & 78.0 & 81.1 & 66.0 & 53.1 & 82.8 & 69.9 & 55.3 & 86.0 & 71.0 \\
SRDC \cite{tang2020unsupervised} & Full & 52.3 & 76.3 & 81.0 & 69.5 & 76.2 & 78.0 & 68.7 & 53.8 & 81.7 & 76.3 & 57.1 & 85.0 & 71.3 \\
HDMI \cite{hdmi} & Full & 57.8 & 76.7 & 81.9 & 67.1 & 78.8 & 78.8 & 66.6 & 55.5 & 82.4 & 73.6 & 59.7 & 84.0 & 71.9 \\
SHOT \cite{shot} & Full & 57.1 & 78.1 & 81.5 & 68.0 & 78.2 & 78.1 & 67.4 & 54.9 & 82.2 & 73.3 & 58.8 & 84.3 & 71.8 \\
\midrule
HR-SHOT (Ours) & Full & 72.1 & 84.6 & 88.4 & 83.6 & 86.7 & 87.2 & 82.6 & 73.4 & 88.5 & 85.3 & 72.3 & 90.5 & 82.8 \\
\midrule
HR-SHOT (Ours) & Cont. & 65.7 & 82.2 & 85.0 & 79.8 & 80.9 & 80.7 & 77.8 & 63.5 & 85.4 & 82.0 & 64.5 & 86.2 & 77.8 \\
ConDA (Ours) & Cont. & 64.4 & 82.2 & 86.2 & 81.3 & 82.9 & 84.0 & 81.3 & 66.6 & 86.4 & 83.5 & 66.0 & 87.1 & 79.3 \\
\bottomrule
\end{tabular}}
\end{center}
\caption{Mean accuracy on the Office-home dataset. The ConDA experiments are performed with a continual batch size of 128 and buffer size of 520 (8 samples per class).}
\label{res:officehome}
\end{table*}
\begin{table*}[htbp]
\begin{center}
\setlength\tabcolsep{1.9pt}
\begin{tabular}{lccccccccccccc>{\columncolor[gray]{0.8}}c}
\toprule
Method & Target & Plane & bycycl & bus & car & house & knife & mcycle & person & plant & sktbrd & train & truck & Per class\\
\midrule
DANN \cite{ganin2016domain} & Full & 81.9 & 77.7 & 82.8 & 44.3 & 81.2 & 29.5 & 65.2 & 28.6 & 51.9 & 54.6 & 82.8 & 7.8 & 57.6 \\
SAFN \cite{xu2019larger} & Full & 93.6 & 61.3 & 84.1 & 70.6 & 94.1 & 79.0 & 91.8 & 79.6 & 89.9 & 55.6 & 89.0 & 24.4 & 76.1 \\
ALDA \cite{alda} & Full & 93.8 & 74.1 & 82.4 & 69.4 & 90.6 & 87.2 & 89.0 & 67.6 & 93.4 & 76.1 & 87.7 & 22.2 & 77.8 \\
CAN+A$^2$LP \cite{zhang2020label} & Full & 97.5 & 86.9 & 83.1 & 74.2 & 98.0 & 97.4 & 90.5 & 80.9 & 96.9 & 96.5 & 89.0 & 60.1 & 87.6 \\
SHOT \cite{shot} & Full & 94.3 & 88.5 & 80.1 & 57.3 & 93.1 & 94.9 & 80.7 & 80.3 & 91.5 & 89.1 & 86.3 & 58.2 & 82.9 \\
\midrule
HR-SHOT(Ours) & Full & 97.0 & 89.2 & 82.8 & 65.3 & 94.9 & 97.5 & 87.2 & 82.3 & 92.0 & 93.6 & 91.9 & 64.0 & 86.4 \\
\midrule
HR-SHOT(Ours) & Cont. & 96.7 & 93.8 & 85.0 & 44.3 & 97.4 & 95.9 & 79.3 & 88.1 & 94.7 & 95.3 & 89.0 & 52.3 & 84.3 \\
ConDA (Ours) & Cont. & 97.0 & 90.4 & 80.9 & 50.0 & 95.2 & 95.7 & 80.3 & 81.9 & 94.9 & 94.2 & 91.1 & 63.9 & 84.6 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Mean per class accuracy on the Visda-C dataset. The ConDA experiments are performed with a continual batch size of 192 and ConDA had buffer size of 96 (8 samples per class).}
\label{res:visda}
\end{table*}
\section{Experimental Setup}
\label{sec:experimentalsetup}
\subsection{Datasets}
We use three popular DA benchmarks for our experiments: Office, Office-Home and Visda-C.
\textbf{Office} \cite{office} is a popular small scale dataset. The dataset has 3 domains, Amazon (A), DSLR (D), and Webcam (W) with 31 object classes of items found in an office environment in each of the domains.
\textbf{Office-home} \cite{officehome} is a medium scale dataset with 4 domains, Art (Ar), Clip Art (Cl), Product (Pr), Real-World (Rw). The dataset has 65 classes of items found in everyday office and home environments.
\textbf{Visda-C} \cite{visda} is a large scale dataset with 2 domains, Synthetic (S) and Real (R). The dataset has 12 classes. The synthetic samples are generated using 3D rendering, and the real samples are taken from MS COCO dataset \cite{lin2015microsoft}.
\subsection{Implementation Details}
In the source model, we replace the ResNet \cite{resnet} backbone of \cite{shot} with HRNet\footnote{\url{https://github.com/HRNet/HRNet-Image-Classification/releases/download/PretrainedWeights/HRNet_W48_C_ssld_pretrained.pth}} \cite{hrnet} to obtain high resolution feature maps. The rest of the network is kept unchanged from \cite{shot}. We use a bottleneck FC layer with 256 units and a batch normalization layer, as shown in Figure \ref{fig:architecture}, followed by a final task specific FC classifier and weight normalization layer, respectively.
We train our network with SGD optimizer with 0.9 momentum. The learning rate for the layers after the HRNet backbone is set to 10 times the learning rate of the backbone. The learning rate for the backbone is set to $\eta_0=1e^{-3}$ for all datasets except for Visda-C which has a learning rate of $\eta_0 = 1e^{-4}$. We also use a learning rate scheduler $\eta = \eta_0 \cdot (1 + 10 \cdot p)^{-0.75}$ where $p$ changes from 0 to 1 as training progresses. We empirically find that $\gamma_1=1$ and $\gamma_2=0.5$ work best for all of the datasets. The number of epochs per incoming target batch for adaptation is heuristically set to 15 for Office-31 experiments, 25 for Office-home experiments, and 3 for Visda-C experiments. Parameter $\rho$ for sample mixup is set as 1.
\begin{table*}[htbp]
\begin{center}
\begin{tabular}{lccccccc>{\columncolor[gray]{0.8}}c}
\toprule
Configuration and loss function & Target & A$\rightarrow$D & A$\rightarrow$W & D$\rightarrow$A & D$\rightarrow$W & W$\rightarrow$A & W$\rightarrow$D & Mean \\
\midrule
HR-SHOT & Full & 98.2 & 97.2 & 80.0 & 99.0 & 80.2 & 99.8 & 92.4 \\
HR-SHOT & Cont. & 95.7 & 90.6 & 73.7 & 96.9 & 76.7 & 99.8 & 88.9 \\
ConDA: Buffer + $\mathcal{L}_{mixup}$ + $\mathcal{L}_{ent}$ & Cont. & 95.0 & 93.1 & 76.7 & 97.4 & 74.9 & 99.8 & 89.5 \\
ConDA: Buffer + $\mathcal{L}_{mixup}$ + $\mathcal{L}_{ent}$ + $\mathcal{L}_{eqdiv}$ & Cont. & 94.8 & 94.7 & 79.1 & 98.4 & 77.2 & 99.8 & 90.7\\
\bottomrule
\end{tabular}
\end{center}
\caption{Performance on Office-31 dataset for various loss functions with buffer. The ablation study for the continual experiments is performed with a continual batch size of 62 and ConDA had a buffer size of 124 (4 samples per class).}
\label{tab:ablation}
\end{table*}
\section{Results And Discussion}
\label{sec:results}
\subsection{Standard DA results}
By replacing ResNet\cite{resnet} backbone with HRNet \cite{hrnet} in the SHOT model \cite{shot}, denoted as HR-SHOT in this work, we find that the UDA performance improves significantly from our baseline method SHOT and outperforms other SOTA methods. In Office-31 dataset, as seen in Table \ref{res:office}, the performance of HR-SHOT is significantly higher than the baseline SHOT. Two of the most challenging adaptations in Office-31 are D$\rightarrow$A and W$\rightarrow$A where HR-SHOT outperforms CAN+A$^2$LP \cite{zhang2020label} by 1.81\% and 2.56\%, respectively. In Office-home dataset (Table \ref{res:officehome}), HR-SHOT outperforms the baseline SHOT with ResNet-101 backbone by a massive 11\%, with high performance gains across all domain pairs over SHOT.
In VisDA-C as shown in Table \ref{res:visda}, HR-SHOT outperforms baseline SHOT by 4.5\%. Also, $truck$ is the hardest class of the twelve classes, and HR-SHOT outperforms CAN+A$^2$LP \cite{zhang2020label} by 3.91\%. These results clearly demonstrate that utilizing an HRNet \cite{hrnet} backbone for domain adaptation can significantly improve the generalization capabilities of the overall method.
\subsection{Continual DA results}
The continual DA results for Office-31 dataset are shown in Table \ref{res:office}. In the continual setting, a buffer of size 124 with 4 slots per class and a continual batch size of 62 are chosen for Office-31. It is notable that with the HRNet backbone, continual HR-SHOT with no buffer outperforms SHOT \cite{shot} by 0.3\%. Furthermore, ConDA outperforms the continual HR-SHOT by 1.8\%. ConDA also outperforms or matches the performance of all SOTA methods except for SRDC \cite{tang2020unsupervised}, even though ConDA has access to only a batch of the target data at a time.
In the Office-home dataset, both continual HR-SHOT and ConDA outperform the existing standard DA methods by a large margin. ConDA outperforms HDMI \cite{hdmi} by more than 7\% on mean accuracy. It is also notable that it achieves the best performance across all the domain pairs.
In the Visda-C dataset, both HR-SHOT and ConDA perform favorably with the SOTA methods. In terms of mean per-class accuracy, ConDA outperforms most of the existing methods, including the baseline SHOT \cite{shot} by more than 1.5\%. While CAN+A$^2$LP \cite{zhang2020label} achieves the best mean per class accuracy in this dataset, ConDA does better in the challenging $truck$ category.
\subsection{Ablation Studies}
We perform ablation studies to demonstrate the impact of various parts of our model on the Office-31 dataset shown in Table \ref{tab:ablation}. For UDA, HR-SHOT outperforms SHOT by more than 3.8\%. However, performance drops by 3.5\% for continual adaptation with HR-SHOT. ConDA with buffer, sample mixup, and entropy loss improves the performance by 0.6\% over HR-SHOT. The addition of our proposed equal diversity loss to ConDA improves the overall performance by another 1.2\%.
\begin{figure}
\centering
\begin{subfigure}[b]{0.37\textwidth}
\includegraphics[width=1\linewidth]{images/office_home_ab_buffer.png}
\end{subfigure}
\begin{subfigure}[b]{0.37\textwidth}
\includegraphics[width=1\linewidth]{images/office_home_ab_batch.png}
\end{subfigure}
\caption[]{Ablation studies on Office-home dataset with varying buffer sizes (top) and varying batch sizes (bottom).
}
\label{fig:ablation}
\end{figure}
We perform further experiments on Office-home to understand the impact of buffer sizes and batch sizes during continual adaptation as shown in Figure \ref{fig:ablation}. To study the impact of buffer size, we consider a fixed continual batch size of 256 samples and 3 different buffer sizes; no buffer, 2 samples per class, and 8 samples per class. Our findings indicate that increasing the buffer length improves performance. ConDA with a buffer size of 8 samples per class achieves 0.7\% better performance than the one with no buffer. Our study further reveals that when the number of samples in the incoming batch increases, ConDA's performance also increases. By increasing the continual batch size from 64 to 256, the overall performance improves by 5.6\%.
\section{Conclusion}
\label{sec:conclusion}
This paper introduces a new paradigm of domain adaptation where target domain data are received continually in batches for adaptation. We introduce ConDA as the first DA method to address such a setting. In ConDA, we selectively store samples in a buffer and replay them with the incoming batches to improve our network's generalization capabilities for the overall target domain. We also use sample mixup technique for data augmentation in the target domain and demonstrate its effectiveness in such a data-constrained situation. We further propose a novel loss function that improves the overall performance of our network.
We hope that this research will lay the foundation for further exploration in continual domain adaptation.
\section*{Acknowledgements}
This research was supported in part by an AFOSR grant.
The authors acknowledge the computational resources made available by Research Computing at Rochester Institute of Technology that helped produce part of the results.
{\small
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\section{Introduction}
Isogeometric Analysis (IgA) was originally introduced in the seminal paper~\cite{Hughes:2005},
aiming to unite the worlds of computer aided design (CAD) and finite element (FEM) simulation.
From a technical point of view, it
is a framework for setting up spline-based discretizations
of partial differential equations.
The key idea is that the spline space is typically first defined on the unit square or the unit
cube and then mapped to the computational domain using
one global geometry function. More complicated domains cannot be
represented by just one such geometry function. Instead, the
computational domain is decomposed into patches, where each of them is
represented by its own geometry function. This is called the
\emph{multi-patch case}, in contrast to the \emph{single-patch case}.
As a next step, the linear system resulting from the discretization
of the PDE has to be solved. This might be challenging as the
condition number of the linear system grows exponentially with the
spline degree, where high spline degrees might be desired because
of their superior approximation power.
While in early IgA literature, the dependence of methods on the
spline degree has not been considered, in the last few years
robustness in the spline degree has gained increasing interest.
Several (almost) robust approaches or approaches with a mild
dependence on the spline degree have been proposed, on the one side for the single-patch
case, cf.~\cite{CollierEtAl:2012,Donatelli:2014a,HTZ:2016,Sangalli:2016,HT:2016} and references therein, and
on the other side as approaches aiming to combine patch-local solvers
to a global solver,
cf.~\cite{KleissEtAl:2012,DaVeigaEtAl:2012,DaVeigaEtAl:2014,HLT:2017} and
references therein.
In~\cite{Takacs:2017}, we have considered a slightly different approach: We
do not aim to combine patch-local solvers to a global solver, but to combine patch-local
smoothers to a global smoother which is used within a global multigrid
solver. In the present paper, we give some additional remarks on an efficient implementation
of the multigrid method, comment on its parallelization and give numerical results.
This paper is organized as follows. First, the model problem and
the discretization are discussed in Sec.~\ref{sec:prelim}. Then, in Sec.~\ref{sec:mg},
we recall the formulation of the multigrid solver. Its parallelization is discussed
in the following Sec.~\ref{sec:parallel}. In Sec.~\ref{sec:num}, we give the results of
numerical experiments and draw conclusions.
\section{Model problem and isogeometric discretization}\label{sec:prelim}
Let $\Omega\subset \mathbb{R}^d$ with $d\in\{2,3\}$ be a bounded computational domain with Lipschitz boundary.
We consider a standard \emph{Poisson model problem}
\[
- \Delta u = f \quad\mbox{in}\quad \Omega\ , \quad
u = 0 \quad\mbox{on}\quad \Gamma_D
\qquad \mbox{and} \qquad
\frac{\partial u}{\partial n} = 0 \quad\mbox{on}\quad \Gamma_N \ ,
\]
where $\Gamma_D$ is a subset of $\partial \Omega$ with positive measure and
$\Gamma_N:=\partial\Omega \backslash \Gamma_D$.
The model problem reads in variational form as follows. Given $f\in L_2(\Omega)$,
find $u\in H^1_{0,D}(\Omega)$ such that
\begin{equation} \label{eq:model}
(\nabla u,\nabla v)_{L_2(\Omega)} = (f,v)_{L_2(\Omega)}
\qquad \mbox{for all} \quad v \in H^1_{0,D}(\Omega)\ .
\end{equation}
Here and in what follows, $L_2(\Omega)$ and $H^1(\Omega)$ are the standard Lebesgue and Sobolev spaces
with standard norms and
$H^1_{0,D}(\Omega):=\{u \in H^1(\Omega)\;:\; u|_{\Gamma_D} = 0\}$.
We preform a standard isogeometric multi-patch discretization as it has been specified
in~\cite{Takacs:2017}. In the present paper, we try to keep
the explanation short and give only an overview.
We assume that the computational domain $\Omega$ is composed of $K$ patches $\Omega_k$ such that
\begin{equation} \label{eq:matching}
\overline{\Omega} = \bigcup_{k=1}^K \overline{\Omega_k}
\quad \mbox{and} \quad
\Omega_k\cap\Omega_l=\emptyset \mbox{ for any }k\not=l\ ,
\end{equation}
where each patch $\Omega_k$ is a bounded and open domain. We assume that the patches are
fully matching, i.e., the intersections
$\overline{\Omega_k}\cap\overline{\Omega_l}$ are either empty, common vertices, common edges or common faces.
Any of the patches is parametrized by a bijective geometry function
\[
\textbf{G}_k :\widehat{\Omega}:=(0,1)^d \rightarrow \Omega_k
:= \textbf{G}_k (\widehat{\Omega})\subset \mathbb{R}^d\ .
\]
Before we define set of trial functions $V_\ell \subset H^1_{0,D}(\Omega)$, we introduce discretizations
living on the parameter
domain~$\widehat{\Omega}$. Let
\[
S_{p,h}(0,1) := \left\{ u \in C^{p-1}(0,1) \;:\; u|_{[hi,h(i+1)]} \mbox{ is a polynomial of degree } p \;,\; \forall_{i=1,\ldots,n} \right\}
\]
be the space of univariate splines of maximum smoothness and the space $S_{p,h}(\widehat{\Omega}):=
S_{p,h}(0,1) \otimes \cdots \otimes S_{p,h}(0,1)$ be the corresponding tensor-product spline space.
The grid size $h$ and the spline degree $p$ might be different for any patch and for any spacial direction;
for simplicity, we do not express that in the notation.
Based on the discretization living on the parameter domain $\widehat{\Omega}$, we define the function
space $V_\ell$ of isogeometric functions living on the physical domain $\Omega$ as follows:
\begin{equation}\label{eq:vh}
V_\ell := \{ u\in C^0(\Omega) \;:\; u\circ \textbf{G}_k \in S_{p,h_\ell}(\widehat{\Omega}) \} \ .
\end{equation}
We assume to have a \emph{fully matching discretization}, which means that the discretizations agree
on the interfaces. A more formal definition of the basis and its discretization is given in~\cite[Sec.~2]{Takacs:2017}.
In Fig.~\ref{fig:decomp1}, a fully matching discretization is depicted, where each node represents one
basis function and therefore one degree of freedom (dof). Note that any of the basis function whose associated node lies
on one patch, vanishes outside of that patch. Any of the basis functions whose associated node lies within
one edge, vanishes outside the union of the edge and the adjacent patches. Finally, any of the basis
functions whose associated node coincides with one vertex, vanishes outside the union of that vertex and
the adjacent edges and patches. The behavior in three dimensions is completely analogous.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.17\textwidth]{decomp1}
\end{center}
\caption{Fully matching discretization}\label{fig:decomp1}
\end{figure}
The Galerkin principle yields the following discretized variational problem. Find $u\in V_\ell$ such that
\begin{equation} \label{eq:model:discr}
a( u, v)= (f,v)_{L_2(\Omega)}
\qquad \mbox{for all $v \in V_\ell$}\ ,
\end{equation}
where
\begin{equation} \label{eq:model:bil}
a( u, v) := (\nabla u,\nabla v)_{L_2(\Omega)} = \sum_{k=1}^K \underbrace{ ( |\det J_{\textbf{G}_k} | J_{\textbf{G}_k}^{-\top} J_{\textbf{G}_k}^{-1} \nabla \widehat{u}_k, \nabla \widehat{v}_k )_{L^2(\widehat{\Omega})} }_{\displaystyle a_k(u, v):= }
\end{equation}
for $\widehat{u}_k := u \circ \textbf{G}_k \in S_{p,h_\ell}(\widehat{\Omega})$ and
$\widehat{v}_k := v \circ \textbf{G}_k \in S_{p,h_\ell}(\widehat{\Omega})$
and where $J_{\textbf{G}_k}$ is the Jacobian of the geometry map.
Using the chosen basis, we obtain a matrix-vector formulation of the discretized problem,
which reads as follows. Find $\underline{u} \in \mathbb{R}^N$ such that
\begin{equation} \label{eq:linear:system}
A_\ell \, \underline{u} = \underline{f}\ .
\end{equation}
Allowing constants that depend on the geometry function, we obtain that
the matrix $A_\ell$ is spectrally equivalent to the matrix $\widehat{A}_\ell$, which discretizes the bilinear form
\[
\widehat{a}\, (u, v):= \sum_{k=1}^K ( \nabla \widehat{u}_k, \nabla \widehat{v}_k )_{L^2(\widehat{\Omega})} \ ,
\]
where, again, $\widehat{u}_k := u \circ \textbf{G}_k $ and $\widehat{v}_k := v \circ \textbf{G}_k $.
\section{The multigrid solver and its extension to three dimensions}\label{sec:mg}
We employ the multigrid solver based on a hierarchy of grids for grid levels $\ell=0,\ldots,L$, obtained by uniform refinement.
Throughout the grid hierarchy, the spline degree $p$ and the corresponding smoothness is kept
unchanged. This yields nested spaces: $V_0 \subset V_1 \subset \cdots \subset V_L \subset H^1_{0,D}(\Omega)$,
which allows to use the canonical embedding $V_{\ell-1}\rightarrow V_\ell$ for the multigrid method;
its matrix representation is denoted by $P_\ell$. Following the usual pattern, we use its transpose
$P_\ell^\top$ as restriction.
One \emph{multigrid cycle} on some grid level $\ell$ consists of the following steps.
\begin{itemize}
\item First, $\nu$ \emph{pre-smoothing steps} are applied, where each reads as follows:
\begin{equation}\label{eq:sm}
\underline{u} \gets \underline{u} + \tau L_\ell^{-1} (\underline{f} - A_\ell \underline{u}) \ .
\end{equation}
The choice of the smoothing operator $L_\ell^{-1}$ and the damping parameter $\tau$ are discussed below.
\item Then, the \emph{coarse grid correction} is performed:
\[
\underline{u} \gets \underline{u} + \tau P_\ell A_{\ell-1}^{-1} P_\ell^\top (\underline{f} - A_\ell \underline{u}) \ ,
\]
where for $\ell>1$, the application $A_{\ell-1}^{-1}$ is replaced by $\mu=1$ (V-cycle) or
$\mu=2$ (W-cycle) recursive applications of the multigrid method on the coarser grid level.
\item Finally, again $\nu$ \emph{post-smoothing steps} \eqref{eq:sm} are applied.
\end{itemize}
As smoother, an additive Schwarz type combination
\begin{equation}\label{eq:additive}
L_\ell^{-1} := \sum_{T} P_{\ell,T} L_{\ell,T}^{-1} P_{\ell,T}^{\top}
\end{equation}
of local smoothing operators $L_{\ell,T}^{-1}$ is proposed, where the dofs are collected based on separating
the domain into \emph{pieces}: patches, vertices, edges and, in three dimensions, faces.
Here, each dof is assigned to exactly one of these pieces, cf. Fig.~\ref{fig:decomp2}.
Certainly, based on such a one-by-one splitting, the matrix $P_{\ell,T}$ is nothing but a indicator
matrix representing the canonical embedding.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.17\textwidth]{decomp2}
\end{center}
\caption{Decomposition into pieces serving as subspaces for the additive Schwarz method}\label{fig:decomp2}
\end{figure}
\mbox{}\\
The local smoothing operators are chosen as follows.
\begin{itemize}
\item For the patch-interiors, the subspace corrected mass smoother as proposed in~\cite{HT:2016} is
chosen as smoothing operator $L_{\ell,T}^{-1}$.
\item For the edges and vertices, in~\cite{Takacs:2017} direct solvers have been
proposed as smoothers, i.e., $L_{\ell,T}$ is
the restriction of the matrix $A_\ell$ to the edge or vertex. To avoid unnecessary
communication, we choose an approximation which can be computed directly.
Using \cite[Lemma~4.1]{Takacs:2017} and \cite[eq.~(4.16)]{Takacs:2017},
we obtain that the restriction of $A_\ell$ to an edge is spectrally equivalent to
\[
L_{\ell,T}:=
\left( \frac{h_\ell}p \right)^{d-1} \mathrm{K}_\ell
+
\left( \frac {h_\ell}p \right)^{d-3} \mathrm{M}_\ell \ ,
\]
where $\mathrm{K}_\ell$ and $\mathrm{M}_\ell$ are the corresponding univariate
stiffness and mass matrices.
Analogously, its restriction to a vertex is a constant in the order of
\[
L_{\ell,T}:=
\left( \frac{h_\ell}p \right)^{d-2} \ .
\]
\item Three dimensional problems have not been considered in~\cite{Takacs:2017}, so we have to discuss how to
choose the local smoothers for faces. If, as for the edges and vertices, again a direct solver was applied,
the overall computational costs would not be optimal anymore. So, again, observe that the
the restriction of $A_\ell$ to a face is spectrally equivalent to
\[
L_{\ell,T}^*:=
\left( \frac{h_\ell}p \right)^{d-2} \mathcal{K}_{\ell} + \left( \frac {h_\ell} p \right)^{d-4} \mathcal{M}_{\ell}\ ,
\]
where and $\mathcal{K}_{\ell}=\mathrm{K}_\ell \otimes \mathrm{M}_\ell
+\mathrm{M}_\ell \otimes \mathrm{K}_\ell$
and $\mathcal{M}_{\ell}=\mathrm{M}_\ell \otimes \mathrm{M}_\ell$
are the corresponding stiffness and mass matrices on the face. For $d=3$, we obtain
\[
L_{\ell,T}^*=
\frac{h_\ell}p \left(
\mathcal{K}_{\ell}
+ \frac {p^2} {h_\ell^2} \mathcal{M}_{\ell} \right) \ .
\]
Here, analogously to the case of the patch-interiors, the subspace corrected mass smoother is used.
Note that the subspace corrected mass smoother is set up such that it bounds the stiffness matrix $\mathcal{K}_{\ell}$
from above, cf.~\cite[eq.~(11)]{HT:2016}. In the present paper, besides a trivial scaling, the
stiffness matrix $\mathcal{K}_{\ell}$ is augmented by $p^2 h_\ell^{-2}$ times
the mass matrix $\mathcal{M}_{\ell}$. So, we have also to augment the local contributions for the subspace corrected
mass smoother, cf. the matrices $L_{\alpha}$ in~\cite[Sec.~4.2]{HT:2016}, in the same way.
\end{itemize}
\section{The parallelization of the multigrid solver}\label{sec:parallel}
The parallelization of the multigrid solver follows the approach presented
in~\cite{DHL}. We use MPI\footnote{Message Passing Interface, see \url{http://mpi-forum.org/}.},
so each processor executes independently
the whole algorithm with its local data until communication is explicitly requested.
We assign each of the patches to one of the processors. So, that
processor holds the values of all dofs that belong
to that patch including its interfaces, cf. Fig.~\ref{fig:0b}. This means that the dofs
on the interfaces might be assigned to more than one processor.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.17\textwidth]{decomp3}
\end{center}
\caption{The distribution of the dofs to the processors}\label{fig:0b}
\end{figure}
A vector that occurs in the algorithm, say $\underline{w}$, is stored either in
accumulated form (Type~I) or distributed form (Type~II).
We say that a vector is stored in accumulated form if each of
the processors holds those parts of the global vector which correspond
to the dofs assigned to the processor.
We denote such vectors by $\acc{w}$.
We say that a vector is stored in distributed form if the
global vector is the sum of the contributions of all the processors. Such
vectors are denoted by $\dist{w}$.
Again, the processor-local contributions of the distributed vectors are
supported only the patches assigned to the processor including their
interfaces.
Note that only certain kinds of operations make sense; so we can add
accumulated and distributed vectors only to vectors of the same type:
\begin{equation}\nonumber
\acc{u} + \acc{v} \rightarrow \acc{w}
\quad \mbox{and} \quad
\dist{u} + \dist{v} \rightarrow \dist{w} \ ,
\end{equation}
cf.~\cite[Sec.~5.3]{DHL}.
As the multi-patch setting is equivalent to a standard approach of non-overlapping domain
decomposition, the overall stiffness matrix is assembled on a per-patch basis, i.e., the bilinear
forms $a_k$ from~\eqref{eq:model:bil} are evaluated separately yielding matrices $A_{\ell,k}$.
Consequently, the global stiffness matrix $A_\ell$ is the sum of
the local contributions $A_{\ell,k}$. This means that the matrix $A_\ell$ is stored in
distributed form, which yields the following mapping type:
\begin{equation}\nonumber
A_\ell \, \acc{u} \rightarrow \dist{w}\ ,
\end{equation}
i.e., $A_\ell$ can be applied to accumulated vectors and the the result of the operation is distributed, cf.~\cite[Sec.~5.4.1]{DHL}.
Similar to~\cite[Sec.~7.2.2]{DHL}, the inter-grid transfer operators satisfy
\begin{equation}\nonumber
P_\ell \, \acc{u} \rightarrow \acc{w}
\quad \mbox{and} \quad
P_\ell^\top \, \dist{u} \rightarrow \dist{w}
\end{equation}
because the prolongation operator has a block-triangular structure as
in~\cite[eq.~(5.9)]{DHL} and the restriction operator has a block-triangular structure
as in~\cite[eq.~(5.10)]{DHL}. The block-triangular structure is obtained because the
following statements hold true:
\begin{itemize}
\item On each vertex, the prolonged value $\acc{w}$ coincides with the coarse-grid value $\acc{u}$ of the same vertex.
\item On each edge, the prolonged values $\acc{w}$ only depend on the coarse-grid values $\acc{u}$ on the same edge and
on the adjacent vertices.
\item On each patch-interior, the prolonged values $\acc{w}$ only depend on the coarse-grid values $\acc{u}$ on the same patch-interior and
on the adjacent edges and vertices.
\end{itemize}
For three dimensions, completely analogous statements hold true.
The global operator $L_\ell^{-1}$ is block-diagonal, where each block corresponds to one piece. Note that
by construction each piece belongs as a whole to one processor or is shared as a whole by
the same processors, so it satisfies both the conditions of~\cite[eq.~(5.9)]{DHL}
and \cite[eq.~(5.10)]{DHL}. This shows
\begin{equation}\nonumber
L_\ell^{-1} \, \acc{u} \rightarrow \acc{w}
\quad \mbox{and} \quad
L_\ell^{-1} \, \dist{u} \rightarrow \dist{w} \ ,
\end{equation}
i.e., this operator can be applied both to distributed and accumulated vectors and it preserves
the type of the vector.
As in any iterative solver, we need to accumulate the vectors of interest in each iterate.
This we denote using the symbol $\mbox{$\Sigma$}$, which maps as follows:
\begin{equation}\label{eq:ops6}
\mbox{$\Sigma$} \, \dist{u} \rightarrow \acc{w}.
\end{equation}
We note that only a communication between the processors holding neighboring patches is required
in order to perform~\eqref{eq:ops6}.
Only the coarsest grid level $\ell=0$ needs some special treatment. Since the focus of the present
paper is set on parallelizing the multigrid solver without changing its mathematical meaning, we
perform an exact global solve on the coarsest grid level.
This seems to be acceptable as it is done only for the coarsest
grid level. So, we are required to
communicate the stiffness matrix between all processors such that every
processor holds a global stiffness matrix.
We set up a direct solver $A_{0}^{-1}$ for this global stiffness matrix, so its application is perform
in the following way
\begin{equation}\nonumber
\mbox{\raisebox{.15em}{$\chi$}}_{glob}\; A_{0}^{-1} \Acc_{glob}\; \, \acc{u} \rightarrow \acc{w} \ ,
\end{equation}
where $\Acc_{glob}\;$ denotes the accumulation of vectors where each processor obtains the global
vector and $\mbox{\raisebox{.15em}{$\chi$}}_{glob}\;$ is the restriction of the global vector to the patches assigned to
the processor. The latter involves only discarding unnecessary data. We obtain
\begin{align*}
\Acc_{glob}\; \dist{u} \to \glob{w} \quad \text{and} \quad \mbox{\raisebox{.15em}{$\chi$}}_{glob}\; \glob{u}\to \acc{w}\ .
\end{align*}
Overall, the parallel multigrid solver looks as follows:
\begin{algorithm}[H]
\caption{Parallel multigrid solver}
\label{alg-mg}
\begin{algorithmic}[1]
\Procedure{Multigrid}{$\ell, \acc{u}, \dist{f}$}
\ForAll{$i=1,\ldots,\nu$} \Comment{Pre-smoothing}
\State $\acc{u} \gets \acc{u} + \tau \mbox{$\Sigma$} L_\ell^{-1} ( \dist{f} - A_\ell \acc{u}) $
\EndFor
\State $\dist{r} \gets P_\ell^\top ( \dist{f} - A_\ell \acc{u}) $
\If {$\ell =1$} \Comment{Coarse-grid correction}
\State $\acc{p} \gets \mbox{\raisebox{.15em}{$\chi$}}_{glob}\; A_{0}^{-1}\Acc_{glob}\; \dist{r} $ \Comment{Exact solver for coarsest grid level}
\Else
\State $\acc{p} \gets 0$
\ForAll{$i=1,\ldots,\mu$} \Comment{$\mu=1$ is V-cycle; $\mu=2$ is W-cycle}
\State $\acc{p}\gets \textsc{Multigrid}(\ell-1,\acc{p}, \dist{r})$
\EndFor
\EndIf
\State $\acc{u} \gets \acc{u} + P_\ell \acc{p} $
\ForAll{$i=1,\ldots,\nu$} \Comment{Post-smoothing}
\State $\acc{u} \gets \acc{u} + \tau \mbox{$\Sigma$} L_\ell^{-1} ( \dist{f} - A_\ell \acc{u}) $
\EndFor
\State \textbf{return } $\acc{u}$
\EndProcedure
\end{algorithmic}
\end{algorithm}
We use our multigrid algorithm as a preconditioner for a standard parallel preconditioned conjugate gradient (PCG)
solver. Note that the multigrid preconditioner already takes a distributed residual and returns an accumulated update. So, the
preconditioned conjugate gradient solver only needs to accumulate data in order to compute the required scalar
products accordingly, cf.~\cite[Sec.~6.3.1]{DHL}.
\section{Numerical experiments}\label{sec:num}
In this section, we present numerical experiments concerning the parallelization of the multigrid
solver. The solver was implemented in C++ based on the G+Smo
library~\cite{gismoweb} and, as already mentioned, the parallelization is performed using MPI.
All numerical experiments have been done using the HPC Cluster RADON1\footnote{We use up to 32 out
of 68 available nodes, each equipped with 2x Xeon E5-2630v3 ``Haswell'' CPU (8 cores, 2.4 Ghz, 20 MB cache)
and 128 GB RAM. More information is available at \url{https://www.ricam.oeaw.ac.at/hpc/}.}.
We present timings for setup, assembling and solving.
The setup costs include
\begin{enumerate}
\item the costs of the setup of the dof-mappers, which describe the
relation between the local dof-indices and the global dof-indices,
\item the costs of the grid refinement and the setup of the inter-grid transfer matrices,
\item the costs of the setup of the piece-local smoothers and
\item the costs of the setup of the coarse-grid solver.
\end{enumerate}
Here, our implementation of item~1 requires that each processor knows about the indexing
of the global dofs. Also for item~4, the information on all dofs is required, however only
on the coarsest grid level. The costs which are typically dominant, i.e., those for assembling
and for solving, are presented separately. It is important to note that assembling does
not require the any kind of communication between the processors. So its parallelization
is trivial. The communication, which is required for the solving phase,
is discussed in detail in Sec.~\ref{sec:parallel}.
\begin{figure}[h]
\begin{center}
\subfloat[The Yeti footprint\label{fig:1}]{
\begin{minipage}{.45\textwidth}
\centering\includegraphics[height=.45\textwidth]{YETI}
\end{minipage}
}
\subfloat[The Fichera corner\label{fig:2}]{
\begin{minipage}{.45\textwidth}
\centering\includegraphics[height=.45\textwidth]{fichera}
\end{minipage}
}
\end{center}
\caption{The computational domains\label{fig:12}}
\end{figure}
We have performed the numerical experiments for two and three dimensions. As
two dimensional domain, we use the Yeti footprint (Fig.~\ref{fig:1}), which has already been
considered in~\cite{Takacs:2017} and which is also a popular domain for the IETI-DP method,
cf.~\cite{KleissEtAl:2012}. As three dimensional domain, we consider the Fichera corner (Fig.~\ref{fig:2}).
This domain
is often considered as extension of the L-shaped domain to three dimensions; the corresponding
numerical experiments show that the proposed method can also be applied to domains without
full elliptic regularity.
\subsection{The Yeti footprint (2D)}
On the Yeti footprint, we solve the model problem
\[
\begin{aligned}
-\Delta u & =50\pi^2\sin(5\pi\ x) \sin(5\pi\ y ) && \quad \mbox{ in } \Omega \ ,\\
u & =0&& \quad \mbox{ on } \Gamma_D \ ,\\
\frac{\partial}{\partial n} u & = 0 && \quad \mbox{ on } \Gamma_N \ ,
\end{aligned}
\]
where $\Gamma_D$ is the outer boundary and $\Gamma_N$ are the four inner boundaries.
The Yeti footprint consists of $21$ patches, which can be seen in Fig.~\ref{fig:1}. Since we need sufficiently
many patches for parallelization, we first split each patch uniformly into $16$ patches, so we
obtain in total~$K=336$ patches. We solve the problem with a conjugate gradient solver, preconditioned
with one V-cycle of the multigrid method. We perform 1+1 smoothing
steps of the proposed smoother. The damping parameter and the scaling parameter (in the subspace
corrected mass smoother) are chosen as in~\cite{Takacs:2017}, i.e., $\tau = 0.25$ and $\sigma = \tfrac{1}{0.2} h_\ell^{-2}$.
In Tab.~\ref{tab:I:it}, we report on the number of iterations required to reach the desired relative accuracy goal
of $10^{-8}$. Here, $\ell$ represents the number of refinement levels and $p$ the spline degree. On the coarsest
grid level ($\ell=0$), the patch-local discretization only consists of global polynomials, i.e., each patch is one
element of the discretization. Refinement is done by uniformly refining the patch-local grids, keeping the number
of patches unchanged. We observe, as in~\cite{Takacs:2017}, that the number of iterates is quite robust in the
grid size and in the spline degree. The presented numbers have been computed with the serial code. The number of
iterates is supposed to be the same if parallelization is applied; however due to some small numerical
instabilities, in some cases the parallel code needs one additional iteration (but never more than that).
Similar iteration counts are obtained for the W-cycle.
In Tab.~\ref{tab:I:strong}, we present the strong scaling results. We fix the grid level $\ell$ and the spline
degree $p$ to two typical values. For $\ell=7$ and $p=4$, we have $5\,768\,189$ dofs and the corresponding
stiffness matrix has $4.6\;10^8$ non-zero entries. For the case $\ell=7$ and $p=8$, the number of dofs
increases slightly to $6\,125\,757$, but the stiffness matrix has already $1.8\;10^9$ non-zero entries.
In the first two rows, we compare the costs of the
serial code and the parallel code. Here, we obtain that the parallel code is slightly slower during the
solving phase which is mainly due to the fact that the parallel code does not assemble the whole stiffness
matrix but works with patch-local stiffness matrices. This allows also to consider the larger problem
with $\ell=7$ and $p=8$, where the serial code caused memory problems.
\begin{table}[t]
\begin{center}
\begin{tabular}{l|rrrrrrr}
\toprule
\multicolumn{1}{l}{$\ell\;\diagdown\; p$} & \quad 2 & \quad 3 & \quad 4 & \quad 5 &\quad 6 &\quad 7 &\quad 8 \\
\midrule
4 & 45 & 42 & 37 & 33 & 31 & 28 & 25 \\
5 & 48 & 44 & 40 & 36 & 33 & 30 & 27 \\
6 & 50 & 44 & 41 & 36 & 35 & 33 & 27 \\
7 & 51 & 45 & 42 & 37 & 36 & 34 & 28 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Iteration counts for Yeti footprint, $K=336$}
\label{tab:I:it}
\end{table}
\begin{table}[t]
\begin{center}
\begin{tabular}{r||rr|rr|rr||rr|rr|rr}
\toprule
& \multicolumn{6}{c||}{$\ell= 7\ ,\;\;p=4\ ,\;\;K=336$} & \multicolumn{6}{c}{$\ell= 7\ ,\;\;p=8\ ,\;\;K=336$} \\
\midrule
& \multicolumn{2}{l|}{Setup} & \multicolumn{2}{l|}{Assembling} & \multicolumn{2}{l||}{Solving} &
\multicolumn{2}{l|}{Setup} & \multicolumn{2}{l|}{Assembling} & \multicolumn{2}{l}{Solving} \\
$\#$ Proc. &$t$&$s$&$t$&$s$&$t$&$s$&$t$&$s$&$t$&$s$&$t$&$s$ \\
\midrule
(serial) & 217.1 & \; --\,\, & 522.1 & \; --\,\, & 4929.5 & \; --\,\, & --\,\, & \; --\,\,& --\,\, & \; --\,\, & --\,\, & \; --\,\, \\
1 & 220.0 & \; 1\;\;\, & 520.0 & \; 1\;\;\,& 5125.9 & \; 1\;\;\, & 549.0 & \;1\;\;\,& 8230.9 & \; 1\;\;\, & 4729.8 & \; 1\;\;\, \\
2 & 80.1 & \; 2.7 & 263.8 & \; 1.9 & 1367.6 & \; 3.7 & 225.4 & \; 2.4 & 4158.5 & \; 1.9 & 1250.8 & \; 3.7 \\
4 & 35.3 & \; 6.2 & 131.8 & \; 3.9 & 399.1 & \; 12.8 & 117.0 & \; 4.6 & 2098.5 & \; 3.9 & 409.9 & \; 11.5 \\
8 & 17.1 & \; 12.8 & 66.0 & \; 7.8 & 109.6 & \; 46.7 & 53.9 & \; 10.1 & 1055.9 & \; 7.8 & 140.2 & \; 33.7 \\
16 & 10.7 & \; 20.5 & 33.9 & \; 15.3 & 40.7 & \; 125.9 & 30.0 & \; 18.3 & 543.4 & \; 15.1 & 59.2 & \; 79.9 \\
32 & 8.0 & \; 27.5 & 17.4 & \; 29.8 & 17.1 & \; 299.7 & 17.7 & \; 31.0 & 275.1 & \; 29.9 & 26.8 & \; 176.4 \\
64 & 7.2 & \; 30.5 & 9.4 & \; 55.3 & 10.6 & \; 483.5 & 12.9 & \; 42.5 & 149.7 & \; 54.9 & 13.7 & \; 345.2 \\
128 & 6.0 & \; 36.3 & 5.1 & \; 101.3 & 4.1 & \;1250.2 & 9.9 & \; 55.4 & 76.2 & \; 108.0 & 7.2 & \; 656.9 \\
256 & 6.3 & \; 34.4 & 3.2 & \; 160.0 & 3.3 & \;1553.3 & 9.5 & \; 57.7 & 51.4 & \; 160.1 & 6.5 & \; 727.6 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Strong scaling behavior for Yeti footprint}
\label{tab:I:strong}
\end{table}
\begin{table}[t]
\begin{center}
\begin{tabular}{r||r|r|r|r|r||r|r|r|r|r}
\toprule
& \multicolumn{5}{c||}{$\ell= 7\ ,\;\;p=4$} & \multicolumn{5}{c}{$\ell= 7\ ,\;\;p=8$} \\
\midrule
$\#$ Proc. & \multicolumn{1}{p{2.5em}|}{$\#$ dofs} & \multicolumn{1}{p{1.5em}|}{It.}& \multicolumn{1}{p{3em}|}{Setup} & \multicolumn{1}{p{3.5em}|}{Ass.} & \multicolumn{1}{p{3.5em}||}{Solving}
& \multicolumn{1}{p{2.5em}|}{$\#$ dofs} & \multicolumn{1}{p{1.5em}|}{It.}& \multicolumn{1}{p{3em}|}{Setup} & \multicolumn{1}{p{3.5em}|}{Ass.} & \multicolumn{1}{p{3.5em}}{Solving} \\
\midrule
4 & 360\,902 & 46 & 1.4 & 9.2 & 7.9 & 383\,262 & 46 & 6.6 & 147.3 & 21.4 \\
16 & 1\,442\,569 & 44 & 2.4 & 9.3 & 8.5 & 1\,531\,977 & 44 & 8.7 & 151.2 & 20.8 \\
64 & 5\,768\,189 & 41 & 7.2 & 9.5 & 10.5 & 6\,125\,757 & 28 & 13.2 & 148.7 & 13.7 \\
256 &23\,068\,573 & 36 & 42.4 & 9.7 & 9.5 &24\,498\,717 & 26 & 54.5 & 153.8 & 20.0 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Weak scaling behavior for Yeti footprint}
\label{tab:I:weak}
\end{table}
In the following rows, we consider the strong scaling behavior. We present in each case the time $t$ in
seconds required for setup, assembling and solving and the corresponding speedup $s$.
We observe that the overall method has good strong scaling properties. As the setup phase consists also
of parts that are not parallelized, we observe this time does not fall below a few seconds. The assembling
phase, which is known to be dominant phase in high-order isogeometric methods, scales almost optimal. Also the
solving phase needs rather little communication and is expected to scale well therefore. Indeed, the speedup
is much larger than what would be expected. The authors think that this might be explained by some extraordinary
caching effects, but here further investigation is required. The extraordinary well behavior of the solver
cannot be explained with changed convergence behavior because in all cases, the convergence behavior is
identical.
In Tab.~\ref{tab:I:weak}, we present weak scaling results. We again fix the grid level $\ell$ and the spline
degree $p$ to two typical values. Here, for the case of $4$ processors, we consider the initial configuration of
$K=21$ patches; in the following rows we consider $84$, $336$ and $1344$ patches. (So, the third row
with $64$ processors coincides with the line with $64$ processors in Tab.~\ref{tab:I:strong}.)
As the setup phase is not fully parallelized, the setup times increase if the number of patches is increased.
Both, the assembling times and the solving times are rather constant and do not indicate a clear tendency.
The solving times also change due to the fact that the required number of iterations decays if the patches
are split up. The computational costs for the global-coarse grid solver is negligible in this example;
for $K=1344$ patches the costs are $0.36$ seconds for $p=4$ and $1.4$ seconds for $p=7$ and for smaller
patch numbers even less.
\subsection{The Fichera corner (3D)}
On the Fichera corner, we solve the model problem
\[
\begin{aligned}
-\Delta u & = 75\pi^2\sin(5\pi\ x) \sin(5\pi\ y)\sin(5\pi\ z) && \mbox{ in } \Omega := (0,2)^3 \backslash [1,2)^3 \ ,\\
u & =0 && \mbox{ on } \Gamma_D := \{ (x,y,z) \in \partial \Omega \;:\; xyz=0 \} \ ,\\
\frac{\partial}{\partial n} u & = 0 && \mbox{ on } \Gamma_N := \partial \Omega \backslash \Gamma_D \ .
\end{aligned}
\]
The Fichera corner consists of $7$ patches, which can be seen in Fig.~\ref{fig:2}, which are uniformly
split into~$K=448$ patches in total. Again, we solve the problem with a conjugate gradient solver preconditioned
with one V-cycle of the multigrid method with 1+1 smoothing steps. Again
$\tau = 0.25$ and $\sigma = \tfrac{1}{0.2} h_\ell^{-2}$ are chosen.
In Tab.~\ref{tab:II:it}, we report on the number of iterations required to reach the desired relative
accuracy goal of $10^{-8}$. We observe, as for the Yeti footprint, that the number of iterates is
quite robust in the grid size and in the spline degree. The presented numbers have been computed with
the serial code. Again, the parallel code yields (almost) the same numbers.
In Tab.~\ref{tab:II:strong}, we present the strong scaling results.
For $\ell=4$ and $p=2$, we have $N=2\,201\,024$ dofs and a stiffness matrix with $2.5\;10^8$ non-zero entries.
The second example with $\ell=3$ and $p=4$ yields $N=596\,288$ dofs and $2.8\;10^8$ non-zero entries.
The timings behave similar as in the two-dimensional case, however the costs of the setup phase are
much larger which can be explained by the fact that the interfaces are much larger.
(For two dimensional problems, the interfaces consist of
$\mathcal{O}(N^{1/2})$ dofs and for three dimensional problems, the interfaces consist of
$\mathcal{O}(N^{2/3})$ dofs.) The assembling times seem to be optimal, whereas the solving times
again behave extraordinary well.
\begin{table}[t]
\begin{center}
\begin{tabular}{l|rrrrrr}
\toprule
\multicolumn{1}{l}{$\ell\;\diagdown\; p$} & \quad 2 & \quad 3 & \quad 4 & \quad 5 &\quad 6 \\
\midrule
1 & 30 & 31 & 31 & 26 & 22 \\
2 & 33 & 32 & 33 & 31 & 28 \\
3 & 39 & 38 & 37 & 33 & 30 \\
4 & 44 & 44 & 42 & 37 & 35 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Iteration counts for Fichera corner, $K=448$}
\label{tab:II:it}
\end{table}
\begin{table}[t]
\begin{center}
\begin{tabular}{r||rr|rr|rr||rr|rr|rr}
\toprule
& \multicolumn{6}{c||}{$\ell= 4\ ,\;\;p=2\ ,\;\;K=448$} & \multicolumn{6}{c}{$\ell= 3\ ,\;\;p=4\ ,\;\;K=448$} \\
\midrule
& \multicolumn{2}{l|}{Setup} & \multicolumn{2}{l|}{Assembling} & \multicolumn{2}{l||}{Solving} &
\multicolumn{2}{l|}{Setup} & \multicolumn{2}{l|}{Assembling} & \multicolumn{2}{l}{Solving} \\
$\#$ Proc. &$t$&$s$&$t$&$s$&$t$&$s$&$t$&$s$&$t$&$s$&$t$&$s$ \\
\midrule
(serial) & 179.4 & \; --\,\, & 260.2 & \; --\,\, & 4980.7 & \; --\,\, & 93.5 & \; --\,\,& 1313.5 & \; --\,\, & 1252.2 & \;--\,\, \\
1 & 198.2 & \; 1\;\;\,& 253.4 & \; 1\;\;\,& 5091.3 & \; 1\;\;\,& 109.7 & \;1\;\;\,& 1985.9 & \; 1\;\;\, & 1073.1 & \; 1\;\;\, \\
2 & 77.7 & \; 2.5 & 127.4 & \; 1.9 & 1355.0 & \; 3.7 & 51.2 & \; 2.1 & 1103.3 & \; 1.8 & 340.0 & \; 3.1 \\
4 & 49.2 & \; 4.0 & 63.5 & \; 3.9 & 395.7 & \; 12.8 & 30.6 & \; 3.5 & 492.2 & \; 4.0 & 110.2 & \; 9.7 \\
8 & 32.9 & \; 6.0 & 32.0 & \; 7.9 & 99.2 & \; 51.3 & 22.8 & \; 4.8 & 214.1 & \; 9.2 & 40.1 & \; 26.7 \\
16 & 28.3 & \; 7.0 & 16.5 & \; 15.3 & 34.4 & \; 148.0 & 27.3 & \; 4.0 & 113.4 & \; 17.5 & 22.7 & \; 47.2 \\
32 & 26.8 & \; 7.4 & 8.4 & \; 30.1 & 12.4 & \; 410.5 & 17.0 & \; 6.4 & 55.5 & \; 35.7 & 8.3 & \; 129.2 \\
64 & 32.2 & \; 6.1 & 5.5 & \; 46.0 & 5.0 & \; 1018.2 & 24.4 & \; 4.5 & 31.2 & \; 63.6 & 6.9 & \; 155.5 \\
128 & 42.3 & \; 4.6 & 2.7 & \; 93.8 & 2.6 & \; 1958.1 & 15.8 & \; 6.9 & 15.4 & \; 128.9 & 3.6 & \; 298.0 \\
256 & 54.1 & \; 3.6 & 1.1 & \; 230.3 & 1.8 & \; 2828.5 & 24.0 & \; 4.5 & 7.9 & \; 251.3 & 3.9 & \; 275.1 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Strong scaling behavior for Fichera corner}
\label{tab:II:strong}
\end{table}
\begin{table}[t]
\begin{center}
\begin{tabular}{r||r|r|r|r|r||r|r|r|r|r}
\toprule
& \multicolumn{5}{c||}{$\ell= 4\ ,\;\;p=2$} & \multicolumn{5}{c}{$\ell= 3\ ,\;\;p=4$} \\
\midrule
$\#$ Proc. & \multicolumn{1}{p{2.5em}|}{$\#$ dofs} & \multicolumn{1}{p{1.5em}|}{It.}& \multicolumn{1}{p{3em}|}{Setup} & \multicolumn{1}{p{3.5em}|}{Ass.} & \multicolumn{1}{p{3.5em}||}{Solving}
& \multicolumn{1}{p{2.5em}|}{$\#$ dofs} & \multicolumn{1}{p{1.5em}|}{It.}& \multicolumn{1}{p{3em}|}{Setup} & \multicolumn{1}{p{3.5em}|}{Ass.} & \multicolumn{1}{p{3.5em}}{Solving} \\
\midrule
1 & 34\,391 & 28 & 0.4 & 4.0 & 1.9 & 9\,317 & 31 & 0.6 & 38.0 & 1.3 \\
8 & 275\,128 & 39 & 1.3 & 4.5 & 4.0 & 74\,536 & 35 & 1.1 & 22.8 & 2.1 \\
64 &2\,201\,024 & 45 & 54.4 & 4.5 & 6.6 &596\,288 & 38 & 24.1 & 28.7 & 6.2 \\
512 &17\,608\,192 & 46 &2071.3 & 5.1 & 11.1 &4\,770\,304 & 35 & 2343.8 & 32.4 & 59.5 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Weak scaling behavior for Fichera corner}
\label{tab:II:weak}
\end{table}
In Tab.~\ref{tab:II:weak}, we present weak scaling results. We again fix the grid level $\ell$ and the spline
degree $p$ to two typical values. Here, for the case of $4$ processors, we consider the initial configuration
of $K=7$ patches. For the following rows, we consider $56$, $448$ and $3584$ patches.
(So, the line with $64$ processors coincides with the corresponding line in Tab.~\ref{tab:II:strong}.)
Again, the assembling times and the solving times do not show
any clear tendency. Only for the last line with $3584$ patches, the coarse-grid
solver causes problems. For the case $\ell=3$ and $p=4$, $51$ of the $59$ seconds required for solving are
due to the global solver on the coarsest grid. Again, the setup costs get dominant if the number of patches
is increased.
Concluding, we have shown that the robust multi-patch multigrid solver
from~\cite{Takacs:2017} can be extended to three dimensional domains and that it converges
well also in this case. We have observed that the multigrid solver can be
parallelized in a natural way yielding very good speedup rates.
Certainly, this is not the end of the story and further improvement should
be considered in two directions. First, the setup phase becomes a bottleneck if many
processors are considered. Here, improvements would be mainly a challenge in terms of
implementation and data management. Second, the coarse-grid problem becomes too large if the
number of patches is increased, particularly in the three dimensional case. To resolve that
issue, it would be necessary to further coarsen the coarse-grid problem or to consider
approximate solvers on the coarsest grid level which certainly would change the mathematical
meaning of the algorithm and could, therefore, influence its
convergence behavior. Finally, further investigation is required
to completely understand the super optimal speedup rates observed in the
strong scaling tests.\\[.25em]
\textbf{Acknowledgments.}
The first author would like to thank the Austrian Science Fund (FWF)
for the financial support through the DK W1214-04, while the second
author was supported by the FWF grant NFN S117-03.
\bibliographystyle{amsplain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,137
|
\section*{}
The existence of a dilaton - a neutral,
scalar field with exponential coupling to matter
is one of the universal predictions following from various
types of string theories and from Kaluza-Klein theories
(see e.g. \cite{GSW}). The dilaton field has an influence on
the cosmological evolution of the Universe
\cite{BER}
and it might modify particle-like
solutions of corresponding theories.
In the present report we are interested in the second aspect of
the problem, namely we will study static spherically symmetric
solutions of the Einstein-Yang-Mills-dilaton (EYMD) theory.
We will consider the EYMD theory defined by the action
\begin{equation}
S=
\frac{1}{4\pi}\int\Bigl(-\frac{1}{4 G}R+
\frac{1}{2}(\partial\varphi)^2-
{{\rm e}^{2\kappa\varphi}\over 4g^2}F^2\Bigr)
\sqrt{-g} d^4x\; , \label{act}
\end{equation}
where $G$ is Newton's constant,
$g$ is the gauge coupling constant and
$\kappa$ denotes the dilatonic coupling constant.
The scaling properties of the action
Eq.(\ref{act}) allow us to put $G=g=1$ without restrictions
in what follows.
The only remaining free parameter in the action is the dimensionless
ratio $\gamma=\kappa /g\sqrt{G}$.
Note that in the limit $\gamma\to 0$ the model described by
Eq.~(\ref{act}) reduces to the
Einstein-Yang-Mills theory \cite{BM,BH}.
In the limit $\gamma\to\infty$ one gets
the Yang-Mills-dilaton theory \cite{LM1} in flat
space.
To study static, spherically symmetric solutions of
EYMD theory
a convenient parametrization for the metric turns out to be
\begin{equation}
ds^2=A^2(r)\mu(r)dt^2-{dr^2\over\mu(r)}
-r^2d\Omega^2\;, \label{interval}
\end{equation}
where $d\Omega^2=d\theta^2+sin^2(\theta)d\varphi^2$
is the line element of the unit sphere.
For the $SU(2)$ Yang-Mills potential we make the usual (`magnetic')
spherically symmetric ansatz
\begin{equation}
W_0^a=0,\quad
W_i^a=\epsilon_{aik}{x^k\over r^2}(W(r)-1)\;. \label{gauge}
\end{equation}
Substituting this ansatz into the action
we obtain the reduced action
\begin{equation}
S_{\rm red}=-\int A\Bigl[{1\over 2}(\mu+r\mu'-1)+
{r^2\over 2}\mu\varphi'^2+{\rm e}^{2\gamma\varphi}
\Bigl(\mu W'^2+{(1-W^2)^2\over2r^2}\Bigr)\Bigr]\,dr\;,\label{redact}
\end{equation}
where a prime denotes $d\over dr$.
The resulting field equations are
\begin{eqnarray}
(A{\rm e}^{2\gamma\varphi}\mu W')'&=&A{\rm e}^{2\gamma\varphi}
{W(W^2-1)\over r^2} \;, \nonumber \\
(A\mu r^2\varphi')'&=&
2\gamma A{\rm e}^{2\gamma\varphi}\Bigl(\mu W'^2+{(1-W^2)^2\over2r^2}
\Bigr)\;, \nonumber \\
\mu'&=&{1\over r}\Biggl(1-\mu-r^2\mu\varphi'^2-2{\rm e}^{2\gamma\varphi}
\Bigl(\mu W'^2+{(1-W^2)^2\over2r^2}\Bigr)\Biggr)\;, \nonumber \\
A^{-1}A'&=&\frac{2{\rm e}^{2\gamma\varphi}W'^2}{r}
+r\varphi'^2\;.\label{eqm}
\end{eqnarray}
The field equations Eq.~(\ref{eqm}) have
singular points at $r=0$ and $r=\infty$ as well
as at points where $\mu(r)$ vanishes.
Inserting a power series expansion into Eq.~(\ref{eqm}) one
finds a $2-$parameter family of regular solutions in the
vicinity of $r=0$
\begin{eqnarray}
W(r)&=&1-br^2+O(r^4)\;,~~~~~~~~~~~~~~
\mu(r) = 1-4b^2{\rm e}^{2\gamma\varphi_0}r^2+O(r^4)\;, \nonumber \\
\varphi(r)&=&\varphi_0+2\gamma {\rm e}^{2\gamma\varphi_0}b^2r^2
+O(r^4)\;,\;\;
A(r) = 1+4b^2{\rm e}^{2\gamma\varphi_0}r^2+O(r^4)\;,
\label{ezero}
\end{eqnarray}
where $b$ and $\varphi_0$ are arbitrary parameters.
Similarly at $r=\infty$ one finds
\begin{eqnarray}
W(r)&=&\pm(1-{c\over r}+O({1\over r^2}))\;,\;\;
\mu(r) = 1-{2M\over r}+O({1\over r^2})\;, \nonumber \\
\varphi(r)&=&\varphi_\infty-{d\over r}+O({1\over r^2})\;,\;\;~~
A(r) = A_\infty(1-{d^2\over2r^2}+O({1\over r^4}))\;,
\label{einfty}
\end{eqnarray}
where again $c,d,M,\varphi_\infty$ and $A_\infty$ are arbitrary
parameters.
In the vicinity of the singular point $r_h$ where $\mu (r)$ vanishes
one finds a $2-$parame\-ter family of solutions which stay regular
at this point.
Under an appropriate choice of the parameters the surface $r=r_h$
corresponds to a regular event horizon.
Globally regular asymptotically flat
(respectively back hole) solutions have to
interpolate between the described asymptotic behavior at $r=0$
(respectively $r=r_h$) and $r=\infty$.
The analysis \cite{LM2,BIZ2} for $0<\gamma<\infty$ yields what could be
expected from the extreme cases $\gamma =0$ and $\gamma =\infty$.
It was found \cite{LM2} that for any value of the dilaton coupling
constant $\gamma$ the EYMD system has:
a discrete family of globally regular (sphaleron type)
solutions of finite mass and a discrete family of (non-Abelian)
black hole solutions.
The solutions in each family are labeled by the
number $n$ of zeros of the gauge field potential $W(r)$.
The solutions are found to be unstable \cite{LM2}.
It is interesting that the black holes have a non-vanishing
Yang-Mills field outside the horizon.
One can interpret the solutions with the horizon as
``black holes inside sphalerons''.
The mass of the solutions in the case of the Einstein-Yang-Mills
theory is of the order of one in natural units $M_{EYM}=1/g\sqrt{G}$.
In the EYMD theory the mass of the solutions decreases with increasing
dilatonic coupling constant $\gamma$ and
for large $\gamma$ goes to zero like
$M_{EYMD}\sim \frac{M_{EYM}}{\gamma ^2}$.
Globally regular solutions with odd $n$ are kinds of sphalerons.
One finds fermion zero modes in the background of these solutions
\cite{GIB,VOLF,LM3}.
One can assign a topological number to these solutions.
It is half integer.
There are some interesting regularities in the EYMD theory.
It turns out that
the ${tt}$ component of the metric is related to the dilaton
$A^2 \mu = e^{2 \gamma \varphi}$ and as a consequence
of this relation the dilaton charge
(parameter $d$ in Eq.(\ref{einfty}))
is equal to the mass of the solution: $d=M$.
A very special situation occurs for the value of
the dilaton coupling constant $\gamma=1$, which corresponds to
the model obtained from heterotic string theory.
It was found \cite{LM2} that for the $n=1$ solution
the parameter $b$ is a rational number, $b =\frac{1}{6}$.
Another regularity found numerically is that an asymtotic coefficient
$c$ in Eq.(\ref{einfty}) is related to the mass of the solution, $c=2M$.
We think these are arguments indicating that the lowest lying
($n=1$) regular solution may be obtained in closed form.
similarly to the ``stringy instanton''
and the ``stringy monopole''\cite{INSTMON}.
Due to the high mass of the solutions the only situation where they
could play a role is in the Early Universe, but at the moment there
seems to be no natural physical scenario where we could make use of
these solutions.
\section*{Acknowledgements}
I am grateful to Dieter Maison for a fruitful collaboration
which led to the results reported in this paper.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,647
|
Nososticta salomonis är en trollsländeart. Nososticta salomonis ingår i släktet Nososticta och familjen Protoneuridae.
Underarter
Arten delas in i följande underarter:
N. s. eburnea
N. s. salomonis
Källor
Trollsländor
salomonis
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,337
|
Q: OutOfMemoryException - GC verbose confirmed a memory leak, what now? I'm monitoring an app whose GC verbose log looks like this:
The graph draws the amount of Used Tenured after the GC runs.
As you can see, there's an obvious memory leak, but I was wondering what would be the best next step to find out which component is holding around 50MB of memory each time the GC runs.
The machine is an AIX 6.1 running an IBM's JVM 5.
Thanks
A: The pattern in the chart definitely looks like a typical memory leak, building up in tenured space over time. Your best shot would be heap dump analyzers - take a heap dump for example similar to following
jmap -dump:format=b,file=dump.bin <your java process id>
and analyze the dump file for example with Eclipse Memory Analyzer.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,647
|
#import <Foundation/Foundation.h>
@class AIError;
#pragma mark - API
/**
These constants identify which API succeeded or failed when calling AIAuthenticationDelegate. The value identifying
the API is passed in the APIResult and APIError objects.
@since 1.0
*/
typedef NS_ENUM(NSUInteger, API) {
/** Refers to `[AIMobileLib authorizeUserForScopes:delegate:]` */
kAPIAuthorizeUser = 1,
/** Refers to `[AIMobileLib getAccessTokenForScopes:withOverrideParams:delegate:]` */
kAPIGetAccessToken = 2,
/** Refers to `[AIMobileLib clearAuthorizationState:]` */
kAPIClearAuthorizationState = 3,
/** Refers to `[AIMobileLib getProfile:]` */
kAPIGetProfile = 4,
/** Refers to `[AIMobileLib authorizeUserForScopes:delegate:options]` */
kAPIGetAuthorizationCode = 5
};
#pragma mark - APIResult
/**
This class encapsulates success information from an AIMobileLib API call.
*/
@interface APIResult : NSObject
- (id)initResultForAPI:(API)anAPI andResult:(id)theResult;
/**
The result object returned from the API on success. The API result can be `nil`, an `NSDictionary`, or an `NSString`
depending upon which API created the APIResult.
- `[AIMobileLib authorizeUserForScopes:delegate:]` : Passes `nil` as the result to the delegate.
- `[AIMobileLib getAccessTokenForScopes:withOverrideParams:delegate:]` : Passes an access token as an `NSString` object
to the delegate.
- `[AIMobileLib clearAuthorizationState:]` : Passes nil as the result to the delegate.
- `[AIMobileLib getProfile:]` : Passes profile data in an `NSDictionary` object to the delegate. See the API description
for information on the key:value pairs expected in profile dictionary.
@since 1.0
*/
@property (strong) id result;
/**
The API returning the result.
@since 1.0
*/
@property API api;
@end
#pragma mark - APIError
/**
This class encapsulates the failure result from an AIMobileLib API call.
*/
@interface APIError : NSObject
- (id)initErrorForAPI:(API)anAPI andError:(id)theErrorObject;
/**
The error object returned from the API on failure.
@see See AIError for more details.
@since 1.0
*/
@property (strong) AIError *error;
/**
The API which is returning the error.
@since 1.0
*/
@property API api;
@end
#pragma mark - AIAuthenticationDelegate
/**
Applications calling AIMobileLib APIs must implement the methods of this protocol to receive success and failure
information.
*/
@protocol AIAuthenticationDelegate <NSObject>
@required
/**
The APIs call this delegate method with the result when it completes successfully.
@param apiResult An APIResult object containing the information about the calling API and the result generated.
@see See APIResult for more information on the content of the apiResult.
@since 1.0
*/
- (void)requestDidSucceed:(APIResult *)apiResult;
/**
The APIs call this delegate method with the result when it fails.
@param errorResponse An APIResult object containing the information about the API and the error that occurred.
@see See APIError for more information on the content of the result.
@since 1.0
*/
- (void)requestDidFail:(APIError *)errorResponse;
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,050
|
{"url":"https:\/\/www.jiskha.com\/questions\/1775050\/Solve-the-integral-x-3-x-2-2-Im-stuck-This-is-my-work-so-far-A-x-2","text":"# Calculus\n\nSolve the integral: \u222b x^3\/(x^+2)^2\n\nI'm stuck. This is my work so far.\n\n\u222b A\/(x^+2) + \u222b B\/(x^2-2)^2\nF(x)=A ln(x^2+2)-B\/(x^2+2)+C\nX^3=A(x^2+2)+B\n\n1. 0\n2. 0\n3. 3\n1. let z = x^2+2 then dz = 2x dx and dx = dz\/2x and x = (z-2)^.5 and xdx=dz\/2\nso x^2 = (z-2)\nnow what you have is\n\u222b (z-2) x dx\/z^2 = \u222b (z-2)dz \/2z^2\n= \u222b dz\/2z - \u222b dz\/z^2\n\n1. 0\n2. 0\nposted by Damon\n\n## Similar Questions\n\n1. ### calc II\n\nExpress the integrals as the sum of partial fractions and evaluate the integral: (integral of) (x^2)dx\/(x-1)(x^2 +2x+1) My work: The above integral is equal to x^2dx\/(x+1)^2 (A\/x-1) + (B\/x+1) + (Cx+D)\/(x+1)^2 = x^2 A(x+1)^2 +\n2. ### Calculus - Seperable Equations\n\nSolve the separable differential equation (dy\/dx)=y(1+x) for y and find the exact value for y(.3). dy\/dx = y(1+x) dy\/y = (1+x)dx Integral (dy\/y) = Integral (1+x)dx ln (y) = x + (1\/2)x^2 + C y = e^(x + (1\/2)x^2 + C) y(0.3) =\n3. ### calc: arc length\n\nfind the exact length of this curve: y = ( x^3\/6 ) + ( 1\/2x ) 1\/2\n4. ### Calc\n\nEvaluate the integral using any method: (Integral)sec^3x\/tanx dx I started it out and got secx(1tan^2x)\/tanx. I know I just have to continue simplifying and finding the integral, but I'm stuck on the next couple of steps. Also, I\n5. ### ALGEBRA 1!PLZ HELP!\n\nUse the Substitution method to solve the system of equations. y - 2x = -5 3y - x = 5 Solve the first equation for y. y = 2x-5 Substitute that for y in the second equation and solve for x. Then use the value of x to substitute back\n6. ### Calculus\n\nI'm doing statistics homework and am stuck on a problem using integration. The problem gives a distribution where for x>1, f(x) = k x^-6. I am then asked to \"Determine the value of k for which f(x) is a legitimate pdf. \" To be a\n7. ### Calculus\n\nSolve the integral (x+3)\/(x-1)^3 So far, I'm stuck. This is what I have thus far: x+3 = A(x-1)^3\/(x-1) + B(x-1)^3\/(x-1)^2 + C(x-1)^3\/(x-1)^3 x+3 = A(x^2-2x+1) + B(x-1) + C x+3 = Ax^2 - 2Ax +Bx + A - B + C x+3 =Ax^2 + 1(B-2A)x + (A\n8. ### Calculus\n\nCan someone check my work and answer? Evaluate the integral from -1 to 0 of (4x^6+2x)^3(12x^5+1)dx My work: let u=4x^6+2x dx=du\/24x^5+2 now we have the integral from -1 to 0 of u^3(12x^5+1)(du\/24x^5+2) Simplifies to the integral\n9. ### math\n\nHow do I derive the secant reduction rule? Integral (sec x)^n dx = Integral (sec x)^(n-2) * (sec x)^2 dx = Integral ((tan x)^2 + 1)^(n\/2-1) * (sec x)^2 dx Doing a substitution with: u = tax x du = (sec x)^2 dx = Integral (u^2 +\n10. ### Physics\n\nSuppose we had a straight tunnel, through Earth's center, to a point on the opposite side of the planet, and used it to deliver mail to the other side. With what speed would our packages pass through Earth's center. So: ag =\n\nMore Similar Questions","date":"2018-10-23 17:17:49","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\": 0.8753241300582886, \"perplexity\": 3027.9420668225916}, \"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-2018-43\/segments\/1539583516480.46\/warc\/CC-MAIN-20181023153446-20181023174946-00545.warc.gz\"}"}
| null | null |
Crossed That off the Whiteboard
Saturday, 30 May 2020 Sunday, 5 April 2020
This week, we're joined again by Adam Richard for a discussion about RTD's early-season two-parters, sidelining the main characters, the military, cloning, Sontarans, and the perils of spending too much time with our families. It all smells very much like The Sontaran Stratagem.
Martha is now engaged to the impressively handsome Thomas Milligan from Last of the Time Lords, who is played by Tom Ellis, who can now been seen in the titular role in Netflix's supernatural police procedural Lucifer.
Sergeant Benton's pretty new replacement in this version of UNIT, Ross Jenkins, is played by Christian Cooke, who was recently one of the suspects in the BBC adaption of Agatha Christie's Ordeal by Innocence (2018).
Take a drink, dear listener. In her TARDIS Eruditorum article on The Time Warrior, El Sandifer explains that Bob Holmes did not create the Sontarans as a second-tier race of Doctor Who monsters; what he created there instead was the character of Linx.
If you're young enough, you might not know that Christopher Ryan — who plays General Staal in this story — first became famous as Mike the Cool Person in the Thatcher-era BBC comedy series The Young Ones. He went on to play Jennifer Saunders's long-suffering ex-husband Marshall in Absolutely Fabulous.
Adam writes for the ABC-TV comedy quiz show Hard Quiz, which has been running in Australia since 2017, and is now in its fifth series.
Nathan is on Twitter as @nathanbottomley, James is @ohjamessellwood, and Peter is still, unaccountably, nowhere to be found. The Flight Through Entirety theme was arranged by Cameron Lam, and the strings performance was by Jane Aubourg. You can follow the podcast on Twitter at @FTEpodcast.
Adam is @adamrichard on Twitter, adamrichard on Instagram and Fabulous Adam Richard on Facebook. His website is at adamrichard.com.au. He can currently be found opining about Doctor Who and Star Trek: Picard on his own podcast Adam Richard Has a Theory.
We're also on Facebook, and you can check out our website at flightthroughentirety.com. Please consider rating or reviewing us on iTunes, or we'll rudely correct your grammar even though you're speaking perfectly idiomatic English.
Our James Bond commentary podcast is called Bondfinger, and you can find that at bondfinger.com, at @bondfingercast on Twitter, on Apple Podcasts, and everywhere else as well.
https://episodes.castos.com/fte/FTE-183-Crossed-That-off-the-Whiteboard-The-Sontaran-Stratagem-.mp3
Download file | Play in new window | Duration: 00:48:26 | Recorded on Saturday, 8 February 2020
The Icy Moral High Ground
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,758
|
0-31 last weekend, earned 17 fir
DOVER, Del. -- Daniel Suarez jumped out of his car and sprinted down the front stretch to grab the checkered flag and take an untraditional victory lap on the track.I had to run a little bit. Thats fine, he said, laughing. When you win, you can run a little bit.Suarez cruised into the second round of the Chase in the Xfinity Series with a win Sunday at Dover International Speedway.He won his second career Xfinity race a day after rain washed out the race and forced a rare NASCAR doubleheader. NASCAR ran its first Xfinity-Cup doubleheader since April 25, 2010, at Talladega Superspeedway.Suarez became the first Mexican-born driver to win a NASCAR national race in June at Michigan. He became a part of NASCAR in 2011 and was involved in its Drive for Diversity and Next programs.Team owner Joe Gibbs has been a stout supporter of NASCARs diversity programs.From a diversity standpoint and all he represents there from a Hispanic standpoint too, I think its huge for our sport, he said. Hes the perfect representative. Hes funny, hes got a good sense of humor and I think he really, really appreciates where he is.Suarez said he was proud to learn in victory lane that he won during National Hispanic Heritage Month. He said he won for all of his fans across North America.Having all of the support that I have had from Mexico and from here in the U.S. and all of Latin America has been super special, he said. I really feel lucky to be in this position and to be able to represent all of them.Suarez led 123 of 200 laps in the No. 19 Toyota and joined Elliott Sadler in the second round of the Chase.Ty Dillon was second, followed by Justin Allgaier, Ryan Blaney and Alex Bowman.With three top NASCAR Cup stars sitting out, the top five was filled with Xfinity regulars. Austin Dillon, Kyle Busch and Joey Logano all had replacement drivers in the Xfinity race to focus on the Cup race scheduled for Sunday afternoon. Regan Smith finished 13th driving for Dillon; Drew Herring was 21st subbing for Busch; and Blaney filled in for Logano.Busch has eight Xfinity wins, Dillon has two and Logano has one this season.Allgaier said that even if the three stars raced, Suarez was still the driver to beat.I dont know that anybody was going to beat him today, Allgaier said. I think it was smart for the Cup guys to sit out. I think two races in one day would have been a strain, mentally.Ty Dillon, Ryan Sieg and Brandon Jones are in danger of being eliminated from the Chase next week at Charlotte Motor Speedway. So is Erik Jones, one of the top rising stars in the sport who entered the Chase as the top speed. Jones finished 16th and was 28th last week.Its pretty embarrassing not to even be in the Chase right now for the next round, Jones said. Its just unfortunate; its just something that I never would have seen it coming. Kyrie Irving Shoes Wholesale . -- Five former Kansas City Chiefs players who were on the team between 1987 and 1993 filed a lawsuit Tuesday claiming the team hid and even lied about the risks of head injuries during that time period when there was no collective bargaining agreement in place in the NFL. Kyrie 2 For Sale . To the surprise of many, it isnt the Wolverines but their in-state rivals the Michigan State Spartans. http://www.kyrieirvingshoescheap.com/ . Having already announced that the race will start May 9 with three stages in Northern Ireland and Ireland and finish in Trieste on June 1, the rest of the route was unveiled Monday. Kyrie 4 Confetti For Sale Cheap . -- The proud fathers huddled near the Dallas Stars dressing room, smiling, laughing and telling stories while wearing replica green sweaters of their sons team. Kyrie 1 For Sale Mens . 1 position. The Mustangs (6-0), who beat Queens 50-31 last weekend, earned 17 first-place votes and 287 points in voting by the Football Reporters of Canada. Western was last ranked first in the country in October 2011. HOUSTON -- The Chicago White Sox lamented their inability to make plays in crucial situations after losing three straight games by one run each to Houston. They finally found a way to capitalize when they had a chance to change the game Monday night and it led to a 4-2 win over the Astros. Dayan Viciedos bases-loaded triple in a four-run sixth inning helped the White Sox snap a four-game losing streak. They also avoided being swept in a four-game series for the first time since 2008. Chicago trailed 2-0 before Alejandro De Aza singled in the sixth and went to third on Adam Dunns two-out single. Paul Konerko pulled Chicago to 2-1 with an RBI single that went between the legs of starter Bud Norris and into centre field. Conor Gillaspie barely beat the throw to first on an infield single that loaded the bases. That brought up Viciedo, whose triple landed on Tals Hill in centre field just out of the reach of a diving Brandon Barnes. All three runners scored to put Chicago up 4-2. "It was a good one for Viciedo to get a hit," White Sox manager Robin Ventura said. "Its just one of those that you hope gets him going a little bit because hes been missing out. And to finally get it, with him doing it, its good to see." It was the second straight game with a triple for Viciedo, making him the first White Sox player to do that since Mark Kotsay in August 2010. "Theres been some rough spots where things havent gone the right way," Viciedo said through a translator. "But I feel like theres some good things, too, and I want to work on the good things Im doing right now. I want to concentrate on those things and become more consistent." Nate Jones (3-4) allowed three hits in 1 1-3 scoreless innings for the win. He struck out three. Addison Reed pitched a perfect ninth for his 20th save. White Sox starter Jose Quintana yielded five hits and two runs in 4 2-3 innings, but didnt factor in the decision for the fourth straight start. He has a no-decision in nine of his 14 starts this season. Jason Castro doubled twice and drove in two runs for the Astros, who had won four straight. Barnes had three hits. Norris (5-7) allowed eight hits and four runs over six innings in his third consecutive loss. "I was in a groove, and that sixth inning was very frustrating," he said. "Im still frustrated about it." It was a rare road win for the struggling White Sox, who have dropped 12 of their last 14 away from Chicago and 14 of 19 overall. The Astros had runners at the corners with no outs in the seventh after back-to-back singles by Barnes annd Jose Altuve.dddddddddddd. But Matt Thornton struck out Castro before Jesse Crain fanned J.D. Martinez and Chris Carter to end the inning. Carter struck out twice to give him 101 this season, which leads the majors. The Astros struck out 12 times in all to extend their total to a major league-high 673. Crain struck out three in 1 2-3 innings for his career-high 27th straight scoreless appearance to tie a franchise record for relievers. He was just happy to help the team end its skid. "We needed to turn things around," he said. "Obviously, weve been struggling and getting this one and avoiding the streak was big and hopefully we start a new win streak and win tomorrow." Ventura raved about Crain. "Its not just easy situations going in there mopping up," he said. "Hes been coming into some real tough situations and coming through for us. Its amazing what hes been doing. I think hes just been locked in, healthy and everything else." Altuve singled with one out in the first and scored on a double by Castro to make it 1-0. Quintana pitched a 1-2-3 second inning before getting into trouble in the third. Barnes singled to start the inning before stealing second base and advancing to third on an error by catcher Hector Gimenez with one out. Castros second double, which bounced off the wall in right field, sent Barnes home to extend the lead to 2-0. Castro has five RBIs, two doubles and two homers in the last three games. Norris looked good early. He allowed a double to Alex Rios with two outs in the first before retiring the next eight batters. Rios singled with one out in the fourth and stole second, but Norris set down the next two batters to end the threat. NOTES: Houston activated OF Justin Maxwell from the 15-day disabled list and optioned OF Jimmy Paredes to Triple-A Oklahoma City after the game. Maxwell, who has been out since April 24 with a broken left hand, will be in Tuesdays lineup. ... Chicagos Dylan Axelrod opposes Minnesotas Mike Pelfrey when the White Sox open a three-game series against the Twins on Tuesday. ... Jordan Lyles is scheduled to pitch for the Astros on Tuesday when they open a series with Milwaukee. Right-hander Alfredo Figaro is set to pitch for the Brewers. ... The Astros signed second-round draft pick RHP Andrew Thurman and four other picks on Monday. Thurman was at Minute Maid Park and worked out with the team before the game. Houston also signed 28th-round pick LHP Jordan Mills, 30th-round pick RHP Jorge Perez, 33rd-rounder 1B Tyler White and J.D. Osborne, who was selected in the 36th round. ' ' '
DOVER, Del. -- Daniel Suarez jumped out of his car and sprinted down the front stretch to grab the checkered flag and take an untraditional victory lap on the track.I had to run a little bit. Thats fine, he said, laughing. When you win, you can run a little bit.Suarez cruised into the second round of the Chase in the Xfinity Series with a win Sunday at Dover International Speedway.He won his second career Xfinity race a day after rain washed out the race and forced a rare NASCAR doubleheader. NASCAR ran its first Xfinity-Cup doubleheader since April 25, 2010, at Talladega Superspeedway.Suarez became the first Mexican-born driver to win a NASCAR national race in June at Michigan. He became a part of NASCAR in 2011 and was involved in its Drive for Diversity and Next programs.Team owner Joe Gibbs has been a stout supporter of NASCARs diversity programs.From a diversity standpoint and all he represents there from a Hispanic standpoint too, I think its huge for our sport, he said. Hes the perfect representative. Hes funny, hes got a good sense of humor and I think he really, really appreciates where he is.Suarez said he was proud to learn in victory lane that he won during National Hispanic Heritage Month. He said he won for all of his fans across North America.Having all of the support that I have had from Mexico and from here in the U.S. and all of Latin America has been super special, he said. I really feel lucky to be in this position and to be able to represent all of them.Suarez led 123 of 200 laps in the No. 19 Toyota and joined Elliott Sadler in the second round of the Chase.Ty Dillon was second, followed by Justin Allgaier, Ryan Blaney and Alex Bowman.With three top NASCAR Cup stars sitting out, the top five was filled with Xfinity regulars. Austin Dillon, Kyle Busch and Joey Logano all had replacement drivers in the Xfinity race to focus on the Cup race scheduled for Sunday afternoon. Regan Smith finished 13th driving for Dillon; Drew Herring was 21st subbing for Busch; and Blaney filled in for Logano.Busch has eight Xfinity wins, Dillon has two and Logano has one this season.Allgaier said that even if the three stars raced, Suarez was still the driver to beat.I dont know that anybody was going to beat him today, Allgaier said. I think it was smart for the Cup guys to sit out. I think two races in one day would have been a strain, mentally.Ty Dillon, Ryan Sieg and Brandon Jones are in danger of being eliminated from the Chase next week at Charlotte Motor Speedway. So is Erik Jones, one of the top rising stars in the sport who entered the Chase as the top speed. Jones finished 16th and was 28th last week.Its pretty embarrassing not to even be in the Chase right now for the next round, Jones said. Its just unfortunate; its just something that I never would have seen it coming. [url=http://www.kyrieirvingshoescheap.com/]Kyrie Irving Shoes Wholesale[/url] . -- Five former Kansas City Chiefs players who were on the team between 1987 and 1993 filed a lawsuit Tuesday claiming the team hid and even lied about the risks of head injuries during that time period when there was no collective bargaining agreement in place in the NFL. [url=http://www.kyrieirvingshoescheap.com/cheap-kyrie-2-shoes-a.html]Kyrie 2 For Sale[/url] . To the surprise of many, it isnt the Wolverines but their in-state rivals the Michigan State Spartans. [url=http://www.kyrieirvingshoescheap.com/]http://www.kyrieirvingshoescheap.com/[/url] . Having already announced that the race will start May 9 with three stages in Northern Ireland and Ireland and finish in Trieste on June 1, the rest of the route was unveiled Monday. [url=http://www.kyrieirvingshoescheap.com/cheap-kyrie-4-shoes-a/kyrie-4-confetti.html]Kyrie 4 Confetti For Sale Cheap[/url] . -- The proud fathers huddled near the Dallas Stars dressing room, smiling, laughing and telling stories while wearing replica green sweaters of their sons team. [url=http://www.kyrieirvingshoescheap.com/cheap-kyrie-1-shoes-a.html]Kyrie 1 For Sale Mens[/url] . 1 position. The Mustangs (6-0), who beat Queens 50-31 last weekend, earned 17 first-place votes and 287 points in voting by the Football Reporters of Canada. Western was last ranked first in the country in October 2011. HOUSTON -- The Chicago White Sox lamented their inability to make plays in crucial situations after losing three straight games by one run each to Houston. They finally found a way to capitalize when they had a chance to change the game Monday night and it led to a 4-2 win over the Astros. Dayan Viciedos bases-loaded triple in a four-run sixth inning helped the White Sox snap a four-game losing streak. They also avoided being swept in a four-game series for the first time since 2008. Chicago trailed 2-0 before Alejandro De Aza singled in the sixth and went to third on Adam Dunns two-out single. Paul Konerko pulled Chicago to 2-1 with an RBI single that went between the legs of starter Bud Norris and into centre field. Conor Gillaspie barely beat the throw to first on an infield single that loaded the bases. That brought up Viciedo, whose triple landed on Tals Hill in centre field just out of the reach of a diving Brandon Barnes. All three runners scored to put Chicago up 4-2. "It was a good one for Viciedo to get a hit," White Sox manager Robin Ventura said. "Its just one of those that you hope gets him going a little bit because hes been missing out. And to finally get it, with him doing it, its good to see." It was the second straight game with a triple for Viciedo, making him the first White Sox player to do that since Mark Kotsay in August 2010. "Theres been some rough spots where things havent gone the right way," Viciedo said through a translator. "But I feel like theres some good things, too, and I want to work on the good things Im doing right now. I want to concentrate on those things and become more consistent." Nate Jones (3-4) allowed three hits in 1 1-3 scoreless innings for the win. He struck out three. Addison Reed pitched a perfect ninth for his 20th save. White Sox starter Jose Quintana yielded five hits and two runs in 4 2-3 innings, but didnt factor in the decision for the fourth straight start. He has a no-decision in nine of his 14 starts this season. Jason Castro doubled twice and drove in two runs for the Astros, who had won four straight. Barnes had three hits. Norris (5-7) allowed eight hits and four runs over six innings in his third consecutive loss. "I was in a groove, and that sixth inning was very frustrating," he said. "Im still frustrated about it." It was a rare road win for the struggling White Sox, who have dropped 12 of their last 14 away from Chicago and 14 of 19 overall. The Astros had runners at the corners with no outs in the seventh after back-to-back singles by Barnes annd Jose Altuve.dddddddddddd. But Matt Thornton struck out Castro before Jesse Crain fanned J.D. Martinez and Chris Carter to end the inning. Carter struck out twice to give him 101 this season, which leads the majors. The Astros struck out 12 times in all to extend their total to a major league-high 673. Crain struck out three in 1 2-3 innings for his career-high 27th straight scoreless appearance to tie a franchise record for relievers. He was just happy to help the team end its skid. "We needed to turn things around," he said. "Obviously, weve been struggling and getting this one and avoiding the streak was big and hopefully we start a new win streak and win tomorrow." Ventura raved about Crain. "Its not just easy situations going in there mopping up," he said. "Hes been coming into some real tough situations and coming through for us. Its amazing what hes been doing. I think hes just been locked in, healthy and everything else." Altuve singled with one out in the first and scored on a double by Castro to make it 1-0. Quintana pitched a 1-2-3 second inning before getting into trouble in the third. Barnes singled to start the inning before stealing second base and advancing to third on an error by catcher Hector Gimenez with one out. Castros second double, which bounced off the wall in right field, sent Barnes home to extend the lead to 2-0. Castro has five RBIs, two doubles and two homers in the last three games. Norris looked good early. He allowed a double to Alex Rios with two outs in the first before retiring the next eight batters. Rios singled with one out in the fourth and stole second, but Norris set down the next two batters to end the threat. NOTES: Houston activated OF Justin Maxwell from the 15-day disabled list and optioned OF Jimmy Paredes to Triple-A Oklahoma City after the game. Maxwell, who has been out since April 24 with a broken left hand, will be in Tuesdays lineup. ... Chicagos Dylan Axelrod opposes Minnesotas Mike Pelfrey when the White Sox open a three-game series against the Twins on Tuesday. ... Jordan Lyles is scheduled to pitch for the Astros on Tuesday when they open a series with Milwaukee. Right-hander Alfredo Figaro is set to pitch for the Brewers. ... The Astros signed second-round draft pick RHP Andrew Thurman and four other picks on Monday. Thurman was at Minute Maid Park and worked out with the team before the game. Houston also signed 28th-round pick LHP Jordan Mills, 30th-round pick RHP Jorge Perez, 33rd-rounder 1B Tyler White and J.D. Osborne, who was selected in the 36th round. ' ' '
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Q: python : use elements from a list as keys to a dictionary in order of list my_lst = ['b','c','a']
dict = {}
for i in my_lst[]:
dict[i] = []
print dict
# prints: => {'c': [], 'b': [], 'a': []}
How can i get dict to be {'b': [ ], 'c': [ ], 'a': [ ]}
with the keys in a similar order as in the list?
A: You're looking for an OrderedDict:
>>> from collections import OrderedDict
>>> my_lst = ['b','c','a']
>>> D = OrderedDict()
>>> for i in my_lst:
... D[i] = []
>>> D
OrderedDict([('b', []), ('c', []), ('a', [])])
>>> print D.items()
[('b', []), ('c', []), ('a', [])]
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The final event of the 5th edition of the Diwan Awards took place on Friday 24 March 2017. The ceremony awarded 37 nominees from the Belgian Moroccan community who are setting an example to inspire the young generations in areas as diverse as medicine, engineering, sports, entrepreneurship and culture, among a total of 13 categories.
Many Belgian politicians, including Prime Minister Charles Michel, have attended the event which also attracts more and more media attention. This is an opportunity for Belgians of Moroccan origin to gather with a sense of pride and to let young generations know that they too can access a better social situation if they make the needed efforts, says initiator of the Awards Fatima Abbach. 'We also want to show models of positive integration to the native Belgians, as immigration represents a real chance for the country to develop and to shine on the international level', she adds.
The Diwan Awards is the most important initiative of the association Diverscity which also publishes a bilingual (Flemish-French) magazine - Aywa - dedicated to inspiring Belgian immigrants. Its success led the way for similar events to be held for the first time in Germany and The Netherlands in the coming months.
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Quilombo dos Palmares of Palmares was de grootste en bekendste quilombo (destijds werd de term mocambo gebruikt) van Brazilië. De quilombo, een nederzetting van weggelopen Afrikaanse slaven, werd tegen 1630 gesticht in de heuvels van de Serra da Barriga in het noordoosten van het land, in de huidige staat Alagoas. Pas in 1695 wisten de Portugezen Palmares van de kaart te wissen.
Geschiedenis
Ontstaan
De slaven die in Brazilië op de plantages moesten werken kwamen vaak in opstand. Middelen daarvoor waren protest, opruiing en vernielingen aan eigendommen van de plantagehouders. Ook vermoordden zij hun eigen kinderen of zichzelf om daarmee de plantagehouders financieel te duperen, want slaven waren duur. Soms lukte het slaven te ontsnappen naar voor slavenjagers moeilijk begaanbare gebieden. Zo ontstond Palmares, een verzameling van dorpen van ontsnapte slaven onder leiding van koning Macaco. Palmares, gelegen langs de rivier de São Francisco in Pernambuco, kon ontstaan doordat de slaven op de plantages gebruik maakten van de chaos rond de aanvallen van de Nederlanders tegen de Portugezen die Brazilië beheersten.
Bestaan
Palmares was in feite een onafhankelijke maatschappij met eigen wetten en levend van de landbouw. De bevolking – op het hoogtepunt enkele duizenden personen – bestond voornamelijk uit Angolezen, hetgeen het karakter van de staat bepaalde. Er leefden ook halfbloedmensen, indianen en enkele gevluchte Portugezen; het was die laatste groep die zorgde voor de opbouw van sterke en slimme verdediging. Het legertje van Palmares trok vaak op rooftocht naar de plantages; niet alleen om te plunderen maar ook om zelf slaven mee te nemen om te werk te kunnen stellen.
Ondergang
De suikerindustrie van Noordoost-Brazilië, destijds de belangrijkste bedrijfstak in Brazilië, was niet blij met het bestaan van Palmares. De grootgrondbezitters pleitten steeds voor het sturen van troepen. In 1644-'45 probeerden de troepen van Nederlands-Brazilië Palmares te vernietigen; hun poging mislukte. Ook de Portugezen faalden steeds. Pas in 1695 slaagden de Portugezen er uiteindelijk in om Palmares te vernietigen.
Noten
Geschiedenis van Brazilië
Historisch land in Amerika
Slavernij in Amerika
Nederlands-Brazilië
Alagoas
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Аві́нда — гірська вершина на Головному пасмі Кримських гір, найвища точка Нікітської яйли. Підноситься у вигляді купола над Південним берегом Криму в районі селища міського типу Гурзуф. Висота над рівнем моря — 1472 м.
Складається із вапняків; розвинуті карстові форми рельєфу. Укрита гірсько-лучною рослинністю, на схилах — розріджені соснові ліси.
Із 1923 року — у складі Кримського заповідно-мисливського господарства.
Назва
Оронім виник способом трансонімізації гідроніма, що походить від іранського ав- «вода» і суфіксу -інда / -унда / -онда / -да на означення місця, в якому багато чогось, тобто за ознакою «місце, в якому багато води». Твірний гідронім міг сформуватися на ґрунті давньоіранського ар- / ab «вода» в скіфському мовному середовищі. На думку В. А. Бушакова, гідронім постав на основі давньогрецького слова на означення місця зборів, що малоймовірно з огляду на неможливість переходу γ у β в грецькій мові або грецької γ в слов'янську в.
Примітки
Джерела
Гори України
Крим. Географічні назви
Розташування на мапі
Гірські вершини України
Горный Крым. Атлас туриста / ГНПП «Картографія», Укргеодезкартографія ; ред.: Д. И. Тихомиров, Д. В. Исаев, геоинформ. подгот. Е. А. Стахова. — К. : ДНВП «Картографія», 2010. — 112 с.
Гірські вершини Криму
Тисячники України
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MIRAMAR BAY
DAVIS BUNN
KENSINGTON BOOKS
http://www.kensingtonbooks.com
All copyrighted material within is Attributor Protected.
Table of Contents
Title Page
Copyright Page
Dedication
CHAPTER 1
CHAPTER 2
CHAPTER 3
CHAPTER 4
CHAPTER 5
CHAPTER 6
CHAPTER 7
CHAPTER 8
CHAPTER 9
CHAPTER 10
CHAPTER 11
CHAPTER 12
CHAPTER 13
CHAPTER 14
CHAPTER 15
CHAPTER 16
CHAPTER 17
CHAPTER 18
CHAPTER 19
CHAPTER 20
CHAPTER 21
CHAPTER 22
CHAPTER 23
CHAPTER 24
CHAPTER 25
CHAPTER 26
CHAPTER 27
CHAPTER 28
CHAPTER 29
CHAPTER 30
CHAPTER 31
CHAPTER 32
CHAPTER 33
CHAPTER 34
CHAPTER 35
CHAPTER 36
CHAPTER 37
CHAPTER 38
CHAPTER 39
CHAPTER 40
CHAPTER 41
CHAPTER 42
CHAPTER 43
CHAPTER 44
CHAPTER 45
CHAPTER 46
CHAPTER 47
CHAPTER 48
CHAPTER 49
CHAPTER 50
CHAPTER 51
CHAPTER 52
CHAPTER 53
CHAPTER 54
CHAPTER 55
CHAPTER 56
CHAPTER 57
CHAPTER 58
CHAPTER 59
CHAPTER 60
CHAPTER 61
CHAPTER 62
CHAPTER 63
CHAPTER 64
CHAPTER 65
MIRAMAR BAY
Discussion Questions
Teaser chapter
KENSINGTON BOOKS are published by
Kensington Publishing Corp.
119 West 40th Street
New York, NY 10018
Copyright © 2017 by Davis Bunn
All rights reserved. No part of this book may be reproduced in any form or by any means without the prior written consent of the Publisher, excepting brief quotes used in reviews.
Library of Congress Card Catalogue Number: 2016955147
Kensington and the K logo Reg. U.S. Pat. & TM Off.
ISBN: 978-1-4967-0830-4
First Kensington Hardcover Edition: April 2017
eISBN-13: 978-1-4967-0831-1
eISBN-10: 1-4967-0831-8
Kensington Electronic Edition: April 2017
This book is dedicated to
Chip MacGregor
For the friendship
The guidance
And the challenge
CHAPTER 1
Connor Larkin stood with the other passengers waiting to enter the Greyhound bus. The nighttime travelers were exactly as he remembered from his early journeys. They were silent and they were weary, and many of them treated the bus as their bed for the night. As Connor climbed on board, he knew he was doing the right thing. He intended to make a complete break. Nothing could be further from his current existence than this.
He took the first window seat where the one next to him was still empty. Connor was six-three and linebacker lean, and he didn't want to crowd somebody who was already settled in for the night run. Three minutes later, a woman who almost matched his size stopped and asked, "You mind?"
"Not a bit," he replied. She must have seen he meant it, for she offered up the best smile she could manage.
Night busses held to an established cadence. Soon as the doors sighed shut, the chatter faded. People settled in as best they could and went quiet. Those who didn't care to sleep talked in respectful murmurs. Even the fretting baby was soon silenced. The bus rumbled its way out of the downtown Los Angeles terminal, stopped twice for additional travelers, then joined the freeway and headed north.
There had once been a time when Connor had loved nothing more than a long bus ride. He had just turned sixteen and was trapped in an imploding family. Connor had not just been escaping. He was building his dreams. He was traveling toward a tomorrow all his very own. He had not thought of those trips in years.
The tires hummed and the night comforted. Connor reveled in being anonymous, at least for a few hours. He had no idea what he would find at the end of this journey. Nor, just then, did he much care. He stared at his reflection in the dark glass to his right, and decided that running away was the first thing that had felt good in far too long.
The woman leaned over and jammed Connor hard against the armrest. She knew what she'd done and grimaced an apology. Connor shrugged in reply. She set a tablet in her lap, plugged in earbuds, and said, "This is the only chance I get to watch my shows."
Connor watched her connect to the bus Wi-Fi and then dial into the number one LA entertainment cable show. That meant everything he was running from flashed up in brilliant color.
The program's anchor was a cable lollipop named Peyton Stein. LA was full of Peytons. Her program and Hello magazine had been granted exclusive coverage of Connor's main event, in return for the agency's publicist having the right to edit everything prior to release.
The image shifted from Peyton's report to a clip of Connor's most recent appearance in a US drama. He played the bad guy, as usual. The thirty-second spot showed him being caught by the hero, who launched into a furious assault that ended with Connor being thrown off a bridge. The camera followed his free fall into the torrent below, then cut back to Peyton's brilliant smile.
Connor should have been accustomed to seeing himself on-screen. His acting coach forced Connor to watch and learn and critique and grow. However, the required detachment was lost to him now.
The next clip came from earlier that same day. Connor watched as he and his fiancée emerged from the intimate engagement party attended by seven hundred of their closest friends. They had rehearsed this next act five times, until the lights and expressions and cadence were camera-perfect. Connor saw his eyes go round as two tuxedo-clad waiters pulled the quilted covering off his engagement present, then shout with delight as his fiancée offered him the gold-plated key to his very own Maserati Ghibli.
Then the woman on the bus noticed Connor watching and pulled the bud from one ear.
Connor's dread rose with the bile in his throat. He knew what was going to happen next, clear as if it was scripted in the slightly stale air. She would cry out his name. She would wake up the entire bus. Then his one chance of escape would be stripped away.
Instead, she turned in her seat, looked straight at him, and said, "You think this is for real?"
"Excuse me?"
"A guy this good-looking marries an heiress, she gives him a Ferrari, and they live happily ever after?"
"Maserati," Connor corrected.
"What?"
"The car. It's a Maserati."
She waved that aside. "Whoever heard of such a thing? I mean, that's not real life, is it? It has to be something they cooked up inside their executive suites. Like, hey, let's find this perfect guy and this superrich girl and we'll have this bash at the Hilton—"
"Beverly Hills Hotel."
"—And we'll invite all the beautiful people, then we'll put it on the TV. And next week we'll show the wedding."
"The wedding is in five days," Connor said.
"See, that's just it. I don't think there is going to be any wedding."
Connor was an actor. He was paid to pretend at emotions. So right now, he nodded slowly, like he was giving the woman's comments a lot of serious thought. He said, "You may be right about that."
"You like, I could turn on the sound and we could watch this together."
"Thanks," Connor said. "That's really kind, but I've seen it already."
Every time Connor glanced over, there he was on the woman's tablet. She kept scrolling back and forth between the various episodes, living the impossible dream. Closing his eyes no longer blocked out the memories. Peyton, the cable lollipop, had a breathless way of speaking as long as the camera was rolling. However, in the seven sessions they'd had together, she had not spoken once with Connor after the lights went off. Not even a hello. Her attitude had said it all. Connor was just another momentary episode, there to suck dry and discard. His fiancée was amazingly adept at handling the LA publicity machine. As a result, people like Peyton treated her as a keeper.
Then he corrected himself. His ex-fiancée.
They had not officially broken up yet. When Connor would fail to show for the rehearsal dinner in four days' time, he assumed the lady would bid him a tearful farewell. In front of the global press. No doubt it would be her finest hour.
* * *
Connor's stop was the juncture where the 101 joined with the Pacific Coast Highway. Steep hills cut jagged slices from the night sky. From here to the ocean was fourteen and a half miles of a single snaking road. Miramar, his destination, occupied a bowl where the cliffs eased back from the sea.
Between Miramar and where Connor stood were two valleys given over to vineyards and avocadoes and rich verdant farmland. He was standing in a parking lot shared by a 7-Eleven, a diner, and a Motel 6. Connor waved to the woman watching him from the window, and waited while the bus pulled away. Now that he had arrived, he had no idea what to do next.
His satchel felt like it weighed two hundred pounds. The night air seemed impossibly heavy. Connor had done nothing for five days except prepare for the next shoot with his beautiful bride-to-be. But he had also not slept. Exhaustion threatened to smother him. He was tempted to take a room for the night, but highway motel chains tended to want troublesome items like IDs. Plus Connor knew he ran the risk of getting stuck there. Just lay on the bed for days, trapped by the vise of so many wrong moves. Connor hefted his satchel and crossed the lot and entered the diner.
The little restaurant had a slow end-of-watch feel. Connor sat at the counter and ate a moderately good meat loaf. But the green beans had been cooked to limp submission and the potatoes were barely okay. Connor had worked in his family's restaurant until he was old enough to escape their constant bickering. He had learned early and well the artwork involved in making a fine meal. The food had been very good and the patrons had remained loyal, at least for a while. In the end, the screeches emerging every time the kitchen doors swung open had driven the patrons away. It felt comforting in a strange and disembodied way to mull over his past while eating. Normally, Connor kept such ruminations buried deep.
Connor was so engrossed in the lost years that he did not notice the cop until he heard someone ask, "You coming or going?"
The counter curved slightly, and the cop had taken a stool three away from Connor, granting him a position where he could study Connor's face. The police officer was ugly as a bull elephant, lumpish and big-boned, with features creased by childhood scars. Connor replied, "Coming, I hope."
The policeman lifted his coffee cup at the waitress' approach, nodded his thanks for the refill, and slurped noisily. All without taking his eyes off Connor. "You got friends around here?"
"Not yet."
He squinted, a sudden tightening of his features. Connor realized he had just seen the guy smile. "What brings you here?"
Such conversations had been a fairly constant part of his early years. A good-looking kid on the road, traveling by overnight bus, cops were always on the lookout for runaways. Connor had learned that honesty was the safest move. He said, "I came for a weekend on one of the ranches up in the hills. The place didn't suit me, so I spent all my free time in town. I didn't want to leave." Connor shrugged. "Now I'm back."
"Which ranch?"
"Hearst Highlands."
The cop whistled. "That place runs, what, a couple of thousand a night?"
"Something like that."
The cop took in Connor's faded T-shirt, worn jeans, woven rope belt, hiking boots. "You were working up there?"
Connor made do with a nod.
The cop picked up his bill, swung his stool around, and stood. "What say we settle up, I'll give you a lift into town."
CHAPTER 2
"Miramar was originally named Castaway Cove, but I guess you already knew that."
"I told you, I've been here just once," Connor replied. "For a weekend. That's it."
"See, the thing is, I'm pretty sure I know you."
The police car was old and smelled of industrial-strength disinfectant. It took dips in the road like a boat in heavy seas. Connor's head bumped against the wire-mesh barrier separating them from the rear seat. His left knee was jammed against the computer station. A shotgun and bully stick were slotted into position between him and the cop. The radio hissed words Connor could not understand.
The policeman went on, "You get in trouble while you were here for that weekend?"
"No, Officer."
"Chief," the big man corrected. "Name's Porter Wright."
Connor did not speak.
"I'm good with faces. I need to be in this job." He stopped at a traffic light and glanced over. His lumpish features were hard as stone. "It'll come to me."
Wright turned off the town's main avenue onto a narrow side street, drove two blocks, then entered a restricted parking area surrounded by hurricane fencing. He pulled into the first slot by the entrance to a nondescript single-story building of painted cinder blocks. He cut the motor, tapped the wheel with his trigger finger, and said, "Guy in my line of work learns to trust his instincts. Mine are telling me I know you. Which usually means I've either locked you up or run across an alert with your photograph. So you can tell me who you are, or we can head on back out to where I picked you up. And I'll let you go, but only if you promise never to show your face—"
"My name is Connor Larkin."
The sheriff twisted his bulk around so as to face Connor. "I know that name. . . ."
"I'm an actor."
Chief Wright grunted. "Sure, now I know. You're the bad guy."
That was good for a chuckle. A lot of people were soon going to be thinking just that.
The cop continued speaking. "This past week, it seems like every time I go home, my wife and daughter are watching some show about your wedding thing."
Connor nodded. "'Wedding thing' pretty much sums it up."
"Is it true what they say, you've died on-screen almost a hundred times?"
"Ninety-seven and counting."
The silence was punctuated by the radio's soft drone. Finally Wright asked, "What on earth are you doing here, son?"
Connor suspected the policeman was very adept at extracting confessions. The soft-spoken question punched him in the heart. He said the only thing that came to mind. "I was hoping maybe I could find a little of what I've lost."
* * *
Porter phoned a guesthouse two blocks farther into town, woke up the proprietor, and said he had a fellow wanting to hang his hat for a few days. The chief hesitated at something the other person said, his gaze tight on Connor. "Hang on a second there." He covered the phone. "You got one of them fake IDs?"
"No."
"If you use your real name, somebody is bound to blow your cover."
Connor's weary brain ran through the myriad names his more recent roles had saddled him with—all of whom, of course, had kicked the bucket. Not a promising start to his visit.
Porter said, "How about we keep things simple. Connor Smith sound okay?"
Connor found the act of being renamed by a cop somehow comforting. "Works for me."
Porter finished the call, then walked him down silent streets. "The owner of a local guesthouse is an old friend. Back in the fifties, the place was used by abalone fishermen operating the big trawlers. I confirmed you've misplaced your ID and will be paying cash."
"Thank you. A lot."
The key was waiting for Connor in the front mailbox. The current owner had torn out walls and turned fourteen cramped motel rooms into seven bright and spacious studio apartments. Connor tried to put some heartfelt thanks into his farewell, but the chief just smiled and said, "Do one thing for me."
"Sure."
"When this all comes out, and you know it's bound to, stop by and let me introduce you to my wife and daughter. You'll make their day."
Connor said he would, then carried the cop's gently stated warning into the shower. He washed off the journey and fell into bed. The sea breeze filtered in through his open window. Everything should have been perfect, but Chief Wright's final words resonated through the dark room. His secret was bound to come out. Connor resigned himself to another night of tossing and turning and a few frantic dreams.
The next thing he knew, it was two o'clock in the afternoon.
Connor was well accustomed to sleeping whenever the hours were available. For everyone but stars, acting meant fourteen hour days or longer. Most actors didn't object because they had so much downtime between shoots. Every good gig was followed by weeks of auditions, meetings, story conferences, classes, and more auditions. But for the past three and a half years, Connor had been working pretty much nonstop. And then came, as the cop put it, the wedding thing.
The studio apartment had a well-appointed kitchenette, with a basket that contained filters and two packets of ground coffee. Connor was too much of a stickler about what he ate and drank to use either sweetener or powdered milk, but many location shoots offered their lesser actors nothing else, so he had grown accustomed to taking his coffee black. He dressed and extracted the wad of cash from his valise. He had stopped by two ATM machines and withdrawn the maximum before running. He entered the motel's office, paid for a week's stay, then asked the manager where he could find an all-day breakfast.
The diner was right where the manager said, on the corner of a street that meandered down the gentle slope and ended at the beachfront. Up here, a dozen or so blocks off the water, the town had an air of genteel seediness. The shops were done up nicely, but without the chichi atmosphere of wealthier towns around LA and San Francisco. Many of the structures dated back to the time when Miramar was home to the world's largest abalone fleet. Few buildings sported more than three stories, and most had a slightly faded, sea-worn air. Connor feasted on a three-egg omelet with spinach and avocado, hash browns, and fresh biscuits with honey. Then he sat in the booth, pretended to study the street beyond his window, and wondered how on earth he was going to fill his days.
He had never been good at doing nothing. Sitting around and studying his navel was definitely out. Connor had worked his entire life. At twelve, he had started as weekend kitchen help in his parents' restaurant. He had loved the work as much as he had hated his parents' constant bickering. Connor had done well enough in his classes to stay under the family radar, and spent every free moment striving to follow his own dream.
It was the dream that had brought him to LA. The dream he had managed to destroy.
That was enough to draw him from the booth. Connor had always been most comfortable when moving at full speed. Pushing ahead meant less time to look back. Looking back only brought regret. Connor paid his bill and left the diner and headed down toward the sea.
There was a great deal to like about Miramar, starting with its location. California's central coast was a rare haven, isolated by distance and winding hilly roads from the frenetic tension that dominated the urban centers. Miramar was almost exactly midway between LA and San Francisco. Both cities had seaside resorts that were closer and filled with people who valued all the things that Connor was fleeing. The result for Miramar was a seaside sanctuary whose residents were quietly satisfied to leave well enough alone.
Connor walked and breathed the salt-laced air. Gulls mewed overhead, and somewhere in the distance a buoy's bell clanged as it rocked with the waves. He was glad to find the place was as nice as he remembered. His unsettled restlessness pressed him on, down the slope, toward the sea.
CHAPTER 3
It took the worst day of her recent life for Sylvie Cassick to realize just how many friends she had.
She stood in the restaurant's kitchen, watching as her assistant chef gutted the day's fish. Porter Wright, Miramar's police chief, stood beside her. On the counter's other side was a narcotics detective from the county sheriff's department. One by one the fish were slit open, carefully inspected, then returned to the cooler.
Did people ever actually have an easy life? Was it like the magazines for some people, where they could design their goals and then watch everything slide into perfect order? Sylvie often wondered about this. She thought it might be nice at some point to meet someone like that. Then again, perhaps she might grab a kitchen cleaver and give in to a case of lethal envy.
There was no room for such thoughts, but here she was anyway, crowded into the corner of her own kitchen. As she watched the detective inspect each fish in turn, she wished she could have everything go her way, just this once.
The restaurant kitchen had once been a second dining room, and the three cooks had worked in a lean-to, where the alley now ran. The former owners had built a state-of-the-art facility, one of the few things they had done right in Sylvie's opinion. The very same owners had turned the upstairs into an illegal casino. Between hands of Texas hold 'em, they had also sealed any number of drug deals. Sylvie had known all this for years, but the detective repeated the stories as he worked. The detective's tone suggested he thought Sylvie should be down in Lompoc, doing five to ten with the former owners.
A steel counter ran the length of the kitchen, dividing the cooking area from the smaller space used by the waiters. Front and center on the shiny steel workstation was the reason behind this morning's ruckus. The eleven bags of white powder glistened in the bright overhead lights, mocking Sylvie's attempts to hold herself together.
People often said she was remarkably calm, the sort of woman who could weather any storm with a smile. Sylvie had heard the words since her teenage years, when she had been hired at age sixteen to serve as the hostess in a fine restaurant. The manager had even thought enough of her appeal to help her obtain a fake ID. These same people also liked to say that despite her magnetic beauty, Sylvie Cassick, at age thirty, possessed an ancient's eyes. Sylvie was amazed at some of the secrets people divulged to her. Making the rounds of her restaurant often drew forth some hair-raising sagas. Recently she asked a couple she had never met before why they felt they could trust her with their darkest secrets. The pair replied that something about Sylvie left them certain she had heard far worse.
Every time Sylvie looked at the eleven kilos of cocaine, she wanted to shriek, pull her hair out by the roots, and prove every single one of those people wrong.
The detective's name was Walker, and they had disliked each other on sight. Walker stood on the counter's opposite side and watched as Bruno, her assistant chef, gutted the last three fish. "Who was it that found the drugs?"
"Asked and answered," Porter Wright replied. Miramar's chief of police was ugly in every way, except the one that counted most. His face was scarred by acne or bad diet or too many hard days. His eyes were piggish and his voice was gravelly. But for some reason, Porter Wright had appointed himself Sylvie's guardian. As a result, Sylvie considered Porter one of the finest men on earth. Porter said, "The fellow gutting the fish found the dope. The restaurant's proprietor made the call."
"Why don't you take a hike, Chief. Let me speak with the lady here without you getting in the way."
"Happy to." Clearly, Porter did not think any more of the detective than Sylvie. "Soon as she either asks me to leave, or she lawyers up."
Walker was compact and tight and viewed everything with a suspicious squint. "You seem very intent on helping her hide something."
Porter shifted his bulk. "You don't want to take me on, son. Not in my town. You really don't."
Sylvie knew Porter had been a cop in San Diego. He had pulled the plug after twenty years, moved north, and taken over a department that some had called lazy and others downright crooked. A bust of this size required Porter to call in the specialists at county level. Porter had sounded like he was apologizing when he told her. Now Sylvie understood why.
Walker demanded, "Who brought you the fish?"
"Now you're getting somewhere." Porter straightened. "Why don't you and I head on down to the suppliers, and let Ms. Cassick get ready to open up."
Walker looked like he wanted to argue, but pulled a card from his pocket and set it on the counter next to the fish. "If you happen to think of anything more—"
"I'll call the chief," Sylvie replied. "Just like today."
Walker cut a dark look at Bruno, and said, "I'll be in touch."
"Long as you contact me first, you're welcome in Miramar anytime." Porter remained where he was as the detective left by the back door. "I heard about that fellow."
"He scares me."
"If Walker comes sniffing around you or any of your staff, you call me before you even tell that guy hello." Porter lifted his gaze. "Bruno, you hear what I'm saying?"
"I ain't saying another word to that guy."
"You'll have to go in and make a statement. Just make sure you don't do it alone. Now tell me about your buddy."
Bruno was from an old-time central coast family, small-boned and narrow featured. For generations, his family had run a boat hunting the abalone; but when that industry faded, Bruno's parents had migrated north to Santa Cruz in search of work. He had gotten involved with drugs as a teen, did a stint in juvie, got out, went back to the dark side, and wound up in Lompoc doing three to five. He had been out and clean for six years. "Carlos and I were best friends as kids. I can't believe he'd do that to me."
Porter asked, "He still ganged up?"
"He swore he was done with all that. But I hadn't seen him in, like, forever."
"His family still live over on Randolf?"
"That's what he said, but who knows if it's true?" Bruno looked like he was about ready to cry. "I'm so sorry, Ms. Sylvie."
Porter answered for her. "You did right, start to finish. Sylvie, walk me out."
She followed Porter through the restaurant and out to where his car was parked. From the neighboring slot, the detective impatiently beeped his horn. Porter turned his back to the man. "Santino supplies your fish, right?"
"Ever since I first opened."
"My guess is, they got a boat that slipped out of port about the same time your Bruno slit open that fish. Anything else you can tell me about Carlos?"
She had been through it all before, but talking helped. "He had an attitude. And the tats. But he worked hard."
"Always first to show up," Porter said. "Helped unload the produce, clean the fish, all that."
Sylvie put her hands to her face. She could feel a headache coming on, the sort of migraine attack that painted her entire world a grim, thunderous gray. "I've been such a total idiot."
"No, you haven't. Straighten up now, folks are watching."
"What am I going to do?"
"Same as always. Open on time. Be the gracious hostess. Make sure your guests have a good meal and take home some happy memories."
"You know people are going to hear about this. What do I tell them?"
"That you're trusting me to handle it."
"I am, you know."
"See?" He gestured her back inside. "Now go talk to your staff. Tell them I said everything's good."
"You don't want to do that yourself?"
"They need to hear it from you."
Sylvie stood there on the sidewalk, watching the man amble over and settle behind the wheel. She waved him off, just another grateful citizen showing respect to the local force. It all could have been so much worse. If Porter was not chief, if he'd had the day off, if a thousand things. Sylvie tried to remind herself of that as she passed back through her dining room and kitchen, and let herself out the rear door. The day could easily have been a total disaster.
The alley had once been a wretched place, especially on a hot afternoon like this one, when the six wheelie-bins were never really cleaned and the tight little lane stank with the residue of a thousand fish dinners. Then Rick, her senior waiter, had come up with the idea of turning a patch of crumbling concrete into an off-duty alcove. The boys had equipped it with cast-off garden furniture, a canvas overhang, and an old wagon-wheel table. They drilled holes in paint buckets and planted a variety of flowers. Now the bins were pressure-washed three times a week, and the alley remained spotless.
Rick was there now, along with Marcela and Bruno and Sandy and Carl, her chef. Sandy, her pastry cook, saw himself as the spokesman whenever the group needed to have a word. Like a lot of locals, Sandy had been born elsewhere, but was now determined to claim Miramar as his own. In Sandy's case, the elsewhere happened to be Swindon, which he assured Sylvie was a British city so ugly it won prizes on four continents. Sandy gestured to where Bruno sat, staring at the concrete by his feet, and said, "Tell the lad he did good."
"Sandy's right," Sylvie said. "I'm proud of you. And grateful."
"See there, lad? There's no way you could have known your mate was bent."
"We all make mistakes," Sylvie went on. "We've all trusted the wrong man at some point."
Sylvie stayed out there long enough for them to know the assurance was genuine. She deeply cared for them, her pack of misfits. Two had done time at Lompoc. Sandy had some past he never talked about. Marcela, the number two waiter, carried her own measure of bad memories. The five bore stains of a very hard road. But they were clean, they were good at their jobs, and they were her misfits.
Rick, Bruno, Marcela, Sandy, and Carl: Five lives brought together by careless fate. Now their afternoon delight was this tight corner of broken concrete and cast-off furniture. Sylvie wanted to tell them what it meant to be trusted with their last fragile threads of hope. But they were not the type of people who put a great deal of store in such words. Sandy would make a joke, Carl would shrug it off. The emotions would be lost. So she remained silent. When Sylvie thought it was okay, she returned inside to face the challenge of pretending this was just a normal day.
CHAPTER 4
Estelle watched Connor Larkin cross the forecourt and enter the office. She had recognized him instantly, and wondered what he was doing here. There had been too many lonely nights where the mindless infotainment shows had brought her both comfort and escape. She followed him into the lobby and stood back, pretending to read a magazine. Under other circumstances, she might have spoken to him. Just then, however, Estelle was too frightened by what she intended to do next. She heard him confirm that his name was Connor Smith and watched him pay cash for a week's stay. All this was curious indeed, since she knew for a fact that his wedding was in four days.
When he had received directions to a local diner, Estelle approached the manager and said, "I'd like to extend my stay, please."
"Let me check." The owner was Mrs. Ware, who possessed an innkeeper's firm no-nonsense attitude. "Yes, we can certainly make that happen. And I'll offer you the long-stay discount."
"Thank you."
"How much longer will you be with us?"
"Two more days, I think. I'm not sure."
"Well, let me know as soon as you can. We can get very full over the weekends." She accepted the credit card, ran it through the machine, and smiled at the otherwise-empty lobby. "That was quite the handsome young man who just left. Wonder whose heart he's going to break next."
Estelle knew the owner was only making polite conversation. At another time, she might have been tempted to reveal what she knew about the man who just paid cash for a week. However, her nerves formed a great lump in her gut, making the simple act of drawing breath an effort. She accepted her card and headed for the exit.
She stepped into the afternoon sunlight, slipped on her sunglasses, and forced herself to cross the lot and head down the sidewalk, just like a normal person out for a normal walk. She had left the room determined that today would be the day. Three times she had started out. This was the farthest she had made it. She had faced many hard events in her life. None of them seemed quite so difficult as this.
Estelle knew where she was going, since the private detective had included photographs with his report. She had gone online and mapped out her directions. She had selected that particular motel because it was within walking distance of her destination. She had everything planned, right down to the moment when she arrived. At that point, her mind simply froze. She had no idea what to say, how to act, whether she should start with an apology—nothing.
She continued slowly down the broad avenue toward the sea. Her legs felt like water. She realized she was not going to make it, so she stepped into a coffeehouse and ordered a latte. She sat in the sun, pretending to watch the street and the pedestrians. Her mind was a complete jumble, but the coffee did its work. Eventually she felt able to rise and continue on.
Estelle spotted the young actor again, seated in a diner across the street. She saw him rise to his feet and leave the restaurant and start down the other sidewalk. She found it remarkable how no one else seemed to recognize him. Maybe people here ignored the latest Hollywood gossip. Or perhaps Miramar was the sort of place where people had the decency to let even actors go about their business in peace.
The man who called himself Connor Smith meandered down the gentle slope, almost directly across the street from Estelle. He showed none of the delight or excitement she had seen recently on the cable show covering his wedding. Connor moved with the slow cadence of having nowhere to go. If he noticed the older woman watching him from the other sidewalk, he gave no sign.
Estelle thought Connor Larkin was right to run away, if that was actually what he was doing. She had seen him on any number of shows and had been drawn to Connor Larkin's characters because she was convinced he was different. He was a man who cared. He was, at heart, a good man. Estelle was certain Connor's fiancée was anything but those things. Kali Lyndon was a vapid woman, a celebrity queen who financed her passion for the spotlight with an inherited fortune. She had never done anything productive in her entire life, as far as Estelle knew. Kali was beautiful in a Photoshopped sort of way, and no doubt she would do wonders for Connor's career. Kali Lyndon, however, was wrong for him.
Estelle halted the mental prattle because she had arrived. She had been so involved in her internal dialogue she had not even noticed where she was. All the fears rushed back in a tsunami that locked her up tight. She could not breathe, much less force herself to cross the street. She steadied herself with one hand on the nearest parking meter and tried to tell herself that this was why she had come. This was why she had hired the detective and gone through nights of soul-searching and flown across the country and . . .
Then she saw Connor Larkin step through the entrance.
From across the street, Estelle heard strands of music from her past. Frank Sinatra sang "Fly Me to the Moon," a melody that pierced her as sharply as a blade through her heart.
Estelle had no choice but to admit defeat and turn away.
She was wrong to have come. She should have left the past buried and at least half-forgotten.
CHAPTER 5
Connor drifted down the sidewalk, still locked in memories about his early days. His mother had returned to her place by the cash register while Connor was in his bassinet. The restaurant's staff had all known Connor's real age when he started working alongside his father in the kitchen. Connor had learned early and well the lessons of haute cuisine. But beneath the restaurant's polish and the money, the family business had radiated a bitter unhappiness. Finally the year he had turned nineteen, it all fell apart. By then, Connor was already traveling the roads, chasing his own dreams.
Then an all-too familiar melody drifted through an open doorway, halting Connor in his tracks.
The restaurant he faced was called Castaways. The building itself appeared to be the oldest structure on the block, a two-story affair with the tall false front of bygone eras. Cast-iron tables and chairs flanked its broad front entrance. The long windows were embellished with flower boxes. The wood was varnished; the glass was polished; the brass handles to the open French doors gleamed. Connor thought the understated elegance suggested a woman's hand at the tiller.
Connor stepped through the open doorway, and fell in love.
* * *
Sylvie walked to the bar and opened the cabinet that held the restaurant's sound system. She put one of her father's favorite albums on the turntable, and stood there listening with her eyes shut. She had often done this in the restaurant's early days, when it was all too much and she thought she would never learn everything required to keep the place up and running.
Her father had been a gentle dreamer. The fact that so few of his dreams ever came true had only deepened his joy when one was realized. He had always called Sylvie the marvel that made every day complete. Even when she was bad, Gareth Cassick insisted that he was blind to all but the delight his daughter brought him. There had been some hard years for them both, for Sylvie was her mother's child as well. That woman had left them when Sylvie was twelve. Sylvie's clearest memories of her mother were mostly strident demands for her father to grow up, support the family, make something of himself, and do a job that mattered. Then her mother was gone, and for a time, everything was good. If not good, at least life had been a great deal quieter. Sylvie's teenage years had brought out a bitter edge to Sylvie's character. She had never been a screamer. Her early years had seen too much of that. Instead, she had developed an acrid sharpness. She could wound with a single word. Her father had simply accepted her tart rage as his due, and loved her through those years.
Gareth would have adored this place, she knew. The light fell through the west-facing windows, painting the old wood with a honeyed gleam. The sea peeked between the roofs like a scattering of blue gemstones. Sandy was in back, doing his magic with pastry and spices. The air carried the gentle incense of promise and good times for all.
Her father had loved the crooners of the fifties and sixties. When she pounded the walls of their home with arena rock, he listened politely. When she condemned his own music, he simply said it was an acquired taste. And acquire it she had, especially on days like this, when she would have given anything to hear his gentle optimism, and have him assure Sylvie that better times were just around the bend. Not yet within sight, but coming to them both . . . and very soon indeed.
She whispered to the empty room, "Pop, I could sure use one of your hugs just now."
* * *
The man who entered her restaurant was too handsome for his own good. Sylvie started to say they weren't open yet and could he return in a couple of hours. Something held her back, though. He fumbled the sunglasses from his face, like his fingers weren't working well. He stood just inside the entrance, where the sun struck him fully, and gave her place a very slow sweep. The light played through his shoulder-length hair, not blond and not brown, but a mixture of both. His face was lean and strong without being cavernous. His hands were what her father would have called cowhand big; they fit well to strong arms and broad shoulders, atop a waist that was almost too narrow. He wore a simple T-shirt and stonewashed jeans and canvas boots. He was, in a word, gorgeous.
Then she saw the tears. His gaze was so clouded she wasn't even certain he could see her approach. Sylvie saw no need to ask if he was all right. Clearly, he was far from okay. In fact, what she actually thought was, this guy's day might have been as bad as her own.
Sylvie asked, "What is it?"
Of all the ways he might have replied, all the words to emerge from that stylish two-day growth, Sylvie would have never expected to hear what he said. Not in a hundred billion years.
The guy's voice was deep and resonant and downright beautiful. He said, "Bart Howard wrote that song in 1954. He was interviewed after it hit number one for six different artists. He said he had trained for twenty years so he could write it in twenty minutes."
Sylvie said weakly, "Excuse me?"
"Originally Howard titled it, 'In Other Words,' " the guy said. "The singer who had been buying Howard's work considered it too cabaret in style and refused to use it. Then Kaye Ballard recorded it."
"On Decca," Sylvie said. "As the flipside to 'Lazy Afternoon. ' "
"Right. Decca wanted Howard to change the lyrics. He fought them, Kaye backed him up, and the song went out like he wanted."
Sylvie said, "Then Peggy Lee recorded it. Nat King Cole. Sarah Vaughan. Brenda Lee. Connie Francis."
The guy said, "Then came 1964 and the Apollo moon program, and Frank Sinatra put it on the album . . ."
"It Might as Well Be Swing," Sylvie said. "What we're hearing."
"Quincy Jones did the arrangement," the guy said. "Count Basie's orchestra backed him."
Sylvie asked, "Why does that make you sad?"
He almost managed a smile. "All that stuff, it used to be really important."
"It still is," Sylvie said.
He looked at her. Really looked. "Can I please have a job?"
CHAPTER 6
When Connor left the restaurant, he headed downhill. He felt impacted at gut level by the woman he had just left. Her image was so clear, she might as well have walked alongside him. Sylvie Cassick held herself with the grace of a former ballerina, slender and balanced and impossibly erect. Her hair was a honeyed trace off pure black, like smoke blowing through a night-clad sky. Her eyes were the gray of morning mist, light and soft and brilliant. Connor put her age at somewhere around thirty, the same as himself or perhaps a year or so younger. Still, there was a timeless quality to the woman, as though she had managed to compress centuries of life into months. She held none of the electric beauty that defined the current Hollywood fashion. Her cheekbones were too pronounced, her gaze too direct, her lips too full. Connor thought she was the loveliest woman he had ever seen.
Not that it should have mattered. At least, not to him.
Miramar's finest shops were clustered along an avenue two blocks off the seafront. The pedestrian area had tiled avenues with elevated concrete basins holding flowers and blooming trees. There were benches for the elderly and a woman in a tuxedo making balloon animals for the children. There was a cupcake factory and an upscale café and shops touting local vineyards. The streetlights were modeled after Parisian gaslights. The entire area covered three blocks, and then Connor was back into the world of well-tended houses on small lots. The sea's presence was strong here, a brisk mingling of salt and seaweed and biting wind. Connor stopped by a tree-lined park and watched the children play. He was falling for the idea that a place like Miramar might truly exist. And even if it didn't, even if his first impressions were nothing more than whispers of dreams long buried, Connor could think of nothing he would like more than to spend his life living the myth.
He had a purpose now, and he was on the clock. He would enter into this new role by using the talents he had developed in Los Angeles. There was no alternative, not if he wanted to blend in and remain unidentified. Which he did. Desperately.
Connor's first big break had come on CSI. They had liked his work so much the producers had offered him roles in the two spin-offs—but only if he could change his appearance so that viewers who had seen him get shot on the first show wouldn't realize he was back from the dead. Connor had worked with a professional stylist who had been in film and television for years. She had taught him a lesson that defined the very best character actors, the people who shifted not just roles, but personalities. She had shown him how no amount of face paint or costumes or cosmetic alterations would help most actors because they were too rigidly defined within themselves. If a performer could learn flexibility at the deepest level of his personality, then the exterior changes merely amplified this shift.
By the time Connor found the shop he was looking for, he had defined his new role.
He wanted a new beginning. What was more, he wanted to deserve it.
* * *
Connor entered the men's store and bought two pairs of black gabardine slacks, a white knit shirt, two long-sleeve white-on-white dress shirts, a mock alligator belt, and black loafers. He paid central coast prices, not Beverly Hills, but the purchases still put a sizeable dent in his cash.
He asked the saleslady for the best hairstylist in Miramar, and she directed him to the corner where the shopping lane met the main road. He entered and asked for whichever stylist could cut him without delay. A young woman with flashing dark eyes and an overtight skirt introduced herself as Lucia and led him back. "What are we after today?"
Connor held his thumb and forefinger about an inch apart. "Trim it down to this length, and part on the left side."
"All this lovely hair." She took a fistful and lifted it away from his neck. "You sure you want me to cut if all off?"
"I have to look respectable," he explained.
"Good luck with that." Lucia smirked at his reflection. "You got 'bad boy' written all over you."
"I'm starting a new job." He checked his watch. "In less than two hours."
"Guess we better get started, then." Lucia led him to the basin. As she shampooed his hair, she said, "Bet it was a woman, she hired you."
Connor smiled. "Guilty."
"You see? That lady, she's not out to respect you. She wants you to walk in, raise the temperature in her office."
"Restaurant," Connor corrected.
"Yeah? Which one?"
"Castaways."
"Oh, that's good. Sylvie is one smart lady."
"You know her?"
"Everybody knows Sylvie. She took a bad place and turned it right around." Lucia toweled him off and led him back to the chair. "Last chance."
"Do it." His hair was darker when wet, the color of sourwood honey in sunlight. He watched her snip away, and hoped it would prove this easy to make the internal shift. "What can you tell me about my new boss?"
"She's got a past. I'm not saying I know what it is—and even if I did, I probably wouldn't tell. But the lady, she's what, thirty? Thirty-one?"
"Something like that, I guess."
"She's old for her age. My grandmother would say her soul's been around for a long time."
Connor thought back to the conversation about music, and the way she had watched him struggle for control. "Have you ever eaten there?"
"Oh, sure. Last time, let's see, it was for my birthday, not the last one, the one before. The food's really good. Like, French."
As Connor watched his new persona take shape in the mirror, his mind returned to that first moment in the restaurant. He shivered.
Lucia noticed and said, "I know, they keep it like an icebox in here. You want, I could ask them to turn the temperature up a notch."
"Thanks, I'm good." He had left LA driven by restless longings for some small fragment of his old passions. That first moment in Castaways had crystalized everything. It was far more than a wonderful conversation with a lovely woman. It opened a door he had thought locked and barred forever.
Connor Larkin had been born to sing, or so he had thought growing up. But obviously he had been mistaken. Nowadays a growing tsunami of fans insisted his real purpose in life was to die.
During his teens, Connor had learned to play piano because he couldn't find anyone to accompany him. All his friends had thought the crooners of the fifties and sixties should be left in the grave. Connor, however, loved the soft melodies and the big bands and the polished sound. Dean Martin was his favorite, but he had learned to sing them all. Frank, Sammy, Tony, the Elvis ballads—an entire lost generation of silky-smooth voices. By age sixteen, Connor had felt destined to join their ranks. Two months before his twentieth birthday, his ambitions took him to LA.
Connor soon had all the gigs he could handle, so long as he played to rooms of drunks and clinking glasses and brassy laughter and fat guys shouting at waiters. Connor had wanted more, though. He wanted the lights. He wanted . . .
Connor had been singing at the silver wedding anniversary of a Hollywood producer, when an agent approached and introduced herself as Ami Chen. She was with CPP, one of the three agencies that had a dead-solid lock on the Hollywood A-list. Only Ami Chen did not work with singers. She handled stars.
Ami liked his voice, but she loved his looks. Connor was rugged and handsome and rakish. According to Ami, the cameras would eat him alive.
Ami arranged for him to take acting lessons between his singing gigs. When she deemed him ready, Ami sent him out to casting calls. By this point, the acting was far more than some sideline activity. Connor had already endured four and a half bruising years making the rounds of music producers and getting squeezed from all sides. LA was filled with wannabees who shone in momentary spotlights, but dwelled in the city's bitter shadows. His gigs paid the rent and bought him a secondhand car, but he basically lived hand to mouth. His four flatmates were all one bad week away from the bus ride back to Iowa or Nebraska or wherever they shipped their broken dreams. The back roads of Hollywood were packed with brittle laughter and tense desperation. Connor was far from the only handsome young face who forgot why he had made the trip west.
Then to his astonishment, Connor was offered an acting gig.
At first, Connor played the guy in the soaps who got it in episode three, after trying to take off with the money and the hero's sweetheart. Even such a small advancement drew bitter envy from his flatmates. Connor had leapt from nowhere straight into speaking roles. When he was hired for a third gig, Connor had no choice but to move out.
Then the medical shows started offering him regular work. As the casting directors put it, Connor made croaking look good.
Then he did a Syfy drama playing a thief of technical secrets who deserved to get it in the end, and did. These were followed by the three CSI shows. That was when the online cult thing started.
When his agent first got wind of his following, neither of them believed it was real. Then Ami started fielding calls from abroad.
Connor did his death-throes in six Mexican telenovelas. He became the most popular American bad guy on Japanese television. He was devoured by wild beasts in India. Twice. He was attacked by a killer scorpion in Australia. He got bitten by a viper in Brazil. When he came back to the States, it was to play the evil genius in that year's e-game smash hit. By that point, Connor Larkin was dying as often as once a week.
Connor was still growing accustomed to the idea that people actually liked watching him get eaten by fickle fate, when he was invited to appear at the Las Vegas Comic Con. The convention organizers informed Ami that Connor's online following fit their demographics. They flew him out first class. He was met at the airport by a limo, given the penthouse suite at Caesars, and paid ten thousand dollars for a two-day gig.
When Connor showed up, he was mobbed.
Three days after he returned to LA, Connor bought a home in the Beverly Flats. He finally accepted what his agent had been saying for almost a year. This was no longer a string of gigs. Connor Larkin had a new career.
* * *
Lucia worked swiftly. She finished by trimming his neck; then she blew his hair dry. Her inspection grew increasingly serious as she gave his length a final check. "You want something in there?"
"Wax, please."
She opened the vial, rubbed it over both palms, her black eyes flashing a warning now. She worked it into his hair, then combed it into place. "I guess you clean up pretty good."
"You're an artist."
Lucia trimmed a final wayward strand; then she waved the scissors in his face. "Don't you go breaking Sylvie's heart, you hear what I'm saying? Else there's gonna be a lot of folks around here, looking to trim more than your hair."
Connor did not say what he thought, which was that the last thing he wanted from this time and place was a woman. Not a casual fling, which was clearly what they were all concerned about, and especially not a serious romance. The very idea filled him with grim humor. Connor thought of all the reasons he had to remain single as he climbed the road back to his little apartment. All the tales he could tell Lucia of other flashing dark eyes.
The problem with bad romance was that Connor's story had not started with Kali Lyndon. She was merely the outcome. Connor had become very adept at clamping down on the hollow ache that such thoughts brought forth. However, no matter how fast he walked the broad avenue, he could not escape the sense of being convicted. Lucia's warning had hit home as fast as a punch on an open wound.
When he passed through the parking area adjoining the studio apartments, Connor spotted the woman he had seen earlier in the office. She was seated in a little fenced area he had not noticed until then, a miniature garden with a couple of shade trees and a Jacuzzi and a small kiddie pool. Connor thought the woman held herself oddly, as though some severe pain kept her slightly hunched to her left.
He started to let himself into his room, but instead he unlocked the door and set down his shopping bags and turned back. The LA habits were part of what had to go. They didn't fit this new role. Connor walked back over and asked, "Excuse me, ma'am. Is everything okay?"
Large sunglasses covered a considerable portion of her face, but Connor had the impression that she came slowly back to reality. She said weakly, "I saw you."
"Excuse me?"
"On the street. You walked into that restaurant."
"Castaways. Right." He pointed back to his open door. "I'm due to start work there in about twenty minutes."
The news twisted up her face momentarily. All she said, though, was "That's nice."
"Ma'am, can I get you anything?"
"I'm . . ." She flicked the fingers of one hand, shooing him away. "I'm right where I deserve to be."
Connor had the distinct impression she had dismissed him. "Well, I'm over there if you need anything."
He crossed the lot, picked up his bags, and entered his room. He showered and shaved, then dressed carefully and returned to the bathroom's full-length mirror. He held himself there, practicing the same technique that had taken him through so many other roles. Clamping down on all the regrets and all thoughts of who he was and who he was not. There was no tomorrow, not for Connor Larkin. Right now there was only this new role. It grew from the tight clump of pain and tension and everything else he would not let himself feel, because just then it belonged to another person. He grew the role from within, by reshaping these emotions and the life energy they represented.
When he was ready, Connor said to his reflection, "This is important. Make it count."
Then he turned to the door, like he had a thousand times before. On set after set, an usher hovered just outside his dressing room door, the director waited with the final words of guidance, then he stepped to where the lights gleamed and the cameras aimed and the people watched.
The big difference this time was that Connor only had one take to get it right.
CHAPTER 7
Sylvie continued with her preparations for what needed to appear as just another ordinary day. Being here in her restaurant, among the people she cared for most in the world, certainly helped. She patrolled all twenty-three tables, adjusting a fork here and a goblet there. She spoke with Gustavo, the new busboy. She phoned Porter Wright, whom Connor had said might be willing to vouch for him. She entered the kitchen and checked the vegetables. She even managed a joke over how the fish did not need any further scrutiny. She discussed the night's specials with Carl, her chef.
The kitchen staff loved these moments, for Carl was the most taciturn man she had ever met. He could go through the most frantic nights, when everyone else was shouting and rushing about, and not speak anything other than "pickup." Plus he would only say that if the ringing of his bell did not bring the waiter running. The kitchen crew loved speaking for Carl, making suggestions about possible concoctions, trying to come up with something outrageous enough to make him object. Her worries diminished in volume. Sylvie was ready to do what she was best at: giving her friends and patrons a memorable evening out.
When she returned to the dining room, Rick was stationed behind the bar, serving their first three customers. Sylvie waited down by the waiters' station. When he joined her, she asked, "Where's Aubrey?"
"On her way. Her mother's had a bad day."
Aubrey was the best bartender Sylvie had ever employed. Her mother, however, was quietly slipping into a fog denser than anything that had come rolling off the Pacific. Aubrey refused to even discuss putting her mother into a home. She had arranged for neighbors to serve as caregivers. Aubrey's system worked fairly well, unless there was a bad day. Such bad days happened with increasing frequency. Sylvie said, "Join us in the kitchen for a second."
When they were all together, she announced a new waiter was starting that night. His name was Connor Smith, and he had come recommended by Porter Wright.
Rick was the one to ask the inevitable. "Is he a con?"
"Porter did not think so."
"He doesn't think?"
"That's what he said." Sylvie pointed to where Carl was filling bowls with his all-day stew. "Go have your dinners. I'll handle the bar. When you're done, somebody bring me a sandwich."
The bar was one of the aspects she most loved about her place. Castaways had originally been built as a saloon and rooming house. The owner had been a retired ship's captain, and had positioned the establishment so he could have an uninterrupted view of the sea. Now, even with so many irregular rooftops segmenting the horizon, it was still a lovely place from which to watch the sunset. The captain had ordered a set of great bay windows to be built facing west, like a landlocked version of the stern windows that would have adorned his private cabin. About half of the original panes were intact. The handblown glass threw wavy patterns of light over the varnished floor and the walnut wainscoting.
The bar ran down the front half of the north wall, curving out slightly like a ship's bow. The waiters' station was a broad space by the eastern corner, separated from the rest of the bar by brass rails. Two paces farther east opened the double doors leading to and from the kitchen. The bar was original, as was the brass footrail. Sylvie had found the four spittoons in a local garage sale.
Half an hour later, Marcela returned and brought Rick with her. The restaurant's headwaiter was tall, lean, and carried himself with a deceptive calm. Sylvie was one of the few people on earth who knew of Rick's carefully hidden past. Most people saw what Rick wanted them to see, a man who walked through life with bulletproof ease, utterly content to be both alone and untouched by the trauma of close relationships.
Marcela set the plate with Sylvie's sandwich on the counter and asked, "This is really all you want?"
On days when he came in early, Carl put a beef-bone stew on low heat and let it simmer for hours. The result was a concoction that had carried Sylvie through many a long night. "Have him put a bowl to one side. Right now, this is fine." She could feel the headache building, still far enough away that there was no actual pain, but her stomach rebelled against the very thought of food. Sylvie ran through what she knew about their new waiter, which was almost nothing.
Rick and Marcela gathered by the bar's serving station, leaning in close enough for their conversation not to be overheard by the growing number of early customers. Marcela said, "That's not much to go on."
"I know. But there was something about him. . . ." Sylvie tried to stow away a sudden smile, and failed. "He likes Sinatra."
"And you like him," Rick said.
"I did. Yes. Why exactly, I can't say."
Rick asked, "He said he was raised in a restaurant?"
"His earliest memories are of his family's place," Sylvie confirmed.
Marcela asked, "And Porter vouched for him?"
"Sort of." Sylvie related her rather curious conversation with the police chief. "He hesitated at first. Then he said that he thought it would make for a good fit. At least for a while."
"Meaning the guy isn't a keeper," Marcela said.
"To tell the truth, I have no idea what Porter meant."
Rick said, "We need another set of hands. Desperately."
"He can't be much worse than Carlos," Marcela said.
"Carlos was hidden away in the kitchen," Rick pointed out. "We're talking about putting Connor Smith on public display."
Sylvie was about to respond when she spotted a sharp-looking guy heading down the opposite sidewalk. The change from the man she had met earlier was so drastic that for an instant she thought she might have it wrong. Connor now wore a proper waiter's white-on-black outfit, except that his shirt had an extra button undone, something she would never have permitted on anyone else. On him, the opening formed a V that accented the triangular shape of his upper body. "Wait. Here he comes now."
Marcela watched him cross the street. "That's him?"
"Yes. Connor Smith."
"You didn't say nothing about him being so hot."
"I didn't notice."
"Then you are one sick lady," Marcela declared.
Sylvie noticed Rick's studied frown, but there wasn't time to ask what troubled him, because two customers were trying to get her attention from farther down the bar. "In that case, you can show him the ropes. Rick, you'll need to seat any early diners. Call Aubrey and tell her we need her on deck pronto."
* * *
Estelle walked along the headland that marked the southern boundary of Miramar Bay. The cliff was part of a seaside park and topped by a triangular meadow. A sunset breeze flattened the grass into a shimmering gold-green plate. The drop-off was marked by a rusting fence and danger signs that clattered in the wind. She walked slowly and wished she was capable of appreciating the beauty that surrounded her.
It was only when she started back toward her car that she noticed the structure nestled within a cluster of stunted pines. The branches were fashioned into a fan-shaped windbreak, and within their shadows stood a weather-beaten shack. What appeared to be a grey stone sculpture stood between the hut and the sea. Then she realized the stone was a podium, and the shack was a chapel. A glass-fronted board was attached to the rear wall, detailing sunrise services twice each month. But this evening the place was hers alone.
Estelle settled into a rear pew and bundled herself more deeply into her coat. The shack was open at the front, so that beyond the dais stretched a glorious view of the dusk-clad Pacific. The wind carried a searing bite. The trees sang a hushed melody to the end of day. For the first time since she had arrived in Miramar, Estelle felt a hint of rightness to her journey.
Her Jack would have loved this chapel. Estelle had buried her husband nine months ago. Jack had been the one who urged her to make this trip. If she had known how difficult it would be, how she felt strangled by all the conflicting emotions, she would have refused his dying request. But here she was, and this lonely chapel at the edge of the world was as fine a place as any to remember why she had come.
When the sun finally settled into the blue-gold waters, and the gulls cried a final farewell to the day's end, Estelle spoke the words aloud. "I need help."
It was not much as prayers went. But it would have to do.
CHAPTER 8
Marcela was slender and energetic, with mocha-colored skin and an abundance of soft, dark curls. Her large eyes seemed to be perpetually seeking a reason to laugh. She had been born in Encinitas, but raised mostly in Sacramento. She was married to an electrician who worked for the local cable company. She loved Sylvie and enjoyed her work so much she and her husband were putting off having kids. All this emerged as she showed Connor his station—the restaurant's worst six tables, lining the back wall, away from the windows. Then she led him through the kitchen doors, two of them, spaced well apart. One opened inward, the other out. Connor's mother had always said this was the first mark of a good restaurant, for there would be times when frantic staff forgot to look before rushing through. None of the kitchen staff even glanced his way as Marcela made a loud introduction.
Marcela led him back into the dining room and around the bar. She showed him the alcove holding the glasses and silverware and coffeemaker and spice bottles. Then an older man came over and Marcela introduced him as the headwaiter, Rick. Connor knew he needed to pay careful attention, that every word and every moment was intended as a test. But just then, his attention was captured by the stage.
Big bay windows, taller than Connor, opened onto the fading sunset. The sea glistened between the rooftops. Gulls wheeled through the sky overhead. The space between the bar and the front windows held a long table with captain's chairs, clearly intended to serve as either a spillover for the bar crowd or to host a single large party. Beyond that, fit snug in front of the bay windows, was a small stage. The left wall contained an alcove shaped like a giant abalone shell, beautifully restored. Tucked inside was a baby grand piano, a Steinway.
Marcela asked, "Something wrong?"
The piano's varnish gleamed ruby-dark in the dusk. Connor said, "I used to play."
Marcela told him, "Back in the day, dance hall girls used to toss their skirts and do their high step up there."
When Connor did not respond, Rick asked, "That makes you sad, seeing the piano?"
"A little." He forced himself to turn away. They were both watching him. The headwaiter was an inch or so shorter than Connor and very narrow. Pronounced cheekbones framed eyes that pierced deep. Connor said, "Music was a dream I let go of."
Rick smiled tightly, as though he understood far more than Connor was saying. "When was that?"
"I haven't touched the keys in seven years." He swallowed hard. "Long enough that I can almost forget the dream ever existed."
Rick and Marcela exchanged a look; then Rick said, "See if the kitchen can use his help." As Marcela led him away, Rick patted his shoulder and said, "Welcome to Castaways."
There was no reason on earth such quietly spoken words would touch Connor's heart. None at all.
* * *
No one spoke to Connor when Marcela left him in the kitchen. Connor knew his tables would be the last to receive customers. On a quiet evening in the middle of the week, he might as well spend it all right here. Being ignored.
A burly red-haired man with a British accent pointed him to a steel bowl filled with fresh spring vegetables and told him to julienne the lot.
Connor waited to see if any utensils would be offered, but no one even looked his way. He gave a mental shrug and asked, "Any chance of an apron?"
The Brit had pecan pies in the ovens separating the pastry station from the main cooking area. The smell of bourbon vanilla was strong. He gestured with his spatula toward a hall at the kitchen's other side. "Pantry."
Connor crossed the kitchen, followed by four sets of eyes. He had no problem with the silence. The stubby hall ended at a steel door, which was propped open, revealing an alley that Connor thought looked remarkably neat. The air coming in was pleasantly cool. A young man with dark hair and tats running down both arms was seated on a cane chair by the entrance, smoking and staring at the gathering dusk. Connor assumed it was the busboy waiting his call to duty.
When Connor started to open the pantry door, he went still for the second time that evening. Directly above the doorway was a segment of varnished wood, on which was branded the message LAST CHANCE SALOON.
Connor felt like he had just collided with a train.
He had no idea how long he stood there. After a time, he felt a hand upon his shoulder. Connor turned to find the red-haired man standing beside him, smiling gently. The Brit asked, "You done time, lad?"
"That's not . . . I haven't. No."
"Ah, well, it's not only inside the cage where the wee dark hours grab hold. You just come along with me."
Connor found himself being led back to the vegetable and garnish station, only this time everyone watched him openly. The Brit introduced himself as Sandy and brought over a paring knife, a sharpening stave, a peeler, and the apron Connor forgot to get for himself. Sandy said, "You know what these are for, lad?"
"Yes."
"Here's a secret you can take to heart. Sylvie doesn't like blood in her veggies. Doesn't make the right impression. So you just wash this lot and hold off until you're certain you won't go peeling any fingers."
"I'm okay."
"Sure, now, of course, you are." Sandy walked back over to the oven and checked his pies. Bruno, a slender dark-haired young man, kept stirring his sauces and preparing plates for the evening rush. Carl, the chef, laid out an array of meats and fish. The computer monitor above the main stove chimed and lit up with the night's first order. Sandy pulled a tray of bread from the warming compartment and slipped it into the main oven. Everybody moved into the next forward gear. All except Connor. He was still getting used to the fact that being slammed at heart level by a simple wooden plaque had somehow earned him a place.
* * *
When his hands steadied, Connor started on the vegetables. No doubt the restaurant had a specialist machine hidden somewhere that normally did this job. However, the chore was a valid way of checking out Connor's kitchen creds, for julienne vegetables were not something normally seen outside of high-end restaurants.
Julienne actually referred to the knife he was using, as well as the vegetables themselves. "Julienne" meant to cut into identical straight segments, similar to large matchsticks. When fried, julienne potatoes were often referred to as shoestrings. The term had been in use for over three hundred years, and first referred to a soup composed of vegetables cut into uniform segments—carrots, beets, leeks, and celery. Added to this would be minced lettuce and sorrel, and onions cut into thin triangular slices. The vegetables were briefly sautéed, and then simmered in chicken stock. Connor could not actually remember learning these things. He had simply known them all his life. Then he had forgotten almost everything.
It was far too simple to say an actor's existence required his single-minded focus, although to a certain extent it was true. Connor maintained a very discerning palate, even when it came to ordering takeout. Los Angeles was a very eclectic city, and the possibilities for ordering in food, or for going out, were endless. Every section of the globe was represented in the restaurants and delis and mobile kitchens. None of this was why he almost never cooked, or spent more than the odd moment reflecting on his family's heritage.
Connor dried the vegetables, peeled the potatoes and carrots, halved the peppers and discarded the seeds; then he trimmed the ends off the zucchini and cucumbers. He cut most of the vegetables into three-inch segments and four-sided blocks. He sliced them lengthwise, fashioning long matchsticks about an eighth of an inch thick. The red and green peppers were trimmed to make matching segments.
His hands stayed busy, which meant his mind was free to roam. He could still visualize the block of wood and the words burned into its surface. They opened the portal to all his carefully repressed memories. Connor had let go of so much from his early life. His villa in Beverly Flats was equipped with a granite and stainless-steel kitchen that had cost over a hundred thousand dollars, which he could not actually afford. Connor rarely even used the microwave. He had bought a used Bösendorfer grand piano, but then shifted it behind the plants in his glassed-in veranda, where it could be both present and hidden. Just like most everything else from his past life.
The Castaways kitchen reached a natural pause at a quarter past eight. The first tables were busy with dessert, the later arrivals had their starters and their wine, and Sandy passed around white ceramic mugs of fresh-brewed coffee and portions of a delicate pastry that contained fresh goat cheese whipped to froth and baked with tiny flakes of Spanish ham. Bruno, the assistant chef, introduced himself to Connor and asked about his family restaurant. Connor found himself trying to explain just how impossible a moment like this would have been, because his parents would have been threatening each other with dire bodily harm. No one spoke for a long moment; then Marcela chose that moment to rush into the kitchen with a new order. Sandy pointed a thumb at Connor and declared, "You want my take, he's a keeper, this one."
But the waitress was too busy for Sandy's banter. "We don't have much choice, at least for tonight. Rick has two tables just arrived, my section's almost full, and Sylvie's had one of her attacks."
"No surprise, given the day she's had," Sandy said.
Connor asked, "Attack?"
"Migraine," Marcela said, reading her order notes, typing furiously, and talking at the same time. "Big one."
"And it's all my fault," Bruno said morosely.
"We've covered that ground already," Sandy said. "So stuff it, that's a good lad."
Marcela said to Connor, "Any more customers show up, we're opening your station."
The kitchen accelerated for the second time that evening. Connor was shifted to garnishes; then he helped out wherever an extra pair of hands was required. He prepped dishes for the washer; he stirred soups; he rolled out extra bread. He shucked four dozen clams. Connor regularly made mistakes. Everything took him too long. The rest of the kitchen staff accelerated to a pace he had lost; but the others evidently saw he was trying, so they did not come down on him too harshly.
Sylvie entered the kitchen and walked over to where he scored a crosshatch pattern into six filets of sea bass, then basted the surface with a sauce of rarified butter and chicken stock and white wine, prior to them being flash baked. She said, "Sorry to drop you in the deep end, but Rick and Marcela are super busy, and we have a table of eight that just walked in."
Connor saw how she squinted against the kitchen's overhead lighting, but no one else said anything. Connor replied, "It's no problem."
"Actually, it is," Sylvie said.
Sandy stopped preparing dough for choux pastries. "Not that Hammond git."
"Enough of that," Sylvie said.
"Phil Hammond's got an assassin's charm, I'll give him that much, but he never leaves the hired help a brass farthing." Sandy pounded the dough, shooting up a cloud of flour. "Great git."
Sylvie tried her best to offer Connor a smile. "Consider this your trial by fire."
CHAPTER 9
There were certain elements of stardom that Connor yearned for. Part of it had to do with money, of course. Connor had been broke since his first day in LA. The rise of his acting career had simply resulted in Connor digging himself farther into the debt hole. Even so, if money and fame were the only draw, Connor would be down on Rodeo Drive, enduring the final fitting of his tux and showing off smiles and excitement for the cameras.
What interested Connor most was the chance to design his role.
All supporting actors were basically there to bolster the star and propel the story. They played the foil, the love interest, the villain, the fiend, whatever. Their screen presence was dictated by the lead character. In television, where the pace was constantly frenetic, secondary characters were expected to know their lines, show up early, hit the mark, and bow out. The word that defined most of the roles Connor played was, straitjacket.
Connor heard the party of eight before he rounded the bar. Five men and two women were playing to the older man at the head of the table. Phil Hammond looked to be in his early sixties and flashed the easy smile of a man well used to center stage. He was in the middle of a story when Connor approached. The seven in supporting roles took their lead from the silver-haired guy and pretended not to notice Connor's arrival. Connor showed them the easy smile of a journeyman actor, and waited as they laughed over a joke he had not heard. Connor kept telling himself that his job was the same as usual, play up to the star.
Phil Hammond was a carefully groomed silver fox. He wore a starched dress shirt open at the neck, gold-and-emerald cuff links, perfect tan, polished nails. He took his time studying Connor, then asked, "Do I know you?"
"It's an honor to serve you, Mr. Hammond. Especially my first night on the job."
"Oh, look," the youngest member of the group said. "Fresh meat."
Hammond smiled the young staffer into silence, then demanded, "Why isn't my girl over here?"
"Ms. Cassick is unwell, sir. What can I bring you, gentlemen?"
"A single malt for me. Rocks." He beamed at the group. "And more water for the horses."
Connor laughed because it was expected of him.
Phil Hammond showed the world a regal graciousness. However, whatever lurked beneath Phil's public mask was enough to keep the seven others fearfully attentive. When Connor brought their main courses, Phil politely asked him to bring him another steak, this one cooked as he had requested, medium rare instead of medium well. Carl made no comment as he put another filet on the grill. Sandy made do with a single muttered comment about the old git. Connor did not respond. He had worked on numerous sets where either the lead actor or the director would have left Phil Hammond in their egotistical dust.
As Connor reentered the dining room, Marcela asked him to stop by the front station when he had a moment. Connor served the plate, then stood motionless until Hammond took a bite and smiled his approval.
Rick was manning the entry when he arrived. Connor asked, "Is Sylvie okay?"
"She will be. Eventually." Rick pointed at the ceiling. "She lives in the apartment upstairs. She's lying down."
Marcela joined them and scowled at Hammond's table. "Old Phil won't leave you a dime."
"Sandy warned me."
"The one nice thing about Sylvie's migraine is she doesn't have to pretend to enjoy old Phil's company," Marcela said.
Rick said, "Just because he stiffs waiters in his own restaurant doesn't mean he's a bad guy." He added to Connor, "He owns a third of this place."
Marcela said, "He's a snake."
"You don't know that."
"I know, all right," Marcela replied. "I just don't know why I know."
Rick asked Connor, "Does that make any sense to you?"
Connor asked, "Doesn't Sylvie have medicine?"
"She took it hours ago," Marcela said. "She says it masks the worst symptoms, but it doesn't make them go away."
Rick told Connor, "Porter Wright just called. The chief wants a late table. He asked for you."
Marcela said, "I can take him if you like."
"No, it's fine. I owe the guy."
Rick said, "Porter waits until the night quiets down and his wife's off duty."
Marcela said, "Carol is a nurse. She's a great lady."
Rick said, "They usually close the place down."
"I don't mind staying late," Connor said.
Marcela said to Rick, "Give the guy my best table. He's earned it."
* * *
Phil's group became steadily louder as the night progressed. By the time Porter Wright arrived, their laughter and shouts punctuated the entire restaurant.
Porter wore his jacket and tie like it belonged to somebody else's wardrobe. His wife was a sharp contrast to the Miramar cop. Carol Wright was athletic and poised and intensely handsome. Her silvery gray hair was cut to draw attention to her striking features. She wore no makeup. She and her husband stopped by one table after another, shaking hands and sharing words. Porter repeatedly frowned at the group beyond the bar. He waited until his wife was seated, then walked over.
The instant Porter appeared around the bar, Hammond went quiet. Porter leaned over, planted one hand on the back of Hammond's chair, and spoke softly. Connor watched Phil Hammond respond with a gracious smile that was utterly at odds with the flush rising from his collar. Porter straightened, greeted several of the bar's patrons, and rejoined his wife.
Ten minutes later, Hammond rose and walked unsteadily toward the exit. Connor stood by the front station and thanked them for coming in. Hammond cast a dark look his way, as though it was Connor's fault that his public mask had been stripped away.
Carol Wright gave no sign that she noticed either the exchange or Hammond's abrupt departure. She had a gentle, knowing smile that she bestowed on Connor like a gift. "Has anyone ever said you look like a movie star?"
"Not recently, Mrs. Wright. Can I tell you tonight's specials?"
"It would be a waste of good breath. I know what I want. Porter, should we suggest this young man take our Celia on a date?"
"Over my dead body," the chief replied.
"Our daughter is struggling to mend a broken heart," she explained, then said to Porter, "I think a night in this young man's company would be just what the doctor ordered."
"If Connor comes within a hundred feet of Celia, he'll need a doctor," Porter assured his wife.
"But you said you liked him."
"As a waiter, sure, I like him fine." He said to Connor, "We'll have the lamb."
"An excellent choice."
Carol said, "Let's order some wine."
"Can't hurt," Porter said. "We walked, I'm off duty, and the night is young."
"I didn't ask for excuses," Carol replied. "I asked for a drink."
Connor said, "I'd like to treat you folks to a bottle."
Carol looked up. "This is new."
Porter said, "Cops aren't allowed to accept bribes."
"You've been a big help," Connor said. "I'd just like to say thanks."
Carol stopped her husband from responding with a hand upon his wrist. "That is very nice of you. Isn't it, Porter?"
"He's still not getting anywhere near our daughter."
"Oh, you. That's very gracious, Connor. We accept."
Rick and Marcela were rushing about, serving other late arrivals. Marcela was the first through the kitchen door. Connor told her what he intended while she punched in the order and apologized to the kitchen staff for making them stay late, as though it was her fault a table had shown up when they did. At first, she gave no sign she had heard him. Then Rick came through the doors, moving faster than Marcela. She said, "Come over here."
"No time," Rick said. "I've got—"
"So do I. And you're going to want to make time for this." When Rick reluctantly moved over, Marcela told Connor, "Say it again."
"It's no big deal. I want to buy the chief and his wife a nice bottle of wine."
"Is that a fact?" Marcela said.
"He's done three big favors for me."
"Has he, now?"
"He found me a place to stay, he gave Sylvie a good word, and he got rid of Phil."
Marcela said, "You get stiffed by your first table, and then you buy the second table a bottle of wine. Not the most profitable way to start your new job."
Rick handed Marcela the keys without taking his eyes off Connor. "Show the new guy around."
"Sure thing." Marcela jangled the keys. "Come on, new guy."
CHAPTER 10
The wall to the left of the entry, just behind the hostess station, was a glass-fronted display case for wine. Connor knew most of these bottles would be filled with colored water and resealed. The bottles on display were drawn from some of California's most expensive vineyards, and were far too precious to be subjected to daylight. The actual cellar was located six steps down the same staircase that led up to Sylvie's apartment. Marcela unlocked the barred entry and stood aside. "The full list is there on the table. You won't be in here, usually, that's Sylvie's job. Or Rick's. But if you do come, be sure and mark whatever you take, and the date."
Connor nodded absently. The chamber was deceptively large, extending back under the kitchen and lined floor to ceiling with shelves. Most of the wine racks were empty. "Where is the rest?"
"This is all we have. Sylvie went broke buying the place and doing it up. Which is why she has old Phil as a partner. Most of what you see here, she bought at auction from another failed restaurant." Marcela scanned the empty shelves. "Sylvie's dream is to build this into something Wine Spectator would write about."
Good wine was one of the lessons Connor had carried with him from his family's restaurant. He could rarely afford to drink any, because he had managed to rack up an elephantine amount of personal debt. However, this was different. Tonight he was after a bottle that showed class, something the Wrights would never buy for themselves. He inspected the few shelves holding French reds, and found what he was looking for midway down the right wall. "This is perfect."
Marcela stared at the bottle, then demanded, "What is your story?"
"I don't . . . This is a good wine."
"Yes, Connor. I know it's good. At a hundred and eighty-five dollars, it better be magnificent. And that's not what I meant."
He hid from the intensity of her gaze by bending over the ledger and making note of what he was taking.
Marcela went on, "You're handsome. You're smart. Sandy says you never did time. You're considerate. You stayed polite to old Phil even when you knew he wasn't going to leave you a nickel."
Connor straightened. "I thought with Phil, you know, it's my first night—"
"We're not talking about what I want to talk about," Marcela snapped.
Connor did not reply.
"What are you doing here, waiting tables in Miramar?"
He faced her because he had no choice. She barred the door with a determined ire. Connor said, "Have you done time?"
"No, and that's still not—"
"But you've got your own reasons, right? I mean, the things you'd just as soon not ever need to talk about. What you've done wrong, why you're here."
Marcela stared at him.
"Not all cages are made of steel," Connor said, repeating Sandy's words.
"So what you're telling me is, you're not telling me."
"Someday. Maybe. Right now . . ." Connor wiped the dust off the bottle. "I'm just trying to get a clear handle on what I've lost. Maybe then . . . I don't know."
"Tell me," she pressed.
If she had stayed angry with him, Connor would have probably deflected. But she was gentle now, and her dark gaze showed a caring nature. Connor had always been a sucker for Latinas with flashing eyes.
He said, "There's something about this town."
"You're right. There is."
"I came here once to escape from a fairly awful weekend up in the hills. Ever since then, I kept thinking about how nice it was, how calm.... I got myself into some trouble last week, and all I could think of was, 'If I could just get back here to Miramar, maybe I might find a way out.' "
Marcela's voice gentled further. "What kind of trouble?"
He traced a guilty script on the bottle's surface. "I finally got what I'd been chasing after."
She held him there for a couple of beats. Long enough for Connor to know if she asked, he would answer. Even if it cost him his chance. But in the end, Marcela stepped aside. "Your table's waiting."
* * *
The wine was a huge success. Connor decanted the bottle at the table, explaining that it was important for an older wine to breathe in order to fully open the flavor.
Carol confessed she didn't know a thing about French wines. Connor warned her, "I could put you to sleep with stuff that doesn't matter nearly as much as whether you like how it tastes."
"Go on, tell."
"Stop me when you've had enough. In 1855, the French government selected sixty-three chateaux vineyards in the Bordeaux regions and gave them a special status, the Cru Classés. The very best of these are called first growths. Lafite Rothschild, Latour, Haut-Brion, and so forth. Nowadays they sell for thousands of dollars a bottle. There are fifteen second growths, fourteen third growths, ten fourths and eighteen fifths." Connor held up the bottle. "Lynch-Bages is a fifth growth, but it's become known as a 'super second' in recent years because the quality has just gone through the roof."
Carol appeared genuinely interested. "Where did you learn about all this?"
"From my mother. She was passionate about everything that went into making a great table."
"And your father?"
"He worried constantly about what things cost. The only thing he liked about good wine was adding it to somebody's bill."
Carol observed, "They didn't get along, then."
"Like chalk and cheese." He set the empty bottle on the table, held the decanter by the base, and poured a small amount into both of their glasses. "Tell me what you think."
There was a singular pleasure to watching their eyes go round from that first incredible taste. Connor stood there a moment longer, observing them with a genuine satisfaction. When he excused himself to go see about their orders, Marcela caught his eye and nodded. Connor entered the kitchen, buoyed by the sense of having gotten something very right.
When he returned with their plates, Carol had reached across the table and taken her husband's hand. She was a handsome woman, strong and solid and deeply in love. Carol saw in Porter what others did not. With a simple molten look, she elevated her lumpish husband to a throne of her own making.
Connor wished them a bon appétit and turned away, his pleasantly attentive mask firmly in place. But the way Carol looked at her husband had drawn Connor down a memory lane littered with women whose names he could not recall.
CHAPTER 11
Connor slept fitfully. Every hour or so, he was jerked awake by the image of that simple wooden sign hanging over the pantry door. By dawn, it felt as though LAST CHANCE SALOON was branded into his brain.
His final dream was about Phil Hammond, sort of. Connor stood by the table and looked down at himself, only thirty years older. He woke up knowing what had troubled him about the guy. Phil was smooth, urbane, and played his role well. Just like Connor. The dream's message was clear enough. This was the best he could hope for? Striving and struggling and finally reaching stardom, so his own team of sycophants could crowd around him at a meal they didn't enjoy, giving the king his due?
He gave up on sleep and dressed. His feet ached and his back felt stiff. His left thumb was blistered and he could not remember when he had burned himself. Despite it all, he actually looked forward to performing in scene two.
A little after seven, Connor walked to the diner and was waved into the same booth he had sat in the day before. Connor caught the cook looking his way. The man was obsidian black, with a froth of gray curls covering his scalp. When Connor nodded, the cook gave him a friendly wave. Then the same waitress came over and told him, "Joey says you get the locals discount. Ten percent."
"Joey is the cook?"
"And owner." The waitress was about Connor's age, but she carried herself like a woman much older. Her expression said everything hurt. Her polyester uniform and pale hose and support shoes made her look shapeless. "How'd you land the gig at Castaways?"
"I asked."
She sniffed. "I asked, too. Like, what, a dozen times. Look where it got me. Working for Joey."
Connor had no idea what to say, except, "Could I have a coffee?"
She walked back to the station, lifted the pot, and filled his ceramic mug. "Is it true what they're saying, you bought the cop a hundred-dollar bottle of wine?"
"Who exactly is saying that?"
"Doesn't matter." She sniffed again. "Don't expect it to do you any good. You step out of line, he'll still lock you up."
Connor was still searching for a response when the elegant woman he had noticed in the guesthouse walked over and asked, "Would you mind some company?"
* * *
After the waitress filled the woman's coffee cup and turned away, Connor said, "I don't think I understood what it means to live in a small town until just now."
Joey called through the pickup window, "What's the matter, you're not eating?"
Connor started to say something about the waitress being too preoccupied to even ask, but he saw her wince and changed his mind. "What's good?"
"I smoke my own turkey sausage."
"I'll have that and two eggs over easy."
"You got it. Miss?"
"Coffee is fine, thank you." When the cook turned away, she said, "I owe you an apology. I was very impolite yesterday."
"Do me a favor. Take off those sunglasses." Connor watched her hesitate, then fumble the oversized glasses from her face. She was an attractive woman in her fifties, but her gray eyes were hollow. She looked like she had not slept in weeks. Connor said, "You don't need to apologize."
"My name is Estelle Rainier."
"Connor Smith. Nice to meet you, Estelle."
She tightened her lips in what might have been a smile. "Anyway, I'm sorry. I had a shock yesterday."
The waitress chose that moment to return. As she refilled their mugs, Estelle murmured, "Thanks."
"No problem." When they were alone, he asked Estelle, "Do you want to talk about it?"
She turned toward the window. "Sylvie Cassick is my daughter."
Connor wondered if this was another part of small-town life. How every bend in the road, every passing hour, carried the potential to punch him in the soul. "Really?"
"Of course, really. You think I would suggest such a thing if it wasn't true?" She picked up her coffee mug, then set it down, untasted. "She's my daughter and we haven't spoken in nineteen years. I didn't even know where she lived. I hired a detective."
"So when I saw you yesterday . . ."
"I've been here four days and the closest I've come to Castaways is across the street. I watched you go inside. I started to follow you, but then I heard that dreadful Sinatra music—"
"Sinatra is many things," Connor replied. "Dreadful isn't one of them."
She studied the steaming mug. "You're right, of course. My ex and Sylvie loved that sort of music. I felt so excluded.... I accused him of making her share his addiction. Which is nonsense. I knew it at the time. But it seems like Sylvie was closer to him than me from her very first breath. I was bitterly jealous."
Connor handed her his napkin and waited until she had cleared her face. "So you heard the music and, what, you left without seeing her?"
"Yes, Connor. I ran away. Can you understand that?"
It was his turn to stare out the side window. "Absolutely."
The waitress set down his plate and announced, "This is on me."
For once, he was grateful for the interruption. "What's your name?"
"Gloria."
He introduced himself, then said, "It's nice of you to offer. But really—"
"Hey, it may not be a hundred-dollar bottle of wine, but still."
Connor showed her his best smile. "That's very nice of you, Gloria, thanks."
When the waitress departed, Estelle asked, "What was that all about?"
"It's nothing. Estelle, why are you telling me this?"
She leaned across the table, her features taut. "I want you to tell me what my daughter is like."
CHAPTER 12
Nowadays Sylvie's life contained few opportunities to watch the sunrise, once her favorite hour of the day. Since opening Castaways, there was always so much to do. However, a migraine and the medicine were always followed by a day where she lived in shadows. Sylvie had built her life on being strong, and she hated this period of enforced weakness as much as the pain itself. She knew from experience that if she reentered full speed too early, she would suffer another attack. So she forced herself to sit by the bedroom's open window and watch gulls dance their silent ballet against the backdrop of shimmering blue. The wind hummed a gentle melody through the cypress and California pines. The clouds sailed great billowing ships across the sky. She felt as though she watched the day through a medicinal blanket.
The old ship captain had lived on the top floor, under the garrets, which now saw duty as her bedroom and walk-in closet and bath. Sylvie often imagined the adventurer seated where she was now, watching sails head into the world from which he had retreated. Sylvie had bought a love seat and matching coffee table, and positioned them so she and Bradley could enjoy a private alcove overlooking the Pacific. That was, of course, before Sylvie had discovered the love of her life had forgotten to mention the wife and three children and Labrador in Santa Cruz. Since that bombshell had landed nineteen months earlier, she went through weekly debates over whether she should burn the love seat in her backyard.
Around eleven, Sylvie descended to the middle floor, entered the kitchen area, and made herself a cup of tea and toast. The entire middle floor was now one great room. The west-facing area had formerly housed four blackjack tables and a trio of roulette wheels. Every shred of that tawdry past had been ripped out. The striped red wallpaper was gone, along with the calico carpet, the fake Art Deco lightshades, and the benches where the fancy ladies had sat and waited to be summoned.
Rick phoned around midday, assuring her he would take over the setting-up responsibilities. Sylvie protested, as usual. Her migraines were part of the restaurant's routine, coming as they did once or twice a month. She liked to think of herself as a woman who thrived on being independent and self-sufficient, but these attacks were a habitual reminder of how much she relied on others. Rick, Bruno, Marcela, and Carl would arrange among themselves to take deliveries and make decisions over the night's specials and do everything she simply did not have the energy to do today.
She drifted downstairs at four-thirty. Her feet seemed scarcely able to find the next step. She made a quick tour of the kitchen, accepted the polite wishes from her staff, then settled onto her stool by the front door. She did not plan on moving one inch more than was absolutely necessary all night.
Ten minutes later, Marcela arrived bearing a carrot cupcake with a single candle. "Happy birthday."
"Shame on you," Sylvie replied. "I had almost managed to forget."
"Hey! Last year, you got your wish, right? You survived that dreadful Bradley. You recovered. You moved on."
Rick stepped up beside her. "We're not supposed to speak his name, remember?"
"Once a year, it's permitted." Marcela lifted the cupcake. "Now wish and blow."
Sylvie looked at the flickering flame. "I still can't think straight. You wish for me."
They both liked that a lot. Rick said, "A guy who deserves you."
Marcela said, "Old Phil gets hit by lightning, run over by a Greyhound bus, and buried in a mudslide. Tomorrow."
Rick said, "You don't think maybe that's a tad overkill?"
"Hey, did I criticize your wish? A guy who deserves Sylvie? Huh. So he walks in and has a meal and leaves? I mean, really."
Sylvie relished having a reason to smile. "So edit the poor man's wish, why don't you."
"You meet the lover who has been looking for you all his life, only he doesn't know it until now. He sees your good heart, and he loves you for the best that you are. He helps you find the happiness you deserve." Marcela lifted her chin at Rick. "Now that's a wish."
Rick was smiling too. "I stand corrected."
"You go stand in the corner, is what." To Sylvie, "Girl, the candle is gonna melt away, you don't blow."
Sylvie did so, tasted a tiny sliver of the icing, then set down the cake and asked, "What do you think of our new waiter?"
"I like him," Marcela said. "Connor doesn't talk about himself, which is nice for a change. But the boy's got a past, I weaseled that much out of him."
Sylvie noticed her headwaiter had lost his smile. "Rick?"
Rick replied carefully, "Connor did not put a foot wrong all night."
"So you like him?"
"I didn't say that."
Marcela flashed genuine ire. "What could you possibly have against the guy? He took on everything the kitchen gave him. He stayed polite, and he handled old Phil with a smile. What, Connor spent too much on the wine?"
"Wait," Sylvie said. "He bought wine?"
"For a table," Marcela said. "To thank the chief."
"Connor bought a bottle of wine for Porter? What was it?"
"Lynch-Bages, 2005," Rick said.
Marcela told their headwaiter, "There is no reason on earth for you not to like him."
Rick's only response was to walk away.
Marcela asked, "What is that all about?"
"Maybe he's got a hunch or something," Sylvie replied. "But why wouldn't he tell us?"
Marcela shrugged. "You figure out what makes any guy tick, you be sure and let me know."
* * *
The restaurant was surprisingly busy for a Wednesday, but it all remained at a distance for Sylvie. A group from one of the Moonstone Beach hotels filled the long table. Locals chose that night to come in for a meal. A business group decided there was nothing they would like more than a good night out to round off their successful meeting. By eight o'clock, every table was full and Sylvie had turned away three late arrivals. Connor handled his six tables fairly well. He was rushed and he made mistakes, but he apologized sincerely and explained that this was his second night after years away from the trade.
Sylvie mostly sat on her leather-clad stool by the host station. She made just one round of the restaurant, and visited the kitchen only twice. She drifted through the hours. The migraine drug always gave her a sense of being disconnected from reality.
Around nine, she began a mental conversation with her new waiter. The handsome man of mystery. Connor Smith. As she watched him, she wondered what it was that troubled Rick so.
The answer was almost audible, certainly clearer than anything she heard from the outside world. Rick is afraid you're going to fall in love with me, Connor mentally replied. And I'm no good.
Sylvie walked to the waiters' station and made herself a cup of coffee as she silently observed, You certainly look like bad news.
I am. Very bad.
Sylvie remained by the coffeemaker, staring out at the night beyond the window. The imaginary Connor asked her, And what would you like for your birthday?
Sylvie did not need to think that one over. The same three wishes as every year. To own this restaurant free and clear. To know the love of an honest man. And to spend another happy hour with my father.
As she returned to her station, she saw Connor smile at a table, brightening the lady's night. And right then, Sylvie realized what Rick had seen.
Connor was just going through the motions.
She could easily have put it down to Connor's newness or her own addled state, but Sylvie felt as though her medicinal distance actually clarified things for once. Connor Smith was not really engaged.
He worked the job and he talked the good talk. However, there was a special quality to a good waiter. They might be aloof; they could be utterly cold in their manner. Still, they conveyed a unique passion about their work. They treated it as a profession. They saw themselves as a vital component of the evening. And that was what an outstanding restaurant was all about. It did not merely prepare good food. It created an experience.
Sylvie kept watching, and became fairly certain two of Connor's tables felt the same. They might not be able to say why, but they were not taken with Connor's act.
And that was exactly what it was.
Rick chose that moment to walk over. "You holding up okay?"
"Marginally." Sylvie gave a fractional jerk of her chin in Connor's direction. "I see what you mean about him. He's . . ."
"Disconnected," Rick finished, and patted her on the shoulder. "Leave this with me."
CHAPTER 13
Connor spent much of the shift thinking back over his conversation with Estelle. He was rushed off his feet, and needed desperately to focus. However, even as the restaurant filled up, he could not get the woman out of his head.
Their time together in the diner had lasted all of ten minutes, maybe even less. Connor had tried to explain that Sylvie had not been feeling well, and as a result they'd only spoken a few words. Estelle would not be put off, though. "You had to have said something. She doesn't just offer a job to any fool who wanders in off the street."
"Thanks so much for the compliment," Connor said. "A wandering fool. That's a new one."
"I've got a lifetime's experience of saying the wrong things." She leaned in closer still. "Please."
He wished he had not asked her to take off her sunglasses. Estelle Rainier's eyes were a stormy gray, one shade darker than her daughter's. Her gaze held a desperation that was borderline frantic. Connor had no choice but to reply to her. "Sylvie is gentle and beautiful in a fractured sort of way."
"'Fractured'?" she said.
"I don't know exactly how to describe it. But I had the impression that she didn't just get a crippling headache out of the blue. It came on because she's carrying some awful burden."
"What is it, do you think?"
"I have no idea. What I can tell you is, the people here care for her."
"What people?"
"Everybody." He related his conversation with the stylist. And the way the two front staff almost cradled her when the pain struck. "Miramar's chief of police came in with his wife. I'm pretty sure he wasn't there just for the meal. He came to check me out."
"For Sylvie."
"Right. Everybody around here thinks the world of your daughter. And something more."
"What?" Her soft plea held a desperate edge.
Connor struggled to put it into words. "She's made a home here."
Estelle's tension and the strength slipped away. Her shoulders slumped. Her features ran like wax. She turned to the window and worked hard not to cry.
Connor did not know what to say. "I'm sorry."
Her fingers trembled as she pushed the sunglasses back over her eyes. She fumbled her way from the booth. As she left, she said, "Don't tell Sylvie I'm here."
"Estelle, maybe—"
She pointed an unsteady finger at his head. "Don't."
Connor had no choice but to say, "All right. Fine. If that's what you want."
"What I want?" Estelle turned away. "I should never have come."
The waitress had clearly been observing them, because as soon as Estelle was out the door, Gloria was standing over Connor and demanding, "What did you tell that poor lady?"
He shook his head. "What she said she wanted to hear."
* * *
At a quarter to ten that evening, Rick pulled Connor into the kitchen. Connor feared he had gotten something terribly wrong. Instead, Rick told him, "I want you to do something for me."
"Sure thing."
"Yesterday was Sylvie's birthday. She mentioned you two love the same kind of music."
Connor watched Marcela slip into the kitchen and step in close. She asked Rick, "Did you tell him yet?"
"I asked."
Connor replied, "Like I said last night, I haven't touched the keys or sung a note in seven years."
Marcela asked, "It's kinda like riding a bike, right?"
"Not at all."
Rick wasn't budging, though. "You see the state she's in. It would mean the world."
"We won't count the bad notes against you," Marcela said. "And neither will she."
Rick took his silence as agreement, and said, "Marcela and I will handle your tables. Let's go set you up."
Marcela patted his arm. "This is going to be great."
Connor thought it was going to be anything but that. Even so, he followed Rick through the bar area and climbed up onto the stage. Rick told him, "We use this mostly for weddings that take over the whole place."
That meant the piano was most likely tuned. Unlike his voice. Connor watched as Rick opened a closet built into the alcove and drew out an electronic drum kit, microphone, stand, and mini-amp. Rick shifted them into position and asked, "Will this work?"
"Rick, man . . ."
Rick guided him onto the stool, positioned the mic stand and drum machine, plugged in the cables, then stepped back and said, "It's all yours."
Connor watched Rick retreat from the stage, then set his fingers on the keys. He had always taken pride in his hands. Connor's reach was as powerful as his grip. He had once been able to play with a gentle smoothness, creating the liquid sort of backdrop that his melodies had required.
Now they rested there on the ivory, ten foreigners to a world they had once claimed as their own.
Connor had no idea how he had gotten himself into this. Either he played or he quit. Just get up and walk out and leave the place behind. Connor had seen that unspoken directive in Rick's gaze. The man had not been making a request.
He said softly, "Test, test." He adjusted the mic, turned up the amp's volume, and set the drum machine for a soft swing beat. He had often used a similar machine. The movements came almost naturally, like he had merely been hibernating for seven long years.
He knew Sylvie liked Sinatra. So he started with one of his favorites, the first he had ever reworked into a style that fit his voice.
Come fly with me.
Let's fly, let's fly away.
It had been so easy to stop. Playing once in a while had meant Connor wasn't growing. And he had always aimed high. It was one of his defining traits. When he switched his attention and his energies to acting, the creative draw to music had faded with remarkable swiftness. Initially Connor had assumed he would keep his music as a sideline. Every time he had sat down, though, all he heard was how far he had regressed.
Connor had always pushed himself hard to improve, grow, meet the challenge of drawing in a large audience. He had developed a style to utilize his strengths and mask his weaknesses. He had learned how to arrange the fifties-era ballads into a style more suitable for today's audiences. He had studied the contemporary singers who had made the songs work for them: Norah Jones, Michael Bublé, Diana Krall, Josh Grogan, to name a few.
All lost.
CHAPTER 14
Connor tried twice to rise from the piano and resume his role as waiter. Both times, though, Marcela ordered him back, saying she could handle everything and he was to stay where he belonged. Thirty minutes in, Sylvie walked over and seated herself at the bar. Aubrey, the bartender, poured her a glass of wine, but Sylvie did not touch it.
An hour and a half later, Connor finished with a melody that had taken Nat King Cole to the top of the Billboard charts. As he cut off the amp and drum machine, Sylvie said, "Come join me."
They talked until two. Or rather, Sylvie talked and Connor listened. Whenever she ran out of steam, she would ask him to play again. He sang a couple more tunes, then rejoined her. Sylvie rose from time to time, closing down the restaurant and bidding the staff a good night. Then she returned and started back where she had left off. Easy and natural. Like they had been enjoying such conversations for years.
She talked mostly about her father. "Pop was a true vagabond artist."
Connor asked, "What does that mean, 'true'?"
Sylvie revealed a special smile in response to Connor's question. "You are the first person in years to ask me that. Most people want to know about the 'vagabond' part. What they're really asking is, how was it for me to live on the road. And the answer is, wonderful and terrible in equal measure. I went to eleven different schools in twelve years. I dressed mostly out of Goodwill hampers. The boys who came flocking around were not the sort that interested me. The girls thought I was weird. Their parents suspected I was a bad influence. I spent most of my time alone."
Connor found it was the easiest thing in the world to reach over and take her hand. He said softly, "True."
"All the wonderful parts of my life start with that word. My father was true to his dream. He painted scenes from the Pacific coastline. I was born just north of the Mexican border. My earliest memories were of playing in the Baja desert. When I was nine, he sold an entire collection of paintings to a gallery. He earned enough to take us north. So far north. We lived for two years in the Alaskan wilderness. Then we traveled back down south, staying six months in one place, a year or so in another." Sylvie gazed back over a wealth of memories, the smile as gentle as her voice. "Pop was a man of quiet happiness. My earliest memories are of the ratty camper, the only home I ever knew before we landed here in Miramar. We'd travel until he found somewhere he wanted to paint. When he hunted out a spot off the beaten track, somewhere with good light and a fair breeze. When he'd put on those albums, I knew we'd found another home."
She spoke of the man with a soft yearning, a hunger so tender it made Connor's eyes burn. He had never imagined what it would be like to have a woman speak his name with such gentle passion. "What about your mother?"
Sylvie pulled her hand away. "She left us when I was twelve. That's all you need to know about Estelle."
Connor did not know what to say, or how to recapture the feeling. He thought about the woman lying alone in the guesthouse, too frightened to come meet the daughter she had abandoned.
Sylvie's voice hardened with her gaze. "Estelle started these one-sided battles when Pop made enough money for us to head north. My mother had always assumed when the money came in, we would buy a home and settle down. My father refused to even discuss it. He would hide himself away in another painting, disappear with his oils and his palette for days on end. When my mother tried to convince me to go away with her, I took off with Pop. We hiked for six days in the Canadian Rockies, and when we came back she was gone. I received, oh, four or five letters. Each time I wrote back and told her if she wanted to talk to me, she could come home. Mom's last letter said that was the problem, she needed the home Pop never gave her. Three weeks later we moved, and I never heard from her again."
Connor nodded quietly, then asked, "Will you show me your father's work?"
In reply, Sylvie walked to the front door and locked out the night. She took his hand and walked him around the paintings on the restaurant's walls, as though it was important for Connor to meet Gareth in his daughter's intimate company.
Gareth Cassick was a troubadour in oils. He painted with too much heart to ever be called great, but there was a colorful wonder, a quiet fervor, that sang through his every brushstroke. Gareth Cassick captured little of the landscape's precision. Nor did he try. His aim was to paint its reflection within his own heart. These images fashioned the likeness of a man who loved the Pacific coastlands with every fiber of his being.
Five of his oils adorned the restaurant's walls. One was of an Alaskan ice floe with iron cliffs as a backdrop. Another depicted a bear fishing the broad mouth of a river in a snowstorm. In the third, a commercial trawler plowed a golden furrow through a placid sunset. But it was the fourth and fifth that captured Connor.
One was of an abalone shell about a foot and a half tall, nestled in the sand, with a wave's froth lapping the upper boundary. The heart of the shell was painted as a sunset. The colors were one step off gaudy, and no doubt the art critics would have called it maudlin. Connor stared at it and felt as though he greeted the painter himself. Perhaps it was how Sylvie stood there beside him, not speaking, not pressing, content to give him all night if he wished. Finally he moved to the fifth, and by far the finest of them all.
The moon rose over indigo hills. The mountains were shaped like a pair of night-clad hands. They sheltered a collection of lights at their heart. The fingers flowed into the sand, and then the ocean. A mist gathered above the effervescent land. Only the moon shone clear.
Sylvie said, "This was the first painting Pop made of Miramar. We arrived here a few weeks before my sixteenth birthday. Pop had become friends with guys working nets on a trawler. They took us out for a night run. On the way into harbor, I told Pop I didn't want to go back on the road. I wanted Miramar to be my home. For the rest of my life. If he left, it would be without me."
Connor took a slow breath. Home.
"He never said, but I've always thought this painting was his way of telling me that he had finally found a place where his bones could rest easy."
* * *
Connor walked back to his room through a dense and chilling fog. But it was nothing compared to the daze he felt. As Sylvie had walked him to the door and bid him good night, she had invited him to share in a sunrise walk. Her invitation had been expressed with a shyness that had touched Connor. Now, as he climbed Miramar's main street, he had the sense of being offered a rite of passage. The evening had woken something inside him. It was such a strange sensation, he could not even name it.
He knew full well he was attracted to Sylvie. A blind man in a coma could see that much. What Connor could not understand was, why now?
Falling for a woman was the absolute last thing that should happen to him now. And it was not because of Kali, his soon-to-be ex-fiancée. If Connor had gained anything from traveling to Miramar, it was the absolute certainty that ending that marriage before it started was the right thing to do.
His footsteps scraped on the pavement, the sound overloud in the silent night. The fog was so thick he could not see much beyond the next streetlight. Time and again, he returned to the same inescapable fact. How could he be right for any woman, or know which was right for him, until he had a handle on who he really was?
One day, he would very much like to have a woman like Sylvie Cassick say she loved him. A woman whose physical beauty was so natural, it emerged spontaneously from her heart, as instinctive as a blooming rose.
Sylvie was a woman who cared so deeply, even strangers wanted to embrace her. Even a broken wretch from the world of film and lies gravitated to her. Even a Hollywood actor who did not have the first idea of who he truly was, a vagabond who could only say that he was lost. Connor was a wandering idiot who had run from all his dreams, simply because they had come true.
Love a woman like Sylvie Cassick? Hopefully. Yes. Someday. When he deserved it.
Love this woman now? The idea was not just absurd, it was dangerous. For both of them.
The fog was so thick Connor almost missed turning into the guesthouse. Then a corner of the sign emerged from the gloom. He crossed the parking lot, let himself into the room, and shut his door against all the vague yearnings that had chased him through the night.
He had not come all this way just to break another woman's heart.
CHAPTER 15
When Sylvie finally slipped into bed, she was not the least bit sleepy. This was another of the migraine's aftereffects. She would be back on a regular schedule tomorrow or the next day. She lay in bed, wide-awake, and rewound the conversation with Connor. Now and then, a refrain from one of the songs Connor had sung for her swam through her mind, and she smiled at the ceiling. His voice was deep and silky smooth. He showed a richness that extended through his entire range. He was rusty, and he made mistakes, and every time he stopped playing, he returned to the bar with the hollow sadness in his eyes. But none of it mattered. Not really.
To call theirs a conversation was entirely wrong. The only time Connor opened his mouth was to either sing or ask a question. Several times, she'd started to inquire about him, but she had stopped. Her words had remained unspoken. Connor's haunted look halted her. The night was simply too fine to make him confess his secrets. There would be other times for that . . . if she wanted.
That was the real reason, of course, why she wasn't sleeping. She was trying to decide whether she wanted to have this move further. On any level.
Which made her invitation for a dawn walk all the more astonishing. She had not been out for a sunrise stroll with anyone since Bradley broke her heart.
She wondered about the great tragic mystery that had brought Connor here. Almost everyone who arrived in Miramar came with baggage. The stories were often sordid and made for spicy gossip. Sylvie did not mind a man with a past. She liked adventure. She liked the aroma of danger. She loved the idea of dancing a lifelong tango with a man whose history was as jagged as her own.
The one thing Sylvie valued most was honesty. The rarest of wines was nothing compared to this. Candor was a distilled elixir that seasoned the finest day and turned even the hard hours into a delicate feast.
Honesty. Frankness. Integrity. It was a shame that so few men understood what those words meant.
She found herself recalling the day she had driven up to the home where the man she had thought was the love of her life lived. She had sat in her car for almost an hour, watching Bradley play Frisbee with two lovely children and a barking dog. The memory still astonished her. How could a man fool her so completely? And why had he done it? The realizations that had wrecked her hopes for love and a family still rankled. How could she have not even suspected that the man lived a double life?
Sylvie rolled over and shut her eyes. She was no closer to answering those questions, and probably never would be. One thing, however, was certain. She was going to have to ask Connor to reveal himself. Not only because she needed to know his secrets, but more than anything, she wanted to see if Connor would tell her the truth.
CHAPTER 16
Despite the short night, Connor rose from his bed before the alarm went off. He was jerked from sleep and five seconds later from the bed itself, drawn upright by the realization that today was the day. As soon as he returned from his walk with Sylvie, Connor was going to contact Los Angeles.
He made a coffee and drank it, standing at the counter of his little kitchenette. He dressed and descended the hill to find Sylvie waiting for him in front of Castaways. She smiled a welcome that appeared strained in the dawn. The mist had retreated somewhat, leaving a glistening blanket over every surface. Together they walked in silence down the winding hill and joined the beachfront lane.
The path was mostly ground asphalt; but where the rocky shoreline took sudden dips, the passage was linked by rough-hewn wooden planks. The walk was almost two and a half miles long, running from the southern to the northern cliffs that defined Miramar's boundaries. The trail dipped and weaved, a lyrical line drawn between earth and sea.
The morning's ocean and sky were both fashioned from the same cold steel. They walked to the northern point without speaking a word, and met no one. Even the gulls gave way at their approach. The central coast revealed its link to the Pacific north this morning. Connor recalled the Carolina fogs he had known growing up, great billowing swaths that settled an expectant hush over the world. This was something else entirely. The air was biting, the wind a soft blade against his exposed skin. A massive swell crashed somewhere out beyond his field of vision, blasting froth across the shoreline.
Sylvie was wrapped in a pearl-gray jacket zipped up to her chin, with a matching woolen cap. She walked with the lithe motions of a dancer, scarcely seeming to touch the earth. When they reached the northern cliffs and turned back, she said, "A penny for your thoughts."
"I was remembering a chef in my parents' restaurant." Connor had no idea where that had come from. For a guy who rarely gave time to his past, he found it remarkably good to share with her. And easy. He felt like he could tell her anything. "I haven't thought of him in years."
"Tell me."
"I started running errands and doing odd jobs around the place when I was eleven. The head cook was this guy, Leonard. My dad nicknamed him 'Spock,' you know—"
"The Star Trek guy played by Leonard Nimoy. Sure."
"He actually looked like Spock's evil twin, minus the pointy ears. He was a convicted murderer—two counts of manslaughter, did twenty years. He was out on parole. Man, could that guy ever cook." He smiled at the memories drifting in the mist. "He loved Louisiana French cuisine, the heavy sauces, the dishes that took all day to prepare, but he basically could do anything. He had an encyclopedic knowledge of spices."
"But he carried shadows."
"You can't imagine. Every conversation with Spock that wasn't about food became a walk on the wild side. Here I was, this little skateboarding kid and in-house mascot. All Leonard needed to do was look my way and my blood froze solid. I heard him tell my dad once, his second year inside he found a book in the prison library about cooking. That book was the only reason he made it out alive."
Sylvie's response was to take his hand. They continued on in silence, content to share the morning's immense solitude
Connor found himself calmed by Sylvie's closeness. The worries and fears waiting for him back up the hill, the call to LA, the questions for which he had no answers, all of this vanished in the drifting fog. All disappeared because of this woman who walked beside him.
The light held a cathedral quality, spilling through the ocean mist like heaven's own stained glass. Connor felt as though everything he glimpsed was made diamond-brilliant by Sylvie's presence. It felt as though she filled a space inside him, opening his senses to a level he had never before known.
They left the path and took a winding staircase down to the beach. The ocean's roar was so powerful he felt it in his chest. The sand was blanketed by clouds of sea froth. He could feel the ocean's chill on his exposed skin. Gulls swept past, eyeing them with a calm loftiness. Otherwise, they had the shore to themselves. At the next set of stairs, Sylvie led him across the hard-packed sand. As they climbed back up to the coastal path, Connor was halted by a sudden desire to kiss her.
Connor might have stood there all day, held by that crystal-gray gaze, as brilliant as the sunrise through the Pacific mist. It was there in her open expression, her need to hear the truth from him. If only he knew what that was.
* * *
On his way back up the hill, Connor stopped by the general store and bought a cheap phone that contained no GPS chip and five hundred minutes. He returned to his room and put on another pot of coffee while he set up the phone.
Connor poured himself a fresh cup and dialed Kali's number. Then he stopped in the act of making the connection. The bedside clock read half past ten. His soon-to-be ex-fiancée never got up before noon, unless she had an engagement of seismic proportions.
He cleared the screen; then he punched in the number for Kali's assistant. Kali ran through PAs at a ridiculous pace. She referred to the males as "Eric" and the females as "Erica." When the voice mail picked up, Connor identified himself and asked for an appointment to speak with Kali at three that afternoon.
Connor walked outside and sat in the narrow doorway, waiting for his heart to slow down. Estelle chose that moment to walk over and ask, "Can a girl buy you breakfast?"
"Not today. I've got . . . things."
"Terrible thing, things," Estelle said, turning away. "Good luck."
"Estelle." He could say it now, because he was facing his own dreaded truth that very moment. "You need to speak with Sylvie."
She did not look back at him. Instead, Estelle lifted her gaze to the blanket of gray. "The worst part of running away is facing the consequences. Wouldn't you agree?"
Connor did not reply or look up as her footsteps receded into the mist. When he was ready, he went back inside and placed the second call.
CHAPTER 17
Ami Chen was a senior talent agent with CPP. The agency's name was formed from the three original partners, now retired, which meant no one even remembered who they were. New actors who yearned to be added to the CPP roster said the name stood for "Careers Pulverized and Plundered." CPP occupied the top six floors of a premier building on Wilshire Boulevard. The six-lane road separated Beverly Flats from Beverly Hills; or as Ami Chen liked to tell Connor, it split the A-list from the hired help.
Ami managed five A-list stars and around sixteen actors of Connor's status. Plus six directors, three producers, and a half-dozen writers, whom Ami referred to as her in-house menagerie. The exact number of character actors depended upon whom Ami had recently dropped. Ami went through her B-list actors like other women went through cupcakes.
There was actually no such thing as a B-list. There was only the A-list and everybody else. Actors like Connor lived on a knife's edge. This fostered a justifiable paranoia, for most character actors were a single slip away from the long bus ride back to Indiana. Ami Chen was only too happy to show them the exit. She called these dismissals her release valve. Every agent in LA ingested a ton of refuse for every successful deal. Ami had once confessed that these firings granted her a momentary sense of power. Connor hoped Ami had shared this secret motivation as a means of telling Connor that he was somehow special. He, however, had never built up the courage to ask her.
Because his number was not identifiable, Connor's call was schlepped instantly to voice mail. This was standard practice in Hollywood, where there simply was no extra second available for the hordes of desperate outsiders. He identified himself, gave his new number, and then went back out to the parking lot. He could not find enough air in his room. Estelle was seated now in the little park behind the guesthouse, a large coffee set on the bench beside her. She glanced over, then away.
Five minutes of pacing, and then the phone rang. Gerald, Ami's assistant, said, "Is it really you?"
"Yes."
"Which of my four hundred messages and counting did you decide to respond to? I'm only wondering, since I've stubbed three fingers reaching out."
"I bolted. I panicked. I had to get out of LA before it was too late."
"Well, that's hardly the most original excuse I've ever heard."
"But it's the truth."
Gerald gave that a beat, then said, "When I couldn't find you, she shouted at me. I hate shouting."
"I'm sorry. Really."
"You sound like you mean it."
"I do."
"Then again, you're an actor. You're paid to sound genuine." There was a strident voice in the background; then Gerald said the words that formed the foundation to Connor's acting career. "Hold for Ami."
Ami Chen was a five-foot-three bundle of fierce intelligence and angry determination. She rammed her way through every blockade Hollywood's film world tried to put in her path. She was considered one of the top agents at making careers. Ami greeted Connor with "Explain why Gerald couldn't track you down."
Not a great one for casual conversation, his agent.
"I ran away," Connor repeated. "I didn't take my computer or my phone. I was too afraid they'd track me. Or that I'd contact Kali in a weak moment."
"Hold one." The line clicked off.
There were any number of reasons for this silence. The first time he had been in Ami's office, Connor had been overwhelmed by the sheer volume of her phone traffic. Ami's granite desk was the size of a small independent island-state and held five computer screens. Two of them were reserved for handling her phones. Gerald fielded all of Ami's calls, then passed those he deemed worthy to the monitor where she could point and click and connect. The neighboring screen held any intel or alert that Gerald might have received in the initial dialogue. Anyone who worked regularly with Ami Chen treated Gerald as a principal ally. Gerald earned more than any junior agent at CPP. It was why he remained in Ami's front office, despite the fact that his boss defined the term "difficult."
The calls Gerald decided to pass through were color-coded. The ones he thought Ami should take instantly were yellow, and those dealing with actual payment were white. Red calls were problems, always with notes attached on screen two. Blues were the ones she took if or when she had the time or felt bored. When Connor had asked what was used for personal calls, Gerald said there weren't enough of those to deserve a color.
The line clicked back on, and Ami demanded, "Are you marrying Kali Lyndon?"
"No."
"This is not some yo-yo exercise, a bit of last-minute jitters that struck in the lead-up to your wedding."
"No yo-yos, and surprisingly few jitters," Connor replied. "At least since I arrived here."
"And where exactly is this 'here' located?"
"A million miles from LA. Farther."
"Strange how you sound like you're in the next room."
"Believe me. I'm not."
Ami did not probe further. She was pure LA. She did not care about such minor issues as love or commitment. Her laser focus was aimed at just one thing.
The bottom line.
Ami was so quiet that Connor thought she had dropped his call. Then she said, "Thinking."
Connor took his phone back outside. Thankfully, Estelle had disappeared. He paced the length of the parking lot. Again.
Finally Ami said, "I have something to tell you."
Connor's heart dropped out of his chest, straight onto the pavement. Splat. He had actually heard Ami use those very words when dumping other actors. Twice. The first time, Connor had been seated in the narrow confines of Gerald's outer office. The second time was in a bar, the night she had confessed the acid pleasure found in such dismissals.
Connor said, "Go on, then."
Ami said, "You are being considered for the primary bad guy in the new James Bond film."
The strength drained from his legs so fast he landed on the pavement. The hand not holding his phone kept him from sprawling backward. "What?"
"It's not a done deal, but they're serious. You'll die, of course, but only after being on-screen for eighteen minutes. Possibly twenty-three, if they go with the script's current draft. They want the film to mark your big century of on-air demises. They think this can be used in the film's advance online promo."
"I thought . . . Never mind."
"You absolutely deserve what you thought I was going to tell you," Ami assured him. "You've been such a bad boy. Have you told Kali?"
"I've asked Erica to set up a call with her at three."
"Let me handle that."
"No . . . I . . . What?"
"I'll prep her PR first, convince them they can use this." Her voice tightened. "You just remember we haven't signed the deal yet."
"Ami, I don't understand a word of what you just said."
"What you're asking for is the kind of service reserved for my A-list. The clients who pay for my place in Bel Air."
"Ami, I haven't asked you for anything."
"Which doesn't change the fact that you need to be handled. Now you apologize to Gerald."
"I already did."
"Do it again, this time with champagne, because Gerald is my handler. Which now makes him yours. So, darling, when Gerald calls, what will you do?"
"Answer the phone."
"Night or day, darling. Whatever he tells you to do, darling, treat it like it's a command straight from God."
It was only after Connor had spoken again with Gerald and sat there on the pavement beneath the steel-gray sky that he realized what Ami had called him.
His agent only had five darlings. Three of them had won Oscars.
CHAPTER 18
Connor left for the restaurant an hour and a half early. His room could no longer hold him. His mind was an electric jumble of conflicting thoughts and emotions. The only word to describe how he felt just then was, disjointed.
Here he was, running away from his life in LA. Only now he was also running from Estelle Rainier, a woman whose most threatening move was to ask questions about a woman Connor hardly knew.
For seven years, he had tried hard not to miss his music. It had been almost that long since Connor had even listened to his favorite artists, for fear of being dragged down into a hopeless reflection of his many wrong moves. Then he had played for much of the previous night, and the clearest impression Connor carried was the smile of thanks from his one-woman audience.
When he had climbed on that midnight bus, Connor thought he had walked away from acting, as well as his soon-to-be ex-fiancée.
But this morning, he had been terrified by the prospect of his agent dropping him. And now he was as excited about the possible Bond gig as anything in his recent life.
He was fleeing a public relations mockery of a wedding. But all he could think of was sharing another dawn with Sylvie.
He repeatedly told himself that the last thing he should be doing was involving himself with another woman. All the while, though, he was held by the memory of those beautiful crystal-gray eyes.
He hurried to a job that had nothing to do with the world he had come to call real. He had slept only a few hours. His eyes felt grainy and his feet hurt.
Still, he was genuinely eager to start his gig, waiting tables at Castaways.
Connor checked his watch. In precisely two hours, his agent would be on the phone handling the situation with his soon-to-be ex-fiancée.
Yet, all he could think about was seeing Sylvie again.
But how was that even possible? What was he going to tell her? Everything she thought about him was wrong, and the prospect of telling her who he really was filled his gut with leaden dismay.
That brought him to the greatest dilemma of all.
Connor had spent enough time in the woman's company to know the question that Sylvie most wanted to ask him.
Who was he really?
Connor had no idea how to respond.
* * *
Sylvie felt as though her day remained sheltered by Connor's quiet strength. The dawn walk had meant more than she could possibly have imagined. It had been a very long time since she had enjoyed that sense of companionship, where words were unnecessary, and the closeness of a man as natural as breath.
During her childhood, sunrise walks had framed Sylvie's happiest times with her mother. Her father often worked late into the night, sometimes falling into bed with the sunrise. Mother and daughter had shared countless mornings, chasing gulls and sharing dreams.
Then toward the end, they had spent those hours arguing. Estelle had revealed her plans to leave on one such walk, a bitterly cold May morning with the Canadian foothills lost to a fog as thick as today's.
Sylvie checked her reflection, but saw most clearly the heartbroken girl in the months following her mother's departure, standing outside their ratty camper, knowing she was not strong enough to reknit the fabric of her family. She recalled the shocking sense of loss she had known, hearing Estelle speak about her need to leave them. That was the way her mother had described her state—as a need. Sylvie had not thought of that in years. Now it was as clear as the cry of gulls through her bedroom window. She remembered how frigid her tears felt on her cheeks. She remembered how it had started to snow, as though the world wept with her.
* * *
Sylvie could not say exactly when Fridays had become cleaning day. There was a certain logic to the timing, however. The weekends were always so rushed; neither the restaurant nor the kitchen was ever fully scrubbed. Monday was their one evening off. The normally slower Tuesdays and Wednesdays often felt as though they would never end. Sylvie spent most Thursdays at the local markets and taking their weekend deliveries. Friday afternoons, everyone showed up early. Carl prepared something special for the Castaways staff and filled the restaurant with the prospect of a fine meal. Today it would be a veal pot-au-feu with baby potatoes coated with olive oil and rosemary and sea salt, then oven roasted. The resulting exteriors were hard and crunchy, while the interiors were cottony and soft, and the flavor exploded with each bite. This was followed up with one of Sandy's trademark desserts, crème brûlée with chocolate-cinnamon biscotti.
Because the scouring took place every week, there was a settled, comfortable routine to it all. Even the brass railings to the outer doors and the bar and the hostess station, the least appealing of all the many jobs, were simply done and left behind. The new busboy's name was Gustavo, and he more or less took every task in his stride. Throughout the initial half hour, as Sylvie helped him remove grease from behind the main stove, she found herself humming snatches from Connor's melodies. He took some of her father's favorite melodies and formed a composite of the modern world. He sang from today's perspective. She could not say it any better than that. The years of not practicing showed in repeated off-key notes, but it had also embedded a rough burr, a sorrow that deepened the emotions. Made it . . .
When she paused to take a new reservation, Rick joined her at the hostess station and asked, "Is Connor coming?"
Sylvie answered, "I forgot to mention it."
Rick shrugged. "I think I met the real guy last night."
"'Real,'" she agreed, thinking that was the word she had been searching for. "You liked his music?"
Rick shrugged. "Not really."
She swiped his arm. "Liar."
"Bruno said it sounded like Whisky a Go Go, only at one-third speed."
"Maybe I was too hasty letting Bruno off the hook for those bad fish." They were both smiling as she said that.
"Are you going to keep Connor on waiting tables?"
The question surprised her. "Of course."
"I just wondered, you know."
"We can't begin to afford live entertainment. Phil would have a seizure."
"Old Phil," Rick said, mimicking Marcela.
"It would be great, you know . . ."
"If Connor could bring that heart to his work." Rick nodded. "I'll have a word with the guy."
* * *
Connor liked how his early arrival was greeted with the simple friendliness of being accepted. Sylvie explained that this was their day to give the entire restaurant a thorough going-over, then assigned him to scour the kitchen's four steel tables. The industrial-strength rubber gloves protected his hands, and the double aprons—one plastic over the white cotton—kept the cleanser from staining his clothes. He hated the acrid stench, but the work was soothing. He had worked all his life. Sometimes it felt as though the need for constant labor was embedded in his DNA. Nowadays when he returned home to Charlotte, his family always chided him for not working hard enough. As though hours of daily labor were the defining trait of a life well lived. His parents had been divorced for over fifteen years, and the restaurant was long gone, but they both still held to this core principle, as did Connor's brother and sister. None of them believed that his acting gigs were true work. They listened to his descriptions of the predawn calls and the weeks of living on four or five hours' sleep, the blistering lights and the acting classes and the rehearsals without pay, the heartbreak and misery that was the fate of most people striving to break in, the bitter jealousy that was the fodder for almost every actor on earth. They heard him out, but they did not believe him. Not really. He could see it in their gazes.
It wasn't that this work was easy, but it was defined. He had a job. He got on with it.
When he was done, he stripped off the gear, found himself a seat on the narrow patio, and ate at the wagon-wheel table with Sylvie and the others. He was one of the team. They joked with him about his choice of music. Bruno asked if he could add a couple of hip-hop numbers. Sandy asked if he knew any by David Bowie. Sylvie caught his eye twice and smiled.
The facts of his double life ate at Connor's heart like acid.
When they finished eating, Rick asked Connor if he'd like a cup of coffee. Something in the question caused all the others to rise and reenter the restaurant, including Sylvie. When they were alone and both cradled steaming white ceramic mugs, Rick said, "Are you sure you want to be a waiter?"
Connor felt that sudden inward jerk, like a corner of his secret covering had just been pried loose. "Of course I do. I'm here, aren't I?"
"Partly," Rick corrected. "You're partly here."
"What's that supposed to mean?"
"You're a smart guy. I think you already know."
Connor tasted the coffee, then set it down on the wagon wheel. The sun was half-hidden behind a corona of high clouds and the air held a biting chill. That was not why he shivered, though.
Rick gave him a chance to respond, then went on, "You don't bring enough of yourself to the table."
Connor found himself fighting back a sudden urge to confess, to tell the headwaiter in Sylvie's restaurant who he was, and what he faced. Instead, he clenched his teeth and swallowed it down.
Rick said, "You're going through the motions. You've got all the moves and none of the heart. You do what it takes to get the job done, and then you walk away. Which is fine, long as people are only interested in hearing the specials and getting their glasses filled and having a plate set down in front of them. But if they come looking for an experience they can share in, if they want a night they'll remember, if they want . . ."
Connor could almost see the words hanging in the air before his face. He finished the sentence in spite of himself. "The best life has to offer."
"We represent their entry to a world beyond normal," Rick agreed. "We put a face on their chance to escape the day-to-day."
Connor sensed that Rick had said all he intended. He was tempted to respond by sitting there, waiting the guy out. See how long Rick would let the silence drag out. The words welled up inside him again, stronger this time. Connor said, "I've spent seven years becoming an expert at going through the motions."
To his surprise, Rick seemed pleased by his response. "What, you think that somehow makes you unique?"
"I don't . . ."
"Someday I'll tell you about all the superficial moves I learned to make." Rick had an ancient's piercing gaze. "Was that true what you said, about being raised in a restaurant, your folks turning it into a battlefield?"
"Every word."
"Did it ever occur to you that's why you disconnect?" Rick rose from the table. "Maybe it's time you moved beyond the past. Try for something new. That's what brought you here, isn't it? The chance to start over?"
Connor opened his mouth, but no words came out.
"There's no future in repeating past mistakes." Rick gestured for Connor to join him. "Come on. It's opening time."
CHAPTER 19
Connor's new phone buzzed just as he was pouring wine for his first table. Connor almost dumped a full glass in the lady's lap. He brought their starters, excused himself, and walked outside. The phone's readout gave no name, and the caller had left no message. But only one person had this number. He hit redial, and Gerald answered on the first ring. "How soon can you get back to LA?"
"Tomorrow, I guess. If I have to."
"Yes, you have to. Do I strike you as somebody who would ask such a thing because I'm curious?"
"Okay, Gerald. Fine."
"Don't you dare give me that tone of voice. I have been slaving away for hours. Now I positively must know where you are."
Connor breathed the name. "Miramar."
"That sounds vaguely familiar."
"It's a small town on the central coast."
"Of course. I hear it's quite lovely. How long is the drive?"
"I have no idea. I came up here by Greyhound bus."
"How utterly dramatic. PR will positively die. I can see the next episode of the cable show now, an opening shot of you boarding the bus—"
"Gerald. Just. Stop." Then Connor heard the chuckle, and realized that had been Gerald's idea of a joke. Connor said, "I can't shoot you. I need you too much."
"Oh, please. You've got to give me my tiny shred of fun, all the trouble you're causing." There was the sound of typing. "The nearest airport is sixty-three miles away. How utterly primitive. All right. I'm sending a car. They'll be there at six o'clock tomorrow morning or I will shoot somebody."
"There's a Motel Six at the juncture of the 101 with the Miramar highway. I'll meet the driver out front. Have him ask for Mr. Smith."
"A Motel Six, my, we are slumming."
"Can I ask why I need to show up?"
"Ami has worked her magic. That's all I'm allowed to tell you. Ta-ta, darling."
When Connor turned around, Sylvie was standing in the restaurant's open doorway. "Everything all right?"
Connor found himself fighting the day's second urge to confess, this one even stronger. But there were customers to be served, and a wall of questions for which Connor had no answers. So he said, "I'm trying to cut off some loose ends."
She nodded, as though that made perfect sense. "Where are they, these loose ends?"
"Down in Los Angeles. I need to travel there tomorrow."
"Saturdays are often our busiest night. Can it wait?"
"No, Sylvie. I'm sorry. It can't."
She hesitated long enough for Connor to know the question before she asked, "Are you coming back?"
"As soon as I possibly can. I'll try and make it back before the restaurant opens tomorrow evening."
"It's a long way to go for a few hours."
"But the hours are important," he assured her. "Vital. If I can't make it back, I'll phone."
"All right, then." A hint of that special smile returned. "I have to cut you some slack after last night."
He did his best to return her smile, though it felt like the day's first lie. "Thank you."
"Only, now you owe me some serious time on the ivories." She gestured for him to follow her back inside. "We're talking hours."
* * *
Porter Wright came in a little after ten that evening. The place was winding down after a typically frantic Friday. The police chief wore his tan uniform, as rumpled as the man himself. He approached Sylvie's station, and whatever he said was enough to blanch her features ash-white. Connor tried not to stare, but he feared she was having another migraine attack.
Marcela followed him into the kitchen, where she explained, "The day you showed up, Bruno found ten kilos of coke. In a couple of our fish."
Connor said, "Get. Out."
"Eleven," Bruno corrected. He hammered his cleaver through a lamb shank and buried the blade into the chopping board. "Eleven keys."
Rick pushed through the doors and stopped by Marcela's other side. His expression said he knew what they were discussing. Marcela went on, saying, "The detective handling narcotics is a real piece of work. I think that's probably behind whatever just ruined the lady's night."
Connor asked, "Porter can't bring the detective into line?"
"Different forces," Rick replied. "Major crimes means the sheriff's department becomes involved."
Connor understood why they were having this conversation. "You want me to play for her again. No problem."
They showed genuine satisfaction. "Hold off until things settle down," Rick said.
Marcela said, "I'll finish your tables."
"Thanks. Listen, I may miss tomorrow's shift. I'm headed down to LA. I'll try to make it back before Castaways opens, but I can't say for certain."
"Sylvie told us," Rick said. "She's arranged for an old pal to take up the slack."
Marcela tightened the distance. "Are you coming back?"
"Absolutely," Rick replied for Connor. "The man doesn't get away that easy."
Sylvie stepped through the doorway. Her features held to that uncommon tight cast. She said to Connor, "Porter wants to have a word."
* * *
Porter Wright offered a politician's greeting, cheery and loud enough to be heard by anybody listening, which was almost everyone. "How are you settling in, Connor?"
"Pretty well, thanks."
"Carol is still talking about you and that wine. Really made for a special night."
Marcela stepped up beside them. "Carol can do that all by herself."
"You got that right." Porter's gaze made a lie of his smile, hard and cautious. "Carol was so taken with you, she actually came up with a little gift all her very own."
Connor sensed there were actually two conversations taking place. "Really, Chief, it isn't necessary."
"First of all, I'm Porter to my friends. And second, did I say she'd given me any choice in the matter?" He turned to Sylvie and asked, "Mind if I borrow the gentleman for a minute?"
"Not long, please, we're busy." But there was a mechanical rote to Sylvie's words.
"I'll cover for him," Marcela said. She added to Connor, "See what happens when you bribe a cop?"
"Couple of minutes is all." As Porter passed Sylvie's station, he touched her arm and said quietly, "Remember what I said. You're not in this alone."
If Sylvie even heard him, she gave no sign.
Connor waited until they crossed the street, and entered an empty side lane, to ask, "What's going on with Sylvie?"
"That business is strictly between her and me," Porter replied, unlocking his cruiser and opening his door. "You should be thankful I'm so good at keeping secrets."
Connor slipped into the passenger seat, fairly certain he knew what was coming down the pipe. Even so, his gut took a swooping dive when Porter said, "Carol knows. She said to tell you, she won't be the one to let this particular cat out of the bag. But there's something you need to see."
Porter reached over, opened the glove compartment, and pulled out an iPad whose pink floral cover said it belonged either to his wife or daughter. "These gadgets are beyond me. But Carol said all I needed to do was turn it on and hit . . . Yeah, here we go."
Within three seconds of the screen flashing to life, Connor knew he saw his agent's work. On one hand, he had to admire it as a stroke of brilliance. On the other, it made him sick to his stomach to watch.
Kali Lyndon was at her styled and coiffed and manufactured best. Her rouged cheeks were adorned with tears that glistened like liquid jewels, which they were. Those tears were a special blend of glycerin and microscopic flecks of diamond dust to catch the spotlights.
She was, Connor had to admit, one stunning and alluring lady. Kali was a modern combination of inherited money, daily trainers, and cosmetic surgery. Rich, fit, and voluptuous.
Her eyes were the giveaway. They were a lovely shade of lavender gray, thanks to special tinted lenses. But there was nothing all the specialists and handlers could do about the empty space down deep. Connor stared into that beautiful face and thought of Rick's words out by the wagon-wheel table. He wondered if his own eyes held any more life or heat than Kali's.
Peyton Stein, the cable lollipop, said, "Kali, you've just received word that Connor Larkin has run away from the marriage. How does that make you feel?"
Kali's response was interrupted twice for practiced sobs. When the camera shifted on the second to show Peyton offering a tissue and a sympathetic look, Connor chuckled.
Preston said, "Your fiancée is working on a broken heart and you're laughing?"
Connor replied, "There's no way Kali could have gotten through that much dialogue in one take."
"Wait, you mean this was rehearsed?"
"And still she missed her cue. So they cut over to Peyton. It gives them the chance to keep what she got right and reshoot the rest."
"Man, that is just cold."
"You have no idea."
Kali was saying, "I know if I could just have ten minutes with Connor, even five, I could change his heart."
Peyton said, "You mean, change his mind, don't you, Kali?"
"His heart has to rule this decision," Kali replied. "No matter how afraid he might be, if he truly loves me . . ."
"And if he doesn't, Kali?"
"He loves me."
"But what can you do? Connor Larkin has run away."
Porter said, "Here's what Carol wanted you to see."
Kali said, "I'm offering a hundred-thousand-dollar reward to anyone who helps me locate my Connor."
They bounced that ball back and forth a couple of times. Connor thought Peyton's shock over the reward was totally overdone, especially when they flashed a phone number at the bottom of the screen. This told anyone with half a brain that the whole deal had been set up in advance.
When the segment ended, Porter stowed the device back in the glove box. "I'd give a moth landing on a bug zapper better odds than you."
"The wheels are grinding down in LA," Connor replied, and gave a swift recap of what had happened earlier that day.
Porter shook his head. "Hollywood might as well be circling out there beyond Neptune."
"A car's picking me up at six tomorrow morning from the Motel Six."
"Might be a good idea if you let me take you there tonight. I'll drive you up and book the room, so there won't be any questions from the front desk."
"I'll pay you back."
"You better believe you're paying."
Connor thought of his promise to play for Sylvie. "It could be late before I'm done here tonight."
"No problem. I'm on duty till four." Porter slipped a card from his pocket and passed it over. "Call me when you're ready."
CHAPTER 20
After closing, Connor hung around and played a few tunes. But his thoughts were two hundred and thirty miles to the south. He felt like the lyrics were just lies set to music. Finally Connor stopped playing, but remained where he was, because Sylvie moved over from the bar and settled onto the bench beside him. She sat there, staring at the empty keys where his hands had been, and asked once more, "Will I see you again?"
"Yes."
"You're not just going to vanish in a puff of LA haze."
"I told you. I plan to work tomorrow's shift. Otherwise, I'll call."
Sylvie responded by wrapping one arm around his waist and settling her head on his shoulder. Connor smelled a mix of fragrances in her hair, long hours and restaurant flavors and a hint of old perfume. Her warmth was as exquisite as her scent. "Have you written any songs of your own?"
"A few. But before I quit playing, I was thinking I'd like to take hits from the seventies, eighties, and nineties and rework them in a sort of signature swing."
Swing purists loathed the very idea. The few times Connor had mentioned it, they had called him a traitor to the cause. He had not spoken of it in years.
All Sylvie said was "You should never let go of your dream."
Connor turned to her, hoping to find some way to say how deeply those words touched him. He was met by her rich gray gaze, deep enough for him to dive into and just keep falling.
Kissing her was the most natural thing Connor had done in a very long while.
The moment felt so right as it happened.
And so utterly, terribly wrong the instant it was over.
Connor gently pried himself off the bench. "I have to go."
"Really?"
"Yes. Now." He crossed the restaurant and unlocked the door. "Good night, Sylvie."
* * *
Connor packed an overnight bag, phoned the chief, then crossed the parking lot and knocked on Estelle's door. Soon as she appeared, he knew Estelle had not been asleep, and that he was right to come.
Connor told her, "Your daughter is one of the finest people I have ever met."
Estelle pushed open the door. "Come in and tell me why."
"No, Estelle."
"Connor, please."
"No!" His pent-up frustration pushed her back a step. "Estelle, you need to go to her."
"I can't."
"You can't let this chance slip away! Sylvie is going through a really rough time. She needs you!"
"I'll only make it worse."
Her tears forced Connor to gentle his tone. "Maybe. But you need to take that chance. For both of you."
Estelle wiped her face. "There's a policeman watching us."
Connor picked up his satchel and stepped back. "Promise me you'll speak to her."
"I'll . . . try."
Connor stood there until she shut and locked her door. He walked over to where Porter stood by his car. "Thanks again for doing this."
"No problem." When they were both seated in the cruiser, Porter asked, "Want to tell me what that was about?"
"No." Connor looked back to where the light framed Estelle's window. "Definitely not."
Porter started the engine and pulled from the lot. "Strange way for you to repay a kindness. Refusing to tell an officer of the law something he might need to know about."
Connor responded, "What did you tell Sylvie that got her so upset tonight?"
"That was a highly confidential matter, and none of your business."
Connor waited.
"Are you suggesting that the chief of police trade information with a man he knows to be operating under false pretenses?"
"Absolutely."
Porter took the valley road out of town. "You first."
"Her name is Estelle Rainier," Connor said. "She's Sylvie's mother."
The road dipped and weaved through the night. "I know I've heard something about that lady. . . ."
"Estelle abandoned them when Sylvie was a kid. They haven't spoken in nineteen years."
Porter swung the cruiser over the final rise, then descended into the farm valley. The fields stretched out to the distant moonlit hills. "Does Sylvie know the lady is here?"
"No. Estelle is afraid to approach her. This, after hiring a detective to find Sylvie."
Porter glanced over. "How'd you get in the middle of that one?"
"I have no idea."
Porter's laugh was a soft, comforting rumble. "You're here in town, what, all of . . ."
"Three days," Connor replied. "Give or take an hour."
"You got a job you don't need—"
"I need it," Connor corrected. "Desperately."
"You're in tight with a girl you definitely don't need. . . ."
Connor knew Porter was waiting for a comeback, but he had no idea how to respond.
Connor's silence only made the chief laugh once more. "Now the lady's mother is all over your case."
"She knows who I really am," Connor said. "Estelle hasn't said anything, but I'm positive she knew from the first moment we spoke."
Porter's laughter bounced around the car. "Carol is going to die, she missed all this."
"You forgot getting rousted by the chief of police," Connor said. "Sort of."
This only made Porter laugh harder. "There ain't no such thing as a sort-of roust. Not with me."
"Your turn," Connor said.
The chief went quiet, then, "You heard about them finding drugs?"
"Eleven keys in the fish," Connor said. "I heard."
"The detective handling the case is not high on my list of good people."
"So I've been told."
"Yeah, well, he's convinced the county prosecutor they should treat Sylvie as their prime suspect. He insists the trawler in question was using Castaways as a conduit for dealers operating around Santa Cruz and Paso Robles."
"You have got to be kidding me."
"I wish. But given the restaurant's history and Sylvie's own past . . . Did she tell you about her upbringing?"
"A little."
"Her dad was arrested a number of times for trespassing, probably parked that old camper on private property and left it there long enough for the owners to bring in the law. Sylvie's got her own sheet—but because she was a minor, it's all sealed. But the records do show how the county repeatedly tried to put her in foster care. There were a couple of court cases, finally dropped because she was fierce about staying up with her schoolwork."
Connor rubbed the sore point over his heart. "She is one amazing lady."
"You got that right." Porter kneaded the steering wheel with two massive fists. "I'd like to wring that detective's neck."
"You told her tonight about the possible arrest warrant because you want her to go ahead and hire a lawyer," Connor guessed. "Which Sylvie can't afford."
"That pretty much sums it up," Porter agreed.
Connor looked back through the wire-mesh screen, out to where the road disappeared over the rise. "There's too much of everything in your town."
Porter glanced over, but he did not speak.
"Too much honesty," Connor went on. "Too many raw emotions. Too many blows coming out of nowhere." Connor turned back around. "Too many reasons to care."
"Welcome to Miramar." Porter slipped into the motel's forecourt and cut the motor. "That's my town in a nutshell."
"Thanks again," Connor said. "For everything."
Porter jerked his chin toward the entrance. "Everything's set up. They've accepted the same story as the Miramar guesthouse because it came from me. You've lost your wallet, you'll be paying cash." As Connor rose from the car, Porter said, "You have to tell Sylvie who you are."
"I know."
"And soon," Porter insisted. "Sylvie needs to hear this from you."
Connor shut his door, then said through the open window, "She's lucky to have you for a friend."
Porter took that as the farewell, nodded, and put the car into gear. Connor stood there as the cruiser pulled out of the lot.
Trucks rumbled along the highway, pulling him away from Miramar. All the people he was coming to call friends. Most especially a woman who deserved to learn the truth about the man she kissed. The mysteries and the questions. All of it belonged to a haven he wished he could stay in a little longer. And come to call his own.
CHAPTER 21
After tossing and turning for a futile hour, Connor rose from his Motel 6 bed and went downstairs to the guests' laundry room. He had once found great comfort in sitting and watching the machines. Tonight, however, the steady rhythm only reflected his churning thoughts. He carried his clean clothes back upstairs and ironed the jeans and T-shirt he intended to wear for the journey south. Finally, around two, he lay back down and eventually drifted away. His dreams were fractured glimpses of a woman's lovely gaze, and lips he probably should never have tasted. But which he yearned to kiss again.
Connor was in the Motel 6 lobby at a quarter to six. After so many disjointed nights, his eyes felt grainy and his thoughts muffled. The night clerk was handing over to his replacement, and their happy chatter drilled at his brain. He stepped through the sliding glass doors into the frigid dawn. He was wondering if he might wake up before he froze, when the car pulled into the lot.
The sight brought both clerks outside. The night clerk was hefty and the girl on dayshift was a slender waif, both were in their twenties, and both completely agog at Connor's ride.
The guy said, "Is that a Rolls?"
"A 1956 Silver Cloud," Connor confirmed. Kali's people had selected it as the car to take her to church, and then launch them into their honeymoon.
The girl said, "That car is bigger than my apartment."
The guy said, "I didn't know they made cars that white."
The girl said, "Are you, like, famous?"
The guy said, "When that cop dropped you off last night, I thought, you know, we had a serious criminal on our hands."
The girl said, "Oh, come on. The Mafia comes to Miramar? Please."
The driver rose from the car and asked, "Is one of you a Mister Smith?"
"That would be me," Connor said.
"Sorry to keep you waiting, sir." The chauffeur scampered around to open his door. "We better get started. You're due on set in less than five hours."
As Connor slipped into the rear seat, the girl said, "Happy trails."
CHAPTER 22
A script in the traditional blue cover embossed with the CPP logo was waiting for Connor. The seat was softest ivory leather piped in gray. The carpet was matching gray silk velvet. There was enough legroom for Connor to stretch out fully. The front seat was backed by a full bar, each item carefully nestled in padded holders. Beside the sterling silver ice bucket was a rack holding bottles of single-malt whisky, signature vodka, Puerto Rican rum, and six hand-cut crystal glasses. The fridge held minis of soft drinks and fruit juices. Connor opened the panel next to the fridge and found a cheese plate, a fruit bowl, and seven sandwiches with rare roast beef and creamed horseradish on rye.
Connor selected one of the sandwiches and poured himself a mug of coffee from the silver thermos. He opened the script and began to read. He went through it slowly and finished in twenty-five minutes. There was a handwritten note on the last page that read, Phone as soon as you're done, Gerald.
Connor asked the driver, "Can I borrow a pen?"
"There should be one in the armrest, sir. Along with a lapel mic and battery pack."
"So there is." Connor started over, this time working out his part's tempo and emotional bearing. After the third reading, Connor was ready to make the call.
When Gerald answered, Connor said, "The bridal Rolls? Really?"
Gerald revealed a laugh like a stork. Ack-ack-ack-ack. "I so wish I could have seen your face."
"At least you didn't order them to tape white ribbons running along the hood."
"I tried, believe me, but the company said the wind would tear them apart. I was very disappointed."
Connor asked, "Did you write this script?"
"Well, I had to. What they sent over was such drivel."
"This is actually very good."
"Now you're making fun of me."
"I'm serious. You've given me a solid set of lines here. The emotional cadence is steady and builds to a genuine crescendo. You ever want a job writing for soaps, let me know. I'll introduce you to my agent."
"Ha. Ha. This is me, not laughing."
"Really, Gerald. Good work."
The genuine compliment left Ami Chen's assistant at a loss for words. He cleared his throat, then said, "Yes, well. There's a downside to all this, I'm afraid."
"Go on, then."
"The episode is going out live."
Connor leaned back in his seat. It all made sense now. The Rolls had been sent so he would show up for his final act in the limo that would have carried them from the ceremony. The script, the urgency, it all came down to this.
Gerald said, "Helloooo?"
Connor replied, "This is actually a very good thing."
"I'm sorry. There must be a problem with this connection. I thought you just said—"
"It means we do this in one go. No outs, no reworks. I don't have to repeatedly dredge up these emotions for all the takes Kali usually requires."
"Ami said you would like it. I didn't believe her. Kali's people shrieked so loud they set dogs to howling in Tijuana."
"I can imagine."
"The cable people are swarming. But really, the attention this is getting. The Internet has been on fire."
"I think I should go in raw," Connor said. "No makeup, no hair, no rehearsals."
Gerald mulled that over. "What are you wearing?"
"T-shirt, jeans, rope belt, canvas boots."
Gerald played the stork again. Ack-ack-ack-ack. "Kali's people will need oxygen and those paddle thingies to restart what passes for their hearts."
"Tell them my rough state will heighten the impact of my lines. Also, it will show how we were never really meant to be together."
"The perfectly groomed lady and the gardener." Ack-ack-ack-ack .
"Is that a yes?"
"I probably shouldn't say this, but it will be a pleasure. You have no idea what a pain they have been."
Actually, Connor knew all too well. "Gerald, I owe you big time."
"I have a penchant for vintage champagne and lavender roses. Four dozen is a nice round number."
"I was thinking," Connor replied, "that it makes up for making me show up in this bridal boat."
Ack-ack-ack-ack. Gerald hung up, still laughing.
CHAPTER 23
Sylvie rose early Saturday and went for another walk along the shoreline path. The lane was crowded today with weekend joggers and cyclists, many of whom puffed out greetings as they passed. Sylvie recognized the faces from her clientele. The day was crystal clear, the sky a pale wash of porcelain blue. Sylvie spent much of the walk telling herself that she should not miss Connor quite so much as she did. She tried not to worry over whatever mystery errand had taken him down to LA, or how it might keep him there.
When she returned home, Sylvie made a fresh pot of coffee and opened her treasure chest. The box had been a gift from Rick and Marcela to mark the restaurant's first anniversary. It remained one of her most cherished possessions. The proper name was a Victorian travel writing desk, and it had been designed for use by adventurers exploring the world's far-flung reaches. The top was slanted and covered with an inlaid leather pad. The front folded out to reveal an inkstand and letter holder. One interior drawer held recipes, the other articles on wines and spices. The larger central space beneath the lid contained ideas for redesigning the restaurant's interior and menus from other restaurants she admired.
She had planned to spend the hours as she often did on Saturdays, leafing through her clippings and printouts of menus, working up a new rendition all her very own. She had a leather-bound notebook where she jotted down ideas, fashioning them over weeks and months until she was ready to spring them on her kitchen staff. Carl was the perfect sounding board. He listened well, remembered everything, and rarely commented with more than a nod. Then he would work through the idea and try it on the staff. Together they would incorporate these comments; and if it worked, they would present it as a weekly special.
Her mind kept returning, though, to that vile detective and the prospect of being brought up on charges of smuggling drugs. She hated the resulting sense of helplessness, how little of this was under her control. She had no choice but to wait it out, and hope for the best. The longer she sat there, the more the cloud grew, skulking dark and heavy on her mental horizon.
Finally she stowed away her half-formed recipes and touched the interior sidewall. It gave a little click and came free in her hand, revealing the box's secret compartment. For the first time in nine months, Sylvie pulled out the bundle of pages folded and tied with a lavender silk ribbon. She had started this wish list as a child. Sylvie loved Miramar, and had no intention of ever living anywhere else. However, she was still her father's child, and the urge to wander remained strong within her.
But she did not want to travel alone.
Her father's company had made the road their friend. She wanted this again. She wanted to journey with someone who shared her hunger for new horizons. She wanted . . .
She unfolded one of the oldest items, a stained and faded pamphlet for Machu Picchu, the mystical ruins in the Peruvian Andes. The words were barely legible now, but it did not matter. She could recite the entire brochure from memory. She turned the fragile pages, and thought how nice it might be to share this dream with Connor.
As she retied her ribbon and refit the wooden sleeve over the secret compartment, Sylvie smiled over the recollection of Connor's kiss.
Sylvie had been astonished by how good it had felt.
How rich the flavor of his lips had been.
How she wanted to taste him again.
How she could still feel the strength of his arms as he held her.
How much she wanted to have him hold her again.
And never let her go.
CHAPTER 24
Connor ran through the script a final time; then he stretched out on the backseat. He assumed the dread prospect of what awaited him at the end of this ridiculous journey would keep him awake, but the interrupted nights and hard work and emotional upheaval that had led up to this moment served as a balm. He was asleep in the space of three breaths.
He dreamed of walking along the Miramar shoreline with Sylvie. It was the same as their time together, only much richer. In his dream, they had been doing this for years, sharing hundreds of dawns. Thousands. He talked about himself and his latest acting gig. He shared from one heart; she received with the same heart. They were that bound together.
He was jerked from the dream by the sound of his phone buzzing. Connor fumbled and pushed himself upright, then dry scrubbed his face and struggled to fit his fractured world back together. The luxurious Rolls-Royce and the silent driver and the script on the velvet carpet all seemed tawdry.
The phone went silent. Connor poured himself a cup of coffee. The dream had seemed so real; the loss of connection to Sylvie left his chest hollow. He tried to tell himself that it was impossible to miss this woman, especially with everything he had waiting for him in LA. However, the dream's impact would not be denied.
When the phone started buzzing again, Connor saw Gerald's number, hit the connection, and complained, "You just woke me from the best sleep of the week."
"Don't you dare take that tone of voice. Kali has been running around here screaming at people. I hate when people scream. That woman has such a voice."
"Tell me."
"And the language. Between her and the director and that Peyton, I was blushing."
"You're enjoying this," Connor stated.
"Oh, all right. This is actually more fun than Mardi Gras. But don't tell anybody I said that." Gerald released another of his trademark laughs. Ack-ack-ack-ack. "When I told Tony what you were wearing, his shriek broke windows in Burbank."
Tony was the show's director. Peyton and Kali both adored him. The behind-camera crew referred to him as Tony the Toad.
Gerald asked, "How far away are you?"
Connor leaned forward and passed on the question, then told Gerald, "The car's GPS says thirty-two minutes."
Gerald asked, "Some cable executive paraded through here half an hour ago. She claimed we are about to knock Days of Our Lives off its ratings perch. I thought Tony was going to have a hernia."
"Sorry I missed that."
"Well, at least you'll be here for the main event." Gerald played the stork a final time, and cut the connection.
Connor opened the central armrest and clipped the mic to the inside of his shirt's lapel, ran the wire down next to his skin, then slipped the battery pack into his back pocket. There was also a fold-out cosmetic mirror, which revealed an uneven two-day growth the color of sunlight through honey. It turned Connor's cheeks as cavernous as his eyes. His hair was tousled; and when sunlight glanced through the side window, he winced. Connor folded the mirror away and poured himself another cup of coffee. He decided his appearance fit the role perfectly.
* * *
Kali Lyndon's father had been a nondescript gentleman who wore his pin-striped suits even at the family dinner table. He had lived in the shadows, shunned publicity, and never had anything bad to say about his only child. Kali's mother died when she was seven, and she was raised by a variety of nannies. Her father's empire continued to grow, even after he confessed he wasn't sure why he felt a need to make more money than he would ever spend. They lived in a fine house in a nice area of St Louis. Her father moved in the circles of power when he was forced to, but he preferred to send his associates and remain the quiet, unassuming man he was. He died of heart failure when Kali was nineteen. At that point, his closely held empire of hotels and shopping centers was estimated to be worth over four billion dollars. He left everything to his beloved daughter.
Kali's attendance at the board of directors' annual meeting remained her only contact with the business her father had built.
So long as Connor played second lead to Kali's star, they actually got on great. Connor found her funny and endearing. He ignored her temper tantrums with the same deaf ease he had shown to any number of stars on set. Kali had never met anyone like him. She claimed Connor was the first man who could handle her. Connor thought it might be at least partly true.
Kali Lyndon's world revolved around playing the poor little rich girl. She was seen and photographed at every star-studded opening. She did the latest, wore the finest, was friends with the hippest. She shone for the cameras. But Kali had a problem. She needed to be fed her lines. So long as she was scripted, Kali was in her element.
Which was how her publicity machine came up with the idea for this wedding.
A fake reality show.
The cable network ate it up.
Kali's estate fronted the coveted ninth green of the Bel-Air Country Club golf course. Her home was nestled in three and a half acres of meticulous gardens and fountains and four swimming pools. The mansion itself covered twenty-three thousand square feet and had four turrets. In addition, there were six garages, a poolhouse, two guest cabanas, and servants' quarters. When asked why a single lady needed such a big place, Kali gave one of three stock answers. Because I can. Because it's fun. Or her favorite, I like getting lost and discovering rooms I've never seen before.
The camera crew and tech support and Tony the Toad were clustered just outside the mansion's main gates. Connor stowed the script away, slipped over to the side facing the cameras, and put on his game face.
Showtime.
EXT. GATES TO KALI'S ESTATE. DAY.
ESTABLISHING SHOT: The pale stone wall extends in both directions, topped by black steel spikes. The only entry is a pair of tall metal gates embossed with the initials, KL. The gates are closed. Two UNIFORMED GUARDS stand by the pillars.
The white Rolls-Royce bearing CONNOR LARKIN pulls up to the gates. Connor's face is visible through the open rear window.
CONNOR
I'm here to see Kali. My name—
GUARD ONE
I know who you are.
Guard One pulls a phone from his pocket. His eyes never leave Connor as he speed-dials a number and speaks softly.
GUARD TWO
Being that stupid is a crime, right?
Guard One cuts the connection and uses a key to open the gates.
GUARD ONE
Some states it's a felony.
The Rolls pulls slowly through the gates.
CLOSE-UP on Connor's face. His expression says this is exactly the reception he deserves.
INT. KALI LYNDON'S OFFICE. DAY.
KALI LYNDON sits at her desk. She is dressed in pastel tights and a pale yellow off-the-shoulder sweatshirt. She looks beautiful, tragic, and utterly vulnerable. A tear rolls down one cheek.
Kali is writing, or at least trying to write. Before her rises a vast pile of engraved wedding acceptances.
CLOSE-UP handwritten on the card directly in front of Kali are the words GIFT: ANTIQUE SILVER SERVICE.
Kali's hand holds a silver pen. She has written a few words on a sheet of her personal stationery, but now she is halted by sorrow.
KALI (VOICE-OVER)
Dear Clarissa, Thank you so much for the lovely gift, which unfortunately I must return because
Connor and I . . .
The voice stops where her hand has frozen.
CLOSE-UP as a tear falls onto the unfinished letter.
INT. ENTRANCE TO KALI'S OFFICE. DAY.
ERICA, Kali's private secretary, knocks on the open door.
ERICA
He's here.
Connor's arrival was tracked by a platoon led by Tony the Toad. The cinematographer was on point, aiming his Steadicam at Connor's face. The second cameraman was hidden behind the left-hand column, tracking the Rolls and showing what Connor saw, the house and the grounds and all the wealth he was walking away from.
Outside the camera's view, everything was in frantic motion. Working on so-called reality television meant learning how to ignore the sound boom hovering overhead and tracking his every motion, the constantly shifting lights, the assistants handling the cables and running interference on Tony the Toad, who was busy hissing into his radio and waving his hands at everybody, especially Connor.
The Rolls pulled around the circular drive and halted by the bottom step. On cue, Kali's latest assistant opened the massive front door and stepped onto the veranda. As Connor rose from the limo, the Steadicam operator shifted position so as to show . . .
A catering truck was parked between the main house and the garages built to look like French stables. Beyond them, a work crew was busy dismantling the wedding tent. At a hysterical cue from Tony, the entire catering staff froze in the process of loading bottles and glasses and silverware back into the truck. They all turned and glared at Connor.
From her position on the top step, Erica greeted Connor with a volcanic scowl. When he reached the top step, she wheeled about and led him through the manor's open door.
The third camera guy and the assistant sound guy and their assistants were all on position just inside the portal. Connor stopped midway across the inlaid granite and marble floor, when his foot hit the piece of tape with his name on it. He gave Kali's palace and the life he was throwing away a long, slow look. This allowed the two outside cameras and their teams to rush around the side of the house, fly through the kitchen door, and tiptoe into positions. At another bat-wing motion from Tony, Connor turned to the sweeping staircase and Erica's burning gaze.
Erica's smoldering walk along the upstairs hall was tracked by camera two. Connor followed at a slower pace, holding to the beat structure laid out in the script. His every step was duplicated by a change of camera angle. One step, a survey of the idyllic life he was tossing aside. Another step, back to the camera focused on his face, close-up on the expression of a man wracked by doubt and regret. Connor had no trouble showing those emotions. They were exactly what he was feeling—only not for the reasons the audience expected.
A final close-up on his face, building on the tension and the guilt and the uncertainty, and then Connor stepped past the assistant's blistering glare and stepped through Kali's office doorway.
The Steadicam operator nudged Connor's ankle, and he shifted slightly to the left, allowing the guy to get a full-on shot of Kali almost falling apart. She made a huge effort, maintained a shred of control, and uttered her first line:
"Why, darling? Why?"
CHAPTER 25
The three mics taped to the ceiling of Kali's office looked to Connor like giant black caterpillars. Metal light-stands stood in both corners behind Connor, bathing Kali in perfect illumination. One cameraman crouched by the window. Another stood on a mini-ladder and shot over Connor's shoulder. Cables snaked all over the floor. Tony the Toad raced back and forth between watching the shoot through the open office doorway and studying the monitors stationed in the guest room next door.
Connor knew his shoddy, sleepless state heightened Kali's polished beauty. It suggested that he and Kali had already broken apart, and really they'd had no business being together in the first place. This was the structure of good drama, where the setting amplified the action and the dialogue.
Connor's lines confirmed what his presence already revealed. He was not good enough for Kali. He could never live up to her expectations. He wished her happiness. He hoped she found the man she deserved....
Despite the crowd and the lights and the hollowness of breaking up for the cameras, Connor thought the drama carried a solid punch.
Like most experienced television actors, Connor had developed a mental clock with a precise second-by-second counter. He could also see when Kali forgot the last bit of dialogue she was supposed to deliver before they left the office. As a result, he completed his line by inserting the first word of her own. In response, she flashed him a tiny hint of the mischievous Kali, the lady who in private moments had referred to the reality charade as a playtime for adults. He had always been genuinely fond of her spirit, and seeing it now added an extra spice to their final exchange. As they left the office hand in hand, and walked the hall and down the stairs and across the palatial foyer, Connor sensed that the entire team was delighted.
They arrived outside the front door within five or six seconds of perfect. Connor had hoped they might have a private moment, but it was not to be. The second and third camera crews had slipped out the side entry and were now ready to track his departure.
The Rolls was gone. His satchel sat forlornly in the middle of the drive.
This was definitely not scripted.
That could only mean they had intended to catch his surprise on live camera. The significance was clear enough. Connor had arrived care of Kali's money. He left on his own steam.
Tony the Toad at work.
Connor gave them five seconds to register his shock for the audience, then turned and looked up to where Kali stood on the top step. He gave his final lines, "Good-bye, Kali. I'm sorry."
"I'm sorry, too. More than you will ever know."
The script had him descending the stairs and walking to the car, where he was to pause for a final, sorrowful glance at the woman and the wealth and the life. Then the camera was to track the car to the gates, pausing there for the guards to offer a caustic farewell. Connor figured it would work just as well if he hoofed it.
Kali, though, interrupted the beat structure by going totally off grid. "Connor, wait."
The sound technician had already started descending the stairs with Connor, and he swung the boom back fast enough to have taken off somebody's head. With a rare flash of genuine emotion, Kali said, "Tell me why!"
Connor was enough of a pro to give the Steadicam operator time to get them both back in frame. He replied, "This is me, Kali."
"I know that."
"No. You know the me I was for you. But this is me. And the truth is, I don't belong in your scene."
"You said you loved it!"
"I did, for a night. Or a weekend. But all of this, the house, the PR machine, the parties, the people—I just don't fit."
"I could change!"
He had to smile. "I believe you would."
"Really, I mean it."
"For a night, sure. Or a weekend. But for good? Kali, I want the life your dad loved."
That stopped her cold.
"The small town. The real people. The simple life."
"You'll stop acting?"
"Not unless I have to. Your dad stayed a businessman. I hope I can do this, too. Hold on to some kind of balance."
Kali struggled with that. "Why are you only telling me now?"
"I just figured it out. That's what came from running away." He climbed back up the steps and kissed her cheek. He tasted salt, and realized these latest tears were real. "Good-bye, Kali. Be happy."
Tony the Toad said, "And we are off air."
CHAPTER 26
As they approached the Saturday afternoon opening time, Sylvie became filled with the sense of a storm building beyond the horizon. She knew from her GP that some patients with migraines had attacks whenever there was a big low coming. Apparently, the most sensitive patients were so impacted by changes in atmospheric pressure they could not get on a plane. Thankfully, this had never been an issue for her. Imagine predicting bad weather with a headache. Even so, Sylvie repeatedly checked the horizon as she prepared for work. But the sky remained blue, the wind mild and off the sea.
It was only when she arrived downstairs that she realized how right she had been about an approaching tempest.
Two of them.
Porter was seated at the otherwise empty bar, cradling a kitchen mug with his two scarred hands. Rick and Marcela stood together on the bar's other side, as though they needed one another's company. They all three shared the expression of funeral directors waiting to greet the recently bereaved.
Sandy came bustling in from the kitchen, bearing a plate of fresh pastries. "They're called petticoat tails. Don't ask me why. My mother swore by them. Said there wasn't a thing that couldn't be put straight by a few of these and a proper cuppa."
"Nix on the tea," Rick said. He reached under the bar and brought out what Sylvie referred to as her hidden stash, a bottle of sixty-year-old Bas-Armagnac she'd found during the renovations. He set a snifter on the bar and poured a liberal splash. "Sylvie, come sit. Now drink."
"That's meant for celebration," she replied.
"The lad's right," Sandy said, pushing the plate toward her, "and have a few of these to mop up the alcohol."
"I'm not the least bit hungry."
Marcela said, "Eat."
Sylvie knew she had no choice, not really. They carried bad news, something so terrible she needed to be dosed. One look at their collective expression convinced her that resistance was futile.
She tossed back the Armagnac and ate two of the tails, or whatever they were. The brandy drew a line of liquid fire down her throat. "All right. Tell me."
Porter waited until Rick refilled her snifter. Then he said, "The prosecutor is bringing formal charges against you. I convinced them to let me bring you in for the arrest and arraignment on Monday, rather than create a scene here. Possession with intent."
Sylvie was very glad for the stool's support. "That's insane."
"They've come up with nothing that will hold up in court," Rick agreed. "Not a single definite lead. This is a face-saving measure. They have to know they can't bring a verdict against you."
"But if I'm indicted I'll be tried and convicted in the court of public opinion," Sylvie protested.
"If people around here didn't care for you as much as they did, I might agree," Porter said. "As it is, this town will definitely think otherwise."
"This is awful."
"It's not good." Rick unbuttoned his shirt pocket and drew out a sheet of paper. "Officially, Porter can't recommend a lawyer."
"Rick . . ."
"Hear me out." He unfolded the sheet and flattened it on the bar between them. "These three are the region's best defense attorneys. I have it on best authority."
Sandy said, "A gutter brawler in your corner is just the ticket."
Rick went on. "I've had an unofficial word with all three. You need to contact them tomorrow. They'll take your call, despite it being a Sunday. Make your choice and sign them on."
Sylvie asked weakly, "How much?"
"All of them ask about the same," Rick replied. "I know because I checked. They'll want a retainer of twenty thousand and another fifty if you go to trial. Which, in my opinion, you won't. Not if you come into the initial arraignment with your guns blazing."
Sylvie pressed both hands to her middle. There were only two places where she could get that kind of cash. Neither was the least bit appealing.
She had falsely assumed such bad times were relegated to her past. Things were different now. She was a citizen of the town she loved. She cared for the town and the people. Somehow that should have been enough to protect her from this sort of calamity. Because that was what it was, really. Everything she made went to paying off the loans she had taken out to first buy and then renovate this beautiful place, her restaurant.
Now she just might lose it.
She realized with a start that they were all silent. Waiting. Sylvie took her time, inspecting each face in turn. It turned out to have been the exact right thing to do at that moment. There in their expressions was the confirmation of all the rightness this place held for her: the caring concern, the friendship, the offering of strength when she was at her weakest.
It granted Sylvie the ability to say, in all truthfulness, "I'll find a way through this."
"Of course you will," Marcela said. "And we'll help."
Sylvie told herself it was utterly unbecoming for a boss to break down and weep, especially in front of the police chief and three of her employees. "I can't tell you what this means. But I couldn't possibly ask you . . ."
"One step at a time," Porter said. "Choose your attorney. Get them up to speed."
"I don't know what to say."
"Good," Marcela said. "At least that's settled."
"Done and dusted," Sandy agreed.
Rick tapped the rim of the snifter with one fingernail, making it ting. "Now get ready for round two."
"What?"
Marcela nudged the glass closer. "Drink."
The only sensation Sylvie had from the two stiff brandies was a muffled distance from her surroundings. She observed the fact that she could not stop leaking tears. She saw how the grim demeanor they shared only grew more severe. The one comforting note was how they all shifted closer to her. It was as though they were determined to shelter her the very best way they could. Their concern only further clouded her vision.
Sylvie wiped her eyes and watched as Marcela drew a tablet from her purse and set it on the bar. Rick reached across the bar and took Sylvie's hand. Sylvie actually had difficulty hearing precisely what Marcela said. Something about Marcela's neighbors, two mad old ladies who loved Marcela and her husband to bits and made them all kinds of superfattening delicacies from their native Argentina. How the two old ladies were addicted to the daytime telenovelas. Only, they had gone on and on, this week, about some celebrity wedding....
Sylvie was not clear on exactly how things proceeded from that point onward. Something about the way they were all clustered about her, so close she could feel their unified strength and support, left her unable to focus intently upon anything.
Then she was struck by the sudden realization that Connor must have traveled south to Los Angeles for this wedding. There had to have been something terrible....
It could only be one thing: a traffic accident.
Marcela's words became a faint buzz. All Sylvie clearly understood was how sorry Marcela was to be the one to have to tell her this.
Sylvie took a hard breath, and asked, "What happened to Connor?"
In reply, Marcela turned the tablet around, brought up the YouTube website, and hit play.
Some artificially bright cable personality was going on and on about . . . Sylvie could not understand what she was hearing. How could this woman be so falsely happy?
Then the scene shifted. And there he was.
The sight of Connor's face appearing in the back of a white Rolls-Royce and speaking to the guards filled Sylvie with an icy calm. When Connor emerged from that ridiculous car and started toward the gaudy palace, she said, "Turn that off."
"I'm so sorry," Marcela said.
"It doesn't matter." Sylvie rose from the stool. "I'm going back upstairs."
"I'll come with you," Marcela said.
Sylvie started to object, then remained silent. She knew the reason they had all gathered about her had less to do with Connor's actions than the betrayal she had endured nineteen months earlier. Connor was a mere shadow of that awful time. She had only known this actor for less than a week. There was no way he could have any impact on her. No. What she felt was merely the sensation of being returned to the nightmare she thought she had finally left behind.
Or so she tried to tell herself.
She started to thank them for coming together and supporting her. But the words remained stuck somewhere inside her. The whole thing was absurd. How could she feel hurt and disappointed by a man she clearly did not know at all? How could Connor possibly wound her?
When Marcela came around the bar and took hold of her arm, Sylvie protested, "I'm fine."
"Of course you are. You're the finest person I know."
Sylvie felt a momentary burning from the words and the concern behind them. But she pushed that away as well. She had gotten through such an awful revelation before. This was nothing. The actor she knew as Connor Smith was of no importance whatsoever.
CHAPTER 27
A Lincoln Town Car waited for Connor around the first bend. When he opened the door, he was astonished to find Ami Chen seated in the back. She was dressed in what Connor had come to think of as standard agent chic—black silk slacks, black one-ply cashmere turtleneck, black suede pumps, dark purple pearls, and matching fingernail polish. Ami Chen was nothing if not predictable in her garb.
She had the laptop open on the central armrest. Her phone earpiece was slightly larger than Connor's thumb and had a two-carat diamond in the lacquered surface, a gift from her husband. She gestured for Connor to enter, then went back to scrolling through a contract on her tablet. "No, no, no, I can live with clause four. The fifth is out of the question, strike it out utterly.... Raymond, I am not making requests here. Shall I proceed? Thank you ever so. The next two clauses are acceptable, the last . . . Hang on and I'll ask." Ami turned to Connor. "Can you ride a bike?"
"Yes."
"I'm not talking about the kind with pedals."
"I figured that. My answer is the same."
Ami was just one-quarter Mandarin, but she liked to say it was the only part that mattered. She held to a distinctly Asian manner of tension and brusque command. Her small form reminded Connor of a black firecracker with the fuse lit. She leaned forward and asked the driver, "Why aren't we moving?"
"I'm still waiting for somebody to give me a destination, ma'am."
She said to Connor, "So tell."
"My house."
"Beverly Drive, south of Wilshire. Now go." To the phone: "Yes, Raymond, he knows bikes. All right. Hang on." She turned to Connor. "Raymond wants to know if you can do some of your own stunts."
"Absolutely," Connor replied. "Who is Raymond?"
"Casting agent and your new best friend. But only if what you claim about bikes is true. Raymond says that if you can't ride well enough for high-speed close-ups, they will void the offer."
"Tell him you'll call back in twenty minutes with everything he needs to know about me and bikes."
"Did you hear that, Raymond? How should I know? Evidence of one form or another, I assume. No, you're certainly not holding. I need to speak with Connor about other matters. No, you are only the center of the universe once we have a signed contract and a deposited first payment." Ami cut the connection. "You have to love the Raymonds of this world. There are too many to shoot."
Connor asked, "That was Bond?"
She pointed to the driver, shook her head, and asked, "Gerald told you about the Web traffic?"
"And the studio executive."
"Your breakup with the lovely Kali Lyndon could be the highest audience ratings the cable network receives this year. In the middle of the day." Ami showed a rare smile, the expression of a contented cat. "How did it go back there?"
"Between Gerald's redraft and the fact that we went out live, pretty good."
"Gerald watched the feed. He said you did quite well. Any regrets?"
"Absolutely. I wish I had never started down that road with Kali."
She surprised him by slipping off her glasses and revealing her piercing gaze. "Is my Connor growing up?"
"I sincerely hope so." He hesitated, then decided there would never be a better time to speak of what had come to him on the ride south. "Can I ask you something?"
"Go ahead and we'll see."
"If there was one thing about my acting that you'd like to have me change, what would it be?"
"Oh, now, that's a dangerous one."
"What, you don't think I can take it?"
"Oh, I suppose it's possible. The question is, do you want to go down that road? After all, you've made a highly successful career for yourself."
"Playing the bad boy."
"A particular kind of bad boy—the one who doesn't care. Your characters are so utterly detached they can laugh as they're tossed off the cliff."
The words struck deep, as Ami clearly knew they would. Connor needed a moment before he asked, "You don't want me to change that?"
"Of course not, darling, not if that is what the role calls for. Your audience adores this. But, well, let's face it. You're coasting. This type of role no longer challenges you."
"So you'd like me to . . ."
"Grow a heart. But only if you can keep playing these uncaring characters when they're called for. Just expand into new directions." She tilted her head, examining her impact. "Was that a horrid thing to say?"
Connor rubbed the skin over his chest. "Only because it's true."
* * *
When the limo pulled into his drive, Connor found himself studying the house with uncommon intensity.
Ami asked, "Something wrong?"
"No, it's just . . ." Connor suddenly felt foolish confessing. "I feel as though I haven't been here in years."
She tilted her head, examining him from a new angle. "What happened to you up there?"
"So much." But now wasn't the time to start on that.
Ami let Connor hold the front door for her, waited while he coded off the alarm, then moved slowly through his home's public rooms. Although Connor was pressed by the ticking clock, he did not rush her. He felt as though he saw it with her, the two of them studying a stranger's house. Then he excused himself and went upstairs for a quick shower and change of clothes.
Connor had bought the house on a whim. A friend who had tried his hand at acting and failed was now a realtor. When Connor had first begun to earn serious money as a television bad guy, the realtor had urged Connor to sink his new funds into this place, and to do it swiftly, because it would not remain on the market for more than a few hours. In fact, the house had not even been officially listed and Connor's was the first of a dozen offers that same morning. His bank manager was well accustomed to the swoop-and-dive incomes of LA actors, and had the mortgage details completed in ninety minutes. Before he even wrapped his head around the idea, Connor owned a home and a mortgage large enough to give him nightmares.
The same realtor referred Connor to another former actor, now a contractor specializing in home reconstructions. The place certainly had needed a great deal of work. The home was late Craftsman, a movement shaped around a rejection of both Victorian foppery and the mass-produced housing that was swamping much of America. The house had been built in the forties, expanded haphazardly in the sixties, again in the seventies, and then once more in the early nineties. The contractor had proposed to tear out much of the interior and bring the entire structure to its original style. After Connor had signed onto another massive load of debt, the contractor had merged the rooms with broad-planked flooring, using oak when the original eucalyptus and teak proved impossibly expensive. The windows were rimmed with stained glass from a local artist, who also supplied all the downstairs light fixtures.
When Connor returned to the living room, Ami announced, "Gerald would positively die to live here. He's bid four times on Craftsman homes and lost."
"I'll have him over for dinner when I get back."
"It would please him no end." She followed him through the living area and into the kitchen. "I am impressed."
"Thanks."
"And surprised. Two words I don't use often." Ami poked him in the chest. "Seeing this place, I think you just might be able to pull off the change in character."
"'Grow a heart,' " Connor said.
"Right." She glanced at her watch. "Now show me what's going to convince Raymond."
"This way." Connor led her through the kitchen and opened the door leading to the garage. When he had returned from his most recent and profitable journey to Japan, Connor had celebrated by buying two new toys. He turned on the lights, used the kitchen panel to reset the alarms, then shut and locked the door.
Ami still stood on the top step, surveying his pride and joy. "Oh, my. Wait, wait, I want to record this. I'll send it as a link to Raymond and blow him out of his argyle socks. All right, tell me what I'm looking at."
"This is a 1989 BMW M6, known in Europe as the M635 CSi," Connor said. "Mine was the next-to-last model to roll off the assembly line. It was sold to a collector, who drove it less than two thousand miles."
Ami smiled from behind her phone. "A perfect bad-boy car."
Connor hoped the camera did not pick up on how those words stabbed him. "Original black diamond-flecked paint, black Recaro leather seats and interior. BBS racing rims. Modified M88/3 engine, the second-fastest BMW ever built."
Ami lowered the phone a notch. "I admit it even makes this agent's heart go pitter-pat. But why should Raymond be convinced of anything?"
"Because the car is not why we're here." Connor stepped to the second bay and swept off the dustcover. "This is."
Ami cried, "What is that?"
"Ducati Multistrada. This S model is the fastest street-legal bike in the world." Connor strapped his satchel onto the back. "Liquid-cooled, twelve hundred cc, ninety-degree twin with desmodromic valve actuation and variable cam timing. Multiplate clutch. Pirelli Scorpion racing tires."
"I have no idea what you just said," Ami told him.
"The people Raymond needs to impress will get it." Connor walked over and hit the control raising the garage door, then pulled a helmet and quilted leather jacket from the wall rack. "Trellis-style tubular steel frame. Sachs shocks. Ducati Skyhook suspension. A pair of three hundred mill Brembo brake discs up front. A single two-sixty in back."
Connor pushed the bike outside, waved Ami to join him, then hit the control to lower the garage door. "The S model comes equipped with a Bosch-designed inertial measurement unit, or IMU. It gauges the bike's lean angle and interacts with the antilock braking and throttle management to bring it out of any high-speed skid."
Ami demanded, "Wait, you're leaving?"
"I promised I'd be back in time for my gig tonight in Miramar."
"What gig?"
"Long story. Not acting."
"I'm your agent. You need to keep me informed of all gigs!"
"Soon." Connor strapped on the helmet, then lifted the face mask and asked, "Are you still recording?"
"Yes."
"Good." He hit the ignition and raised his voice. "One last thing. Zero to sixty in three seconds flat."
CHAPTER 28
In the five months since Connor had last been on his bike, he had often dreamed of riding. The dreams always began with the sound. There was nothing like it on earth. The structure of a racing bike drew the rider down to a high-kneed crouch, the best position for an aerodynamic passage. The sound was monstrous. In his dreams, Connor first heard the bellow from a distance. The shifting gears took the bike through curves and dips; then Connor swooped in, drawn into the position of control. Or rather, as close to control as any human could be atop this much force. His dreams carried him through a few heart-swooping moments, before the end came. Connor either lost control or simply steered his bike partly around a corner and then turned away from safety and flew out over the cliff. His dreams always had the same end.
Connor had started riding at seventeen, using earnings from his music to buy an old Suzuki. He had ridden racing bikes owned by friends. He knew the danger rush, and he was ready for something that took him to the edge.
That became the problem.
There were secret clubs all over the LA basin. The website passwords were given out only after the group personally accepted the newcomer. They usually met after midnight, with a route planned out in advance. Two, three hundred miles of empty moonlit terrain.
They flew. Fifty, sixty bikes, racing for the pure unadulterated rush.
Most racers spoke in a slow, careful cadence, totally at odds to their action on the bikes. They didn't care where their mates came from. They never discussed the outside world. They rarely asked who the other riders were. They were uninterested in what jobs others held, what else they owned. When they met, there were only three topics.
The machine.
The road.
The speed.
Focused.
Connor had seldom tested his limits in the daytime, much less on a road as public and monitored as the 101. So he kept it to the limit observed by most high-end cars, just under eighty-five, for the first hour. Then he came upon three other racers. Two rode BMWs; the third was a woman on the new Harley racing beast. They exchanged a quick set of hand signals, enough to show that Connor belonged and needed to hurry.
They joined together . . . and they flew north.
Early on, Connor had realized that like the majority of racing bikers, he lacked the fear factor that served as an inhibitor for most people. The only time Connor became afraid was after. When he stepped off the bike, and came to a physical halt, and his heart still had wings, and his mind was perched out there . . . beyond. Only then did he realize how close to the final boundary he had come. It left him feeling exquisitely alive and achingly vulnerable.
At least it had once. Then about five months ago, everything had shifted.
The moment of change had been as definite as the crystal stars and the frigid night that had surrounded him. Connor had stood on top of the cliffs overlooking Rancho Mirage. He had made the solitary climb with his lights off, the silver moonlit asphalt illuminating curves with no guardrail. The ride done, Connor had stood there by his ticking bike and realized that he had not gone looking for a thrill. He had taken the ride without lights, alone, because life no longer held meaning. Live or die, he did not care anymore. The night had become filled with the jarring recognition of how much he had lost. The hollow void at the core of his being had grown so vast it threatened to consume him totally. He had ridden slowly back to LA, parked in the garage, pulled on the dustcover, and had not touched his bike since.
Until now.
Connor did not race to Miramar in order to be there in time for his job.
He was chasing after some small shred of everything he had lost.
The answer was there. In Miramar. He was absolutely certain. The town held his last chance at life.
CHAPTER 29
Sylvie made it easily up the stairs to her apartment. Marcela hovered just behind her, like she expected Sylvie to collapse at any time. The idea was ridiculous. How could anyone possibly think that a man she had only known for four days could impact her in such a way? Even so, her mind felt partially disconnected from her legs, which remained somewhat reluctant to obey her instructions. When she entered her apartment, Sylvie looked around in confusion. Now that she was here, she could not remember what had seemed so important to bring her upstairs.
Marcela asked, "Do you want to lie down?"
"Of course not," Sylvie replied calmly. Talking seemed like such a bother. She started toward the chaise lounge by the front windows.
Marcela continued to shepherd her. "What can I get you? Another brandy? Something to eat?"
"Tea would be nice," Sylvie decided.
"Tea it is."
Sylvie waited as Marcela bustled about and tried to fill the apartment's empty spaces with her chatter. When Marcela brought the tea, Sylvie said, "Connor is engaged."
"He was until a few hours ago," Marcela corrected.
"He was going to be married. What kind of man cheats on his fiancée the week of their wedding?"
"A total louse," Marcela said sadly.
"I feel like a complete idiot." Sylvie lifted the cup, set it down. "Did you see how beautiful that woman is? And famous? And—"
"A total airhead. I saw."
"Connor was toying with me."
"I liked him," Marcela said softly. "A lot. So did the others."
"We liked Bradley, too."
"No, girl. You liked Bradley. Everybody else watched and worried."
"Why am I only hearing this now?"
"We told you, like a hundred times. You were too much in love to hear."
The care and concern in Marcela's gaze almost undid her. "You are a dear, sweet friend."
"Is that your way of saying you'd like to be alone?"
"It is indeed."
Marcela set Sylvie's phone by the saucer. "You need, you call."
"Thank you."
When Sylvie heard the downstairs door click shut, she rose and walked unsteadily to the corner that served as her office. She unplugged her laptop from the charger and brought it back to the chair. Before sitting down, she opened the front windows.
Just in case.
She had no trouble finding the YouTube link. The latest so-called episode, the one that had gone out live on the cable channel, was less than three hours old and already had more than two million hits. She watched the entire tawdry incident; then she found the link to the previous installment. She sat through the engagement party and Connor's disappearance and Kali Lyndon's tearful appeal; then she turned off the computer.
She forced herself to drink the tepid tea. There on the blank screen, Sylvie found herself watching a replay of events from nineteen months ago. She overlaid the cable personality's absurdly cheerful voice and relived the last time a man had pretended to be someone he wasn't. The last time Sylvie had allowed herself to fall in love.
She had been with Bradley for eight blissful months. He traveled a great deal, he claimed for his work. He kept a small apartment overlooking the northern cliffs. Or so he had claimed. She had since learned that it actually was owned by a former friend. They had met when he had entertained clients at Castaways. Bradley had returned the next night to continue their conversation. He was handsome, intelligent, a good listener, beguiling.
Everything he had told her had been a lie.
Sylvie had only learned the truth when regular customers, a couple from Santa Cruz who spent almost every weekend at Miramar, observed her with this man from their hometown. This man whom she loved and planned to marry. This man, they revealed, who was nothing like the person he claimed to be.
And now it had happened all over again.
Only this time, things had not gone nearly as far with Connor. Instead, her friends had come to her rescue. She turned and looked over to the silent kitchen, and the empty space where Marcela had stood. There was no telling how long it would have taken her to discover the truth, had it not been for those dear friends.
Then, far in the distance, she heard a motorcycle.
The engine made a distinctive racing roar. The noise was exquisitely refined. The driver kept the revs up to a screaming level as he powered through the curves leading into Miramar.
Long before the engine's bellow rushed down Main Street, before it slowed and burbled to a stop below her window, Sylvie knew it was Connor.
Sylvie rose from her chair and looked out the window just as he pulled off his helmet and unlimbered from the ride.
She stared down at him for a long moment. As he eased himself to a full upright position, a pair of ladies on the opposite sidewalk slowed for an appreciative glance. His hair glinted dark and honeyed in the late-afternoon light. His weary features looked sculpted. He was not merely handsome. He carried the magnetic quality of a star.
Not that it mattered any. Sylvie turned from the window and uttered the same words she had spoken as she'd risen from the car and crossed Bradley's front lawn, "Let's get this over with."
CHAPTER 30
Connor was still vibrating from the ride when Sylvie appeared in the doorway. As usual, he only felt the impact of the ride and his tucked-in position now that he was off the bike. His thighs and lower back ached somewhat. His neck and shoulders were stiff. He stretched back, then forward. The adrenaline rush granted the moment a crystal precision.
This meant that the instant Connor saw Sylvie's face, he knew it was over.
There was little to her outer appearance to suggest anything wrong. She wore a lovely pearl-gray jacket and slacks, clearly ready to greet the weekend crowds. But there was no welcome for Connor. No greeting.
"Sylvie, I'm so sorry—"
"Don't be silly. What on earth do you have to apologize for?"
He felt as though the ground continued to move beneath him. Thankfully, the bike's handlebars were there for him to grip for stability. "All this has been such a terrible mistake."
Her smile was merely a professional reflex. "I couldn't agree more."
Connor could see that discovering who he was had erased something very fragile inside her. Something very precious was no more. "If I could go back and stop—"
"Oh, please." Her voice was almost cheerful. "There's no need for such drama."
Connor breathed in and out. He tasted the salt in the air. He heard the gulls cry. His clarity was sweet agony. This, he knew, was how it felt to break a heart.
Two of them.
Sylvie seemed to find exactly what she wanted in his silence. "There was nothing real between us. So there is nothing for you to apologize for. Not really."
Connor did not speak because he had nothing to say. His body hummed to the realization that he loved her, and that she would never be his.
"Please don't come back here anymore."
Connor saw Marcela and Rick both watching from the restaurant's front window. Sandy and Bruno stepped up behind them. It was only fitting that they be there to watch him crash and burn.
Sylvie scalded him with another cool smile. "Have a good trip back to LA."
She turned and walked back into her restaurant.
CHAPTER 31
There was no reason why Connor should choose to push his bike the three and a half blocks uphill from Castaways to the guesthouse. But something about the fragile quality of his heart seemed to require this labor. By the time he rolled into the parking lot, he was puffing hard. He set it onto the kickstand; then he just stood there. His hands rested on the controls. His limbs still vibrated, but he suspected it was more from the impact of Sylvie's words than the road. He could feel the aftereffects from his toes to fingertips.
Connor had no idea what to do next. Stay in Miramar or return to LA? The question had no meaning.
Then Connor spotted the light framing Estelle's window.
He walked over and stood by the door, long enough for the night to gather strength. He had no logical reason to involve himself further, but this was the only act that felt the least bit right.
He knocked softly and waited.
When Estelle opened the door, he knew the answer before he asked, "Have you spoken with her?"
"I tried. I told you I would." Estelle's voice was scarcely a murmur. "I failed. Again."
Connor had no idea what to say.
"I'm leaving tomorrow. Would you . . ." Estelle breathed hard. "Tell me about my daughter. Please."
Connor nodded slowly. Coming here made sense now. "I'll meet you in the coffee shop around the corner. Fifteen minutes?"
Connor crossed the lot and took the satchel off the back of his bike. He showered and changed into his only remaining clean shirt, the white knit pullover he had bought for the job. He put on the second pair of new black gabardine trousers; then he settled on the edge of his bed to put on socks and shoes. The leaden weight of his heart threatened to plant him there. Connor tried to tell himself that it was ridiculous to hurt this badly. He had only known the woman for four days. What was going on?
The only answer he came up with, no matter how little sense it made, was to walk down to the café and settle into the seat across from Estelle.
She asked, "What will you have?"
"Americano, black." When she returned and set the cup before him, he cradled it in both hands. He saw Sylvie in the steaming liquid. He tried to describe Estelle's daughter, and failed. So he started at the beginning. Not of his own journey, but from the moment he had heard the music and let himself be drawn into Castaways.
Connor described his own feelings, along with the events. He left nothing out. He named every song he had played for Sylvie, and did so in proper order. He related what Sylvie had told him about her past. The only thing he left out was the cold manner in which Sylvie had spoken about her mother. Connor did not once feel that he was breaking confidences. This was her mother. But as he reached the point where they had kissed, he finally realized the true purpose behind the talk, at least for him. Connor was trying to determine the precise moment he had fallen in love.
As he related that horrible confrontation this afternoon, as he described being gutted on the street, Connor reached the answer to at least one mystery.
His heart had been captured from the very first moment he had laid eyes on the lady.
Estelle drew him from his painful reverie. She reached across the table and took hold of his hand. When he looked up, he found himself met by a truly sympathetic gaze.
Estelle said softly, "You poor kid."
Connor clenched his jaw against the sudden upwelling. The power of his emotional upheaval was shocking. He was a man known for feeling nothing. Yet, he was on the verge of sobbing here in public.
Estelle went on, "I suppose you'll be heading back to Los Angeles. I'd like to give back my rental car and catch a lift to the airport."
Connor swallowed hard. "I'm on my bike."
"It's been a while, but I used to enjoy the occasional ride."
As he started to agree, he found himself struck by a realization.
He needed to stay in Miramar.
Connor had no idea why he felt so certain, but to have clarity of any sort was far more important than logic. Nothing about his entire sojourn in this place was sensible. He said, "Don't go."
"I told you. I've been here eight days and the closest I've gotten to my daughter is across the street." She smiled sadly. "At least that was true until tonight."
The passing moments only intensified the awareness that he was not going anywhere. Not yet. And neither should she. "Stay one more day."
"What for, Connor?"
"I have no idea," he replied. "But I think it's important."
CHAPTER 32
The one thing Connor could say for certain about Miramar, it was not a place made for easy nights. The best sleep he'd had all week had been in the limo riding south. Despite his hard days and broken nights, he spent most of those dark hours pacing. Sylvie's act of final closure was only partly to blame for his restlessness. There was a hunger to understand what was happening—a need to see himself moving beyond the confusion. He had no reason to stay in Miramar and every reason to leave. Down in LA, he was freed from the shackles of a PR-driven marriage. His star was on the rise. His agent had laid down a challenge for growing his abilities as an actor. And yet . . .
Every time he opened his door and glanced at his bike, he grew increasingly certain that his time here was far from over.
Around midnight, he lay down and slept a few hours, repeatedly jerking awake from images strewn like leaves blown by a careless wind. A little after four in the morning, he rose and put on a pot of coffee. He took his mug out to the bench where Estelle often sat. The night was utterly still, the stars a great silver sea. An owl greeted him. A car passed. Otherwise, the hour was his.
It seemed as though the idea grew with the dawn. Connor remained where he was, long after the chill worked into his bones. The concept was so fragile he was not certain he could shape it into words. He feared if he rose he might allow the doubts and sorrow to seep in and wreck it. So he waited until the tremors shook him; then he went inside and took a long, hot shower. He lay back down and must have dozed off, because daylight greeted him when he opened his eyes.
Connor rose and stretched and made a fresh pot. He stood by the kitchenette and studied his room. Today was to have been his wedding day.
Their honeymoon was to have been a three-week extravaganza. Week one was to have been spent in a mahogany palace built on stilts above its own private atoll in the Maldives. For week two, they were booked into a seven-hundred-square-foot tree house in the virgin forests of New Zealand. Week three was in the royal suite of the Paris Ritz.
Connor looked around his little studio. The recent redecorations had not removed its basic flaws. Beneath their coats of fresh paint, the concrete walls dimpled and cratered. He could easily have reached up and shifted the popcorn ceiling tiles. The air conditioner rattled. His view through the old slatted window was over the central parking area. The soundproofing between rooms was almost nil.
There was no place Connor would rather be.
What was more, he knew what he needed to do next.
He still could not make sense of it all. And the heartache from yesterday's confrontation with Sylvie rendered him hollow. This new direction felt right, though. He clung to that simple fact. It had been a long while since he could say that about any of his moves.
Connor poured a fresh mug and went through his clothes from the previous day. He found the police chief's card in the back pocket of his ride-stained jeans.
Porter Wright answered, "Do you know what time it is?"
"A quarter past nine."
"A quarter past nine, on my Sunday off," Porter corrected.
"I need to ask you something, and I don't think it should wait."
Porter sighed. "Hang on a second." There was a rustling sound, a pause, then, "All right. What is it?"
"That problem you told me about—Sylvie and the drugs. Is that still an issue?"
Both Porter's sleepiness and his ire completely vanished. "Why are you asking?"
"I want to help."
"Connor . . . you need to understand, whatever you do won't repair the damage."
He had already reached this conclusion, but having Porter slam the door shut on his last shred of hope was tough. "You heard about yesterday?"
"The town's favorite lady discovers her mystery waiter is a Hollywood star on the lam from his own wedding. She strips him bare on Main Street. He pushes his two-wheeled red rocket up to the guesthouse and disappears inside. Yeah, I heard. I expect most of the town spiced their Saturday dinner with speculation over why you're still here."
"I told you. I want to help Sylvie."
"You mind if I ask why?"
It was the first time Connor had spoken the thought aloud. "I can't make things right. I accept that. But I want to try and make things better."
Porter went quiet. Then, "Can you come up with twenty thousand dollars?"
The simple answer was no. Connor was as deep down in the debt hole as he had ever been. Just then, it seemed as though his attitude toward money was basically one more splinter from a broken life. Connor saw no need to tell the cop what he was thinking, so he merely asked, "This is for Sylvie's lawyer?"
"She needs one, and I know for a fact she's dragging her heels because of the money issue. Sylvie is afraid she'll wind up owing Phil Hammond more than she already does. Rick and Marcela are trying to raise the money in town. A lot of folks will probably want to help out, but there's not much time to go find them."
"Give me a couple of hours," Connor said. "I have an idea."
"Son . . . I don't think she'll accept any help coming from you."
Connor opened his door and looked over to where Estelle sat, isolated and lonely in the sunlight. "As far as Sylvie is concerned, I won't be doing a thing."
CHAPTER 33
Estelle agreed to Connor's idea while he was still in the windup. "I'll do it."
"I can help."
"You already are. I've received a . . . What is the word for unexpected funds dropping into your lap?"
"Every actor's dream," Connor replied.
"'Windfall.' That's the word. My second husband recently passed away. He had an insurance policy he never told me about. I want to use these funds to pay for Sylvie's lawyer." When he started to protest, she said, "This is important to me."
Porter kept them waiting by the phone for almost three hours. Then the chief called to report the attorney would give them twenty minutes of his Sunday afternoon, but only if they could work to his schedule. Connor gave Estelle his helmet and made do with wraparound shades to protect his eyes from the wind. The two-lane roads held to Sunday afternoon sleepiness. Connor pushed it hard. The thirty-nine-mile ride from Miramar to San Luis Obispo took thirty-one minutes. When Estelle slipped off the rear seat, all she said was "Oh, my."
"You need a minute?"
"No, I'm . . . Actually, that was rather fun."
"You have a lovely smile," Connor observed. "You should show it off more."
As they entered the attorney's office building, Estelle replied, "Funny, I was thinking the same about you."
Sol Feinnes was a soft-spoken teddy bear. His broad features showed a gentle demeanor, but the hand that swallowed Connor's held a well-padded strength. He led them through a warren of empty offices and settled them into a conference room overlooking a tree-lined street. "I'd offer you coffee, but my wife and secretary both assure me my cooking is offensive."
Estelle said, "We're fine, thank you."
"Then I'll come straight to the point. We must establish a clear means by which you came to know about a pending arrest that has not yet been made public."
"Two nights ago, in front of the entire restaurant, Chief Wright warned Sylvie this might be happening," Connor replied.
"This is Sylvie Cassick, the defendant." Sol made swift notes on the cover of a manila folder. "And her restaurant is Castaways, correct? Good. Which brings us to the next point. What exactly are your interests in this matter?"
"This is Estelle Rainier. She is Sylvie Cassick's mother."
"That will certainly satisfy any questions the judge might raise. And you are . . . ?"
"An interested third party," Connor replied. "For all intents and purposes, I am not here."
"Your name?"
"Connor Larkin."
Feinnes leaned back in his seat. "The actor."
"Correct."
"You have an ID?"
Connor passed over his driver's license.
"Mr. Larkin, you are here because . . ."
Connor found it impossible to respond. The answer that came to mind was . . . he was learning how to care. Which made no sense whatsoever. Thankfully, Estelle came to his rescue. She set her check on the attorney's desk and replied, "Connor is part of a group that wants to help Sylvie."
Sol Feinnes examined the check. "Ms. Rainier, you are covering this from your personal account."
"I am, but Sylvie isn't to know."
Sol Feinnes paper-clipped the check to the file's inside flap. "And this is necessary because . . ."
"Sylvie might not accept this help if she knew I was involved."
The attorney examined her over the top of his reading glasses. "May I ask why?"
"We haven't spoken in quite some time."
He made another quick note. "All right. I think it is safe to assume that the DA is moving so swiftly because they intend to blindside my new client. Their case is at best a charade, and at worst an abuse of the legal process. They are apparently being pressed to make an arrest. Sylvie Cassick's role is to play the rabbit they pull from their legal hat."
Connor asked, "So you can stop this?"
Feinnes kept his gaze on Estelle. "My goal tomorrow will be to annihilate them both. I intend to thrash them in a highly public manner."
Connor thought the gentle manner of his speech added an extra flavor to his words. "Excellent."
Feinnes slipped a document from the folder and passed it across the table. "This names me as Ms. Cassick's legal representative. Have her sign and scan them and email as an attachment to me here. I will notify the judge and arrange for a private meeting in chambers in advance of the formal questioning and arraignment. Ms. Cassick should be at the courthouse no later than ten-thirty. You, Ms. Rainier, are obviously welcome to attend."
"And the trial?"
"I can assure you, madame, that if I have my way tomorrow, there will be no trial. What they are intending is outrageous. My objective will be to make the judge agree with me, and then respond with savagery. Unless they spring some astonishing bit of new evidence to support their case, by the close of business tomorrow this entire episode will have vanished in a noisy blast of cordite."
Estelle showed Connor another rare smile. "It sounds as though we came to the right place."
CHAPTER 34
Sylvie kept the list of attorneys on the counter that separated her kitchen from the dining area. She resisted the temptation to shift it to her desk, where it could more easily be ignored. That had been one of her father's favorite tactics. Gareth's response to every unpleasantness was the same. Whenever a court summons or an eviction notice or an IRS demand or another letter from child services landed in their lives, Gareth stuffed it in the cupboard over the ratty camper's sink and did his best to forget it existed. Once her mother had left, Sylvie had taken over handling all such issues. This was a big reason why she had grown up as fast as she had. Settling in Miramar had basically been her responsibility. At sixteen, she had obtained a fake ID and started waitressing. The funds had mostly paid for their apartment.
All those hard days, all that determination and sweat, had brought her here. Standing by her kitchen window. Trying to decide which attorney to phone. Knowing the expense would probably mean losing outright control of Castaways. Of course, that was the real reason she was tempted to follow her father's example. Shove the list of attorneys in a drawer and pretend the problem could not touch her.
She was dressing for the Sunday rush and still circling the paper on her cabinet when Porter phoned. "You called one of those lawyers yet?"
"Still trying to decide," Sylvie replied, which was at least partly the truth.
"Don't do anything just now. This may be taken care of."
"You mean, they're not going to arrest me?"
"I wish, but no."
"Then what's happened?"
Porter sighed. "If I'd known this call was going to be so tough, I would've refused. I should have."
"You are making no sense at all," Sylvie pointed out.
"Your mother has arranged for a lawyer. There, I've let the cat out of the bag."
"What?"
"I don't know how to say it better than I already did."
"But . . . I haven't spoken to her in years."
"I know all that. But she's here. She heard about this mess. She offered to help."
"Wait . . . my mother is in Miramar?"
"Right now, she's on her way back from San Luis Obispo. She met with Sol Feinnes, the first name on my list. He's agreed to take your case. Estelle has some papers you need to sign and scan and email back to his office."
Only when Porter hung up did Sylvie realize she was seated on her kitchen floor.
* * *
As Connor lifted his hand to knock, Estelle opened the door and said, "I can't go through with this."
Connor stepped back. "I understand."
The absence of any argument deflated her, as Connor had hoped it would. Estelle wrung her hands. "What if I've got it all wrong?"
"Then you leave. But at least this way, you will have known you did what you came to do."
"That's what I've been trying to tell myself all day."
Connor remained where he was, two steps removed from her tension and fear. "Here's what I propose. I will walk you down the street. We'll stick to the same side as Castaways, because I can walk with you almost to the doorway and she won't see me."
"You won't leave?"
"No, Estelle. I'll stay close by for as long as it takes." He spoke in the calm monotone employed by most good directors, keeping their own egos contained, allowing the actor to insert the required emotional energy. "I'll find someplace I can watch the doorway without being seen. When you're done, come out and start back up the street. I'll join you and walk you back here."
She lifted her hands and pressed hard against the bridge of her nose. Then, "How do I look?"
"Just fine. Great, in fact." And she did. Estelle's hair was like her eyes, a shade or two darker than Sylvie's. She had bound it back, but left two long strands to flow over both shoulders. Her silver-and-turquoise jewelry caught the gray strands in her hair. She wore rough silk trousers the color of jeans, with an Indian-print sweater.
Her features were bloodless, and her gaze terrified.
Connor reached for her hand. "Ready?"
"No." However, she took his hand just the same.
Her fingers felt like ice.
CHAPTER 35
When the woman appeared in the doorway of Castaways, Sylvie's first thought was, there was not room in this one weekend for another such shock.
Sylvie instantly recognized Estelle. Her mother's silhouette released a torrent of conflicting emotions. Then Estelle stepped forward, out of the light, and there she was. Nineteen silent years erased in one step.
Sylvie had no idea what to say. She certainly wasn't going to speak the word "Mom" or "Mother," the very idea was repulsive.
"Hello, Sylvie."
When she had told Marcela and Rick what was happening, Sylvie had the distinct impression that they already knew. Sylvie could only assume Porter had told them. That was another thing she did not want to deal with just then. Miramar was not a place for keeping secrets—she had known that since her early days. For the first time in years, Sylvie wished it were otherwise.
When Sylvie remained silent, Marcela walked over and asked, "Hello, I'm Marcela. You must be . . ."
"Estelle Rainier."
"Very nice to meet you, Estelle. Can I get you something?"
"No, thank you."
That galvanized Sylvie into action. "Marcela, take over and seat the early tables, please." Sylvie was glad to find her legs remained capable of functioning. "Let's step outside."
Estelle hesitated, as though reluctant to leave the place, or perhaps she feared Sylvie was going to order her away. Sylvie had to admit that was tempting. Instead, she led Estelle over to the wrought-iron tables and chairs occasionally used by bar patrons or customers waiting to be seated. Estelle said, "Your place is lovely. It is yours, I take it."
"Mostly." Sylvie seated herself, grateful for the emotional numbness. At least the weekend's endless shocks had that one positive effect. She observed, "You seem so calm."
"I was thinking the same about you," Estelle replied. She glanced around, as though searching the perimeter. "I was a total wreck before I arrived. Now I feel . . ."
"Detached," Sylvie replied. "Observing this from a distance."
"Yes. Exactly. I've been here in Miramar for nine days, trying to work up the courage. This after paying a detective to track you down."
Every word seemed filtered through a mental confusion. Sylvie's strongest desire was to demand that Estelle take back the money she had paid the lawyer. Was it even possible that Estelle thought she could buy herself into a relationship with the daughter she abandoned? However, something in Estelle's face left Sylvie unable to utter those words.
Instead, Sylvie asked, "Why did you come?"
"To apologize, Sylvie. I won't say I was wrong to leave. I honestly don't think it would have been possible for me to have stayed."
Sylvie had no idea what to say.
"I kept writing you until six letters in a row were returned. I knew Gareth had moved on." Estelle's lovely features were creased by old sorrow. "I had no way of finding you. It was the most painful thing I have ever endured, losing that last thread of contact with you."
For years, Sylvie had imagined this reunion. These mental confrontations had always involved rage, tears, shouting, fury, and rejection. Instead, she heard herself say, "Can I offer you something?"
"Thank you so much, dear. That is very kind. But no." Estelle paused. "Am I allowed to call you 'dear'?"
Sylvie replied slowly, "I don't know."
"Then I won't." A silence settled over the table. A breeze drifted up the street, spiced by the sea. A car passed. Somewhere a young girl laughed. Finally Estelle said, "I am so sorry, Sylvie. So very, very sorry."
Sylvie could not work out a response. Should she say it was okay? Offer forgiveness, when she felt nothing of the sort? What was the protocol for meeting a mother who had abandoned her?
Estelle asked, "Can I see you again?"
The initial shock was wearing off. In her absence, Sylvie had given this woman an image much larger than life. She was the mother who had abandoned her family. But Estelle was nothing at all like Sylvie had imagined. She was an attractive woman, both familiar and a stranger at the same time. She looked kindly. And sad. And surprisingly ordinary.
Sylvie realized Estelle was still waiting for her response. "Yes, all right. Tomorrow."
"Thank you, Sylvie." Estelle reached into her purse and set a folded document on the table between them. "Your attorney needs you to sign these. Scan the signature pages and send them back tonight as an attachment. You need to be at the courthouse tomorrow at ten-thirty. His name is Sol Feinnes. He seems to be a good man."
There was no way she was going to taint this moment with some false expression of gratitude. "I'm having trouble accepting your help."
"I understand that, but you need a lawyer to represent you in this. Sol hopes he can make all this vanish before the actual arraignment."
Sylvie nodded slowly. "I suppose I should thank you."
"No, Sylvie, you shouldn't." Estelle rose from the table. "Thank you for speaking with me. It means the world. I . . . Until tomorrow."
CHAPTER 36
Monday morning, Estelle suggested to Connor that they take her rental car down to San Luis Obispo. She thought Connor's motorcycle might draw the wrong sort of attention. Connor did not object. He seemed incapable of arguing. He merely asked if Estelle was certain that he should come at all.
The longer Estelle was in his company, the more certain she became that Connor Larkin truly loved her daughter. Yet, he completely accepted that Sylvie would have nothing more to do with him. He clearly felt that he deserved nothing more.
Even so, here he remained. Deeply concerned. As involved as Sylvie would permit him to be.
Estelle was actually more comfortable with San Luis Obispo than Miramar. The beachside enclave was too isolated for her taste, too set apart from the rest of life. San Luis Obispo served as the commercial center for much of the central coast. People came here to shop the big stores, get their teeth fixed, and jump through all the legal hoops of modern life. Cal Poly was based here. A spiderweb of roads spun out from the city, joining together a wide array of towns. As a result, San Luis Obispo reminded her of her previous home in Wilmington, North Carolina. Both held to the vibrant, go-ahead energy of much larger communities.
If Connor noticed any of this, he gave no sign. He did not speak much during the journey. Estelle pulled into the parking garage adjacent to the county courthouse and parked next to a police cruiser. Estelle recognized the Miramar police chief as he rose from the car, talking on his phone. The policeman waved in their direction and headed off. When she cut the motor, Connor said, "I'll wait here."
"What if the lawyer needs something?"
"Give me your phone." He coded in his number, then handed it back. "Just make sure Sylvie isn't around before you ask me to come over."
His attitude baffled her. "Connor, I really think—"
"This is how it has to be." He checked his watch. "You better hurry."
The courthouse had the nondescript functional style of many California government structures built in the eighties. Estelle entered by the main doors, passed through security, and found Sol Feinnes already in discussion with Sylvie. Estelle thought her daughter looked exhausted. The police chief was talking with two people Estelle recognized from the restaurant. He motioned for her to join them.
Estelle walked over and greeted them with, "It's so nice that Sylvie has friends like you at such a time."
Porter introduced himself, then asked softly, "Where's our boy?"
"In the car."
Marcela asked, "Who?"
"Doesn't matter." Porter gestured to where the lawyer was still talking intently with Sylvie. "Sol has requested a meeting in chambers. If the judge agrees, Sol hopes they can wrap this up without it actually moving to a formal arraignment."
Estelle said again, "Sylvie is very fortunate indeed."
Porter pointed to a uniformed deputy who had stepped into the hall and waved to Sol. "Looks like we're on."
* * *
There was nothing for Sylvie and the others to do except wait. Rick and Marcela went off in search of coffee. Estelle settled next to her daughter on a wooden bench.
Sylvie said, "I didn't sleep a wink."
"Hardly a surprise." Estelle found herself recalling what she had not thought of in years. Such events had marked her own worst times, being tracked down by some court official and dragged into yet another bureaucratic nightmare. As far as the courts were concerned, they had been homeless. That meant too many judges treated them like cannon fodder. All Estelle said was, "I'm so sorry you have to go through all of this."
Sylvie directed her words down the long, empty corridor. "If I was able, I would have refused your offer of help."
Estelle nodded. "I understand."
"Why are you here? I mean . . ."
"Why now and not before? I buried my late husband nine months ago." Estelle watched her daughter work through the news that she had both married and lost a man Sylvie had never known. "Jack made me promise to try and find you. He'd been after me for years. I always said too much time had passed, that you wouldn't want to see me. . . . Jack was a very good man."
"So this was his dying wish, you coming here?"
"Yes, in a way, I suppose that's true." Estelle hesitated, then decided Sylvie should hear the rest. "The money for your attorney came from a life insurance policy I didn't even know Jack had. He would be very happy to know how it's been used."
Sylvie leaned her head against the wall. "What a weight."
Estelle had no idea how to respond. She sat there. Breathed in and out. Beside her daughter.
Sylvie asked, "Do you have other children?"
"No. Jack and I wanted. We tried. But . . . no."
The silence lingered. Finally Sylvie asked, "Where are you staying?"
"The motel up the street from your restaurant."
"Is it . . . nice?"
"It's fine."
Sylvie took so long in shaping the next words, Estelle could guess what was coming. "Is . . . he still there?"
Estelle decided if a little white lie was ever needed, this was it. "If you mean Connor Larkin, I can't say for certain where he is."
Sylvie sighed. Shut her eyes. Sighed again.
In that instant, Estelle realized that Sylvie was in love with Connor.
Estelle took advantage of her daughter's closed eyes to inspect Sylvie. Perhaps this was what it meant to be a mother, having an ability to see with utter clarity the emotions that Sylvie tried so hard to deny.
Rick and Marcela returned bearing coffee and a box of Dunkin' Donuts. Marcela said, "We were on our way to the cafeteria when old eagle eye spotted this across the street."
"Grease and sugar and caffeine," Rick said. "Nothing better for the blues."
"Nothing legal, anyway," Marcela said. She opened the box and held it out to Sylvie. "I know you're on a permanent diet. I asked for broccoli donuts, but they were fresh out."
Estelle found their feelings for her daughter very touching. It confirmed everything Connor had said about the powerful effect her daughter had on others. They set everything aside in order to be there for her. They expected nothing in return. And this included Connor. Estelle blinked away the burn and smiled her thanks as she selected a donut.
Sylvie ate a mouse-sized portion of donut, just enough to show appreciation for the gesture. She set the remainder on the bench beside her, checked her watch, and pulled out her phone. "Excuse me a minute."
As she stepped away, Estelle asked, "What is she doing?"
"The delivery van should have arrived," Rick replied. "The fleet went out last night."
"Fleet?"
"Fishing. The seas were calm, so there should be . . ."
Sylvie chose that moment to turn back and say, "Bruno says there's been an excellent catch of halibut today. I told him to buy the lot. We'll do a trio of daily specials."
Estelle listened as the three of them discussed possible recipes. Her daughter showed a remarkable mixture of fragility and strength. Despite everything she faced, Sylvie maintained her calm poise. Estelle clamped down on a sudden upsurge of emotions. She had never been more proud of anyone in her entire life.
When Sylvie seated herself once more, Marcela said, "Call Bruno back, tell him to be sure and check the fish for packages. Sorry. Terrible joke."
Sylvie managed a narrow smile. "Awful. You should be ashamed."
"Mortified," Marcela agreed, winking at Estelle.
"Heads up," Rick said, pointing to where Sol appeared in the doorway. "Here comes the posse."
Estelle had the impression that a dark cloud accompanied the attorney down the hall toward them. Sol stopped in front of Sylvie and said, "I have some bad news."
Estelle remained distinctly separate as Sol Feinnes related how the judge refused to even consider his request to drop the charges before they were officially filed. Instead, the judge accepted the prosecutor's claim that there was adequate grounds for proceeding, and again refused when Sol demanded they reveal there in chambers what supporting evidence they might hold. Estelle watched as Rick and Marcela moved in closer as Sol explained the arraignment process, Sylvie's need to respond with a declaration of not guilty, and the probable bail requirement. Estelle trailed well behind them as they moved down the corridor and entered the courtroom. She seated herself in the back row, maintaining a clear distance. When Sylvie glanced back, Estelle smiled briefly. Since her daughter did not motion for Estelle to join them up front, she remained where she was.
How did she feel about this? Being here, living the dream she had carried for so long. Despite all the hardship this moment carried, Estelle was still playing a role in Sylvie's life. Not even the barriers that separated them could impact what she felt, which was . . .
Joy.
CHAPTER 37
Connor did not mind sitting inside the parking garage. The place was quiet and dark as a man-made cave. Occasionally a tire squealed or someone walked past, but no one noticed the lone male in the Chevrolet parked nose out. His recent days had been so intense and his nights so interrupted. Having a chance to reflect was very useful. The mysteries did not plague him so fiercely, though the answers remained unclear. Even so, Connor had the distinct impression that he was closing the door on his years of wrong moves.
If only the turning did not hurt quite so much.
Half an hour later, Porter arrived, bearing a steaming cup of coffee. Connor unlocked the door and Porter slipped inside. "Black, right?"
"Yes, thanks. Shouldn't you be inside?"
"I said my piece and decided to make myself scarce. They're in court. Things did not go like we hoped. Sylvie's being formally charged with felonious possession with intent to distribute."
"That's nuts."
"Everybody agrees with you, except the three who matter—the judge, the prosecutor, and the detective handling the case. Sol intends to lodge a formal complaint. But it's all after the fact now."
"Sylvie isn't in there alone, is she?"
Porter glanced over, then away. "Rick and Marcela are with her. Estelle's camped out at the back of the courtroom."
"What happens now?"
"She'll get bonded out and get on with her life." Porter tapped his fingers on the side window. "Something is going on here. I can smell it."
"What possible good could come from charging an innocent woman with a felony?"
Porter nodded slowly. "That's the question we need to answer."
Connor liked being included in the hunt for a solution. "I still want to help. Long as you keep my name out of it. I don't want to embarrass Sylvie or add to her troubles in any way."
"Which it would, if she ever found out you're involved in this." Porter gave him the sort of stare perfected by cops, level and direct and unflinching. "You're sure you don't want to head on back to the bright lights and the big city? Forget about a lady who doesn't want to have anything more to do with you?"
Connor felt no need to hide the truth. "I can't help but hope, even when I know it's useless. But that's not why I'm staying." He shrugged. "Miramar is where I need to be right now."
Porter just waited.
"I've been sitting here, trying to remember the last time I helped somebody without expecting to get something in return."
"Feels good, doesn't it?" Porter opened his door, started to climb out, then said, "Why don't you come up to the house tonight and join us for dinner."
"Thanks, Porter, I'd like that a lot." He was struck by an idea. "Is it okay if I bring Estelle?"
Porter seemed to find that humorous. "Always room for a lady with a story to tell. Last house on Little Bear Road. Six o'clock."
Reflecting on his odd mixture of emotions, Connor watched the police chief lumber away. Porter was right. It felt better than good to help Sylvie out.
Even when it hurt.
CHAPTER 38
Sylvie had been home less than an hour when Rick called to her from downstairs. She walked over to the top of the stairs and said, "I thought I told you to go home."
"Good thing I didn't," he replied. "Harold Reamus just showed up."
She winced at the thought of adding another burden to her day. Harold served as Phil Hammond's attorney. "What does he want?"
"No idea. I tried telling him he could call and make an appointment like everybody else. He says he only needs a moment of your time."
Sylvie was tempted to agree, but she knew it would only be putting off the inevitable. "I'll be right down."
"After the day you've had," Rick protested.
The attorney stepped up beside Rick. "Which is precisely why I'm here. At Mr. Hammond's personal request."
Rick snorted softly, then stepped aside as Sylvie descended the stairs. Harold Reamus was dressed as always, in a Brooks Brothers suit and narrow club tie and round gold spectacles and polished cordovan shoes. He thanked Rick and smiled as Sylvie led him over to the bar. Everything was very normal about Harold. However, Sylvie often suspected that given the right motivation, the top would spring open to his tight little box, and out would pop the evil clown.
She asked, "Would you like something?"
"Thank you, I won't be staying long."
Rick pushed through the kitchen doors, saying, "I'm just in here if you need anything."
When the door sighed shut, Harold said, "Mr. Hammond has heard of your current dilemma and wishes to offer you his full support. Which is quite considerable, I assure you."
"That's very kind, but—"
"Please hear me out. He asks that you reconsider your choice of Sol Feinnes as your legal representative in this matter."
"How did you hear about this?"
"Your legal troubles are now public record. As I was saying, Feinnes is quite adequate for most of his local clients, but his expertise is extremely limited. He has handled less than a dozen drug-related trials."
"I didn't have anything to do with drugs!"
"Which is precisely what Mr. Hammond said when he heard. It is a sham situation, but the ramifications are dire. If you are convicted, you face a long incarceration and the loss of your restaurant."
Hearing her worst fears stated in such a calmly precise manner left her nauseous. "Why are you here?"
"Mr. Hammond wishes to provide you with the services of the finest trial attorney in California. A man at the top of the state's legal empire, with the power to call a battery of witnesses and deal this case a crushing blow. In exchange, he merely asks that you reconsider his offer of four months ago."
"'Merely' sell him a controlling interest in my restaurant," Sylvie said. "'Merely.'"
"You continue to miss the big picture," Harold said. His voice was as slick as his hair, a tight sheen that was completely immune to Sylvie's growing ire. "Mr. Hammond is expanding into large-scale hotel and restaurant ventures. He wants you to run the entire division."
"See, Harold, that's where you and Phil miss my big picture." Sylvie swept a hand around the restaurant. "This is what I want to run. For the rest of my days. Not Phil's empire. This."
"Alas, Ms. Cassick, your current attitude risks costing you everything. Not just this restaurant. Your life here in Miramar. I urge you to reconsider your position on the matter."
Sylvie swallowed hard, shoving her immediate reaction down deep. Phil Hammond remained her partner in Castaways. There was nothing to be gained by telling his attorney that she found the prospect of working for him repulsive.
Harold seemed pleased by her silence. He slipped from the stool and said, "Thank you for your time. I'll see myself out."
Sylvie was still standing at the bar when Rick's head appeared through the kitchen doorway. "Everything okay in there?"
"Sort of." She actually felt a lot better than she might have expected. Having Phil's lawyer lay things out had eliminated that option. She would never work with the two of them. It simply was not in her genetic makeup. But that was not what held her at the bar. Sylvie felt captured by the most foolish, nonsensical, impossible thought.
She wished she could talk with Connor. She missed him terribly. She positively ached for the chance to see Connor again.
Which was ridiculous, considering that the Connor she wanted hadn't really existed in the first place. He was just a figment of her imagination, wasn't he? So pathetic . . .
She entered the kitchen, chatted with her staff, then ordered them all to go out and enjoy what was left of their day off. Sylvie assured them she was fine, and knew they did not believe her. Her reflection as she locked the front doors showed deep lines that could well last the rest of her life. She worried over what her friends and employees must have thought of this wild-eyed woman who saw them off with a Kabuki mask of a smile. No wonder they had all stayed as long as they did. Sylvie thought her reflection did not look quite sane.
She walked around, turning off the last lights and setting the alarms. Then she seated herself at the bar. Just another lonely woman, staring at another empty night. She felt the weariness like a weight that bound her into place.
Her gaze came to rest upon the piano. She heard the soft refrain of a man inviting her to fly away. Singing the same words her father had hummed for years. The melody came to her now, sung by a man whose arms she could still feel.
Sylvie forced herself to rise and turn away and climb the stairs and prepare herself for bed. The melody did not leave, nor the voice, nor the memory of his kiss.
CHAPTER 39
Porter Wright's home was on the eastern side of the ridge separating Miramar from the farming valleys. His home occupied a natural plateau two-thirds of the way up the slope. The ledge covered five or six acres, most of it given over to a fenced pasture holding three saddle horses. One in particular caught Estelle's eye as she pulled into the drive, a palomino with a snow-white tail and mane. A young woman in her late teens or early twenties was currying a dappled gelding as they cut the motor. She waved an easy hello and called something lost to the evening breeze. Estelle waved back, then turned toward the policeman and his wife standing in the door. "Thank you so much for letting Connor bring me along."
The house was a comfortable ranch that carried the easy grace of Porter's wife. They grilled steaks on the bricked patio positioned between the house and the paddocks. The hillside below was densely wooded with cypress, eucalyptus, and California pine. A fragrant wind whispered and sang through the branches. Sparks rose and joined with the stars overhead. The daughter's name was Celia, and she had her mother's natural strength and easy manner. From several things that were said during the meal, Estelle guessed Celia was recovering from her own bout of loving the wrong man.
Mother and daughter peppered Connor with questions. Nothing was said about the canceled wedding or the woman he had left at the televised altar. Instead, Connor spoke of life as a journeyman actor. When pressed, he described some of the stars he had worked with. Whatever awe the young woman might have felt over being in the company of a man she recognized from television was soon lost. Connor talked of how actors at his level were excluded from the places and parties where the A-list gathered. This meant he only knew them from work, and even there the real stars maintained barriers designed to keep others out. In Hollywood, he explained, even the most casual contact could result in someone handing over a script or pitching an idea.
Celia had seen Connor in a number of his episodic death spirals. Connor showed a detailed and precise recollection of each set and story line. Toward the end of their meal, he had them all laughing over a drama he had recently filmed outside Delhi. In the final scene, he was supposedly bitten by a fake scorpion. Only, a real scorpion had crawled into his bed and Connor almost didn't make it out alive.
They spoke a little about Sylvie and the legal proceedings. Estelle could see how hollow Connor's expression turned even hearing her daughter's name. However, she had no other place to turn for real information. Porter explained about the debts Sylvie had run up while remodeling the restaurant, and how Phil Hammond had come in as a silent partner. At the time, it had seemed like pennies from heaven, Porter said, but Phil's charm had not lasted. There had been several run-ins, most recently when Phil had declared his intent to purchase the restaurant outright. Estelle had many more questions, but the evening's mood had been darkened by the discussion. She did not object when Celia asked Connor if he'd like to see the horses.
Together they left the house and walked to the paddock railing. The three horses trotted over, snuffling their hands for treats. Celia ran to the stables and returned with a sack of apples. Porter sliced off segments and handed them out. Estelle had never spent time around horses. She found their size and obvious strength a little frightening. Yet, they acted like eager children, stomping their feet and nudging the family when they weren't fed fast enough.
Estelle watched the four of them smile as Connor fed the white-maned palomino and rubbed its nose and declared, "This is the most beautiful animal I have ever seen."
Celia asked, "Do you ride?"
"No, and I probably should learn. There are a lot of roles that require it."
"I could teach you," Celia said.
Estelle found it very touching, how they both then glanced over, seeking Porter's approval. All the chief said was "Long as there's no quid pro quo on the man's two-wheel rocket."
"It's called a Ducati," Connor said.
"Call it whatever you like. It's still off-limits. Celia knows what I think of motorcycles, don't you, daughter?"
"We were talking about horses," Celia replied.
Connor said to her father, "No bikes."
Carol pulled her cell phone from her pocket and said, "Celia, sit on the railing. Connor, go stand beside her. Okay, Celia, put your arm around his shoulders. Great, now smile."
Celia said, "I can think of about a dozen ladies who are going to keel over in a dead faint when they see this."
Porter said, "Connor, unhand my daughter."
Celia said, "Daddy, how often does a girl get her very own Hollywood star to help her forget a guy who did her wrong?"
"Connor's helped you all he's going to. Now climb down."
As they walked back to the house, Carol pulled her husband on ahead. Estelle held back, so she was able to observe and overhear as Celia said, "Tonight has been great."
"For me as well," Connor said. "I feel . . ."
"What?" When Connor did not reply, Celia nudged him. "Haven't you heard? Miramar sunsets are made for sharing secrets."
"I feel like I've found a home." Connor sounded subdued. "It woke me up last night. I haven't felt that way about a place in a long time."
"What about LA?"
"I have a nice house," he agreed. "But this is different. I came up here looking for, I don't know. Peace, maybe. A way out. Something. What I found was . . ."
"Tell me."
"A second chance. Which is a really big challenge for me. It means learning how not to make the same mistakes all over again." He lifted his gaze to the sunset vista. "I'd really like to stay here."
"You mean, like, buy a house?"
"I'm so far in debt, buying a shed is pretty much out of the question. But, yeah, I'd love to find a place I could call my own." He addressed his words to the night's first stars. "I don't suppose you folks know of a place that might be on the market."
Only then did they realize that Carol and Porter were both watching and listening. Porter said to Connor, "Why don't you and me take a little drive."
Celia said, "I'm coming, too."
"You don't even know what we're—"
"You want to show him the Kaufmans' place."
Carol laughed. "She's got your number, honey. Looks like we're all coming."
CHAPTER 40
Porter's personal ride was a Chevy double-cab pickup with a six-liter diesel. Connor sat up front and the three ladies were comfortable in back. They descended to the valley floor, then drove another winding lane up the opposite ridge. Connor did not say anything during the ride. Estelle was beginning to think that this was his natural state. She decided the silence suited him.
The road ended at a set of tall metal gates. Porter opened his glove box and drew out a key, which caught the light in a remarkable fashion. When he saw Connor's expression, Porter held it up and said, "Gold alloy."
"He and his wife moved in here on Valentine's Day," Carol explained. "He had the keys specially made."
"Is that romantic or what," Celia said.
Porter checked a sheet of paper, then rolled down his window and punched a code into the keypad. Lights came on along the graveled drive and the house's forecourt. Porter took it slow, granting them time to inspect a Japanese garden replete with fountains and several groves of miniature cypress and fruit trees. The entire fenced-in plateau was less than an acre, and its size suited the meticulous garden. The home itself was simple in the extreme, a long flat-roofed rectangle that stretched the entire length of the mini-plateau. The interior held to that same Oriental-inspired concept. Everything was done with a rough-hewn precision. Hand-painted shoji walls were framed in thick redwood beams. The floors were teak and tatami. Carol touched a panel by the entrance, and seasoned wood shutters rolled up. The rear wall was almost entirely glass. The western view showed the full sweep of Miramar and the Pacific.
The home was fully furnished, and appeared to Estelle as though the owners had stepped away for an evening. A lovingly polished Baldwin baby grand stood in the front parlor's far corner.
They let Connor take his time. He wandered through the place on his own. Every time he came into view, and then vanished down another corridor, the Wrights exchanged smiles.
Finally Connor emerged and said, "I'm listening."
Carol said, "The Kaufmans were our dearest friends. Last year, Jamie had a stroke. They moved to Minneapolis, where their daughter and her family live."
Celia said, "They call every week. We exchange gossip, then they talk about putting this place on the market."
Carol said, "Selling their home means accepting that Jamie will not get better."
The words only seemed to make Connor sad. Estelle asked, "What's the matter?"
In response, Connor turned to Porter and said, "You want me as your neighbor?"
Celia walked over and hugged the actor. "Daddy's right," she said. "You're our kind of people."
* * *
On the drive back across the valley, Estelle's mind returned to Sylvie and the dilemmas her daughter faced. When they pulled up in front of the Wrights' home and Porter cut the engine, Estelle asked, "What happens with Sylvie now?"
"You were there in the courtroom," Porter replied. If he found anything odd about her abrupt change of subject, he gave no sign. "The judge agreed with Sol's request for a speedy trial. He had an opening in his calendar starting next Wednesday."
"That poor girl," Carol said.
Estelle felt the dampening effect her words had on the entire car. "I'm sorry for bringing it up like this."
"You're her mother," Carol said. "It's the most natural thing in the world."
Estelle asked, "So Sylvie needs to come up with the other payment, what's the word?"
"'Retainer,'" Porter said. "Fifty thousand dollars."
Connor shifted in his seat, but he did not speak.
Estelle said, "I can manage a little over half of that."
Celia said, "The whole thing is just crazy. Why are they picking on her?"
"That question has me wondering, too," Porter said.
"You're the chief of police, Daddy. Go out there and stop it."
"My powers end with the arrest, honey. It's in the court's hands now."
Celia huffed and crossed her arms. "Well, I don't like it."
Carol said, "None of us do, sweetheart."
Connor shifted again, like the seat could barely contain him.
Porter asked, "You got something you want to say, spit it out."
"I've got a lot that needs saying," Connor replied. "Only not just yet."
* * *
Estelle sensed there was more to Connor's intense silence than the house and the evening with a fine local family. She waited until they said their farewells and were driving back into Miramar to ask, "You liked the Kaufmans' place, didn't you?"
"What's not to like," Connor said. "It's beautiful, and the view is nothing short of stupendous."
"Why didn't you ask the price?"
"I need to work out some things first. If I ask, they'll have to talk with the Kaufmans. And that means the couple will have to confront some very hard questions."
"That's very considerate of you," she said. When Connor did not respond, she asked, "Can you afford it?"
"I'm paying a pretty sizeable mortgage on my place in LA," he replied. "I could sell it and move up, then rent somewhere when I have gigs. Or . . ."
"Or what?"
"I'm up for a big role."
"Big enough to pay for a place like that?"
"It's not just the one role. If I land it, and if I do well by it, my career elevates to a whole new level."
"Can I ask what the role is?"
He hesitated, then said, "I'm being considered for a major part in the new Bond film."
She slowed and put on her blinker and pulled to the side of the road. The news was not what had stopped her. Connor seemed to be utterly disconnected from his words. She struggled to find the right thing to say, and settled on, "When did this come up?"
"Apparently, it's been in the works for a while. I received confirmation that it might be moving forward while I was down in LA." Connor rolled down his window and looked up to where the darkened house was invisible against the night. "I could make a home there."
It hit her then. Sitting by the side of the main road leading into Miramar, Estelle realized what Connor was thinking. Any place he called home would remain empty without Sylvie.
And not only that.
She knew what she had to do.
Estelle saw with utter clarity the act that would bind her to her daughter and take a step toward healing the rift of nineteen years.
If only she could figure out a way to make it happen.
CHAPTER 41
When the phone rang on Tuesday morning, Connor was drawn from the most remarkable dream. He had difficulty recalling where he was, what day, why he was sprawled on this lumpish bed, why his hand couldn't find the phone and make the buzzing stop....
"Hello."
"I woke you, didn't I?"
"Gerald?"
"I'd apologize, but I'm not the least bit sorry."
"What time is it?"
"Almost nine. Don't they have clocks up there?"
"I slept until nine o'clock?"
"Well, apparently so."
"Give me five minutes to put on coffee."
"Give Ami five minutes and you won't need any."
"Wait, Gerald. Just wait." He heard Gerald squawk, but he put the phone down, anyway. Half a minute later, he picked it back up and said, "I can't talk to Ami without pants."
Gerald played the stork again. Ack-ack-ack-ack. "A half-dozen ladies on this floor just entered meltdown. Hold for Ami."
Ami greeted him by saying, "The Bond gig is almost yours."
Connor threw open his door and made a barefoot and shirtless circle of the parking lot.
"Are you there?"
"Trying to find enough air to say thank you."
"They want a screen test. I told them no way, you've played enough roles of this kind for them to forgo such ridiculousness. But they insisted. You'll be happy to know I exacted my pound of flesh."
"Tell me."
"They have agreed to a pay-or-play clause in your contract."
A screen test was standard ops for anyone but the biggest stars, and even they might agree to one if the role was out of their standard mode. To have Ami negotiate anything in return for Connor testing meant he was, for the first time in his life, being treated as a star.
Pay-or-play was another item restricted for top actors. Once the contract was signed, Connor would be paid even if the film was never made.
"Ami . . . thank you so much."
"That's my boy."
Connor scrubbed his face, trying to force his mind to wake up. "Before the shoot, I need to work through the script with my coach."
"They'll scream over the very idea of an outsider seeing their precious screenplay."
"Can you make it happen?"
"Stay by the phone. It looks like the test will be set for tomorrow morning. Gerald will call with the time and contact your coach. The studio will insist you go through the screenplay on set. No way will they allow you to take it home. I assume you can find your way back to Los Angeles?"
"In a heartbeat."
"Now that I've seen your bike, I almost believe you. Come on down this afternoon. They will want you on set bright and early."
* * *
Connor showered and dressed. He grabbed a pad and pen and walked over to the diner. Gloria was busy with other clients, and motioned for him to take the booth by the window. Connor waved a greeting to Joey, the cook and owner, and ignored the looks cast his way. Gloria came over with the pot of coffee, poured him a mug, and said, "Joey wants to know if cooking your breakfast will get us part of the hundred-thousand-dollar reward."
"Time's run out on that one."
"Shame."
A shrug at the cook grinning through the kitchen portal, and that was it. Gloria took his order and drifted away. Attention at the other tables returned to whatever had occupied them before Connor showed up. He was left alone.
Connor opened his pad and started writing. The Bond gig was a huge feat. It could potentially elevate him into the rarified status of a character actor who was also a bankable star. There were only a few such people in each generation. Stanley Tucci had been one of Connor's favorites since childhood, precisely because he accomplished what Ami had challenged Connor to do. Only it was now, after the upheaval of the previous few days, that Connor was beginning to realize what it actually meant. Such an astonishing range of roles would only remain believable if the emotions were real. Connor wanted to use this role as a target, and then begin aiming for the same breadth. And flexibility. And heart.
But that was not what his list was about. At least, not directly.
He worked through breakfast. Over a last cup of coffee, a familiar voice asked, "Mind a little company?"
He waved Estelle into the booth. "Not at all."
She smiled at Gloria, said coffee would meet her every need, then asked Connor, "What are you doing?"
"Trying to cement everything I've gained here. I've gotten through the next hoop on this Bond gig."
"So you'll be going back to Los Angeles."
"If the earlier Bond films are anything to go by, they'll build their interiors and custom sets both at the MGM lot in Century City and Pinewood Studios in London. But the on-location work will be shot all over the place."
"Is that answering my question?"
He smiled across the table. "Not really."
"So what happens next?"
"I'm heading back to LA for a screen test."
"And so the list?"
"Right."
"Will you buy the house?"
"If I can manage the cost, if it's available, if we can work things out with my bank." Connor glanced down at his notes. "But the house . . ."
"It's not what the list is about. I understand. You want to anchor yourself in the changes you have started working through here in Miramar."
He liked that about Estelle, the ability to grasp the unspoken, and gently nudge her way into the sensitive areas. "I'm trying to build a list of next steps. I don't want to get back into the rush and the grind and the hype, and then one day discover I've lost it all again."
"You won't."
"I wish I could be so sure."
"Let me be sure for you. You're going to make it."
Connor saw she was studying the large blank space at the center of his list, the place where a woman's name might have become the focus of actions and questions. If she was willing to even speak with him. If he had any idea what to write.
When Connor remained silent, Estelle said, "We have to help Sylvie. We can't let her lose the restaurant."
"I agree." Connor leaned back. "I am in debt up to my eyeballs. Otherwise, I'd—"
"No, Connor. It's sweet of you. But I'm thinking, well . . ." She took a long breath. "I was wondering if maybe we should hold a silent auction, where Sylvie's friends donate prizes."
Connor had a remarkable experience of hearing not just Estelle's words, but the formation of his own question. It was the one that had remained unasked, because he had no idea what exactly he wanted—other than being with Sylvie, which he knew was not going to happen. The issue that remained there in the empty portion of his list was: what should he do? Estelle's idea was not what rocked him back in his seat. He still did not know the answer to his dilemmas. But for the first time, he saw clearly the question he needed to ask next.
What could he do as a means of healing the rift? That was the issue he had to focus on. How could he apologize in a way that went beyond words and actually revealed . . .
His heart.
Connor realized she was still waiting for his answer. He said, "It's a great idea."
"You really think so?"
"Estelle, it's better than great. Everybody I've met around here cares about Sylvie. A silent auction would give them a chance to show her they're in this with her."
Estelle allowed her own uncertainty to show. "But raising fifty thousand dollars . . ."
"Right. The prizes need to be big."
"Something that will get people talking."
And that was it. The first part of his answer slipped into being as he spoke. Fully formed. Ready for action. Suddenly the booth was no longer able to contain him. He rose from the table and signaled to Gloria for the check. He looked down at Estelle, who was clearly taken aback by his abrupt departure. "Make one of the prizes that I'll play."
"Play?"
"Music. A private concert. Ask Rick or Marcela, they'll explain. It's what drew me into the restaurant that first day. I've loved swing ballads since before I could talk. My mom used to put on one of Nat King Cole's albums when I wouldn't stop crying."
"That's it," Estelle whispered. Her eyes glistened. "This is . . ."
Connor nodded. When Estelle did not speak, he finished the thought for her. "A headline event. Tell you what. See if a reporter for the local rag wants to come down and interview me in LA while I'm doing the Bond screen test. Use that to announce your silent auction."
Estelle wiped her eyes. "You really are a good man."
"No, I'm not." He folded up his half-finished pages. "But I'd like to be."
CHAPTER 42
When Connor entered the police station, a couple was shouting somewhere in the distance. The uniformed woman at the front desk smiled a pinched welcome and said that Porter was busy with a complaint. The woman introduced herself as Maud. She was in her late forties or early fifties, and had the strong, well-worn look of a woman very comfortable in her own skin. She told Connor, "You can go on back or you can sit here and have me pepper you with questions."
Connor was still getting used to the small-town attitude of familiarity. "How about I sit here and quietly work on something?"
"Oh, I seriously doubt that will happen. Celia and my daughter are besties. At breakfast, I heard all about your evening. There is nothing in the world that would please my daughter more than to know you and I had ourselves a real heart-to-heart."
"Do I have any say about my life becoming an open book around here?"
"Not in Miramar. How about I ask one question, then I leave you alone?"
"Doesn't sound like I have much choice." Then two voices, one male, the other female, rose to a near bellow. "What's going on back there?"
"Domestic," Maud replied. "The next-to-worst part of our job."
Connor had no interest in knowing what was worse than enduring that level of rage. "So ask."
"Are you going to buy the Kaufmans' place, and if you do, will you leave LA totally behind?"
"That's two questions, but I'll answer, anyway." As if he had any choice. "I loved the house, but I am up to my eyeballs in debt already. And the house looks extremely expensive. So I have to think very carefully about my next step. Which will probably include going to my bank manager on my hands and knees. As for LA, I want to keep acting. What that means about where I live, I haven't gotten that far yet."
"You like Miramar." She held up one hand. "That's a statement, not a question."
"I like it very much. Everything except the speed with which gossip spreads around here."
"Haven't you heard? Gossip is a small town's version of reality TV."
A beefy woman and a stumpy, red-faced man came storming down the corridor and across the front office and out the main doors. Porter followed them, his expression weary. "Remind me why I didn't just shoot them both."
"Lompoc State Prison is nasty. Plus, we'd miss you when they locked you up." Maud winked at Connor. "And Celia would go off to LA on the back of Connor's bike."
"All good reasons." He asked Connor, "You waiting for me?"
"I'd appreciate five minutes."
"Are you here to register a domestic complaint?"
"Not today."
"Then come on back."
Porter's office was fairly nondescript, highly functional, and uncluttered. The photographs lining one wall were all of his daughter and wife. Connor slowed enough to view Celia as she grew from a smiling toddler to a lovely young woman. When he turned around, Porter waved him into a seat and said, "Those two ladies make even the worst hour manageable."
"They are two great women."
"Yeah, Celia's lucky she took after her mom."
"You must have a lot of bad hours in this job."
"Some days. But I can't imagine acting is all bluebirds and buttercups." He shifted his weight, causing the chair to squeak. "What's on your mind?"
"Phil Hammond," Connor replied.
"Old Phil."
"What's his story?"
"Not much I can tell you beyond Miramar's city limits. You know the story about him and Sylvie?"
"Marcela told me a little. He owns a minority share of Castaways, right?"
"And two of the beachfront hotels. He's a developer from Santa Cruz. That's pretty much it." Porter picked up a pen and spun it between his fingers. "I don't deal in rumors, you understand."
"Sure."
"I don't know anybody who's willing to offer a good word about the man."
"Is he a criminal?"
"When he offered to come in beside Sylvie, I checked because she asked me. The answer is, he's never been arrested. More than that, I can't say."
"Has he ever tried to buy Castaways outright?"
Porter spun the pen between his fingers. "Where are you going with this?"
"Just wondering."
"Well, the answer is, old Phil doesn't ask. He tells. And, yes, about four months back he told Sylvie he wants to take over Castaways. He's offered to make her vice president of his hotel and residential group, or some such. Major pay raise. Lots of perks. Sylvie needed about twelve seconds to turn him down. She tried to be polite about it. Phil didn't take it well."
Connor sat there a minute, trying to fit the puzzle together. He liked how Porter clearly felt no need to push or pry or run on to the next thing. It was oddly comforting, even seated in a chair that was still warmed by the warring couple. "Miramar is nothing like what I expected."
"What do you mean?"
"I thought I'd come here, have a quiet few days, get my head sorted out, go back, rejoin my old life, like that."
"And that hasn't happened."
"There is not one single item that's worked out like I expected," Connor replied. "Starting with the idea that Miramar is calm."
"But you like it here."
"I do. A lot."
"Mind if I ask why?"
"I've found exactly what I needed," Connor replied, rising to his feet. "Thanks for your time."
"Don't mention it. You just be sure and let me know if you ever move past wondering about old Phil."
"You bet."
Porter waited until Connor was almost to the door before adding, "I thought we were going to be talking about the Kaufmans' place."
"We will. Soon. I hope."
"You're wondering about that, too, huh?"
"Oh, no," Connor replied. "I'm way past that."
CHAPTER 43
Estelle left the diner and drove her rental car down to the southern seaside cliffs. She would have preferred to walk, but she could almost hear the ticking clock. Water beaded on her windscreen as she pulled into the parking area. She left the car and walked out to the point. A lazy mist drifted over the water, like clouds too heavy to rise to their proper station. Pillars of sunlight fell here and there, their reflection off the water almost blinding. After a time, she walked back to the open-fronted chapel. A middle-aged couple were seated on the back row. Their eyes were closed and they gave her approach no notice. Estelle settled on a bench where the roof sheltered her somewhat. Her thoughts drifted with the mist.
She had no need of Connor's written list, though she admired him for making the effort. She had carried her own dilemmas for so long they felt imprinted in her DNA. This cliffside refuge seemed an ideal place to prepare herself for what she intended to do next.
Estelle knew she was about to step over an invisible line. This was no longer about helping her daughter from a safe distance. She proposed to take an active role in Sylvie's world. Estelle had no idea how her daughter would react, or whether she was doing the right thing at all. There was every chance she was making a terrible mistake.
When she felt as prepared as she might manage on this tumultuous day, she rose from her place and stepped out to where she would not bother the other penitents. Her whispered words were a simple repetition of the same fractured plea. "I need help."
As she returned to the car, she was certain she had been right to come.
* * *
Estelle phoned Marcela from the cliffside parking area. When the waitress came on the line, Estelle asked, "Could you please spare me ten minutes of your time?"
"What's this about?"
"I have an idea about how we might help Sylvie."
"She sure needs it. Look, I told Rick I'd help them prep today. Let me go check in, then we can meet."
"Could you please not mention this to Sylvie?"
"Oh, sure." The woman's good humor shone over the phone. "There's no one who likes being helped less than your daughter. Tell you what, why don't you go up to the café by your motel and I'll meet you there. Less chance of being caught out by the house detective."
Estelle drove back up the hill, parked by her studio, then walked to the café. By the time Marcela arrived, Estelle's nerves had almost taken control. She confessed, "This is probably a bad idea."
"Why don't you buy me a latte and let me decide."
When Estelle brought back the drinks, she took a hard breath and launched straight in, but Marcela did not allow her to get past the first few sentences. "Hold it right there."
Estelle nodded miserably. "I told you it was silly."
"You want to hold a silent auction and ask the town to chip in with prizes and then come and bid? And all the money goes to covering Sylvie's legal fees, so she doesn't have to sell out to old Phil?"
"Something like that. I suppose I—"
"Girl, this is brilliant."
"Really?"
"Of course, really. The whole town is already talking. There are a lot of people who would love nothing more than a way to do something. How much does Sylvie need?"
"Fifty thousand dollars. On top of the twenty I already paid."
"Wow." Marcela sobered. "That's a lot."
"Tell me."
"You'll need some major prizes. Something big enough to pull in the valleyites."
"Sorry, who?"
"That's what we call the rich people who have the estates back behind the ridgeline. They're a big part of why Castaways is a success. You'll need a real whopper to get their attention."
"Connor has offered to play."
"Play?"
"As a prize. A private concert with Connor Larkin, star of the next James Bond film."
"Wait . . . Our Connor?"
"He's headed down to LA this afternoon for the screen test. A reporter from San Luis Obispo will be meeting with him on set. Connor offered them a scoop in return for promoting the silent auction."
Marcela surprised her then. She turned her face to the late-afternoon sunlight and sat there in silence for a time. Estelle thought the woman defined Latina beauty, the honeyed skin, the full lips, the soft curls, the dark and soulful eyes. Finally Marcela said, "It's such a shame."
Estelle nodded, but did not speak.
"Sylvie won't even talk about him anymore."
"Which is why we need to keep this quiet until . . ."
"Until it's too late for Sylvie to call it off," Marcela finished. "When were you thinking about holding the event?"
"This coming Monday. I've spoken with the police chief."
"Porter. He's great."
"Right. He's arranging for us to use the town hall. It doesn't give us much time to put together the prizes and get word out."
Marcela bundled her things together and rose from the table. "Leave it with me."
But midway across the café, Marcela came to an abrupt halt. Estelle asked, "Something wrong?"
Marcela turned. "I've just had the most delicious thought."
"'Delicious,' as in, 'bad'?"
"Oh, no. It's a lot worse than that. This is just terrible."
"Do you want to tell me?"
"Best you don't know. It'll give you, what's the word?"
"'Deniability.'"
"Right." Marcela shivered. "This is going to be fun."
CHAPTER 44
For once, Connor remained the model driver his entire journey south. Twice he was passed by other bikers, who slowed to observe him and then fly on, clearly thinking unkind thoughts about a rider who could afford a Ducati and apparently didn't know what to do with it. But Connor liked having the time to put everything away, except for the road and the daylight and the ride. The drive forced him to focus down on the moment. He had the sense of developing a clarity that he could then apply to other things.
Twice he pulled off the highway and leaned against his saddle and watched the ocean. Both times he drew out his list and worked through a couple more points. When he arrived home, Connor parked the bike in his garage and sat there, waiting until the next item crystalized in the sunset. He went inside and showered and changed, then rode down to his favorite deli. He did not so much work through the meal as reflect on what he had written, and what it all meant. Then he returned home and went to bed. Twice in the night, he rose and studied the list and thought about his next moves. There was no sense of pressure, or really even of a possible next move. Just the same, the fact that he was thinking about these things left him at ease in his own skin. Given his state just a week or so earlier, Connor considered it a remarkable achievement.
The next morning, he pulled into the MGM studio gates at seven-thirty. Connor gave his name to the guard and was directed to a nondescript office building about halfway down the central lane. When he entered, his coach was nowhere to be found. Instead, Gerald said, "Don't you dare give me that look."
"Where's Mavis?" Mavis was his acting coach and a dear friend.
"Sick. Chest. No voice. I've heard cement mixers that sounded better."
"Gerald, what are you doing here?"
"I'm trying to rescue the moment, is what. Although Ami couldn't possibly survive a day without me, she's agreed this is important enough to try. And you are thrilled that I'm here. Go on. Say it."
Connor knew complaining would get him nowhere. "Thank you, and thank Ami. It's good to have a friendly face in my corner."
"On the next take, you can try and put a little more feeling into your gratitude." Gerald pointed Connor toward the aide waiting with visitor badges. "Go. Scoot. We have to hurry and get started before the suits upstairs change their minds."
* * *
The studio would only release the full script after Connor signed a confidentiality form and agreed to work alongside Christopher, one of the film's assistant directors. Christopher dressed like most film school snobs—stovepipe pants and black shoes with absurdly long toes, like they were about to curl up and turn purple and fit the clownish gestures that Christopher made constantly. His dress shirt had French cuffs, which flopped like starched wings around his fingers because Christopher wore no cuff links. His hair was waxed into a precise bird's nest. Christopher greeted every comment Connor made with an eye roll and a dismissive smirk. Normally, working alongside a film school snob made Connor's days crawl as slowly as a receding glacier. Today, however, Gerald handled Christopher with astonishing ease. Gerald made little insider jokes that left Connor completely baffled. If they had been speaking Farsi, he would have found more reason to laugh. Gerald asked Christopher for his opinion at every step. Gerald treated the AD's responses as solid gold. Christopher gradually left his arrogant attitude behind and worked with them.
When they had completed three read-throughs, Connor declared himself satisfied. Christopher left to phone the director and make arrangements for the set and crew. Connor and Gerald remained seated at the back of an unused soundstage, surrounded by the normal assortment of crates and cables and light stands and mic booms.
Gerald said, "You were almost human there at the end."
"I am in the presence of a master," Connor said.
"I hope you were watching," Gerald replied. "You can't expect me to show up every time they throw an AD at you. And the higher you go in this business, the more often they'll assign you a Christopher."
Gerald was a study in physical contrasts. He was aged in his late forties and wore his graying hair cropped military short. He was not unattractive as much as simply unremarkable. Connor had never seen Gerald wear anything other than conservative three-piece suits. He had once told Connor that nothing hid a multitude of sins so well as a tailored waistcoat. Today's outfit was gray herringbone, offset by a starched shirt so glaringly pink the shade could almost be called violent. His wingtips were polished to a mirror shine and his fingernails were always buffed. Gerald could easily have passed as a midlevel accountant, fussy and precise and short-tempered.
"You are as good a coach as I've ever worked with," Connor replied. "And I'm not just talking about the script."
Gerald was still forming his response when the makeup artist came for Connor. The role called for him to have been disfigured by a previous assault, one where acid had scarred the left side of his face. The film's producers wanted to view Connor in full character, and Connor had overridden Gerald's objections and readily agreed. This level of cosmetic makeover was new to him. Connor wanted to see how it affected his speech and his facial expressions. He suspected it would result in a more stifled emotive response, a sort of uniform deadpanned expression. A good deal of their work that morning had centered upon fashioning a character that suited the scar.
Christopher returned, accompanied by a reporter and photographer from the San Luis Obispo Tribune. After the introductions were made, the AD announced that the director had become tied up in a meeting with studio executives and could not make the shoot. Christopher fumbled the news so badly, Connor was certain he had known about this all along.
Most actors loathed working with an AD in control of a scene. With many high-powered directors, the AD was little more than a glorified gofer. Shooting a screen test was the director's way of offering the AD a bone. The problem was, doing a scene with Christopher risked having the director insist upon a redo. Connor could see that Gerald was preparing to baste the AD and serve him up well done, but Connor cut him off. "Actually, I'd prefer to work with you."
Both Gerald and the AD gaped at him.
Connor went on speaking. "All the director told you was, he wants me to be suave and deadly. And scarred. Right?"
Christopher jerked a nod. "Pretty much."
"Okay. We've crafted this into an urbane killer whose good looks were part of his allure. His trademark. Then came the last gig before the film opens. This was the first time my character ever failed. And the failure cost him his beauty. So why doesn't he go in for surgery and have it repaired? Because the scars constantly remind him of failure's cost. It's brilliant. It's a role I can grow into. Thank you."
Christopher looked from one face to the next, as though expecting someone to laugh and reveal it was all a joke. When the room remained silent, he said feebly, "I had better go make sure everything is ready on set."
When the door sighed shut, Gerald lifted his hands and applauded silently. Connor smiled, then turned to the reporter and asked, "What would you like to talk about?"
* * *
The shoot lasted almost three hours, far longer than was really required. Christopher insisted on multiple takes and constantly fussed over everything. Connor grew weary of the process, and the heavy facial makeup began to itch terribly. Even so, he found the delays actually fit the event. Connor chatted with the reporter; he positioned himself for the news photographer; he introduced himself to the behind-camera crew. Most important of all, he experienced what it was like to work at star level.
When they were done, Connor was standing in the MGM parking lot, seeing off the reporter and her photographer, when his phone rang.
Porter said, "You've made me go and break my own rules. I've meddled in something that's none of my business. Again. Carol says it's just part of growing old, but I have my doubts."
Connor replied, "You do realize you're not making any sense."
"I went ahead and asked the Kaufmans if they're the least bit interested in selling."
Gerald stepped into Connor's field of vision and pointed at his watch. Connor lifted one finger. Wait. "What did they say?"
"They're not ready to sell. Yet. But they'll rent. For a year."
Connor watched a golf cart hum past, an aide driving a star to the set. He felt himself become distanced from everything that surrounded him, the studio lot and the screen test and Gerald's impatience and the electric aftereffects of an on-camera performance. "Tell them yes."
"Don't you even want to know how much?"
"Is it a fair price?"
"More than, for around here."
"Then I accept. Who can do up the lease?"
"Carol got her realtor license a while back."
"Ask her to get things ready, I'll swing by tomorrow." Connor thanked the chief, then cut the connection and said to Gerald, "There's something I need to show you."
"Ami is dying back there. She hates answering her own phone."
"Gerald, you really want to see this."
* * *
Connor drove back from the studio with Gerald following in his own car. He opened the front door, cut off the alarm, then stood back and let Gerald take it all in. The man moved with the slow, drifting walk of an art lover entering a new gallery.
Ten minutes later, Gerald found Connor in the kitchen. "It's lovely, and I am terribly jealous."
"Thanks." Connor opened the door to what appeared to be a second pantry. Inside was a wine room, crafted when the subcontractor redid his home, and one reason why the kitchen had cost him so much. The chamber was almost completely empty. Connor had nicknamed the room, Someday. He asked, "Will you take a glass?"
"If it's nice, I might even have two." Gerald showed a rare smile. "It's a perfect day for a stroll back to the office."
"White or red?"
"Red is too somber. We're celebrating." Gerald lifted his phone. "Ami just texted. The studio and director both love your take. Houston, we have liftoff."
"In that case, we need something special." Connor selected one of the few really fine bottles he owned, a gift from a star who had shone more brightly because of Connor's work. He opened the Montrachet, filled one glass, and poured a trace into a second. He lifted the second and said, "Here's to meeting new challenges."
Gerald clinked glasses, drank, then declared, "Marvelous. You're not imbibing?"
"Can't. I need to head north. You should take this bottle with you, share the rest with Ami."
"Not on your life. This little bottle and I are soon going to be best friends." Gerald sipped again, hummed a note, then said, "All right. I'm listening. Who do you need murdered?"
Connor replied, "I want you to come live here."
Gerald knocked over his glass. "I'm terribly sorry."
"Don't worry about it." Connor reached for a towel. "It's only wine."
"A Montrachet? Only wine?" Gerald watched Connor refill his glass. "Could you possibly ease me into what is coming next?"
"It's hard to explain."
"Well, at least try."
"I'm not happy here. I mean, in LA. At least I haven't been."
"So you've found your nirvana on the central coast?"
"Not hardly. I just feel . . ."
"What?"
"Like I've finally started asking the right questions."
Gerald drained his glass. Set it down. He tapped the rim with one fingernail. As Connor poured a refill, Gerald asked, "So that means what, exactly?"
"Friends have found me a house I'd like to rent up there. And possibly buy. If the current owners decide to sell, and if I can afford it."
"With the Bond money."
"Right. But for now, it's a tryout. And to swing it, I need to rent this place." Connor gestured toward the house. "Ami mentioned your love of the Craftsman style. Find out what this would rent for. Knock off twenty percent. Let's do a twelvemonth trial run."
Gerald took exaggerated care in setting down his glass. "I positively insist upon paying you fair market value."
"Come on, Gerald."
"Oh, all right. Ten percent below. You are such a hard bargainer."
"Write up a lease. Include a clause that if I sell, you get right of first refusal."
Gerald turned and stared at the sunlight falling upon the sliding glass doors. After a time, he said, "When Ami took you on, I told her I thought she was making a terrible mistake. Ami said you merely needed one thing to bring your full potential to the table."
"A heart," Connor said.
"Actually, she said you needed a reason to move beyond your comfort zone. But I suppose a heart will do." Gerald smiled at the sunlit glass. "When I tell her she was right and I was wrong, Ami will crow. I hate crowing."
Connor gave that a beat, then said, "I need to ask your help with something."
"It's not often I utter this word and actually mean it. Anything."
"I need a detective. Somebody who can hunt out dirt on a person, and fast."
"Is the individual you want investigated here in LA?"
"Central coast."
"Let me make a few calls."
"Thanks." Connor slid over an envelope he had prepared while Gerald had toured the home. "Keys to the house and alarm codes."
Gerald looked down, but did not touch it. "You don't mean to say that I can move in now?"
"Whenever you like. Long as you don't mind me coming back for my things. I'm packing a couple of cases and hitting the road."
Gerald was still standing there, staring at the envelope, when Connor returned from the bedroom, set his suitcases in the Beemer's trunk, and set off. As he drove north along the freeway, he reveled in the sensation of having gotten at least one thing right.
CHAPTER 45
On Thursday, Connor signed the lease and then slowly toured the Kaufmans' home. They wanted to rent their place with all the furnishings, right down to the antique Baldwin baby grand piano. While Connor was making a mental list of the things he wanted to bring up from LA, his new phone rang. So few people had the number he was obliged to answer. "This is Connor."
"I hear you need a hunter."
The voice was heavily accented and carried a raw, dark edge. Connor asked, "You're a detective?"
"Your guy said you needed information fast."
"My guy," Connor said, "was right."
"Then no detective." The man was definitely Slavic of some persuasion, with a gravelly voice scarred by vodka and harsh black tobacco. "Your detective, he is most interested in building up the hours."
"Hours I don't have," Connor agreed. "You're Russian?"
"Me? No, man, I'm from Alabama."
"Do I get a name?"
"Sure. Jones. How's that?"
"Jones works for me. How much do your services cost?"
"That's between me and your guy. He says call, I call."
"I don't like involving him in my business."
"What business. This is a favor I do for a friend. Nothing more. No payment, no paper trail. I do this, I go back to being a happy Jones in Alabama. Now tell me what you need."
Connor realized he had just two choices: accept the man's terms or hang up the phone. He said, "There is a guy in Santa Cruz. At least I think he's based there. His name is Phil Hammond."
"Spell the name." When Connor had done so, Jones said, "There are many Hammonds in this city. Nine with first names Phillip or P. What else can you tell me?"
"He owns part of two motels on Ocean Drive in Miramar." Connor spelled the name.
"He a big shot?"
"Phil definitely likes to think so."
"Okay. Here. Phillip Jackson Hammond. CEO of Hammond Enterprises. They are . . . Okay, they own parts of many companies. He's rich, this guy. Big bucks. Wait, let's have a look at his IRS filings.... Okay, last year he paid Uncle Sam one point three mil."
"Do you see among his holdings a restaurant in Miramar—"
"Castaways. Thirty percent."
"That's our guy," Connor said. "But why would somebody that rich worry about owning a restaurant?"
"You like, I give you his number, you make the call. Otherwise, you ask questions I can answer."
Connor paced down the main hall and across his living room.
"You still there?"
"Thinking." He rounded the kitchen's center island, then retraced his steps. "What else does he have going on around Miramar?"
"Okay, that I can work with. Hold on." There followed several minutes of furious typing. Then, "Lots of activity, this guy. He thinks this activity is hidden. You meet this Hammond fellow, you tell him to hide better next time."
"What?"
"Oh, yeah. This company owns another, and that company has permit requests, okay, so many. Two new hotels, one oceanfront and a big development inland. Complex with name of Rancho Santa Maria. Hotel spa. Two hundred houses. Big."
Connor felt a subtle adrenaline rush. It was nothing like the nuclear high he experienced on his bike, or before the first take of a new film project. This was a different sensation, something so unique it took him a moment to fashion a name. He was moving toward an answer to the core mystery plaguing Sylvie. He was doing right by a good woman, a lady who deserved better. Connor said, "Tell our guy you just earned your check."
"That's all you need?"
"Oh, no. We're just getting started."
"Then you make a terrible negotiator, saying that now." But he sounded like he was enjoying the exchange. "Me, I listen to clock. It says, ticking."
CHAPTER 46
Sylvie passed the next three days wrestling with a dilemma. She was tempted to retreat into the sort of haze that her father had lived in much of his life. Throughout her teenage years, as Sylvie had gradually taken on more and more responsibility for structuring their home and lives, she had often accused him of living in a blind bubble of his own making. But now, she thought that perhaps she had gotten it completely wrong. Maybe there was a point beyond which an individual simply could not cope, and her father's tolerance level for crises had simply been lower than most people's. Whatever the reason, his response to the bad times had been to immerse himself in his work. But instead of using his art to express his anger and frustration, as did so many of his fellow artists, he used his palette to heal his heart. Sylvie wished she could speak with him now, discuss the clarity that had come through her current impasse. No doubt her father would have enjoyed hearing her apologize. He relished anything that closed a distance between them. He would have heard her out and said something about how much he admired her. Then they would have selected an album, and their home would again be filled with the smooth sounds of swing.
The music that Connor had performed so very well.
Throughout Tuesday, Wednesday, and Thursday, as she debated over her next step, Sylvie found it impossible to dislodge the man from her thoughts.
Each day, Porter came by to check on her as she was opening up the restaurant, then again as she ushered out the last patrons. Sylvie had to fight against the burning urge to ask about Connor.
Sol Feinnes called daily and assured her he was doing everything possible, which was not a great deal.
By Friday, Sylvie had the distinct impression that many of the Miramar locals were caught up in some secret drama. Everywhere she went, she felt eyes tracking her. But her internal debate isolated her. She had a decision to make. Whatever it was that had people so worked up would just have to wait.
As Sylvie descended the stairs on Friday evening, she finally accepted the inevitable. Thankfully, the front room was empty, so no one was there to see her as she walked to her favorite painting.
She stood there a long time, studying the hands fashioned from the sea and the hills and the night. She remembered the day he painted it, and what it meant to know they would both be calling Miramar their home.
She reached out and lifted the painting from the wall. "Sorry, Pop."
Sylvie carried it back upstairs and set it on her desk, where she could visit with it at the end of the shift, and again over morning coffee. Gradually growing used to the idea that one more fragment of her former world was gone.
CHAPTER 47
When Sylvie returned downstairs, the empty space on the wall was mirrored by the void at heart level. She found Rick and Marcela and Aubrey and their stand-in waitress, clustered around the bar. They straightened at her approach, and must have seen something in her face because Rick demanded, "What's the matter?"
She knew she couldn't tell them without bawling. That she was going to sell her favorite painting, the only one worth real money, in order to pay for a lawyer to protect her from going to jail for a crime she didn't commit.
But she was also not going to lie to them. These were her friends. They were as close to family as she had these days. They deserved to know.
Only not now.
"More of the same," was all she said. "If you don't mind, I'd rather not go into details."
"Sure thing," Rick said. "Where's the painting of the bay?"
"I moved it upstairs. Just temporarily." That was most certainly true. "What were you talking about when I came in?"
"We've invited Estelle for dinner," Marcela replied.
"What? Here?"
Now that the news was out, they seemed genuinely thrilled by the prospect of Estelle coming for a meal. Sylvie found herself unable to process their reaction. She endured their chatter as long as she could, then excused herself. She walked to her hostess station, gripped the sides of her podium, and inspected the night's bookings. Sure enough, there it was. A reservation for one, booked by Marcela, with the note that she wanted Estelle to have her finest table.
* * *
Throughout that busy Friday evening, Sylvie grew increasingly certain that she was the clientele's primary topic of conversation. She felt eyes track her every time she passed through the restaurant. She saw half-hidden smiles and heard discussions quickly stifled at her approach.
Then Estelle appeared, standing in the doorway. Sylvie thought she had prepared herself. Even so, crossing the floor required a special effort. "Welcome to Castaways."
"Thank you so much." Estelle was dressed in a pleated skirt and jacket of pale gray with narrow lavender stripes so subtle they were almost invisible in the restaurant's lighting. "This is such a thrill. I can't tell you how delighted I was to hear you wanted me to dine here. I've so wanted to see what you've created."
Sylvie was still digesting the news that Marcela had included her in the invitation, when the waitress rushed over and embraced the older woman. "You're here! Great! I've got an order waiting. See you!" And she was gone, leaving Sylvie to follow with an embrace of her own. Then she picked up a leather-bound menu and started to lead Estelle through the restaurant.
Only to be halted by yet another surprise.
Everyone seemed either to know Estelle or know about her. Table after table rose and introduced themselves. Time and again, Sylvie saw groups of locals smile and invite Estelle to join them. Sylvie realized the subtle excitement was rising up now, revealing itself. She had no idea how involved Estelle had become in this community.
Sylvie felt distinctly threatened by this realization. Miramar was her town. Her home.
By the time she seated Estelle by the window, Sylvie felt as though the power of choice had been taken from her.
She stood and smiled as Marcela went through the night's specials. Estelle then said, "You decide. They all sound wonderful."
Sylvie said, "The fish is especially nice."
"I would love that."
"What about some wine?"
"Just a glass, please."
Sylvie went to the bar and returned with something, she had no idea what, a glass of the first open white she had seen. She then stepped back as Rick stopped by Estelle's table. Sylvie returned to her hostess station and watched Aubrey go over and shake Estelle's hand. Then Marcela returned with a starter. The three of them laughed over something.
Sylvie's resentment grew steadily as the night progressed. Why did Estelle have to come to Miramar, now of all times? Sylvie had spent her entire life building a home for herself here. And now these people all assumed she would make room for the woman who had abandoned her? This was her decision. Not theirs. And certainly not Estelle's.
Sylvie did her best to suppress the evening's hidden tempest. Every time Sylvie returned to Estelle's table, Rick or Marcela or Aubrey was already there, chatting away like old friends. This was good in a way, as it allowed Sylvie to stand beside them, poised and smiling, humming little notes in response to a conversation she could not hear over the noise in her head.
As closing hour approached, Estelle rose from her table. Sylvie walked over and asked, "Are you sure you won't have dessert?"
"No, thank you. I'm not one for sweets." She turned and surveyed the restaurant. "Sylvie, this place of yours is simply marvelous."
"You're very kind." As Estelle was granted a final round of farewells from her staff and the few remaining customers, Sylvie held to her smile and waited. Finally she walked Estelle through the main doors and out into the night. When they were alone, Sylvie asked, "So what are your plans?"
The night shattered.
Sylvie could almost hear the sound of crystal breaking. She saw the broken shards appear in Estelle's gaze. "I just meant . . ."
"I know what you meant." Estelle's voice had resumed the soft sorrow it had held on that first awful meeting. "I'll leave Tuesday. In four days. If that's acceptable."
"Stay as long as you like," Sylvie said feebly.
"Thank you. That is very kind. I think Tuesday." Estelle's voice strengthened with each word. Only now it carried a flat, metallic note. "That way you have one less distraction going into the trial, yes?"
"Whatever you think is best."
"Good night, Sylvie."
She stood and watched Estelle climb the hill. Logic told Sylvie she'd done the right thing at the right time. But she was chased back inside by an echoing refrain. How much she wished she could take back what she had said.
Only when she locked up for the night did Sylvie realize that she was the only person Estelle had not embraced in farewell.
CHAPTER 48
Two hours later, Sylvie carried her turmoil upstairs. She had a dozen perfectly good reasons to let the situation with Estelle remain as it was. A hundred. The days were already too full. She needed to focus on getting through this crisis and finding some way to pull her life back on track. Maybe then, she could reconnect with the woman who had abandoned her. Right now, the only thing she really needed . . .
Sylvie seated herself in front of the painting. She missed her father with a visceral longing. He rarely had offered any answer to the problems they had faced. And there had been a lot of problems. But he had always made her feel better. His painting glowed softly in the room's lights, a gentle communication passed through all the years and experiences that separated him. Sylvie knew he would want her to make peace with Estelle. This made her feel even worse about how the night had gone.
As she climbed the stairs to her bedroom, she found herself wishing yet again for the chance to talk things over with Connor. The sheer impossibility of the yearnings did not free her from the simple fact that she missed him. Despite everything. She wondered what it would be like to rely on a man's strength again. She had been alone for so long, faced so much on her own, gone so far. Sylvie tried to tell herself that it was ridiculous to wonder about such things with Connor. The man who had lied to her about almost everything. The man who could not be trusted. Especially now.
* * *
But that night Sylvie found no peace in sleep.
She dreamed that she was downstairs in the restaurant. Once or twice each week, she had a nightmare where she stood at the hostess station and searched for a reservation that she herself had taken and then forgotten to write in the book. Sylvie would start to tell the group of eight or ten or even an entire wedding party that she had made a mistake, and there was no room for them. Then she'd glance down and realize she had forgotten to put on pants.
Tonight's dream was very different indeed.
To start, the restaurant was almost empty. For another, she stood by the long table and stared at the stage. Her mother was seated in a chair by the tall bay windows. She smiled as Connor played the piano and sang.
Sylvie almost recognized the melody, but she could not place it. Estelle nodded her head and tapped her fingers in time to the song. They were so wrapped up in the moment they did not even notice Sylvie. It was just the two of them, and they were happy.
Abruptly Sylvie became lanced by the conviction that they both knew she was there, but chose to ignore her.
There was nothing she wanted quite so much as to join them. As she started forward, another person leapt in between her and the stage.
Sylvie faced herself. She was an angry, vindictive, bitter woman who looked thirty years older. Aged and desiccated and filled with a lonely, burning rage. The fury left no room for any goodness. Or love.
The second Sylvie fought with vicious ferocity. The harder they struggled, the more Sylvie wanted to be up there. With the two of them. However, she could not break through. She could not . . .
When Sylvie woke up, she was standing in the middle of her bedroom floor.
CHAPTER 49
A brisk wind blew from the northwest when Sylvie emerged Saturday morning. Normally, these were her favorite days. The air carried a special bite that was only found along the Pacific Northwest. The cloudless sky was a piercing blue, the sunlight strong enough to defy the day's chill. Sylvie loved taking a few indulgent hours where she could revel in a place that was, for her, unique in all the world.
But after such a fitful night, Sylvie rose to a breakfast of regrets. She knew she had to act. She walked up the main road to the guesthouse and found her mother seated on a bench, reading the newspaper. Estelle started at her daughter's appearance and hid the paper away, like she had been discovered doing something wrong.
Sylvie said, "I've come to apologize."
"There's no need."
"I shouldn't have spoken like I did. I . . ."
Estelle studied her for a long moment, then asked quietly, "Would you like to sit down?"
Sylvie settled onto the bench and searched for something more to say. But now that she had apologized, she felt deflated.
"I was sitting here, wishing I could speak with my Jack," Estelle said. "I need to tell him he was right to make me come and find you."
Sylvie opened her mouth to shape words, which she would probably never speak, about wishing to know a man she had not even been aware existed.
Estelle turned to her. "My darling daughter, you never need to apologize to me for anything. And certainly not for last night. The very thought is, well, it's absurd is what it is. I abandoned you."
Sylvie watched as Estelle swung about and aimed her face at the sun, as though drawing in enough strength to maintain control. Sylvie wished she knew how to reach across the impossible distance and offer comfort. But all she could manage was, "There's so much in my life right now."
Somehow, for once, she had said the right thing. Estelle calmed enough to say, "And the very last thing I want to do is add to your burdens."
"You're not," Sylvie said. And only as she shaped the words did she realize they were true. "Really. I'm fighting against . . ."
When Sylvie could not find the proper word, Estelle offered, "Shadows?"
Sylvie could not think of a better word, so she nodded agreement. It wasn't exactly what she'd had in mind, but somehow the word fit comfortably in the space between them.
They settled back and sat through a most uncommon time. The sun warmed the silence, and even knitted together the wounds, at least enough for Sylvie to say, "Would you like to take a drive with me?"
* * *
The journey to Paso Robles took just under an hour. Sylvie spent much of the time regaling Estelle with stories from her teenage years. Back then, the regional farmers' markets had been a highlight of Sylvie's week. San Luis Obispo, Santa Maria, even occasionally Santa Cruz, all of these had shaped her knowledge of the region she called home. Her favorite was Paso Robles, an inland town with two very distinct faces. The old section dated from the same era as Miramar's founding. Back then, Paso Robles had been a center for the state's cattle and sheep industries. Nowadays the local agriculture was dominated by grapes. The central coast was California's second-largest wine producer after Napa.
The region was also home now to thriving organic farms; as a result, Paso Robles held a prosperous, go-ahead air. The central coast had not been quite so hard hit by the drought as farther inland or to the south. The result was a lingering trace of the easy high-octane life that had characterized Californian farming communities until the reservoirs dried up. Shopping at these markets had been Sylvie's introduction both to the region's oldest families and to healthy cuisine. She had obtained her first waitressing job from a chef she had met here in the Paso Robles market. From that experience, she had been drawn into the artistry of food.
Her father had painted any number of the markets. Many old-timers still held on to sketches and pastels Gareth had traded for produce. Nowadays their children greeted Sylvie as one of their own.
Sylvie's ride of choice was a Ford F-150, with a covered rear compartment and two oversized coolers. As she parked in the market's overflow lot, she asked, "Is it bad, my talking about times with Pop?"
"I doubt very much that a rule book has ever been written to cover this situation," Estelle assured her.
"I mean, do you mind?"
"It hurts," Estelle confessed. "A little. But I also like it very much. You're filling the empty spaces."
The oddest things seemed to impact Sylvie these days. She needed several minutes to swallow down the emotional burn. They were almost at the market's periphery when she managed to say, "I want you to stay."
Estelle stopped, causing the foot traffic to veer around them.
"Don't leave on Tuesday," Sylvie told her. "Even if you were really planning to. I mean, before I said what I did. Leaving now would only make it worse, going into the trial." She gave herself a mental kick. "That sounds so selfish."
"No, Sylvie. It sounds so human," Estelle replied. "You are facing one of the worst episodes of your adult life. You are absolutely justified in being selfish."
"Then you'll stay?"
Estelle said slowly, "There's something I need to tell you."
"What?"
"Let's wait until we've enjoyed this outing a little. Please. It's such a special moment for us both."
"So . . . it's bad, what you are going to tell me?"
Estelle bobbed her head from side to side. "It's hard for me to say. But it needs to be discussed. Afterward, if you ask me again, I'll be here with you through the trial."
"But I have to tell you a second time that I want you to stay," Sylvie said, wanting to get it right. "After you tell me your secret."
"Right."
"Which you won't say now."
"Right again."
"What if I insist?"
"Don't," Estelle pleaded. "This is a dream come true for me. Just let's enjoy this time together, all right?"
In truth, Sylvie was grateful for the chance this offered to leave the painting in the truck's rear compartment and pretend the day did not hold such a hollow ache. As they started walking toward the market, Sylvie said, "I hate knowing a secret is out there worse than anything."
"Of course you do." Estelle slipped her arm through Sylvie's. "It's part of being a woman."
CHAPTER 50
Everyone in the Paso Robles market, traders and customers alike, seemed to know about Sylvie's problems. She and Estelle were stopped every few steps by yet another person who assured Sylvie of his or her support. Sylvie had no idea how to respond, especially when these folks included Estelle in their greetings. Word had spread so fast, she assumed it had something to do with the secret Estelle insisted upon putting off.
They passed the art gallery that formed the horrid purpose for this journey. The owner was a regular at Castaways and had repeatedly offered to buy the painting in Sylvie's truck. Sylvie's step turned leaden at the thought of carrying the oil inside. Then they were past it, and another couple greeted them, and Sylvie found herself in no great hurry to do anything more than enjoy an hour with Estelle.
At the market's heart was a makeshift food court, with over a dozen food trucks offering everything from Szechuan to Honduran specialties. From a family that remembered serving Sylvie as a child, they selected tortillas of spicy pulled chicken and coriander. They had found a table somewhat removed from the others and ate in companionable silence. Then Estelle bundled up the waste and said, "Thank you for sharing this with me. It means more than I can say."
Sylvie took that as her signal. "Will you tell me whatever it is you've been holding back?"
"I'll do better than that," Estelle replied. "I will show you."
It seemed to Sylvie as though Estelle moved with exaggerated care. Every gesture carried a sense of dramatic tension. Sylvie realized Estelle was struggling against her own anxieties as she drew a newspaper from her purse, unfolded it, and set it on the table between them.
On the front page of the San Luis Obispo Tribune, Connor Larkin stared up at her from four different photos. One showed a terrible scar covering almost half his face. Then she read the caption, though the words seemed scrambled in her brain. Something about a starring role in the new James Bond film.
Even more surprising than being confronted by Connor, and realizing that Estelle not just knew but had participated, was Sylvie's internal reaction. She was not nearly so shocked as she might have expected. All the smiles and furtive discussions made sense now.
So, too, did the dream.
She stared down at Connor's pictures and read how he was backing a regional appeal to cover her legal costs. She read the response from Sol Feinnes, who declared the entire court proceedings a travesty, as bad a case as he had ever seen in his three decades as a trial lawyer. Then there was the paragraph quoting Estelle, who expressed an unqualified confidence that with the support of Sylvie's friends along the central coast, her daughter would emerge triumphant.
She lifted her gaze and watched the market crowds stream past their table.
For one incredible, impossible moment, it felt as though her father was seated there with them. Sylvie stared across the table at Estelle, the woman whose efforts meant she might just possibly be able to rehang the painting back on the restaurant wall.
Sylvie confessed, "I had a secret reason of my own for this trip. I have one of Pop's paintings back in the truck. My favorite. A gallery owner here in Paso Robles offered me forty thousand dollars—"
Estelle said sharply, "Don't you dare."
"I thought I didn't have any choice."
"Well, you do." Estelle reached across the table. "You have faced so much. You have done so well. Now you need to let others help you."
"That's very hard for me," Sylvie said, "letting others be strong."
"Well, of course it is. You had no one else to count on for so long except yourself." Estelle's face crimped tight. "And I am largely to blame for that."
It would have been so easy to remain silent, to allow the harsh truth to punish Estelle for abandoning her. But just then, in that sunlit moment with her father so close she could almost hear his voice, Sylvie found it easy to say, "But now you're here."
Estelle squeezed her hand once, then let go, leaned back, and used her napkin to clear her eyes.
Sylvie lingered there, knowing all along that there was a fourth person who needed to join in their moment. Finally she spoke the word aloud. The name she had vowed never to utter. "Connor."
"He's a good man," Estelle said. "Yes, he's made mistakes. A lot of them. But he's trying to make amends."
Sylvie found a delicious pleasure in fashioning the name again. "Connor. He lied to me."
Estelle nodded. "Yes. He did."
"I hate liars worse—"
"Worse than living alone?"
Sylvie took a long breath. "There was a man . . ."
"I know. Bradley. That wretch."
"Who told you?"
"Rick and Marcela. I'm so sorry."
"The two of them are in on this?"
"Up to their eyeballs." Estelle smiled. "They are wonderful."
"Yes, they are." She resisted the urge to speak his name again. Instead, she said, "I don't know what to think."
"I understand."
"You say that a lot."
"Do I?" Estelle toyed with her cup. "My late husband Jack was a very good man. He helped me see that the first thing that I had to do, if I was ever going to know any sort of happiness, was to forgive myself."
"That must have been hard."
"Very."
"I'm sorry. That sounded terrible. I didn't mean you . . . Actually, I don't know what I meant."
Estelle's smile was mostly sad, but not entirely. "You don't know, you can't imagine, how wonderful it is to be sitting here and talking to you."
CHAPTER 51
On Friday, Connor hired a U-Haul and drove back down to Los Angeles. He finished loading up on Saturday, tucked his bike into the rear hold, signed Gerald's one-year lease, accepted the man's fumbled thanks, and joined the heavy northbound traffic with a smile on his face. On Saturday night, he slept in the Kaufmans' former home. Twice he woke in the night and padded through the rooms, taking it all in.
On Sunday, he was joined by forty-seven of his new best friends.
What happened was this.
At dawn, he went out to the east-facing Japanese garden and discovered a rough-hewn wooden bench tucked into an alcove of miniature cedars. The bench was situated so the sunrise would reflect in the ornamental pond. Now and then, Connor caught glimmers of liquid gold flashing in the water as giant koi rose and joined in his salute to the new day.
Then he went back to bed and slept until almost noon. He woke and stretched and showered, filled with a luxurious sense of languor. Connor drove his Beemer into town and enjoyed a late brunch at the diner. When Gloria asked a couple of questions about Hollywood, several tables turned to hear his response. It all seemed to carry the idle curiosity of small-town life.
Connor left the diner, crossed the street, and stopped by the guesthouse office to thank the matron and check out. When he entered the rear parking area, he was greeted by a dozen or so people milling about the small patch of green. The lot was jammed with cars and pickups. Estelle waved and continued with her conversation as Marcela walked over and declared, "I don't know whether to shoot you or hug you."
"I deserve the first, but prefer the second," Connor replied.
"Don't we all." Marcela shook her finger at him. "You bad boy. Why didn't you say something?"
"You saw the shape I was in when I showed up here." Connor saw others shifting in closer. "I needed . . ."
"A reality check?" Marcela offered.
"A chance to step away and take a good look at all the wrong moves." He asked, because he needed to, "How is Sylvie?"
"Hanging in there. Barely." Another finger shake. "You hurt a good woman."
Connor dropped his gaze. Nodded to the pavement. Guilty as charged.
Marcela stepped forward and hugged him. The woman's embrace was as intense as her smile. "That's for everything you're doing to make things right."
It burned his throat, but he said it, anyway. "If only I could."
She cocked her head, with her dark eyes glistening. "Is it true, you're moving into the Kaufmans' place?"
"Signed the lease Thursday, slept there last night." Connor could not ignore the people any longer. "What's going on here?"
"We were planning to use Estelle's studio for sorting the auction goodies and making preparations."
Connor observed, "You'll never fit all those people into her studio. Much less the items."
"I know. Our auction has grown from a minnow to a whale. But we can't get into the town hall until tomorrow." Marcela started ticking off her mental list. "We need to set up the program. We've got volunteers supplying lots of finger food, but they need a kitchen. The beverage service that supplies the restaurant is donating a ton of drinks. We've got almost a hundred items to be auctioned, and turned down almost as many. We need to set up—"
"You can use my place," Connor offered.
And that was all it took. Five minutes later, he was at the head of a car train that stretched back almost half a mile. Estelle was seated beside him, and spent the journey describing her time with Sylvie in the market. When they arrived, they jammed the drive and spilled down the narrow lane. Marcela took charge with the ease of a born general, showing the newcomers the same mix of brisk authority and warm greeting that endeared so many at Castaways.
Connor drifted around for a time, lending a hand here and there, feeling overwhelmed by the number of people and the frenetic activity. An hour after their arrival, he stepped outside to take a call from his Alabama-Russian detective.
"You say check in, so I check," Jones reported. "So far, I find hints but nothing definite."
"But where there's smoke," Connor replied.
"Indeed much smoke. This Hammond fellow, he probably lights many fires. Legal, not legal, I'm thinking this doesn't matter much to him. Only what he can get away with."
Connor felt the renewed sense of electric tension building at gut level. "I need evidence. I need it now."
* * *
When Connor returned inside, he found Rick and Carol and Celia and the Castaways kitchen crew had arrived and were busy supervising a new contingent of worker bees in preparing dips and whatever finger food could be left in the refrigerator overnight. Someone set up a portable grill on his back deck. Others began passing around plates of hors d'oeuvres, while the kitchen crew complained bitterly over their food being stuffed down the throats of every lazy Joe who invaded Connor's new home.
A silent auction was basically a reason to hold a giant party, or so it seemed to Connor. Each item would have a bidding sheet attached. People were expected to mill about, eating and drinking and bidding against one another. According to Marcela, the only way to keep a silent auction even barely civil was to assign each person a secret number and to ban all guns.
By the time people started leaving around late afternoon, every flat space in Connor's home was decked out in donated prizes and platters of the goodies that couldn't be crammed into his oversized fridge. His front walk was lined with crates of soda and beer. As Marcela and Rick left for work, they warned him not to drink more than twenty of either. That was not a problem, since he hated both. The wine delivery was late and would now be taken straight to the town hall, along with washtubs of ice. Connor wandered through the rooms, bemused by the absence of people. The air still vibrated slightly from all the noise that had vanished.
Then he spotted Estelle weeping.
She had drawn one of the deck chairs over to the veranda's far corner, beyond a pair of waist-high stone planters that marked his bedroom windows. Connor hesitated over whether he should disturb her. Estelle managed to give sorrow an elegant air. Something about the way she held herself left him fairly certain she knew he was there, so he asked, "Do you need something?"
To her credit, Estelle saw no reason to apologize for her tears. "I am positively overwhelmed with joy."
Connor settled himself on the planter's edge. He was so moved he needed a moment to try and find a response that suited, at least a little. "My life in LA was pretty much defined by solitude. Even when I was surrounded by people, I was alone."
Estelle used both hands to clear her face. "And now?"
Connor sat and watched lights illuminate the streets of his new hometown, and thought of a painting in a restaurant he would probably never visit again. "This is the first housewarming party I've ever had. The fact that it's all been done for someone else makes it even better."
Estelle gave that a moment, then asked, "Will you do something for me?"
"Anything," he replied, and meant it.
"I need to warn you up front, I'll probably bawl my eyes out."
Connor nodded. "I've had some experience at making good women cry. At least this time it'll be because they asked."
Another swipe of the face, then, "When Sylvie was still tiny, her father sang her to sleep almost every night with the same tune. It became the first song she ever sang to me." Estelle took a very hard breath. "I don't think I can even say the name."
"Just tell me the artist," Connor said.
Her chin quivered, but she managed, "Dean Martin."
He knew the one she meant. The song had remained forlorn and forgotten for almost twenty years. Then Dean Martin was fooling around in the studio with the song's author, Ken Lane. They had just one hour of recording time left, and Martin was a song short on his new album, Dream with Dean.
Martin had always loved this tune, and suggested they wing it. No arranging, no discussion. Lane was on the piano, with three pals who'd been hanging around the studio on guitar, bass and drums. They finished in one take. They had forty-seven minutes still on the clock, but neither felt like playing it through again.
Listeners loved it so much, Martin took the unheard-of step of recording the same song again on his next album. Only this time, he did so with full orchestration and chorus. His label, Reprise Records, was so excited by the result, they renamed the album after the song. When released as a single, it knocked the Beatles' A Hard Day's Night off the number one position in the Billboard 100. Everybody Loves Somebody Sometime remained there for eight weeks and went on to become Dean Martin's defining sound.
Connor rose to his feet and held out his hand. "It will be my pleasure."
CHAPTER 52
Sylvie somehow managed to navigate her way through the busiest Sunday her restaurant had ever known. Almost every table shared the suppressed excitement of people about to burst from holding on to the news of a surprise party. And though she knew about tomorrow's silent auction, Sylvie continued to be surprised. Dumbfounded, in fact.
To outsiders, Californians could seem a superficial, even dismissive people and culture. The reason for this was simple: California was constantly being inundated by newcomers. Its larger cities were filled with people who came from somewhere else. The vast majority still classified that "somewhere else" as home. The only way small-town California remained even partially intact was by being insular. They weren't unfriendly. They weren't closed. It just took time and effort to become accepted as a true member of the local community.
As a result, it was hard to put down roots in this western land, and harder still to be accepted. Nowhere was this more true than along the central coast. The locals nicknamed their region the Middle Kingdom, as in halfway between Los Angeles and San Francisco. It was a world unto itself and beholden to neither.
Sylvie had never felt so bonded to the region or its people than that Sunday. Castaways was jammed with locals. The bar was three deep. Everyone was there, in his or her loudest casual fashion, to show that she was one of Miramar's own.
And there on the wall was her father's painting of Miramar Bay. Right where it belonged . . .
Because of these raucous, pushy, opinionated locals who knew all her secrets and loved her just the same.
And because of a mother who had abandoned her, and had given Sylvie every reason to never forgive or forget. But who now was here, offering support in what would otherwise be a very dark hour indeed.
And because of a man Sylvie had every reason to mistrust.
But Sylvie held it together because that was what a gracious hostess did. To all the people who entered, Sylvie warmly welcomed them to Castaways.
Where they belonged.
CHAPTER 53
On Monday morning, Connor woke just as dawn painted its first strokes upon the sea. He was drawn from slumber by the sound of his own playing. Before he had moved to Los Angeles, such early-morning dreams had been frequent occurrences. As he made his coffee and carried his mug into the parlor, Connor tried to remember the last time he had dreamed about music. Years.
He sat down at the Baldwin baby grand. The handmade instrument was considered by some to be the world's finest for small venues and homes. The sound was warmer and smoother than some professionals cared for. Both the Bösendorfer and Steinways had a cleaner, crisper finish to the keys, but the Baldwin created a more welcoming resonance. As Connor drank his coffee, he fashioned a one-handed melody from the dream's lingering traces. Connor set his mug on the floor by his bench and began to play.
The song that woke him was "Something," written by George Harrison and released on the Abbey Road album. It was the only song by Harrison to ever reach number one on the Billboard charts, and Lennon claimed it was the finest song the Beatles ever created. The song was subsequently covered, or rearranged and released, by over a hundred other artists. It reached the top ten another two dozen times. Arranging this song placed Connor in the company of Elvis, Ray Charles, James Brown, Shirley Bassey, Smokey Robinson, and Joe Cocker. Over the next two hours, Connor slowly restructured the song to fit his own voice.
By the time he was satisfied and played it through from beginning to end, he had the sense of taking a giant step toward fashioning his own style. Even so, when Connor rose from the piano a little after eleven, he knew none of this day's accomplishments had really been about the melody.
He made a sandwich and carried it out to the veranda. He stood looking out over the town and the cove and the shimmering Pacific. Connor knew this was why he had come to Miramar. It had never been about the music. It was all about regaining the ability to dream.
As he returned inside, Connor wondered if there was any chance Sylvie would attend the silent auction. If not, Connor hoped that word of this song would get back to her, and she would understand the message behind the words. That Connor was sorry. That he was trying to do as she had said, and hold on to his dreams. That he would never be leaving Miramar. No matter how far his acting career might take him, Connor had found his home.
He was rinsing out the coffeepot when his cell phone rang. "This is Connor."
"Jones here. Remember me?"
"My Alabama Russkie. Sure thing."
"Russkie. Please. Such offensive talk. I am proud to be Ukrainian."
"My sincere apologies. How are tricks?"
"Tricks is an excellent word. I have information. You decide about the tricks."
"So tell."
As he listened to Jones's report, Connor felt the zinging skyrocket explosion of having gotten another something very right.
CHAPTER 54
Estelle found Monday to be an almost impossibly good day.
Even the simplest of acts carried an uncommon sparkle. The air was alive with an excitement no one bothered to hide any longer. She ate breakfast at the diner's largest table, surrounded by eleven of her new best friends. From there, they went straight to the town hall, where they found another dozen people already busy with preparations.
People treated Estelle like a glorified manager. Like she knew what she was doing. Like she belonged. Trestle tables were carted in. Red and green felt coverings were tacked into place. Prizes were sorted, bidding sheets were stapled, lists arranged, jobs appointed, bars set up. There were signs, bunting, balloons....
Suddenly, at four, it was all done, or, at least, the core workers had shooed out all the others so they could pretend to work at last-minute details and, in truth, just take a moment to revel in a party that had not yet started. She and Marcela and Rick found a quiet corner by the stage. Marcela asked, "Where's Connor?"
"Handling something with Porter," Rick replied.
Marcela asked, "Does it have anything to do with our party?"
Rick wiggled his hand back and forth. "Yes and no."
"In that case," Marcela said, "it has to wait until tomorrow."
"No way," Rick replied. "This has to happen right now."
"Why is that?"
"I can't tell you."
Marcela said to Estelle, "Does that sound like a man or what?"
But Estelle was busy studying the hall's decoration, the tables lining the room, the prizes, the signs. "Three days ago, this was just some vague idea."
"Nothing brings people together around here better than a party for a good cause," Marcela said.
Rick's phone rang. He excused himself, and then walked away. Estelle said, "If only we could get Sylvie to come."
"Oh, she's coming," Marcela assured her. "She just doesn't know it yet."
"Sylvie might not agree."
"Oh, she may think she has a reason to stay away," Marcela said, "but she's coming anyway."
Estelle resisted the urge to hug the lady. "You're saying Sylvie's feelings about it don't matter?"
"They matter, all right," Marcela replied. "They just don't matter enough."
Rick shut his phone, walked back over, and announced, "Bingo."
"Nice idea, wrong night," Marcela replied. "Whose canary did you swallow?"
"Our boy has come through again," Rick said.
Estelle asked, "You're speaking of Connor?"
"None other." Rick's phone chirped. He checked the readout. "Perfect. Sol is five minutes out."
Estelle asked, "What's going on?"
"This is Connor's doing," Rick replied. "I'll let him tell you. Right now, we need to find us a very quiet corner."
CHAPTER 55
Rick led Sol Feinnes down the rear corridor and into the mayor's office. Sylvie's attorney was taken aback to find himself confronted by Connor and Porter and Estelle and Marcela. Porter greeted him by saying, "You're late."
"I've spent twenty minutes crawling along the road into town. The traffic was unbelievable. Half the population of the entire central coast must be coming to this silent auction. Am I under arrest?"
Porter demanded, "Did you break any laws getting down here?"
"Not that I'm aware."
"Then you're okay in my book."
Sol surveyed the five faces, all of whom appeared to be holding tightly to a barely suppressed mirth. "Will somebody tell me what's going on?"
Porter used a thumb to direct attention at Connor. "This guy has something to tell you."
"The guy in question being the star whose face is on the front page of my newspaper."
"Right."
"You look better without the scar, by the way."
Porter nodded agreement, and said to Connor, "Tell him."
"I hired a sort of detective," Connor began.
"I'm not sure 'sort of' is a legally recognized term."
"I guess you could call him a researcher," Connor said.
"Slippery word, researcher," Sol observed.
"This is definitely one slippery guy," Connor agreed.
"I'm assuming this means whatever he's uncovered will not be admissible in a court of law."
"You'd be better off presenting the judge with week-old squid," Porter agreed.
Sol said to the chief, "And yet you're here."
"You betcha," Porter said. "Wouldn't miss this for the world."
Rick said, "He's actually responsible for getting your dance partner to show up."
"Excuse me?"
"Phil Hammond," Porter said. "He's the reason we're talking."
"What does that snake have to do with anything?"
"Hammond is minority owner of Castaways," Connor said. "Recently he tried to buy it outright. Sylvie refused. I had this idea, maybe Hammond was somehow connected to everything that's been going on. So I hired this . . ."
"Researcher," Rick chimed in.
"The man has uncovered some serious dirt," Porter said.
When Connor finished laying it out, Sol Feinnes blinked slowly. Then he declared, "This isn't dirt. This is solid gold."
"Told you," Marcela said.
"It's also radioactive," Sol continued. "I couldn't bring this into a courtroom if I was wearing a lead-lined suit."
"But you can use it as a lever," Porter said.
"Oh, my yes." Sol revealed a truly wicked grin. "You say Hammond is coming?"
"He's due any minute," Porter confirmed.
"Then we better plan fast." Sol rubbed his hands together. "This is going to be fun."
CHAPTER 56
Phil Hammond was born to play the prince. He had refined an ability to obtain whatever he wanted without raising his voice. It was a role he had perfected over many episodes, until it became his signature performance. He proceeded down the town hall's back corridor trailed by two nervous young assistants and Harold Rhemus, Phil's bespectacled attorney. Hammond entered the mayor's office with a regal calm. He surveyed the six who waited his arrival with utter disdain. "Could someone please tell me what could not wait until normal business hours? I'm due to speak at an event in Santa Cruz."
Sol was seated in the mayor's chair. Porter stood directly behind him. The other four observers had chairs drawn from the conference room next door. Connor and Rick sat on one side of Sol, Estelle and Marcela were on the other. They faced a lone empty chair on the desk's other side. "Mr. Hammond, I'm Sol Feinnes."
Hammond was dressed in a tailored jacket with a herringbone weave. Striped dress shirt with white collar and cuffs. Massive gold watch. Flash tie. Italian loafers so soft they could probably be rolled up like socks. Phil demanded, "And precisely why is that important to me?"
Hammond's pin-striped attorney said, "Sol is an attorney based in San Luis Obispo."
Hammond flicked an imaginary bit of lint off his sleeve. "Same question."
Sol asked, "Won't you sit down, Mr. Hammond?"
"Thank you, but I won't be staying that long."
Sol said, "Yours is certainly a familiar name to me, sir. I've been at the receiving end of several messes that you initiated."
"Shame you didn't learn to steer clear," Hammond said.
Sol said, "We have a problem and we were hoping you might be able to advise."
"What's your name again?"
"Sol Feinnes, sir."
"I sell advice, Feinnes. I give nothing away." He turned away. "I think that's more than enough wasted minutes."
"That's actually why I've been asked to speak for these interested parties," Sol said. "We are absolutely willing to pay you the going rate."
That stopped Hammond. "Exactly what rate did you have in mind?"
Porter said, "You get to stay out of jail. Now sit down."
* * *
Porter Wright's words were enough to keep Phil Hammond in the room, but he stood behind the chair, arms crossed, fuming.
Sol Feinnes gave no sign he noticed Phil's ire. "We are locked on the horns of a dilemma. You see, our research has turned up the most remarkable set of circumstances. Wouldn't you agree, Porter?"
"Remarkable," Porter agreed. "That's the word."
"The closest casino to Miramar happens to be in Chumash tribal territory. This particular casino, as it happens, has a silent partner. One PH Enterprises, based in Nassau. It's been hard for us to determine who actually owns the company, since that information is carefully guarded. But even the Bahamians can be forced to divulge identities if our federal authorities present evidence of potential illegality—"
Hammond turned around and said to his two young aides, "Out. Now."
When the door clicked shut, Sol continued, "Our research has turned up the most remarkable set of gambling debts."
"Remarkable," Marcela agreed. "I love that word."
"We have obtained documentation regarding the casino's outstanding loans to three individuals. A certain judge in the San Luis Obispo regional court. A county prosecutor. And a detective assigned to the sheriff's—"
Harold Rhemus said, "These accusations would never stand up in a court of law."
Estelle said, "Sort of like the accusations you've leveled against my daughter."
"My client has no direct involvement—"
"Stow it," Connor said.
Sol said, "We have dates. We have amounts. We have interest accruing at the astonishing rate of twenty-six percent per annum."
"Astonishing," Marcela said.
"Remarkable," Porter said.
Harold Rhemus said, "No way could this so-called evidence lead to a valid prosecution. You'd be laughed out of court."
"Who said anything about going to court?" Sol held to an almost convivial air. "We're far beyond such an unseemly and time-consuming action. What we intend to do is take what we know to the FBI and the IRS."
"Tomorrow," Porter said.
Hammond dropped heavily into the chair.
"You see, Phil, may I call you Phil? Our Chief Wright, along with the help of several associates, has pieced together a very interesting concept.... Explain it to us, Chief, why don't you."
Porter said, "You arranged the coke, the drop, the whole shooting match."
"No attorney on earth could have put it more succinctly," Sol said. "Wouldn't you agree?"
"We know you're behind the Santa Maria development. We know you've hidden your involvement behind the same company that owns the Chumash casino," Porter said, "You needed a front for your projects. A local who would put the town at ease. You needed Sylvie. But she wouldn't play your game, so you decided to buy her out."
Estelle said, "You were going to play carrot and stick with my daughter. Pay her a salary, give her a title, let her keep her restaurant as your little sideline. But only if she did what you said. First you had to get her where you wanted."
Porter said, "It's bothered me since the beginning, how we weren't able to identify even one other company or individual who was receiving drugs by sea. I mean, what's the point of going to all that trouble for just eleven keys? Then it hit me."
Estelle said, "You ran a scam on my daughter. You filth."
"Mere suppositions," the attorney scoffed.
"Oh, it's far more than that," Sol corrected. "Our evidence has uncovered the employment records showing that Carlos, the individual who planted the keys and has subsequently vanished, formerly worked in the casino's security detail."
Estelle said, "You're going down."
"Old Phil," Marcela said, "you've stiffed your last waiter at Castaways."
Sol said, "I'm sure the state investigators would be fascinated with the information regarding the gambling debts of your three friends."
Marcela said, "The best friends money can buy. Right, Phil?"
"No doubt your three friends will be convinced to turn state's evidence against you." Sol offered a cat's smile, all teeth and malice. "Especially when the authorities present them with the alternative of watching their careers ground down to fragments that couldn't be found with an electron microscope."
Phil's attorney clearly knew it was time to ask, "What do you want?"
"We have a few things in mind." Sol glanced over. "Estelle, would you care to lead off?"
CHAPTER 57
Sylvie's Monday crawled by.
There was no longer any pretense of normalcy. Rick or Marcela or Bruno or Carl usually slipped in the back door at some point, ensuring no urgent duty had arisen to cloud their day off. But today Sylvie repeatedly checked downstairs, and no one came.
By four that afternoon, Sylvie had run out of excuses to hide away. She went downstairs and made herself a cappuccino on the machine in the waiters' station. She drank it at the bar, staring at the silent piano in the abalone-shaped alcove. The restaurant was filled with an expectant air, like the energy in a silent room before a big party. Sylvie could almost hear the laughter.
Around five, people started streaming past the windows fronting the street. Sylvie checked her reflection in the washroom mirror; then she let herself out of the restaurant. She was mildly disappointed that neither Rick nor Marcela had thought to come walk with her. After all, it was her party, at least in some respects.
But within ten paces of her front door, Sylvie found herself surrounded by smiling faces and laughter and easy chatter. She could not have named most of those who walked alongside her. Nor did it matter. Not really.
* * *
Miramar's town hall stood six blocks farther inland from Castaways. The old-town area, with its raised sidewalks and mock gaslights, gave way to a series of modest buildings that housed the city's offices and one of two supermarkets. The town hall was a fifties-era gray clapboard structure with a broad front veranda of seasoned redwood. The floors and walls and high ceiling of the vast main room formed an echo chamber, such that the crowd's growing din greeted Sylvie as she entered.
That evening, the meeting hall had been cleverly divided into two sections, both of which were packed with people and items waiting to be auctioned. Three tables staffed with volunteers fanned out by the entrance, where each newcomer was handed a number printed on a three-by-five card in exchange for credit card details. It was done in such a friendly and smooth fashion that even the lookie-loos became potential bidders long before they entered the real arena.
Most of the hot-ticket items were placed in the front section, well removed from the exit. The hall's two portions were split by a long cash bar, staffed by a dozen cheerful volunteers and offering a vast selection of donated finger food.
Ringing the walls were balloons and banners urging the growing crowd to help one of their own.
People kept rushing over, laughing and excited in their welcome. They chattered about the prizes, the night, the awful trial. Actually, Sylvie heard very little of what they were saying. Their voices and their embraces formed one great wave of affection that kept surging over her.
When Marcela appeared, she pretended not to notice Sylvie's misty-eyed condition. "Glad you could make it on your own steam. I was about to come up with the six-wheeler and drag you down."
"You're saying I didn't have a choice in the matter?"
"Of course you did. Whether you came on your own steam or mine." Marcela beamed. "You need a drink before we mingle?"
But at that moment, the lights dimmed and Porter Wright stepped to the microphone at the center of the stage. He nervously cleared his throat, then said, "I've been assigned duty tonight, don't ask me why."
The crowd gave him a welcoming cheer. As it subsided, he went on, "We're here to help one of our own. Where's Sylvie? I saw her come in. . . ."
A voice called, "She's back here!"
"Glad you could join us. Folks, give Sylvie a big hand, let her know how much we care for the lady of the hour." He paused for a renewed cheer. "Most of the time when rotten things happen to people for all the wrong reasons, there's not much we can do but stand with them through the dark hour. But in this case, we can really make a difference. So dig deep and buy what you don't need."
Porter was almost off the stage, when his wife called up, "Aren't you forgetting something?"
Shamefaced, he stepped back to the mike and said, "I told Carol they should've gotten somebody up here who knew what he was doing."
"You're doing just fine, honey," Carol said. "Now give them the good news."
"Right. Unless you've been asleep for the past week, you know Connor Larkin. . . ."
Sylvie felt her heart take wings as the crowd applauded. The upsurge of emotions threatened her ability to maintain a calm façade.
"Connor Larkin is fresh from being signed on to play the bad guy in the new James Bond picture. . . ." Porter had to stop a second time.
As he waited, Connor stepped onto the stage. This time, Sylvie felt the applause squeeze her heart with a fierce yearning. She had not expected to feel such a clamor of conflicting emotions. She was the only person in the hall not cheering. Somebody patted her shoulder. Sylvie knew eyes were on her. However, she could not smile, or move, or lift her hands and even pretend to join in.
Porter said, "Connor has agreed to give a private concert for the highest bidder, and here's a sample of what the winner will enjoy."
Sylvie stayed where she was, silent and immobile, desperate to hear him sing one more time.
The lights dimmed a trace more, the audience went still as a sleeping infant, and Connor sang:
Something in the way she knows,
And all I have to do is think of her.
Sylvie knew without a doubt that Connor was singing to her. His music instantly pulled her back to a night of fairyland bliss, when strong arms held her and she had believed in love again....
* * *
Connor released a sigh as strong as winter's tempest as Sylvie left the hall. He had handled some tough roles in his acting career, but nothing so difficult as completing his song and smiling his thanks to an audience he could no longer actually see.
Determined to launch the auction on a high note, he shut his eyes to the burn of lost hope and began the final song. He told himself that he had done all he possibly could, and now it was up to Sylvie. All of that helped, at least a little.
Even so, as he rose and accepted their raucous applause, Connor reflected that now he had a gut-level understanding to draw from, when he was next called upon to die beneath the lights.
CHAPTER 58
As far as Sylvie was concerned, the week that followed was one long series of unexpected bright spots.
Early Tuesday morning, she asked Estelle to meet her in the café by the guesthouse. As soon as they were seated, Sylvie declared once more, "I don't want you to go."
"And I don't want to leave. My darling daughter . . ." Estelle lifted three fingers to her trembling lips. She took a long breath, clenched her hands together on the table, and did her best to smile. "There. I said it."
Sylvie asked, "How did it feel?"
"Awesome." Estelle must have seen the conflict in Sylvie's features, for she did not ask how Sylvie felt hearing it. "May I come by the restaurant?"
"Can you . . . Of course."
"There is no 'of course' to anything we're doing here." She smiled then, the joy almost palpable. "You must tell me if I become a nuisance. Otherwise, I might stop by each evening for a glass."
"And a meal."
"I will do no such thing. I will merely sit at the bar and I will watch you." The lips trembled through the smile. "And I will be the happiest and proudest mother on the planet."
* * *
On Tuesday afternoon, Sylvie heard from the Castaways linen supplier that Phil Hammond had bought a teacup at the silent auction.
For thirty thousand dollars.
When the delivery van pulled away, and everyone in the kitchen refused to meet her eye, Sylvie demanded, "Did anybody know about this?"
"I might've heard something," Bruno replied. "I bought a vintage T-shirt."
"Rick?"
"I tried to buy half a dozen things," he replied. "But I got outbid every time."
When she turned to Marcela, her friend snapped, "I didn't get a chance to bid, since I spent all night trying to find you. Where did you go, by the way?"
Sylvie had no choice but to reply, "I need to check on tonight's bookings."
"You do that," Marcela huffed. As Sylvie left the kitchen, Marcela said, "She moves almost as fast as my husband did when I asked him to bid on Connor's concert."
It was only later, when the town's deputy mayor and his wife confirmed that Phil's check for thirty grand had already cleared, that Sylvie realized Marcela had succeeded magnificently in changing the subject.
On Wednesday morning, Sol Feinnes woke her at half past six. "Sorry to call so early. But I needed to catch you before you left for San Luis Obispo. The trial has been postponed."
Sylvie had no idea how she felt about a delay. "What does that mean?"
"I try not to deal in rumors. We should know something definite before very long. Two days at the most. You sleeping all right?"
"Surprisingly well, considering."
"Glad to hear it. Hang in there, friend. I'll be back in touch as soon as I know anything for certain."
Most of the other revelations seemed timed to opening hour, when they had the most impact. At least that's what Sylvie told Estelle when she settled at the Castaways bar that evening. Sylvie related how, twenty minutes before she unlocked the front door, Rick announced that an anonymous bidder had paid seventy thousand dollars for Connor's private concert.
Estelle sipped at her glass of white wine and observed, "For seventy thousand dollars, I'm surprised the buyer didn't ask for a teacup to match old Phil's."
"You knew about this," Sylvie realized.
"I might have heard something somewhere."
"You didn't think it might be worth mentioning? Since that one bid covers my entire legal bill?"
Estelle had the remarkable ability to sip her wine without lowering the level in her glass. "Why did you leave the silent auction Monday evening?"
"Why . . . You know full well. I ran away. And don't change the subject."
"I wouldn't dream of doing any such thing." Estelle pointed at the door. "You have customers."
* * *
On Thursday afternoon, Phil Hammond's attorney called to say he was stopping by, and that he needed ten minutes of her time. Sylvie was tempted to put him off, but she decided she'd rather hear the bad news now, rather than have another reason to fret through a sleepless night.
Harold Rhemus arrived an hour later, during a tidal wave of early clients. Phil's attorney appeared both nervous and sweaty in his suit and button-down blue shirt and the restaurant's only tie. She seated him at the long front table, and then was surprised when Estelle and Marcela and Rick shifted over to surround her. At first, Sylvie thought they had come to offer support, but then she saw smiles being stifled. This only added to the confusion.
Sylvie asked, "Can I get you something?"
"I'm not staying, thank you." It was only when he adjusted his spectacles that Sylvie realized Rhemus was trembling. Regardless of how her staff was almost dancing in place, Sylvie knew Phil and she knew this lawyer. They had no connection to good news. Whatever had brought him here, it had to be bad.
"Mr. Hammond has elected to withdraw all participation in food services," Harold said. "As a result, he wishes to divest himself of his interest in Castaways."
"I-I'm sorry . . . What?"
Rhemus opened his attaché case and drew out a manila folder. The tremors rose to his voice as he went on, "Mr. Hammond understands that he has run up quite a large bill. He is offering you his share in Castaways in exchange for all outstanding . . ."
The attorney was halted by Estelle and Rick and Marcela, who were entering into a somewhat clumsy three-way jig. He cast them a dark glance, then slid the folder in front of Sylvie. "It's all there in black and white. Sign both copies with a notary as witness. Return both to me for Mr. Hammond's signature."
Aubrey popped the champagne cork just as the attorney fled from the restaurant.
CHAPTER 59
Connor took the week at a steady and unhurried pace. The canceled three-week honeymoon left a very convenient hole in his schedule. His new home had not been lived in for almost a year, and he found a multitude of problems that needed urgent attention. Every day was filled with work, sunup to sundown, but he set the pace. He listed the next day's chores over his solitary dinners. He spent his evenings watching all the old Bond films, right back to Goldfinger. He also intended to study at least three movies by each of the current stars and the director and the cinematographer. Each night he went to bed early, slept well, and rose ready for more of the same.
Saturday was Porter's day off. He drove Carol and Celia over, selected a deck chair on the rear veranda, and lost himself in one of Lisa Jackson's suspense novels. Estelle arrived a half hour later and joined mother and daughter in a thorough housecleaning. When Connor offered to help, he got three versions of the same tirade, which he basically translated as, If Connor was any good at cleaning up, they wouldn't be here in the first place.
Connor retreated to the rear patio and asked the police chief, "Why do I get the impression they're not telling me something?"
Porter shook his head without taking his eyes off the page. "Nope. I am not getting volunteered again."
"What are you talking about?"
"You want answers, you go ask the ladies."
"I'm asking you."
Porter glanced over the top of his half-moon reading glasses. "And I am telling you that for once I am not re-upping."
"I have no idea what you just said."
"Watch carefully." Porter turned the page. "This is me enjoying my day off. Now, unless you're going to offer me a fresh cup of coffee, I'd appreciate your not aiming another word in my direction."
* * *
Almost half of the patio's flagstones had become dislodged, probably from some tremor the town had slept through. Time and the previous weekend's foot traffic had crumbled some of the corners. Connor refit them with a special cement the Home Depot guy had said was blended for flexibility. He worked through one bucket, long enough to grow completely hot and bothered. Porter ignored him with a cop's stony intent. Finally Connor gave up and walked back to the sliding doors leading to his bedroom. The ladies had already finished up in there, so he showered and dressed in clean shorts and T-shirt.
When he entered the kitchen, he found the trio arrayed against him, all lined up behind the counter.
Estelle said, "We have some news."
"I sorta figured that."
Carol pointed him to one of the stools on the counter's other side. "You're going to want to sit down." When he did, she went on, "Your private concert has been bought for seventy thousand dollars."
Connor looked from one lady to the next. Behind him, he heard the sliding door open. He asked, "Is this a joke?"
Celia asked, "Do we look like we're joking?"
"How long am I supposed to play? A year?"
Carol said, "Sol Feinnes bought it."
"I don't . . . Wait . . . Sylvie's lawyer has refused payment?"
"He didn't refuse anything," Porter said. "He's given it back."
Carol said, "Here's where you ask about Sol's one condition."
Celia said, "We should have a drumroll for this next part."
Carol said, "He bought the concert so you would play for Sylvie."
Celia brought her hands together. "And cymbals."
Carol said, "This was Estelle's idea."
Celia said, "Correction. Her brilliant idea."
Estelle said, "You're too kind."
"No she's not," Celia said.
Carol said, "Sol is making the donation on the condition that you play for Sylvie."
Connor replied, "Sylvie won't want that."
Estelle smiled. "You just leave that with me."
CHAPTER 60
The tectonic plates beneath Sylvie's world continued to shift right through the weekend. As she was planning Saturday's specials, Porter entered the kitchen and made a slow circuit, shaking everyone's hands and speaking a few friendly words. Any other season of her life, Sylvie would have found a gentle gratitude in how Miramar's chief of police could be so comfortable around two convicted felons and her other miscreants, including herself. Sylvie Cassick's own record was clean only because her juvie files remained sealed.
However, given the fact that she was just days removed from her delayed felony trial, all Sylvie could manage was to keep her lunch down.
When Porter finally arrived at where she stood frozen to the kitchen's polished concrete floor, he said, "Let's go up front."
Only when he took hold of her arm was she able to move. As she passed through the doors, she glanced back in time to watch Bruno and Carl exchange high fives. Sylvie considered the action almost treacherous.
Porter led her over to the bar and said to Aubrey, "Give us a few minutes."
"No problem." Aubrey added her own disloyal smile to the day's strangeness and departed.
Porter then surprised her by pulling his phone from his shirt pocket and punching in a number. He watched her intently as he waited. Then he said to the phone, "We're good to go." Porter passed over the phone and said, "It's for you."
She needed two hands to lift the phone. "Yes?"
"Sol Feinnes here. I'm happy to inform you that all charges against you have been dropped."
Sol continued to talk for a while about papers and such. She tried hard to listen, but the mist over her eyes had somehow managed to affect her hearing as well. When he went quiet, Sylvie made a total hash of her thanks. Then she handed back the phone, took a few unsteady breaths, and finally managed, "Is this really real?"
"As real as it gets."
"But what if, you know, they come back?" She knew she sounded like a nine-year-old asking about the boogeyman, but she could do nothing about it.
"Actually, that's why I'm here." Porter shifted his lumpish bulk on the stool, leaning closer still. "The sheriff's detective has been reassigned. He's now on guard duty at Lompoc Men's Prison."
"What does that mean?"
"It means there are a lot of others who are very upset about the case against you ever having gone anywhere near a courtroom." Porter was clearly enjoying himself enormously. "The prosecutor has been reassigned to the lovely desert resort of Barstow. And this morning, the judge in question announced he's taking early retirement."
Sylvie gave a very tight sigh, a quick in-and-out breath, like she was recovering from a wound and needed to see how far she could stretch the scar tissue.
Porter liked her silence enough to ease off his stool, lean over, and kiss her on the cheek. He pocketed his phone, patted her on the shoulder, and left without another word.
CHAPTER 61
On Sunday evening, Miramar's mayor and half the town council stopped by Castaways with the auction's official tally. The events had run up a surplus of 104,000 dollars.
"That's not possible," Sylvie told them.
"I absolutely agree." The mayor was a rawboned woman who ran one of Miramar's two veterinary services. It was said she could still a bucking horse with one bark. "But I've gone through the numbers myself. Twice."
They were seated at the long front table. Behind them, the restaurant was filled with the normal weekend clamor. Her staff rushed about, borderline frantic. Sylvie had a dozen things that urgently needed her attention. But just then, all she could accomplish was to sit upright. "What do you want from me?"
"We met with Estelle this afternoon. For all intents and purposes, she's the woman responsible for the whole shooting match. She agrees with us that the money is yours to do with as you please."
"I don't . . . No. Absolutely not. You're not handing me that mess."
"I absolutely am." The mayor's seamed features rearranged themselves into a vast grin. "It's not every day I get to argue with somebody about writing them a check."
"I can't accept that money!"
"Nobody expects you to. You just need to decide which local charities are getting an early Christmas." The mayor rose from her seat. "One thing can't wait. Estelle is still out twenty thousand dollars. She's making noise about how she doesn't want anything paid back. I need you to talk some sense to that lady."
* * *
An hour later, Sylvie found a brief quiet moment and phoned Estelle. To Sylvie's surprise, when she insisted that Estelle take back the funds, her mother simply asked, "Are you sure?"
"About this, absolutely."
"All right. I just want you to know that I was glad to be there when you needed me." There was a shaky breath; then she added, "This time."
CHAPTER 62
Late Monday morning Estelle walked along Miramar's central avenue, down past Castaways and on toward the sea. She exchanged greetings with a few people, but she refused to allow anyone to slow her progress. She did not have much time, as she had agreed to volunteer at the animal shelter and there were still a few details to complete regarding the day's main event. But before all that she wanted a few moments in her little seaside haven. And like any good penitent, Estelle felt a need to arrive there on foot.
When she reached the beachfront road she turned left and climbed the gentle slope. She crossed the parking area and took the path through the clifftop park. There was a breathless hush to the air, neither any wind nor the faintest ripple to mar the ocean's surface. The Pacific stretched out in blue-gold majesty to join with the cloudless horizon. The air was a mix of sunlit heat and the water's biting chill. The result was a champagne headiness to her every breath.
Estelle retreated to the open-fronted chapel and was pleased to find it empty. She seated herself and stared at the sunlit vista and silently acknowledged the true significance of the day's events. More was at work, she knew, than her being accepted by the town her daughter called home. More too than helping Sylvie reunite with a man desperately seeking to become someone who deserved her daughter's love.
Miramar had granted Estelle what she could not have brought herself to even ask for. Her past was full of reasons why she should never be granted her silent wish. She remained fully aware of all her many wrongs, including the fractured prayers she had formed while seated right here.
Even so, Estelle had come to discover this town's secret gift. And not just her. Miramar had offered it to three people who yearned for this above all else.
A second chance.
When she was ready, Estelle rose from the bench and whispered the shortest prayer of all.
"Thank you."
* * *
On Monday, Sylvie finally reached the decision she had been working toward since the moment Connor had arrived onstage. Only she had needed this long to accept the truth.
She was going to see Connor again.
The question was, how should she make that first huge step?
Her decision only strengthened the whispered refrain of wanting it all to just go away. Simply because she had realized that she needed Connor did not erase the mental arguments. The fears brought up worries for which she only had one answer. She really, really wanted to see him again.
Even so, the internal conflict kept her from taking the desired step. Sylvie went through the motions all through her Monday. Sylvie pretended that she was going to have a normal evening off, eat her usual solitary meal, and join Humphrey Bogart and Ingrid Bergman in their safe black-and-white world. She puttered about the apartment in cutoffs and a raggedy T-shirt. Why shouldn't she? After all, nobody at Rick's Café in Casablanca would care what she wore.
But the lonely hours only clarified the truths she had wrestled with all week. She had fled the auction precisely because she had known even then that she was going to take this step. She was going to invite Connor back into her life.
Nothing about this was safe, or easy. There would be no half step with this man. How could there be? She was already in love with him. Now that she was coming to terms with this undeniable fact, it felt as though she had loved him since the very first moment he had walked into Castaways. And stood there in the doorway, staring blindly about, lost to the song and the chance that they might together actually fly to the moon.
She ran herself a hot bath and spent time on her makeup, though she had no idea what to do with herself once all the preliminaries were out of the way. Thankfully, just as she finished dressing, Estelle phoned and asked, "Are you busy?"
"I have no idea." Sylvie found herself wanting desperately to tell her mother about her growing desire to start anew with Connor, or at least try. If only she knew how. That, of course, meant admitting that Estelle was becoming her friend. Another incredible component of this amazing week. "Is there any chance we could meet?"
"That was actually why I called. I have something I need to talk with you about. I'm volunteering at the animal shelter. The woman who's supposed to be on duty has sprained her hip. Could you come up here?"
"Why don't I just wait until you're finished?"
"Because . . ." Estelle's giggle sounded like an excited young girl's. "Because of a hundred different reasons. Marcela is on her way over to pick you up."
Sylvie started to ask what Marcela had to do with anything, but then she heard a car's horn through her front window. She glanced down and said, "Marcela is already here."
* * *
When Sylvie appeared in the front door of Castaways, Marcela greeted her by saying, "What took you so long?"
"I was down in ten seconds flat." Sylvie watched Marcela do an excited two-step by her car door. "You look like a child at Christmas. What's going on?"
"You'll see. It's a surprise."
Sylvie set the alarm and locked the door and complained, "I hate surprises."
"Don't I know it." Marcela slipped behind the wheel and started the car. "Get in. This one won't wait."
"I need a week off," Sylvie objected. "Calm, no explosions."
"And you're going to get it," Marcela promised. "I have you down for early next year."
They drove a mile farther from the ocean and parked in the lot behind the county buildings. They rose from the car and crossed the street, heading for the town's animal shelter. Sylvie asked, "How did Estelle even hear about this place?"
"This is the mayor's number one project," Marcela replied. "They got to talking about using some of your money for renovations."
"It's not my money," Sylvie countered. But she could see Marcela getting ready to argue the point, and waved it aside. "All right, yes. It's my money until I give it away. So the mayor and my mother know each other now."
"Your mother's become the second most popular person in town, after you." They entered the shelter and asked the volunteer on front-office duty, "Where's Estelle?"
"Seeing to our newest arrivals," the woman cheerfully replied. "Second door on your right. Hi, Sylvie. Prepare to have your heart stolen right out of your chest."
Marcela pushed through the door marked CLINIC, and halted in front of the steel diagnostics table, which held . . .
A basketful of kittens.
Nine of them.
Four were gray fluffballs, soft as smoke and about the size of Sylvie's hand. The other five were calicos, with matching white socks. They mewed and crawled and squirmed. One of the gray furballs lifted its front paws at Sylvie and cried to be picked up.
"Aren't they the most adorable things you've ever seen!" Estelle lifted the gray cat and handed it to her daughter. "Don't you want to take this one home with you?"
Sylvie lifted the kitten to her cheek. "This is the sweetest form of bribery I've ever known."
"Oh, we didn't bring you here for this," Estelle said. "We want to talk with you about Connor."
Sylvie started to put the kitten down, but it chose that moment to purr and nestle beneath her chin. She managed, "Connor?"
"Let's review the situation," Marcela said. "A major hottie has basically been dropped into your life."
"Yes, Connor lied to you," Estelle added. "Yes, you had every reason to respond as you did."
Marcela went on, "So how does he react when you turn him to ashes in public? Does he try to argue his way out? Does he get all defensive and yell at you?"
Estelle said, "Connor does his best to change. He apologizes with his every action."
Sylvie started, "Actually, that's what I wanted—"
"We're talking now," Marcela said. "Your job is to listen."
Estelle said, "My darling daughter, Connor Larkin is head over heels in love with you. It's time you accepted that the feeling is mutual."
Marcela said, "Let's not forget the fact that he's incredibly good-looking and a Hollywood star."
"Not yet," Estelle said. "But he will be once the Bond film is released."
Marcela added, "Did we mention that Connor's moving up?"
Sylvie forced herself to settle the kitten back in the basket, where it looked up at her and mewed plaintively. "Connor Larkin is moving to Miramar."
"He signed a lease on the Kaufmans' place," Estelle said. "He's making this his home."
Sylvie looked from one shining face to the other. Her mother and her best friend were here for one reason. They wanted her to be happy. Sylvie swallowed against the burn and managed, "You are both so precious to me."
Her mother's laugh held a songbird's quick note. She wiped her eyes with the hand not holding a kitten, then said, "This is supposed to be your day for dreams come true. Not mine."
Marcela recommenced her dancing in place. "Now tell her the best part."
CHAPTER 63
When the three ladies arrived back at Castaways, every window in the supposedly closed restaurant gleamed softly. Sylvie thought the place looked like one of her father's paintings. She sat taking tight, little breaths, flooded with an impossible mixture of excitement and fear. As Estelle cut the motor, Sylvie said, "I really don't want to go in there alone."
"If that's what you want, of course we'll stay," Estelle said.
"You expected this?"
"I wouldn't say, 'expect,' " Marcela replied from the rear seat. "More like . . ."
"We thought this might be your response," Estelle replied. "Daughter, do you want to do this? That is the real question."
"Say you do," Marcela pleaded.
Sylvie breathed once again. "More than I know how to say."
Marcela bounced up and down in her seat. "Yippee."
Estelle said, "Carol and Celia have helped with the arrangements. They asked me to tell you that they would like to come join us. But only if you want."
Sylvie saw the front door open and Connor step into view. "I suppose . . . all right. Yes."
Marcela said, "Yippee again."
CHAPTER 64
Connor stood in the doorway and waited. He could see the three ladies were deep in some serious confab. His stomach was too filled with electric butterflies for him to even smile, much less walk over. He knew without the tiniest fraction of doubt that Sylvie was having second thoughts. He personally felt she should receive a medal for making it this far.
He remained where he was, willing to stand there all night if necessary. He had no idea what might make things worse, or even blow this second chance right out of town.
All afternoon, he had become increasingly convinced that this was, in fact, just that, a second chance at making his dreams come true.
That is, if the lady actually did decide to rise from the car and cross the street and enter her restaurant. And let him stay. And listen to him try his best to put his feelings into song.
Connor was alerted to a shift in the evening's currents by the sound of a pickup truck door opening. Until Celia bounded out, Connor had not even been aware that she and her mother had hung around.
She called to him, "Sylvie says we can stay for the party!"
And just like that, Connor knew it was going to be okay.
* * *
Sylvie felt as though she floated across the street and through her restaurant's front doorway. She had read about how some women drifted upon clouds as their lovers led them across the ballroom floor. Here she was just coming home. There was no orchestra, no gleaming chandeliers, and certainly no bevy of uniformed waiters ready to leap at her every whim.
Here there were just her friends. Four ladies who only wanted the best for her.
It was already the finest night of her life.
The area behind the bar was mostly dark. Celia and Carol and Connor had positioned a few candles here and there, but the real surprise awaited her as she rounded the bar. The long table was aflame. A gleaming silver bowl was filled with water, and a dozen candles floated in it. The serving platter was surrounded by flowers and ivy. More flowers and candles traced a pattern around the stage's perimeter. Two tall iron stands stood behind the piano, turning the abalone-shaped shell into a varnished rainbow.
Sylvie glanced at her father's painting on the wall. The stage and the painting looked like twin patterns.
For a brief instant, she had the sensation that her father was standing there beside her, inspecting them with her. She allowed herself to be guided forward, grateful for the help just then, because it had become rather hard to see her way.
Connor stood to one side as the ladies seated themselves. His position was that of a formal waiter, silent and attentive and holding to a respectful distance. Once they were comfortable, he went back around the bar and returned with an ice bucket that held two bottles of Dom Pérignon champagne. Another trip and he brought out five glasses, then plate after plate of hot and cold appetizers. He filled their glasses, and then climbed onto the stage.
All without saying a word.
Sylvie's chair was angled slightly, so she could both watch Connor and look out the bay windows, over the rooftop view of her hometown. She observed Connor as he lit the candelabra and set it on a protective cloth atop the keyboard. He seated himself, then started to play.
His first tune was one made famous by Ella Fitzgerald, "Into Each Life Some Rain Must Fall." Her father had played it constantly in the weeks after Estelle's departure. Sylvie had not thought of the song in years. She knew an odd sense of comfort when Estelle leaned forward and cupped her face in her hands. This time, thankfully, the distance between them was not so great. In fact, it seemed natural to reach over and settle her hand upon Estelle's shoulder.
When the song ended, and they all applauded, her mother straightened and turned to Sylvie and whispered, "I had no idea it would be this hard."
Sylvie replied softly, "I'm so glad you're here."
Connor shifted smoothly into Nat King Cole's signature song, "When I Fall in Love." Sylvie found herself carrying on a most intense conversation that rose and fell with the melody. She knew the time would come when she shared her thoughts with Connor. But just then, it was important to discuss this with herself. How the birthday wishes had all come true, even the one of sharing an hour with her father, who drifted just beyond the candle's reach.
More important still, however, was that she had been reminded of how important it was to dream, to ignore all the reasons life offered to give up on hope, and to yearn for the impossible. Believe it was possible to know such things as boundless joy, and the love of a good man.
The moon nudged over the northern coast, and a soft breeze spiced the air with a hint of the Pacific. Sylvie thought the night was gentle as her father's brushstrokes, and loved how her mother kept hold of her hand.
Connor segued into one of Tony Bennett's greatest hits, The Way You Look Tonight.
CHAPTER 65
The others were long gone.
Sylvie had joined Connor on the piano bench. He played a little, but mostly they just sat. Connor felt as though they were both getting used to the way they fit together. Communicating at the level of bone and sinew and hearts.
Though he feared the words would get in the way, still he knew he had to say them. He spoke for the first time that night in something other than a melody. "Sylvie, I'm so very, very sorry."
She nestled closer still. "You said that already."
"Did I?"
She touched the keys, though not hard enough to make a sound. "Over and over."
He started to turn, afraid and yet sensing it was time. He was both glad and grateful when she turned to meet him, like two dancers who had practiced the move for years.
They kissed. For Connor, it felt as though the kiss was meant never to end. He leaned back and cupped her face in his hands. Then he let himself fall into her gaze.
Then they kissed again.
He knew if he was ever to write his own lyrics, it would be about this. How a kiss could come and go, and still be part of forever.
A READING GROUP GUIDE
MIRAMAR BAY
Davis Bunn
The following discussion questions are included to enhance your group's reading of Miramar Bay.
Discussion Questions
**1.** Have you ever thought about starting over? Have you ever considered just leaving your life behind and beginning again in another city, or in a small town like Miramar Bay? What would be good and bad about this? What would you miss? What would you be excited about?
**2.** What do you think about Connor Larkin's decision to hide out in Miramar Bay, using an assumed name? Is this an act of courage or cowardice?
**3.** Connor has everything that most people think they want—fame, fortune, talent, a bright future. Why is he not happy at the start of this novel?
**4.** What do you think of Connor's fiancée, Kali? Is Connor being fair or unfair to her?
**5.** What is it about Miramar Bay that draws Connor? Can you understand the appeal of a place like this?
**6.** The restaurant that Sylvie owns and runs is called "Castaways". Why do you think that Sylvie and her father chose this name? Do you think it remains an apt name?
**7.** What is Estelle looking for when she comes to Miramar Bay? Do you think that she knows what she's looking for . . . and does she eventually find it? What do you think of how she handled things in the past with regard to her first husband and her daughter? Does she deserve forgiveness? If you had been her daughter, would you be able to forgive her?
**8.** Several characters refer to Sylvie as a strong person. Do you agree with this assessment? Where does her strength come from?
**9.** If you could have your own private concert with Connor, what would you ask him to play for you?
**10.** Do you have any dreams or talents from earlier in your life that you would love to get back to? What is stopping you?
Don't miss the next warmhearted, wise, and
wonderfully moving novel from the
internationally bestselling Davis Bunn . . .
FIREFLY COVE
Available January 2018
Read on for a preview. . . .
CHAPTER 1
MAY 1, 1969
Most people said Lucius Quarterfield wore a name bigger than he deserved.
As Lucius and his sisters were passed from aunt to grandparent to cousins, family members had often said it to his face. In the forties and fifties, California's central coast was a vibrant farming region with an aggressive go-ahead attitude. Strong men tilled the earth and raised robust families. Lucius Quarterfield was a nice enough boy, quiet and watchful. But the families who took in Lucius and his sisters knew he would never amount to anything. The bullies gradually grew tired of picking on Lucius. Some even slipped into guardian roles, when it suited them. Mostly Lucius grew up being ignored. His quiet nature made that all the more possible. He lost himself in books and schoolwork, though he was careful to hide his passions. He was a cautious fellow by nature, with a zeal for numbers.
The one thing that had come easy to Lucius was success. It did not make up for all the misery and loneliness, but it certainly made it easier to bear.
This particular doctor's office had always struck Lucius as a restful place, which was extremely odd, because most of life's problems had centered on doctors. But Nicolo Barbieri was different from many in the medical fraternity, who assumed a ridiculous superiority and lied to young Lucius with their smiles. Nicolo Barbieri's family was among the original Italian immigrant clans who had moved from Tuscany to till the California earth as tenant farmers. A generation later, they had scraped together enough money to buy land of their own, and planted one of the early central coast vineyards. Nico had fought against the tradition-bound family's wishes and studied medicine. Perhaps as a result, Nico Barbieri was a brusque man without a comforting bone to his body. His patients either adored him or found another doctor. "You're dying, Lucius."
"So what else is new." Lucius buttoned his shirt and pushed himself off the doctor's table. He always perspired when being examined, a leftover effect of all the pain doctors had caused him growing up. "I've been dying for twenty-two years."
"Your heart reminds me of a garbage disposal working on a spoon. I should put you in the hospital and run some tests."
"The tests will tell you what we already know."
Barbieri fished a cigarette from his shirt pocket as he slipped behind his desk. "Are you truly so cavalier about death?"
"You've been telling me I'm dying since I was seven years old," Lucius replied. "And don't light that."
"Sorry. Bad habit." Barbieri stuck the unfiltered Camel cigarette back into the pack. "This is different. Are your affairs in order?"
The room suddenly chilled enough to turn his skin clammy. "You've never asked me that before."
"Never felt the need. Are they?"
"Pretty much. I'm negotiating a new deal. Should be finished next week."
"Lucius, you don't need the stress of another deal. Your heart can't take it. And I know for a fact you don't need the money." Barbieri opened the patient folder and began making notes. The file was almost three inches thick. "Bad ticker, weak bones, half a lung."
I have this, Lucius thought, knotting his tie and pulling it tight. I have today.
As though Barbieri could hear his unspoken reply, he said, "You've made the best you could of a thin life. Now go out and enjoy yourself. While you still have time."
Dr. Barbieri's waiting room always appeared half-full. The patients changed, but the setting remained the same. The adults leafed through old copies of Life and Look and National Geographic, while the children played with toys made sticky from hundreds of little hands. Rooms like this had been one of the few constants in Lucius's early life. He used the phone in the nurses' station to call his banker and cancel the day's meeting. The banker was a longtime acquaintance and Lucius was an important client, so he did not complain when Lucius told him to reschedule the meeting with the seller's lawyers. As he hung up the phone, Lucius caught sight of himself in the mirror behind the weight machine. He was five feet eleven inches tall, when he held himself fully erect, which seldom happened. Lucius was underweight and his posture was awful. His cheeks had become sunken during his early bout of pleurisy and never filled back in. The childhood illness had cost him his sense of taste. Smells were vague entities, like words spoken in some foreign tongue. Eating was a troublesome task. His hair was a mousy brown and limp as old noodles. His eyes . . . Lucius turned away. He rarely bothered with his appearance, even when buying clothes.
Lucius Quarterfield was twenty-eight years old.
The nurse asked, "When does the doctor want to see you again?"
"He didn't say."
Her hand hovered over the appointment book. "Are you sure, Lucius?"
"Not a peep. Maybe he thinks I'm all better."
Her smile carried all the false cheeriness of his childhood. "I'm sure that's it."
Lucius drove his brand-new Chevrolet Impala north from San Luis Obispo. He had not been back to Miramar in almost a year, though for a time he had traveled this road every week. He was not a man given to holidays and easy living. Recently his only days off had been when he was unable to rise from his bed. Otherwise his every waking hour was spent making money. When the doctor said he should take some time off and enjoy himself, this journey was the only thing that came to mind.
Lucius had never much cared for his name, which to him sounded like it was made for a guy with one lung, bad bones, and a poor heart. He had always preferred Luke. He considered Luke to be a hard name, full of the go-ahead spirit that burned with volcanic fury inside him. Three days after his sixteenth birthday, Lucius Quarterfield had taken his meager inheritance, borrowed everything his two older sisters were willing to loan him, and bought a vacant lot a block off Santa Barbara's Fifth Street. Even then his sisters recognized their brother had something most people lacked, a fire his ravaged and weakened body could scarcely contain. Lucius had strung plastic multicolored flags and buntings from the trees, paid a builder to clear the earth and lay down gravel, erected a motheaten army surplus tent, and put nine road-weary used cars up for sale. In the twelve years since then, Lucius had built an empire that contained eleven dealerships selling some blend of the GM lines. He also owned another four businesses that combined used cars with farm equipment. His two sisters had long since moved to Florida, as far from their father's memory as they could get, and lived in houses bought with Lucius's earnings. They sent him Christmas cards and phoned when they needed something. Lucius did not blame them for their distance. Their family situation had not been one to forge strong emotional bonds.
His life was embedded in the road. Lucius was free here. He could unleash the Impala's big V8 and let the car be strong for him. Lucius rarely indulged in past regrets. But the doctor's words cast a magnetic force over the morning, drawing in one potent memory after another. His sisters had loathed their father and revered their mother. Their father had been a stone - mason and a nasty drunk. Lucius's few memories of his father had been of a burly giant, massive in every way, who had hulked over the dinner table like a bear in a graying T-shirt and suspenders. But he never laid a hand on his own family. The sisters claimed they had all remained shielded by their mother's remarkable grace, so strong it kept peace in the house long after she had died birthing Lucius. The closest he ever came to his father, at least in theory, was when they both went down with pleurisy three days after his sixth birthday. His father had died, while Lucius was delirious with fever. Lucius had spent the next few years being passed around among various relatives, until his sisters had managed to find jobs and get a place of their own. Of course he took care of them now.
The drive from San Luis Obispo to Miramar took just over two hours. The road was fairly awful in places, but Lucius did not mind. The authorities were always talking about building a proper county route, linking Miramar to the new highway. State Route 1, also known as the Pacific Coast Highway, had officially been opened the previous year, but the steep hills surrounding Miramar had forced the coastal thoroughfare inland, isolating this little oceanfront haven. Most of the locals were of two minds over building a better link to the outside world. Some wanted growth, while others feared all the bad things they remained sheltered from. And in 1969, there was certainly a lot of bad to avoid.
At the top of the hour Lucius turned on the radio to catch the news. It was more out of habit than anything. Virtually none of what he heard had any bearing on his very constricted world. The launch of Apollo 10 was approaching, and this final run before man landed on the moon was the only bright spot in a series of grim tidings. The Basques in Spain had earlier set up a new guerilla army called, of all things, ETA, and their outlaw government were demanding the right to form a nation of their own. British troops arrived in Northern Ireland to reinforce the local constabulary, and the Catholics referred to them as invaders. Harvard's administration building was taken over by Students for a Democratic Society. A ferocious battle had erupted in Vietnam at someplace called Hamburger Hill. The news ended and the announcer introduced a song by Sly and the Family Stone off their new album, Stand. Lucius liked the music well enough, though it made him feel more isolated than ever from the sweep of current events and all the good things other people his age were enjoying. He wasn't sorry when the hills closed in and the signal faded.
At the final approach to Miramar, Lucius pulled off the road and parked where a cluster of California pines offered a masking shadow. He peered across the street at the smallest of his dealerships, a Buick-Chevy-Olds he had acquired two years and eleven months ago. Nowadays his banker was pressing him to sell the place and put his money in a region with stronger growth. But Miramar held a special place in his poorly functioning heart.
During Lucius's first visit there, the old man who sold him the dealership had regaled him with legends dating back to the Wild West heyday of abalone fishing and Mexican banditos and the occasional gold prospector. Back then, the rough and frigid waters had earned the town its original name, Castaway Cove. Around that same period Miramar had latched onto a very odd claim to fame. Stay there for a while, so the tales went, and you might be given a second chance. Second chance at what, Lucius had asked. He had instantly regretted his question, for the old man had taken on the smug look of someone offering a secret of supposedly great worth, but which Lucius already knew was bogus. The old man had then replied, "Whatever it is that you most want to try your hand at again."
Lucius had smiled over the fable, signed the purchase papers, handed the old man his check, and two hours later had fallen head over heels in love.
Whatever else he might think about the town and the lady, Lucius had known it had nothing whatsoever to do with Miramar's fable. For the event was singular. As in, the one time in his short, hard life he had ever known for himself what love actually meant.
Lucius liked to spend a few days loitering around every new acquisition. He called it "kicking the tires." Everyone was on his best behavior, at least at first. But Lucius fit so naturally into quiet corners that gradually the employees relaxed and slipped back into their routines. Lucius learned a great deal in those early days, mostly about money. As in, where the most revenue was generated. Where changes were needed. Where potential profits were being missed. Over his solitary evening meals, Lucius made notes in a script so precise his aging secretary described it as human lithotype. With each new dealership, Lucius brought in the employee he had come to trust the most, named him president of that particular location, and gave him a ten percent share, with an option to purchase another ten percent. Loyalty among his employees was fierce, turnover almost nil. Putting his plans down on paper gave spice to his otherwise tasteless dinner.
One of the first things Lucius noticed about the Miramar dealership was how the salesmen mostly ignored the new vehicles. They clearly made higher commissions pushing used cars. This suggested they were bilking the former owner out of part of his share. On that fateful day Lucius took up station at an empty salesman's desk, blocked from view by a gleaming new Buick Riviera. It was a car he especially liked, with the newly redesigned GM engine and a luxury velvet finish to corners that before had sharp and dangerous edges. As of this year vent windows were a thing of the past, replaced by air-conditioning made standard on all Buicks. It was altogether a beautiful machine, as far as Lucius was concerned. This made the way the salesmen clustered together in the used-car lot all the more irritating. Quiet, silent Lucius Quarterfield was mostly ignored.
Which was when the young woman planted herself directly in front of him and declared, "I hate cars, don't you?"
"Excuse me?"
"Positively despise them." She was far too lovely to be spending time with the likes of him. Tall and willowy, she had a face that was filled with an electric fire that sparked through her wavy auburn hair. Her eyes were alight with a mischievous emerald gleam as she went on. "Great humping metal beasts just looking for an excuse to bellow."
"That bellow is why I'm here," Lucius replied. "I consider it the finest music on earth."
She pulled over a chair and seated herself so close, their knees almost touched. "How positively dreadful."
Normally, he was so shy around women his own age that he could scarcely speak. Today, however, he heard himself reply with ease, "For years I've wanted to own a Jaguar, simply because I love the way their engine sounds."
"Then why on earth don't you buy one?"
Her matter-of-fact tone surprised him. "How do you know I can afford it?"
"Don't be silly. Everybody knows you just bought this business and paid cash."
"Who is everybody?"
"All of Miramar, of course." The way she spoke made it sound like, All the known universe and beyond.
"Why do you think I'm talking to you?" she asked.
"I have no idea."
"Because you're fabulously rich and ever so mysterious, of course. I'm Jessica Waverly, by the way."
"Lucius Quarterfield."
"Are you really? I've never met a Lucius before, much less a Quarterfield. You haven't just stepped out of a Charlotte Brontë novel, by any chance."
"I don't know who she is."
"You can't be serious."
"I haven't read a novel since my sixteenth birthday. I left school and went to work. Since then I've hardly had time to read everything I need for my business."
She gave a cheery shrug. "In that case, I shall just have to educate you. Your name should obviously belong to some great strapping stable lad who goes around tossing cows for a living. You don't, I suppose."
"I can state with absolute certainty that I have never tossed a cow in my entire life."
"What a pity. What with owning your own business, and being stinking rich and loving to make these awful motors bellow, if you also tossed cows, I fear my father would marry me off in a flash. That's him over there, by the way, trying to convince my dear mother that he needs a new car more than his next breath."
"Then I'll just have to go out and toss my first cow this very afternoon," Lucius replied. And just like that, his heart was lost to the woman whose fire was as merry as his own was morose.
Lucius relied constantly on his objectivity and his logic. Both of which he lost completely whenever in the company of Miss Jessica Waverly. She referred to their relationship as Pride and Prejudice, the title of her favorite novel. Jessica refused to say which aspect referred to him. She remained stubbornly blind to his many frailties. She insisted that her parents positively adored Lucius, when he could see they were growing ever more alarmed by his calls and visits.
Jessica's father was Miramar's only dentist. Jessica had served as his assistant ever since her mother had developed problems with her joints. Her father had pressured Jessica to attend dental school and take over his practice, but Jessica was incredibly stubborn in her capricious manner. She claimed to have no intention of ever working that hard, not in school and certainly not for the rest of her professional life.
Their disapproval of Jessica's lack of ambition and their love of cars were the only two items that Lucius's and Jessica's fathers agreed on.
Jessica confessed in their next time alone that it was neither dentistry nor working alongside her beloved father that made her so passionate about her job. Jessica was, in her own words, born to touch hearts. She saw the fear and discomfort the patients carried as opportunities to share with them her special brand of happy comfort. When she had spoken the words, Lucius had the distinct impression that Jessica expected him to scorn her for aiming too low. She sat there with her chin angled up, defiant and already hurting from what she thought he might say. In truth, Lucius was so moved by her willingness to share the illogicality of her beautiful heart, he needed several minutes to speak at all. When he did, it was to confess the impossible words of love for the very first time.
When her father finally demanded that Jessica address what he called "the Quarterfield situation," Jessica stubbornly insisted to Lucius that it didn't matter, none of it did. Even when Lucius knew full well that Jessica's father had issued an ultimatum to his only child. And in the process had broken her heart.
But when they next met, Jessica did not want to talk about that. She wanted to discuss what color his Jaguar should be. Lucius replied, "I will never buy such a car. But Jessica—"
"Then I just suppose you'll have to sweep me away on your yacht."
"I don't own a rowboat and I can't swim. About your father—"
"Oh, never mind him. I think fifty feet is a nice round number for a boat, don't you?"
"I won't buy a yacht for the same reason I will never own a Jag. I don't like to draw attention to myself."
For a brief instant she sobered. "I understand that."
"Do you really?"
"Actually, I've spent much of my life perfecting the ability to hide in plain sight."
"But why? You're . . ." He started to say "healthy," which sounded like he was sizing up a prize steer. So he said what he really thought, which was "beautiful."
She tilted her head. "Do you really think so?"
"You are quite simply stunning," he replied. "You take what little breath I have completely away."
"See? Why should I let Daddy pester me when a fabulously wealthy man with a lifetime of secrets, and who thinks I'm lovely, is going to buy me a fifty-foot yacht?"
"I will not do any such thing."
"Oh, well, never mind." She waved it aside. "As if beauty ever mattered. All the beautiful ladies in Jane Austen's novels die of the pestilence, alone and ravaged by cruel fate."
Her sudden changes in direction left him dizzy. "Really?"
"Well, no. But they should have."
"Jessica, why do you want to spend time with me?"
She cocked her head and showed him an expression of utter amazement. "Why, because you need me, silly."
It was only later, as he drove back to his lonely house outside Santa Barbara, that he realized she had succeeded in doing precisely what she had intended all along. Which was to tell him about the situation with her father, then avoid talking about it an instant longer than necessary. She was, Lucius decided, the most adroit negotiator he had ever met.
Three days later, Lucius journeyed to Miramar once more. Jessica had made a picnic, her determined method to ignore the fact that he had come with a very definite purpose in mind. He allowed her to take off his socks and shoes and roll up the trouser legs over his pale shins. They set up the picnic by the southern cliffs. The sea was calm, the sunlight fierce for early May. A trio of beachfront eucalyptus offered a perfumed shade. They walked, hand in hand, down the beach, as far as Lucius felt comfortable. Half a mile north, they stopped and watched two young girls and a spaniel race joyfully along the shore, chasing gulls. Jessica released his hand so as to slip her arm around his waist. She settled her head upon his shoulder. Lucius felt the warmth and strength and life in her vibrant form, and was grateful when the breeze tossed her hair into his face and blocked his tears from view.
After the picnic all he wanted was to stretch out beside her and have her rest her face upon his chest. Her every smile almost broke his resolve. But he had to be strong. For her.
"Jessica. Sweetheart."
"You've never called me that before." She rose to a seated position and tucked her knees under her dress and wrapped her arms around her shins. "Is this to be a serious discussion?"
"I fear it must." He shaped the words, very slowly, allowing it to linger on his tongue. And in his heart. "I love you so much."
Her eyes grew huge. "You're saying good-bye." Her breath caught on the last word, like a hook's barb had become lodged in her throat. "Aren't you."
"I love you . . ." He needed a hard breath to dislodge his own barb. "Too much—"
"There's no such thing as 'too much,'" she whispered.
"—to make you my widow."
"Silly you." She tried to laugh. But her tears refused to give her enough air. "We're all dying. Every day is just one more step toward the final end."
"Not this soon," Lucius replied. "You have years left to live."
"You don't know that."
"I have weeks. A few months. Perhaps only days."
"You don't know that, either," she said, but the words were mangled now. She refused to stop talking, not even when her every word was forced around sobs that convulsed her entire frame. "Shall I tell you why I love you, Lucius? Because you are the loneliest man on God's green earth."
It was true. The truest thing he had ever heard. So true he was silenced. His reasons for doing as he did vanished in the flood of her sorrow. But not his resolve. All he could do was sit there, the distance between them impossibly wide. Even when she pleaded desperately for him to reach over and offer the comfort that only his arms could give.
When he did not speak, Jessica went on talking. "You need me to give you what you will never have without me. And that, my dearest beloved, is joy."
That is true as well, he wanted to tell her. He could not even have named the flavor of ecstasy until their first conversation. Even now, when he was filled with bitter regret, he knew he was saying not just farewell to her, but to any shred of happiness. Any hope of bliss.
His silence defeated her. She started crying too hard to help him gather up their belongings. When the car was packed, he came back to where she sat, staring blindly at the Pacific, helped her to stand, and supported her weight back to the parking lot. One cripple helping another.
The first comforting sign that he had done the right thing came when he pulled up in front of her parents' home. Jessica had recently taken an apartment in town. But Lucius did not want to take her back there, both because he did not want her to be alone just then, and because he did not want to test his broken resolve. As he rounded the final corner, he saw Jessica's mother standing on the front walk, waiting for them. There was no way the older woman could have known what had just taken place, yet there she was, watching worriedly as Lucius pulled up. When he cut the motor, Jessica fumbled blindly for the latch. Finally her mother opened the door and enfolded Jessica in a comforting embrace.
Lucius watched the two women climb the steps and enter the house and seal him out. Only then did he rise from the car and open the trunk and lift the remnants of their last day together. He started up the drive, carrying the blanket and the hamper and her shoes. The garage door opened, and Jessica's father emerged. Jessica's father set the burdens on a shelf by the door leading into the house. Then he unbent enough to take Lucius in a strong embrace. Lucius was so shocked he did not know how to react. He stood there, frozen in place. He had only shaken the man's hand twice, and one of those was the day he had sold the man his new Buick. Then Jessica's father released him and stepped back and shut the garage door. All without saying a word.
That had taken place eleven months ago. Today was the first time since then that Lucius had returned to Miramar.
Author photo: © Paul Wheeler
DAVIS BUNN is a New York Times bestselling author with more than seven million books in print in sixteen different languages. Born and raised in North Carolina, Davis left for Europe at age twenty. After completing graduate studies in economics and finance, he began a business career that took him to more than forty countries all over the world. David came to faith at age twenty-eight, while living in Germany and running an international business advisory group. He started writing two weeks later. Since that moment, writing has remained both a passion and a calling. His first novel, The Presence, released in 1990, became a national bestseller. Davis went on to become a New York Times bestselling author, writing suspense, thrillers, and historical fiction, and reaching both a Christian and a mainstream audience. He has been honored with four Christy Awards, most recently in 2013. A soughtafter speaker on the art of writing, Davis serves as Writer in Residence at Regent's Park College, Oxford University. He divides his time between England and Florida. For more information, see: www.davisbunn.com
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"redpajama_set_name": "RedPajamaBook"
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{"url":"https:\/\/burttotaro.wordpress.com\/","text":"## WAGS @ Pomona, 13\u201314 November\u00a02020\n\nThe Fall 2020 edition of the Western Algebraic Geometry Symposium (WAGS) will be held on November 13-14, 2020, via Zoom. You can register here. Pomona College is serving as host. The meeting will include an online poster session; please register by November 6 if you would like to present a poster.\n\nImage: Bookstore cat who works at Magic Door Bookstore IV in downtown Pomona.\n\nFiled under math\n\n## Morse theory on singular\u00a0spaces\n\nMorse theory is a powerful tool in topology, relating the global properties of a smooth manifold X to the critical points of a smooth function on X. In this note I want to consider the possibility of Morse theory on singular spaces. Some of this dream can be made to work in algebraic geometry, where it helps to analyze the Hilbert scheme of points in new cases.\n\nThis is related to my joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, Maria Yakerson on \u201cThe Hilbert scheme of infinite affine space.\u201d The connection with Morse theory appears in my paper Torus actions, Morse homology, and the Hilbert scheme of points on affine space.\n\nMorse theory studies a smooth function f on a closed Riemannian manifold X using the gradient flow associated to f. That is, from any point on X, move in the direction in which f is (say) decreasing fastest. In the limit, every point of X is attracted to one of the critical points of f.\n\nFrom the modern point of view known as \u201cMorse homology,\u201d a central part of the theory is to compactify the space of orbits of the gradient flow. The key point is that any limit of orbits of the gradient flow is a broken trajectory, a chain of orbits that connect a sequence of critical points with decreasing values of f.\n\nThis situation has an analog in algebraic geometry. Consider a projective variety X with an action of the multiplicative group T = Gm. (Over the complex numbers, this group can also be called C*.) Then the orbits of T on X are analogous to the gradient flow lines in Morse theory. In particular, for every point x in X, the \u201cdownward limit\u201d of its orbit, limt\u21920(tx), is a T-fixed point of X (analogous to a critical point in Morse theory). Over the complex numbers, this T-action can actually be identified with Morse theory for the \u201cHamiltonian function\u201d on X.\n\nJust as in Morse theory, I show that every limit of T-orbits on a projective variety X is a broken trajectory, a chain of orbits that connect a sequence of T-fixed points. An interesting point is that this works without assuming that X is smooth. So this is a possible model for Morse theory on singular spaces.\n\nI give an application to the Hilbert scheme of points on affine space. Namely, let Hilbd(An) be the space of 0-dimensional subschemes of degree d in affine n-space. And let Hilbd(An,0) be the (compact) subspace of subschemes supported at the origin in An. I show that, over the complex numbers, these two spaces are homotopy equivalent. (Computing the cohomology of either space is a wide open problem, in general.) The proof uses the algebraic version of Morse theory described here, using the action of T on Hilbd(An) coming from the action on An by scaling. I hope to see more applications: torus actions are everywhere in algebraic geometry.\n\nImage: A chubby Tom Kitten in a broken pair of pants, from The Tale of Tom Kitten, by Beatrix Potter.\n\nFiled under math\n\n## How to get started with the Hilbert\u00a0scheme\n\nNow that we can attend seminars all over the world, beginning algebraic geometers may be encountering the Hilbert scheme everywhere. At first glance, however, the idea of the Hilbert scheme is so capacious that it can be hard to grasp.\n\nSo, in this post, I want to sketch a path that an interested reader (or student seminar) could follow in beginning to understand the Hilbert scheme. (Some of the links may require your library to have access.) The topic came to mind because of my current joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, and Maria Yakerson, \u201cThe Hilbert scheme of infinite affine space.\u201d\n\nFirst, the general definition: for a projective variety (or scheme) X over a field, the Hilbert scheme of X classifies all closed subschemes of X. The existence of the Hilbert scheme (as a union of projective schemes) reflects a basic feature of algebraic geometry: families of algebraic varieties are parametrized by an algebraic variety (or scheme). But then we have to find ways to analyze the Hilbert scheme in cases of interest.\n\nTo start the path of reading, there is an excellent introduction by James McKernan, a 10-page set of MIT lecture notes. It discusses a special case, the \u201cHilbert scheme of points\u201d, with several examples.\n\nThe general construction of Hilbert schemes, due to Grothendieck, is outlined in several places. Perhaps the easiest to read is chapter 1 of Harris and Morrison\u2019s book Moduli of Curves. The most technical step of the proof is not included there (roughly, the fact that there is a uniform bound for the equations of all subschemes of projective space with a given Hilbert polynomial). It might be reasonable for learners to come back to this step later. A standard reference for this step is chapter 14 of Mumford\u2019s book Lectures on Curves on an Algebraic Surface.\n\nAlthough the Hilbert scheme is hard to understand in full detail, there is a clear \u2014 computable! \u2014 description of its Zariski tangent space at any point, even where it is singular. Namely, if S is a closed subscheme of a projective scheme X over a field, then the tangent space to Hilb(X) at the point [S] is H^0(S, N_{S\/X}), the space of global sections of the normal sheaf. Computing this group in examples is essential for getting to grips with the Hilbert scheme. As a start, you can look in chapter 1 of Harris-Morrison or chapter 1 of Koll\u00e1r\u2019s book Rational Curves on Algebraic Varieties.\n\nFinally, for many applications, it is important to go one step further and understand the \u201cobstruction space\u201d as well as the tangent space for the Hilbert scheme. This is a great setting in which to learn deformation theory. Roughly, the obstruction space tells you the number of equations needed to define the Hilbert scheme near a given point. There are many possible introductions to deformation theory; let me recommend Sernesi\u2019s book Deformations of Algebraic Schemes. Section 3.2 addresses the Hilbert scheme, with examples and exercises.\n\nThere is a vast literature on Hilbert schemes in particular settings, such as the Hilbert scheme of points on a surface. But I hope what I\u2019ve said is enough for you to start exploring.\n\nImage by @kernpanik; license CC BY-NC-SA 4.0.\n\nFiled under math\n\n## WAGON (WAGS Online), 18-19 April\u00a02020\n\nThe Western Algebraic Geometry Symposium (WAGS) is going online and worldwide! For more details and to register, see the conference webpage.\n\nImage lifted from @littmath.\n\nFiled under math\n\n## Late-night motives* \u2014 MoVid-20: motivic video-conference on 15 April 2020 via\u00a0Zoom\n\n*The first talk is 1 am Pacific time.\n\nTo register, please send an email to denis.nardin@ur.de with the subject \u201cMoVid-20\u201d.\n\nThe schedule in Central European Summer Time (aka time in Germany) is as follows.\n\n9:45-10:00 Conference opening\n10:00-11:15 Tom Bachmann\n11:15-12:00 coffee break\n12:00-13:15 Marc Hoyois\n13:15-14:30 lunch break\n14:30-15:45 Maria Yakerson\n15:45-16:30 coffee break\n16:30-17:45 Denis Nardin\n\nHere are the titles and abstracts.\n\nTom Bachmann: Pullbacks for the Rost-Schmid complex\nLet F be a \u201cstrictly homotopy invariant\u201d Nisnevich sheaf of abelian groups on the site of smooth varieties over a perfect field k. By work of Morel and Colliot-Th\u00e9l\u00e8ne\u2013Hoobler\u2013Kahn, the cohomology of F may be computed using a fairly explicit \u201cRost-Schmid\u201d complex. However, given a morphism f\u200a: XY of smooth varieties, it is in general (in particular if f is not flat, e.g. a closed immersion) unclear how to compute the pullback map f\u200a*: H*(Y,F) \u2192 H*(X,F) in terms of the Rost-Schmid complex. I will explain how to compute the pullback of a cycle with support Z such that f-1(Z) has the expected dimension. Time permitting, I will sketch how this implies the following consequence, obtained in joint work with Maria Yakerson: given a pointed motivic space X, its zeroth P1-stable homotopy sheaf is given by \u03c03P13X)-3.\n\nMarc Hoyois: Milnor excision for motivic spectra\nIt is a classical result of Weibel that homotopy invariant algebraic K-theory satisfies excision, in the sense that for any ring A and ideal I\\subset A, the fiber of KH(A) \u2192 KH(A\/I) depends only on I as a nonunital ring. In joint work with Elden Elmanto, Ryomei Iwasa, and Shane Kelly, we show that this is true more generally for any cohomology theory represented by a motivic spectrum.\n\nDenis Nardin: A description of the motive of $Hilb(A^\\infty)$\nThe Hilbert scheme of points in infinite affine space is a very complicated algebro-geometric object, whose local structure is extremely rich and hard to describe. In this talk I will show that nevertheless its motive is pure Tate and in fact it coincides with the motive of the Grassmannian. This will allow us to give a simple conceptual description of the motivic algebraic K-theory spectrum. This is joint work with Marc Hoyois, Joachim Jelisiejew, Burt Totaro and Maria Yakerson.\n\nMaria Yakerson: Motivic generalized cohomology theories from framed perspective\nAll motivic generalized cohomology theories acquire unique structure of so called framed transfers. If one takes framed transfers into account, it turns out that many interesting cohomology theories can be constructed simply as suspension spectra on certain moduli stacks (and their variations). This way important cohomology theories on schemes get new geometric interpretations, and so do canonical maps between different cohomology theories. In the talk we will explain the general formalism of framed transfers and\nshow how it works for various cohomology theories. This is a summary of joint projects with Tom Bachmann, Elden Elmanto, Marc Hoyois, Joachim Jelisiejew, Adeel Khan, Denis Nardin and Vladimir Sosnilo.\n\nImage: Still from The Third Man (1949, dir. Carol Reed).\n\nFiled under math\n\n## Happy Birthday to this peevish\u00a0two-year-old\n\nMarch 31, 2020 \u00b7 9:55 pm\n\n## WAGS @ Pomona,\u00a0POSTPONED\n\nWAGS at Pomona is being postponed until the Fall term.\n\nThe Spring 2020 edition of the Western Algebraic Geometry Symposium (WAGS) will be held at Pomona College on 3\u20134 April 2020. You can register here. Financial assistance is available and can be requested via the registration process.\n\nImage: Bookstore cat who works at Magic Door Bookstore IV in downtown Pomona.\n\nFiled under math, travel\n\n## The bees and the birds at\u00a0UCLA\n\nGetting emails from central administration is often a joyless part of university life, but today\u2019s email was interesting, informative, and even delightful.\n\n==========\nIn the Spring and Summer it is common to find young birds away from their nests. Often concerned students or staff will think the birds need rescuing, but it depends. Below is a note from one of our bird researchers on campus and a helpful article from Audubon. A couple photos attached from campus showing nestlings versus fledglings. Please share this information with your departments, and let them know that if they are not sure they can call or text me: [xxx.xxx.xxxx]. Our office works with a network of wildlife rehabilitators and can help transport wildlife in need of rescue. In addition, with everything starting to bloom, you may see bees around campus. See below for instructions if you discover a hive or swarm.\n\nBee Information:\nSometimes bees can swarm or build hives on campus. UCLA recognizes the critical role pollinators like bees play in our food system and ecology. We work with a company to ensure that bees are live captured and relocated and not killed. See photo attached of a swarm removal from on a car. If you discover a hive on campus or swarm that needs removal please call Facilities Management Trouble call at [yyy.yyy.yyyy], they will coordinate the response.\n\nBird Information:\nOne of our bird researchers on campus notes: \u201cKnowing when a young bird is \u201csupposed\u201d to be out of the nest vs. when it\u2019s not supposed to is key when dealing with these sorts of situations. It\u2019s important to remember that nests are unsafe places to be; it\u2019s easier for a predator to kill four chicks that are in the same cup of sticks and hair than four chicks that are in four different parts of their parents\u2019 territory. As a result, parents will push their chicks out of the nest before they\u2019re fully able to fly, and take care of itself.\n\n* If you have to chase after the chick to catch it, it\u2019s old enough to let it\u2019s parents take care of it outside the nest. If it\u2019s out in the open, or in a dangerous place try herding it to the nearest shrub or other protected place. Mom and dad know where the fledgling is and will feed it discretely.\n* If the bird is sitting upright and is alert, it probably has recently left the nest. Check the wings, if the wings are fully or partially feathered (as opposed to being in gray-looking sheaths), it\u2019s old enough to be outside of the nest. If it\u2019s out in the open, or in a dangerous place, you can move it to a place nearby with greater safety.\n* Most \u201cbaby\u201d birds you find will fit in above. In both cases, they are where they need to be. Even if it looks like they\u2019re abandoned, they aren\u2019t, the parents are just making sure not to lead predators to their offspring. Trying to rescue it means a lot more work and stress for you and the wildlife rehabber you take it to, when the parents will almost certainly do a better job for free.\n\nSo: when should you interfere with nature?\n\n* Very young nestlings (ie mostly naked, no or few feathers, can\u2019t sit up, appears helpless). It might have fallen out of the nest; if so look around to see if you can find the nest. If so, put it back, if not, follow the directions in the article cited below, and then bring the nestling to a licensed rehabber.\n* The chick is obviously injured (ie broken wings or legs.) In this case, bring the chick to a licensed rehabber. If you know or suspect the chick was grabbed by a cat, bring it to a rehabber immediately! This is because cat mouths are breeding grounds for all sorts of nasty bacterial that kill birds, and any bird that has been exposed to cat teeth needs to be given antibiotics ASAP.\n\nSome additional information can be found in this helpful article from Audubon, When You Should\u2014and Should Not\u2014Rescue Baby Birds: https:\/\/www.audubon.org\/news\/when-you-should-and-should-not-rescue-baby-birds\n\nThank you,\nNurit\n\nNurit Katz\nChief Sustainability Officer\nExecutive Officer of Facilities Management\nUCLA\n\nImages are (from top to bottom) junco juvenile, bee removal, junco nestlings, fledgling starlings.\n\nFiled under UCLA\n\n## SoCalAGS @ UCSD, 29 February\u00a02020\n\nThe next meeting of the Southern California Algebraic Geometry Seminar (SoCalAGS) will take place on Saturday, 29 February 2020, at University of California, San Diego. Speakers are:\n\nFabio Bernasconi (Utah)\nNathaniel Bottman (USC)\nDavid Stapleton (UCSD)\nMaria Yakerson (Regensburg)\n\nPlease register. It\u2019s free and helps us to demonstrate appetite for this seminar.\n\nImage via @uscg: \u201cIn 1945, Salty, the mascot at Air Station San Diego, became the first cat to take part in a rescue mission when she stowed away with her kittens on an amphibious reconnaissance plane just before it took off to rescue a pilot who had gone down at sea. #NationalCatDay #SARcat\u201d Look carefully for the second kitten!\n\nFiled under travel\n\n## What if J. Peterman wanted you to learn commutative\u00a0algebra?\n\n1983.\n\nThe crisp leaves as I waited for the Dinky. Atiyah-Macdonald light in my hand. That was the fall I became intoxicated with power as I shrank open sets to points and revealed the structure of the universe. By the holidays I\u2019d acquired the whole set of moves. Cracking the nut of a Noetherian ring. Folding and unfolding integral extensions. Localizing to a point with a gimlet eye. Flipping from rings to spaces, spaces to rings, with the ease of Evert working the baseline.\n\nMath 215A. Commutative Algebra. James Cameron. Not that James Cameron, but just as life-changing.\n\n\u2014 Guest post","date":"2021-03-02 08:39:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.4919571876525879, \"perplexity\": 1274.168644374892}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178363782.40\/warc\/CC-MAIN-20210302065019-20210302095019-00583.warc.gz\"}"}
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package io.apiman.gateway.engine.beans.exceptions;
/**
* Exception thrown when a policy tries to get a component from the context
* but the component doesn't exist or is otherwise not available.
*
* @author eric.wittmann@redhat.com
*/
public class ComponentNotFoundException extends AbstractEngineException {
private static final long serialVersionUID = 8430298328831765033L;
/**
* Constructor.
* @param componentType the component type
*/
public ComponentNotFoundException(String componentType) {
super("Component not found: " + componentType); //$NON-NLS-1$
}
/**
* Constructor.
* @param componentType the component type
* @param cause the exception cause the root cause
*/
public ComponentNotFoundException(String componentType, Throwable cause) {
super("Component not found: " + componentType, cause); //$NON-NLS-1$
}
}
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Home / Energy / U.S. rig count up 215 from November 2021; 70 counted in state
U.S. rig count up 215 from November 2021; 70 counted in state
By: Journal Record Staff November 29, 2022 0
Rigs drilling for oil or gas in Oklahoma this month totaled 70. A year ago, 44 rigs were counted in the state, according to Baker Hughes. (Photo by Worksite Ltd. on Unsplash)
As November reaches toward its end, there are 784 rotary rigs drilling for oil or gas across the United States, according to industry watcher Baker Hughes.
In its most recent report, dated Nov. 23, Houston-based Baker Hughes said the combined number of oil and gas rigs increased by two from its previous count total.
A year ago, the rig count total in the U.S. was 569.
Rigs on land in the U.S. last week totaled 764. That was 212 more than were counted in the same period in 2021. There were three rigs counted in inland waters and 17 offshore. Rigs drilling for oil in the U.S. totaled 627, an increase of four from the previous week; there were 155 counted drilling for gas, a decrease of two; and two were classified as "miscellaneous" rigs.
Traditional vertical wells being drilled totaled 23, while there were 47 identified as directional and 714 identified as horizontal.
The rig count in Oklahoma increased by one from the previous week to total 70, Baker Hughes reported. A year ago, there were 44 drilling rigs counted in the state.
There were 372 rigs counted in Texas on Nov. 23. There were 105 in New Mexico, 66 in Louisiana, 38 in North Dakota, and 22 each in Colorado and Pennsylvania. Fewer were counted in Alaska, California, Ohio, Utah, West Virginia and Wyoming.
In reporting major basin variances, Baker Hughes said 352 rigs were in the Permian, 69 in Haynesville, 71 in the Eagle Ford, 39 in Marcellus, 42 in Williston, 29 in Cana Woodford, 20 in DJ-Niobrara, 13 in Utica, six in Ardmore Woodford, two in Arkoma Woodford, two in Barnett, four in Granite Wash and two in the Mississippian.
Baker Hughes counted 194 rigs drilling in Canada, a decrease of seven from the previous week.
The total rig count for North America was pegged at 978, a decrease of five from the previous week's total of 983. A year ago, the North American rig count total was 740.
Baker Hughes has issued the rotary rig counts as a service to the petroleum industry since 1944.
Baker Hughes 4:30 pm Tue, November 29, 2022
Journal Record Staff
Tagged with: Baker Hughes
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,576
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Q: Materialized view is not refreshing in Postgres 10.4 I am importing a table from remotely located database say DB1 using DBlink and creating a materialized view in Postgres (DB2). This is working fine but at the time of refreshing materialized view updated data from source table (DB1) is not reflecting the changes in Postgres materialized view.
Please find the steps below.
CREATE MATERIALIZED VIEW IF NOT EXISTS schemaDB2.test_view
AS
SELECT *
FROM dblink('foreign_server','select col1,col2 from schemaDB1.tablename') AS t1 (col1 varchar,col2 varchar);
commit;
CREATE UNIQUE INDEX IF NOT EXISTS dummyindex ON schemaDB2.test_view (col1);
REFRESH MATERIALIZED VIEW CONCURRENTLY schemaDB2.test_view WITH DATA;
Any help is appreciated.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,372
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Revised and updated paperback edition of the national bestseller (now in its fifth printing), and "definitive guide to the art of living well" (Charlotte Empey, editor-in-chief of Metro), including a new foreword by Debbie Travis and updated sections on business etiquette and everyday entertaining.
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In a fantastic follow-up to his national bestseller, The Butler Speaks, Charles MacPherson lays out the essentials of entertaining and business etiquette in this small, easy-to-follow guide. Now in its third printing, The Butler Speaks has become a go-to resource on household management, manners, and personal style. With a combination of his best tips from the first book and a wealth of new information, The Pocket Butler is the perfect basic overview for anyone looking for that extra edge in business and in life.
In addition to revisiting topics such as how to present a business card, shake hands, or set a table, The Pocket Butler offers advice on e-mail and text messaging etiquette, how to set up a modern greeting line in a boardroom, foolproof menu plans for every entertaining scenario, and much more. In his signature unfussy and approachable style, Charles shows how modern manners are more important than ever before, not only for those just entering the workforce, but for all professionals at any stage in their careers.
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,660
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#region Copyright
#endregion
#region
using Autodesk.DesignScript.Runtime;
using Dynamo.Models;
using System.Collections.Generic;
using Dynamo.Nodes;
using Kodestruct.Common.Section.Interfaces;
using ds = Kodestruct.Common.Section.SectionTypes;
using dm =Kodestruct.Common.Mathematics;
#endregion
namespace Steel.AISC.Composite
{
public partial class CompositeSteelShape
{
[IsVisibleInDynamoLibrary(false)]
protected CompositeSteelShape()
{
}
private ISliceableSection section;
[IsVisibleInDynamoLibrary(false)]
public ISliceableSection Section
{
get { return section; }
set { section = value; }
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,945
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\section{Application to the two-dimensional transverse field Ising model} \label{sec:d2Ising}
To confirm the advantages of the SEs,
we apply our formulation
to the two-dimensional transverse field Ising model
on the square lattice,
defined by the Hamiltonian
\begin{align}
\hat{H}_N = \hat{X}_{N(0)}
\equiv - J \sum_{\langle i, j \rangle} \hat{\sigma}_i^z \hat{\sigma}_j^z
- g \sum_{i} \hat{\sigma}_i^x.
\end{align}
Here, $\langle i, j \rangle$ denotes
the nearest neighbor sites $i$ and $j$.
We take the coupling $J$ positive,
and measure energy in units of $J$.
\subsection{Inapplicability of conventional ensembles}
This model has an ordered phase
for small $g$ and large $\beta$ $(=\Pi_{(0)})$ \cite{DeGennes1963,Elliott1971}.
The order parameter is given by
\begin{align}
\hat{X}_{N(1)} &= \sum_i \hat{\sigma}_i^z.
\end{align}
Hence, it is necessary to employ an ensemble
with macroscopically well-defined order parameter
instead of the GCE given by
\begin{align}
\mathrm{gc}(\bm{\pi};\bm{x}) = \pi_{(0)} x_{(0)} + \pi_{(1)} x_{(1)}.
\end{align}
However, it is impossible to construct the MCE
because $\hat{X}_{N(0)}$ and $\hat{X}_{N(1)}$ do not commute with each other when $g \neq 0$.
We will show in Section~\ref{sec:d2qIsing}
that the SE solves this fundamental problem,
and successfully describes the first-order phase transition.
We also show in Sec.~\ref{sec:d2cIsing} that
the SE has advantages
even when this fundamental problem
is absent (i.e., when $g=0$).
\subsection{Zero transverse field} \label{sec:d2cIsing}
We first consider the case of $g \to 0$, in which
the model reduces to the classical Ising model and hence
the fundamental problem mentioned above is absent.
We will confirm that the SE is advantages over the GCE even in this case.
\subsubsection{Methods}
We choose the parameter space of $\bm{\kappa}$ as
$K=(0,\infty)\times\mathbb{R}$, and $\eta$ as
\begin{align}
\eta(\bm{\kappa};\bm{x}) = \kappa_{(0)} x_{(0)} + \frac{1}{2} \lambda (x_{(1)}-\kappa_{(1)})^{2}. \label{eq:d2cIsing_eta}
\end{align}
As described below, this choice of $\eta$ is convenient
for classical systems which undergo the first-order phase transition.
In order for the order parameter $x_{(1)}$ to have a definite value,
the convexity of $\eta$ as a function of $x_{(1)}$ must be strong enough.
For finite size systems,
this condition is particularly important
because the concavity of the Boltzmann entropy $s_N$ is broken
near the first-order phase transition region
\cite{Challa1988_PRA,Yoneta2019,Bixon1989,Labastie1990,Gross1990,Gross1997,Litz1992,Nielsen1994,Reyes-Nava2003}.
Since the second order derivative of $\eta$ with respect to $x_{(1)}$ is $\lambda$,
the strength of the convexity of $\eta$ can be freely adjusted by tuning the value of $\lambda$.
That is, for sufficiently large $\lambda$,
$s_N-\eta$ becomes strictly concave,
and the order parameter has a definite value.
The inverse temperature $\beta = \Pi_{(0)}$
is given directly by $\kappa_{(0)}$.
In fact, using Eq.~\eqref{eq:tdforce-formula},
we have
\begin{align}
\kappa_{(0)} = \Pi_{(0)}(\bm{x}^\eta). \label{eq:d2cIsing_Pi0}
\end{align}
Therefore,
by taking $\kappa_{(0)}$ to a certain value,
we can investigate the properties of the equilibrium states
with the inverse temperature at that value.
Furthermore, again using Eq.~\eqref{eq:tdforce-formula},
the thermodynamic force conjugate to $X_{(1)}$
can also be easily calculated from $x_{N(1)}^\eta$ as
\begin{align}
\Pi_{N(1)}^\eta = \lambda (x_{N(1)}^\eta-\kappa_{(1)}). \label{eq:d2cIsing_Pi1}
\end{align}
Using these formulas, one obtains the thermodynamic forces
without differentiating the thermodynamic functions numerically.
We calculate the statistical-mechanical quantities in the SE
using the replica exchange Monte Carlo calculations \cite{Hukushima1996}.
The acceptance probability can be easily computed
using the Metropolis algorithm \cite{Metropolis1953}
in almost the same manner as in the GCE.
In the SE, the replica exchange method works well
even in the first-order phase transition region,
as in the case of the ensembles
with macroscopically well-defined energy
in the temperature-driven first-order phase transition region
\cite{Kim2010,Schierz2016}.
As argued in Section~\ref{sec:parameter},
the equilibrium state changes continuously in $\bm{\kappa}$
even in the phase transition region.
Therefore, the exchange of configurations between adjacent replicas
is accepted with high probability.
By contrast, in the GCE, the different phases are separated
by the free energy barrier \cite{Schierz2016}
and the equilibrium state changes discontinuously in the parameters of the ensemble.
This degrades greatly the replica exchange method in the GCE.
In the SE, since the order parameter is made to have a definite value by the ensemble,
boundary conditions can be imposed arbitrarily.
For numerical simulations,
we here employ periodic boundary conditions.
By this choice of the boundary condition, the surface effects,
which are of the same order as the effects of the phase interfaces,
can be eliminated.
Consequently, the SE enables us to study the phase interfaces
without suffering from the surface effects.
\subsubsection{Results}
Figure~\ref{fig:d2cIsing_L-var} shows
the $N$ dependence of the variance of $\hat{x}_{N(1)}$ in the GCE and the SE.
In order to compare them in the first-order phase transition region
(where $\Pi_{(0)} > \frac{1}{2} \log (1+\sqrt{2}) \simeq 0.44, \Pi_{(1)} = 0$),
we focus on the equilibrium state with $\bm{\Pi}=(0.45,0)$.
Then we take $\bm{\pi}=(0.45,0)$ and $\bm{\kappa}=(0.45,0)$
so that $\bm{\Pi}_N^\mathrm{gc}=\bm{\Pi}_N^\eta=(0.45,0)$
from Eqs.~\eqref{eq:d2cIsing_Pi0} and \eqref{eq:d2cIsing_Pi1}.
It is seen that in the case of the GCE the variance remains finite even in the thermodynamic limit.
This implies that the GCE gives a statistical mixture of macroscopically distinct states
which have the same values of the thermodynamic forces.
In fact, as shown in the insets of Fig.~\ref{fig:d2cIsing_L-var},
there are two typical spin configurations for the GCE,
which are macroscopically homogeneous states
with positive $x_{(1)}$
and those with negative $x_{(1)}$.
By contrast, in the case of the SE, the variance vanishes in the thermodynamic limit as $O(N^{-1})$.
This implies that the SE successfully gives a phase coexistence state,
as can also be seen from the insets of Fig.~\ref{fig:d2cIsing_L-var}.
We have thus confirmed that the SE is a microcanonical-like ensemble
and gives the phase coexistence state.
\begin{figure}
\centering
\includegraphics[keepaspectratio, width=\linewidth]{d2cIsing_L-var_insets.pdf}
\caption{$N$ dependence of the variance of $\hat{x}_{N(1)}$ in the GCE and the SE for $\bm{\pi}=(0.45,0)$ and $\bm{\kappa}=(0.45,0)$.
Insets show typical snapshots of the Monte Carlo simulations
(white represents $\braket{\hat{\sigma}}_i^z=+1$
and black represents $\braket{\hat{\sigma}}_i^z=-1$).
Two snapshots are shown for each ensemble.}
\label{fig:d2cIsing_L-var}
\end{figure}
In order to illustrate how the convexity of $\eta$ controls the variance of $\hat{x}_{N(1)}$,
we plot the variance as a function of $\lambda$ in Fig.~\ref{fig:d2cIsing_lambda-var}.
It is seen that the variance becomes smaller as the convexity of $\eta$ becomes stronger.
\begin{figure}
\centering
\includegraphics[keepaspectratio, width=\linewidth]{d2cIsing_lambda-var.pdf}
\caption{Relation between the strength of the convexity of $\eta$ defined by Eq.~\eqref{eq:d2cIsing_eta} and the variance of $\hat{x}_{N(1)}$ for $N=32^2$ and $\bm{\kappa}=(0.45,0)$.}
\label{fig:d2cIsing_lambda-var}
\end{figure}
Figure~\ref{fig:d2cIsing_ordered_tdf-op} shows the relation
between the thermodynamic force
(thermodynamic magnetic field $-\Pi_{N(1)}^\eta/\Pi_{N(0)}^\eta$)
and the order parameter $x_{N(1)}^\eta$,
in the ordered phase for the GCE and the SEs.
Again, we take $\pi_{(0)}=\kappa_{(0)}=0.45$.
When the GCE is used,
$x_{N(1)}^\mathrm{gc}$ is given as a function of $\pi_{(1)}=\Pi_{N(1)}^\eta$,
which cannot be a multivalued function,
changing monotonically and continuously.
Due to the large fluctuation,
the singularities of thermodynamic functions at the first-order phase transition region
are smeared significantly \cite{Stump1987,Huller1992,Huller1994}.
This makes it difficult to identify the phase transition and to determine its order.
This should be contrasted with the results of the SEs,
which show that $x_{N(1)}^\eta$ is a multivalued, S-shaped function of $\Pi_{N(1)}^\eta$
for sufficiently large $\lambda$.
This is caused by the concavity breaking of the Boltzmann entropy
due to the effects of the phase interface \cite{Challa1988_PRA,Yoneta2019}.
It is confirmed that, by using the SE,
thermodynamic anomalies are correctly obtained
and the phase transition can be detected directly.
It is also seen from Fig.~\ref{fig:d2cIsing_ordered_tdf-op}
that the curve becomes insensitive to the magnitude of $\lambda$ for sufficiently large $\lambda$ ($\gtrsim 1$).
This means that the statistical-mechanical quantities in the SE are physical,
independent of the detail of the ensemble \cite{Yoneta2019}.
\begin{figure}
\centering
\includegraphics[keepaspectratio, width=\linewidth]{d2cIsing_ordered_tdf-op.pdf}
\caption{Relation between the thermodynamic force and the order parameter in the ordered phase for the GCE and the SEs associated with $\eta$ given by Eq.~\eqref{eq:d2cIsing_eta} for $N=32^2$ and $\pi_{(0)}=\kappa_{(0)}=0.45$.}
\label{fig:d2cIsing_ordered_tdf-op}
\end{figure}
\subsection{Finite transverse field} \label{sec:d2qIsing}
To demonstrate that the SE
enables microcanonical-like simulations
and gives phase coexistence state in quantum systems,
we consider the case of $g \neq 0$.
We take the strength of the transverse field to be moderate, $g=1$.
\subsubsection{Methods}
We choose the parameter space of $\bm{\kappa}$ as
$K=(0,+\infty)\times(-\infty,+\infty)$,
and $\eta$ as
\begin{align}
\eta(\bm{\kappa};\bm{x}) = \kappa_{(0)} x_{(0)} + \kappa_{(1)} x_{(1)} - 2 \nu \log (x_{(1)}+1), \label{eq:d2qIsing_eta}
\end{align}
where $\nu$ is an appropriate positive constant.
Though this $\eta$ is not a polynomial,
all the results of Section~\ref{sec:ensemble}
hold also for this $\eta$
(see Appendix~\ref{sec:derivation}-\ref{sec:invariance} for details).
This choice of $\eta$ is particularly convenient
for quantum systems which undergo a first-order phase transition
because,
as shown in Appendix~\ref{sec:d2qIsing_method},
the density matrix can be approximated as the product of local operators,
and therefore the SE can be numerically constructed easily even for quantum systems.
Since the second order derivative of $\eta$ with respect to $x_{(1)}$ is
$2\nu/(x_{(1)}+1)^2$,
the strength of the convexity of $\eta$ can be freely adjusted by tuning the value of $\nu$.
That is, by taking sufficiently large $\nu$,
we can make the order parameter to have a definite value
even in the first-order phase transition region.
The inverse temperature $\beta = \Pi_{(0)}$
is given directly by $\kappa_{(0)}$.
In fact, using Eq.~\eqref{eq:tdforce-formula}, we have
\begin{align}
\Pi_{N(0)}^\eta = \kappa_{(0)}. \label{eq:d2qIsing_Pi0}
\end{align}
Therefore, by taking $\kappa_{(0)}$ to a certain value,
we can investigate the properties of the equilibrium states
whose inverse temperature takes that value.
Furthermore, again using Eq.~\eqref{eq:tdforce-formula},
the thermodynamic force conjugate to $X_{(1)}$ can also be easily calculated from $x_{N(1)}^\eta$ as
\begin{align}
\Pi_{N(1)}^\eta = \kappa_{(1)} - \frac{2\nu}{x_{N(1)}^\eta+1}. \label{eq:d2qIsing_Pi1}
\end{align}
We calculate statistical-mechanical quantities in the SE
using the minimally entangled typical thermal states (METTS) algorithm \cite{White2009,Stoudenmire2010}
extended to the SE as follows.
We take the quantization axis along $z$-direction
and the classical product state
\begin{align}
\ket{s} \equiv \bigotimes_i \ket{s_i}
\quad (\hat{\sigma}_i^z \ket{s_i} = s_i \ket{s_i}).
\end{align}
Then we produce METTS of the SE that is defined by
\begin{align}
\ket{s;\eta} \equiv \frac{1}{\sqrt{\braket{s|e^{-N\eta(\hat{\bm{x}}_N)}|s}}} e^{-\frac{1}{2}N\eta(\hat{\bm{x}}_N)} \ket{s}.
\end{align}
Due to the completeness of the classical product states,
$\hat{\rho}_N^\eta$ can be decomposed into a convex mixture of $\ket{s;\eta}$
with the weight $\displaystyle p(s;\eta) \equiv \braket{s|e^{-N\eta(\hat{\bm{x}}_N)}|s}$:
\begin{align}
\hat{\rho}_N^\eta
\propto \sum_{s_i=\pm 1} p(s;\eta) \ket{s;\eta}\bra{s;\eta}.
\end{align}
Therefore, the expectation value in $\hat{\rho}_N^\eta$ can be calculated
by averaging the expectation value in $\ket{s;\eta}$,
which is sampled according to the weight $p(s;\eta)$.
The numerical simulations are performed
for the square lattice of size $L_1 \times L_2$ with
periodic boundary conditions along $L_1$-direction
and open boundary conditions along $L_2$-direction.
\subsubsection{Results}
Figure~\ref{fig:d2qIsing_histogram} shows
the histogram of the number of samples
as a function of $\braket{s;\eta|\hat{x}_{N(1)}|s;\eta}$
for the GCE and the SE.
In order to compare them in the first-order phase transition region,
where $\Pi_{(0)} \gtrsim 2.15^{-1}$ and $\Pi_{(1)}=0$ \cite{Nagai1987},
we focus on the equilibrium state with $\bm{\Pi}=(0.5,0)$.
Hence, we take $\pi_{(0)}=\kappa_{(0)}=0.5$
so that $\Pi_{N(0)}^\mathrm{gc}=\Pi_{N(0)}^\eta=0.5$
from Eq.~\eqref{eq:d2qIsing_Pi0}.
Furthermore, we take $\pi_{(1)}=0$ and $\kappa_{(1)}=2\nu$
(indicated as filled circles
in Fig.~\ref{fig:d2qIsing_ordered_tdf-op}).
Then, from Eq.~\eqref{eq:d2qIsing_Pi1},
both the GCE and the SE give the density matrix
where the order parameter $x_{(1)}$ is zero
in the thermodynamic limit.
For the GCE, the histogram shows a double peak structure.
This implies that
the density matrix given by the GCE is
a statistical mixture of two single phase states
which have the same values of the thermodynamic forces.
In fact, as shown in Fig.~\ref{fig:d2qIsing_snapshots} (left),
typical METTS of the GCE have
macroscopically homogeneous order parameter profiles
with positive or negative $x_{(1)}$.
By contrast, in the SE,
it can be seen that the order parameter has a macroscopically definite value.
As a result, as shown in Fig.~\ref{fig:d2qIsing_snapshots} (right),
large domains are formed in typical METTS of the SE.
In addition,
comparing the order parameter profiles in the two typical METTS of the SE,
we confirm
that
the SE is composed of the states
which are macroscopically identical
except for the spatial arrangement of the coexisting phases.
As can be seen from thermodynamics,
the spatial arrangement of the coexisting phases
is not determined by the values of the additive quantities
(of the total system) alone.
Therefore, the SE is consistent with thermodynamics.
\begin{figure}
\centering
\includegraphics[keepaspectratio, width=\linewidth]{d2qIsing_histogram}
\caption{Histogram of the number of samples
as a function of $\braket{s;\eta|\hat{x}_{N(1)}|s;\eta}$
for $L_1=5,L_2=16$ and $\bm{\pi}=(0.5,0), \bm{\kappa}=(0.5,2\nu)$.}
\label{fig:d2qIsing_histogram}
\end{figure}
\begin{figure}
\centering
\includegraphics[keepaspectratio, width=\linewidth]{d2qIsing_snapshots.png}
\caption{Order parameter profiles in typical METTS of the GCE (left) and the SE (right)
for $L_1=5,L_2=16$ and $\bm{\pi}=(0.5,0), \bm{\kappa}=(0.5,2\nu)$
(color map, where yellow represents $\braket{\hat{\sigma}}_i^z=+1$
and black represents $\braket{\hat{\sigma}}_i^z=-1$).}
\label{fig:d2qIsing_snapshots}
\end{figure}
Figure~\ref{fig:d2qIsing_ordered_tdf-op} shows the relation
between the thermodynamic force and the order parameter.
Again, we take $\pi_{(0)}=\kappa_{(0)}=0.5$.
As in the case of the classical limit $g \to 0$,
$x_{N(1)}^\eta$ is a multivalued function of $\Pi_{N(1)}^\eta$ in the SE,
whereas
it is a monotonous and continuous function in the GCE.
This is due to the presence of interfaces separating the coexisting phases
as shown in Fig.~\ref{fig:d2qIsing_snapshots} \cite{Challa1988_PRA,Yoneta2019}.
\begin{figure}
\centering
\includegraphics[keepaspectratio, width=\linewidth]{d2qIsing_ordered_tdf-op_m.pdf}
\caption{Relation between the thermodynamic force and the order parameter in the ordered phase
for the GCE and the SE associated with $\eta$ given by Eq.~\eqref{eq:d2qIsing_eta}
for $L_1=5,L_2=16$ and $\pi_{(0)}=\kappa_{(0)}=0.5$.
The filled circles indicate the points corresponding to the parameters of Figs.~\ref{fig:d2qIsing_histogram} and \ref{fig:d2qIsing_snapshots} for the GCE and the SE, respectively.}
\label{fig:d2qIsing_ordered_tdf-op}
\end{figure}
From these observations, we conclude that
the SE successfully gives the phase coexistence state
with phase interfaces,
whereas the GCE gives a classical statistical mixture
of single phase states.
\section{Numerical implementation of the SE associated with $\eta$ given by Eq.~\eqref{eq:d2qIsing_eta}} \label{sec:d2qIsing_method}
We calculate statistical-mechanical quantities in the SE associated with $\eta$ given by Eq.~\eqref{eq:d2qIsing_eta}
using the METTS algorithm.
To obtain the METTS,
one needs to apply $e^{-\frac{1}{2}N\eta(\bm{\kappa};\hat{\bm{x}}_N)}$
to a random product state.
Choosing $\eta$ as Eq.~\eqref{eq:d2qIsing_eta},
this can be easily done as follows.
Consider the general case
where $\hat{X}_{N(0)}$ and $\hat{X}_{N(1)}$ can be written
as the sum of local observables:
\begin{align}
\hat{X}_{N(0)} &= \sum_{i=1}^N \hat{h}_i,\\
\hat{X}_{N(1)} &= \sum_{i=1}^N \hat{m}_i,
\end{align}
where $\hat{h}_i$ and $\hat{m}_i$ are observables located around site $i$.
Then it follows that
\begin{align}
- \frac{1}{2} N \eta(\bm{\kappa};\hat{\bm{x}}_N)
= \frac{1}{\delta} \left\{
\sum_{i=1}^N \delta \hat{\gamma}_i
+ N\nu\delta \log \left( \sum_{i=1}^N \hat{m}_i + N \right)
+ \sum_{i=1}^N \delta \hat{\gamma}_i
\right\}
+ \mathrm{const.},
\end{align}
where $\hat{\gamma}_i \equiv - \frac{1}{4} (\kappa_{(0)}\hat{h}_i + \kappa_{(1)}\hat{m}_i)$.
Therefore, by using the Trotter-Suzuki formula \cite{Suzuki1976},
$e^{-\frac{1}{2}N\eta(\bm{\kappa};\hat{\bm{x}}_N)}$ is decomposed
into a product of local operators as
\begin{align}
e^{-\frac{1}{2}N\eta(\bm{\kappa};\hat{\bm{x}}_N)}
\propto \left\{
e^{\delta \hat{\gamma}_1} \cdots e^{\delta \hat{\gamma}_N}
\left( \sum_{i=1}^N \hat{m}_i + N \right)^{N\nu\delta}
e^{\delta \hat{\gamma}_N} \cdots e^{\delta \hat{\gamma}_1}
\right\}^{1/\delta}
+ O(\delta^2). \label{eq:METTS_Trotter}
\end{align}
Especially when $N\nu\delta$ and $1/\delta$ are both integers,
the right hand side of Eq.~\eqref{eq:METTS_Trotter} can be obtained
by simply multiplying the local operators repeatedly.
In particular, by choosing $\nu$ so that $N\nu$ is an integer with many divisors,
it is possible to extrapolate the statistical-mechanical quantities to $\delta \to 0$
while taking $N\nu\delta$ as an integer,
as shown in Fig.~\ref{fig:METTS_Trotter}.
Hence, the SE can be numerically constructed easily even for quantum systems.
\begin{figure}
\centering
\includegraphics[keepaspectratio, width=0.5\linewidth]{Trotter.pdf}
\caption{Trotter step dependence of the statistical-mechanical quantity
for the one-dimensional transverse field Ising model
($\hat{X}_{N(0)} = - J \sum \hat{\sigma}_{i}^z \hat{\sigma}_{i+1}^z - g \sum \hat{\sigma}_i^x,
\hat{X}_{N(1)} = \sum \hat{\sigma}_i^z$)
in the SE for $N=50, J=1, g=1$ and $\nu=3.6, \bm{\kappa}=(1,5)$.}
\label{fig:METTS_Trotter}
\end{figure}
\section{Derivation of the results in Section~\ref{sec:ensemble}} \label{sec:derivation}
We here derive the results presented in Section~\ref{sec:ensemble}.
When $\hat{\bm{X}}_N$ do not commute with each other,
we will show in Appendix~\ref{sec:noncommutative}
that the statistical-mechanical properties becomes identical
in the thermodynamic limit
between the SE for $\hat{\bm{X}}_N$ and that for $\check{\bm{X}}_N$.
Therefore, it is sufficient to consider the case
where $\hat{\bm{X}}_N$ commute with each other.
In the noncommutative case,
$\hat{\bm{X}}_N$ in the following equations shall be read
as a commutative set
$\check{\bm{X}}_N$ that approximate $\hat{\bm{X}}_N$.
For this reason, we can assume, without loss of generality, that
$\hat{\bm{X}}_N$ commute with each other.
This allows us to use the asymptotic approximation such as Laplace's method.
Although the argument based on the asymptotic approximation
is not mathematically rigorous, its validity was supported
by numerical calculations \cite{Yoneta2019}.
Regarding the thermodynamic forces,
we will prove their formulas without relying on the asymptotic approximation,
in Appendix~\ref{sec:derivation_tdforce}.
\subsection{Probability distribution}
We examine how $\bm{x}$ distributes in $\hat{\rho}_N^\eta$.
Let $f$ be an $m$-variable function.
Then
\begin{align}
\mathrm{Tr} \left[ f(\hat{\bm{x}}_N) e^{-N\eta(\hat{\bm{x}}_N)} \right]
&\simeq \int d\bm{x} f(\bm{x}) e^{N [s(\bm{x})-\eta(\bm{x})]}.
\end{align}
Since $s$ is assumed to be continuously differentiable,
$\bm{x}_\mathrm{max}^\eta$,
where $s-\eta$ takes the maximum,
satisfies
\begin{align}
\Pi_{(i)} (\bm{x}_\mathrm{max}^\eta)
\equiv \frac{\partial s}{\partial x_{(i)}} (\bm{x}_\mathrm{max}^\eta)
= \frac{\partial \eta}{\partial x_{(i)}} (\bm{x}_\mathrm{max}^\eta).
\label{eq:x-max-eta}
\end{align}
Thanks to the strict convexity of $s-\eta$ at $\bm{x}_\mathrm{max}^\eta$,
$e^{N [s-\eta]}$ has a sharp peak
even in the first-order phase transition region.
Then we have
\begin{align}
\mathrm{Tr} \left[ f(\hat{\bm{x}}_N) e^{-N\eta(\hat{\bm{x}}_N)} \right]
&\simeq f(\bm{x}_\mathrm{max}^\eta) e^{N [s(\bm{x}_\mathrm{max}^\eta)-\eta(\bm{x}_\mathrm{max}^\eta)]}. \label{eq:tr_f-rho}
\end{align}
\subsection{Genuine thermodynamic quantities}
Using Eq.~(\ref{eq:tr_f-rho}),
we have
\begin{align}
\psi_N^\eta &= \eta (\bm{x}_\mathrm{max}^\eta) - s(\bm{x}_\mathrm{max}^\eta) + o(N^0), \label{eq:asymptotic-approx_psi}\\
x_{N(i)}^\eta &= x_{\mathrm{max}(i)}^\eta + o(N^0). \label{eq:asymptotic-approx_x}
\end{align}
Therefore, we obtain
\begin{align}
s_N^\eta
= \eta\left(\bm{x}_N^\eta\right) - \psi_N^{\eta}
= s(\bm{x}^\eta) + o(N^0).
\end{align}
This is Eq.~\eqref{eq:entropy-formula}.
Using Eqs.~(\ref{eq:x-max-eta}) and (\ref{eq:asymptotic-approx_x}), we also obtain
\begin{align}
\Pi_{N(i)}^\eta
= \frac{\partial\eta}{\partial x_{(i)}}\left(\bm{x}_N^\eta\right)
= \Pi_{(i)}(\bm{x}^\eta) + o(N^0).
\end{align}
This is Eq.~\eqref{eq:tdforce-formula}.
Here, since $\displaystyle \frac{\partial \eta}{\partial x_{(i)}} = \frac{\partial s}{\partial x_{(i)}}$ only at $\bm{x}_\mathrm{max}^\eta$,
replacing $\bm{x}_\mathrm{max}^\eta$ with $\bm{x}_N^\eta$
yields the difference of $o(N^0)$.
\subsection{Laplace approximation} \label{sec:derivation_Laplace}
In particular,
in the case where $s-\eta$ is strongly concave and twice continuously differentiable,
$e^{N [s-\eta]}$ behaves as the Gaussian distribution
peaking at $\bm{x}_\mathrm{max}^\eta$,
with the small variance $O(N^{-1})$.
Therefore, the $2n$-th central moment of $x_{(i)}$ is $O(N^{-n})$.
Consequently, the $\eta$ dependence of $\psi_N^\eta$ and $\mathrm{Tr} \left[ f(\hat{\bm{x}}_N) \hat{\rho}_N^\eta \right]$ are $O(N^{-1} \log N)$ and $O(N^{-1})$, respectively.
\section{Proof of Eq.~\eqref{eq:psi-s}} \label{sec:genLeg_proof}
Using Eq.~\eqref{eq:s-psi}, we have
\begin{align}
s(\bm{x})
&\leq \eta(\bm{\kappa};\bm{x}) - \inf_{\bm{x} \in \Gamma} \left\{ \eta(\bm{\kappa};\bm{x})-s(\bm{x}) \right\} \\
&= \eta(\bm{\kappa};\bm{x}) - \psi^\eta(\bm{\kappa})
\end{align}
for all $\bm{\kappa}$.
Then we have
\begin{align}
s(\bm{x}) \leq \inf_{\bm{\kappa} \in K} \left\{ \eta(\bm{\kappa};\bm{x})-\psi^\eta(\bm{\kappa}) \right\}.
\end{align}
On the other hand, using Eq.~\eqref{eq:s-psi}, we have
\begin{align}
&\inf_{\bm{\kappa} \in K} \left\{ \eta(\bm{\kappa};\bm{x})-\psi^\eta(\bm{\kappa}) \right\}\\
&\leq \eta(\bm{\kappa};\bm{x})-\psi^\eta(\bm{\kappa})\\
&= \eta(\bm{\kappa};\bm{x}) - \inf_{\bm{x} \in \Gamma} \left\{ \eta(\bm{\kappa};\bm{x})-s(\bm{x}) \right\}\\
&= \eta(\bm{\kappa};\bm{x}) - \eta(\kappa, \bm{x}_\mathrm{max}^\eta(\bm{\kappa})) + s(\bm{x}_\mathrm{max}^\eta(\bm{\kappa}))
\end{align}
for all $\bm{\kappa}$.
The condition~\ref{cond:D} implies that there exists $\bm{\kappa}$
such that $\bm{x}_\mathrm{max}^\eta(\bm{\kappa})=\bm{x}$,
then it holds
\begin{align}
\inf_{\bm{\kappa} \in K} \left\{ \eta(\bm{\kappa};\bm{x})-\psi^\eta(\bm{\kappa}) \right\} &\leq s(\bm{x}).
\end{align}
Therefore, we have Eq.~\eqref{eq:psi-s}.
\section{Derivation of Eq.~\eqref{eq:tdforce-formula} without the asymptotic approximation} \label{sec:derivation_tdforce}
In this section,
we give another derivation
of the formula for the thermodynamic force,
Eq.~\eqref{eq:tdforce-formula}.
A great advantage of this proof is
that it works even in the case
where $\hat{\bm{x}}_N$ do not commute with each other
because it does not rely on the asymptotic approximation.
Suppose that the SE is outside the phase transition region,
and the GCE
\begin{align}
\mathrm{gc}(\bm{\pi};\bm{x}) = \sum_i \pi_{(i)} x_{(i)}
\end{align}
describes the same equilibrium state as the SE
in the thermodynamic limit
when $\bm{\pi}=\tilde{\bm{\pi}}$.
In particular, the following holds:
\begin{align}
\lim_{N\to\infty} \bm{x}_N^\eta
= \lim_{N\to\infty} \bm{x}_N^\mathrm{gc}(\tilde{\bm{\pi}}). \label{eq:tdforce-woLaplace_0}
\end{align}
Since $\displaystyle \lim_{N\to\infty} \bm{x}_N^\mathrm{gc}(\tilde{\bm{\pi}})$
is assumed to be outside the phase transition region,
the grand canonical thermodynamic function $\psi^\mathrm{gc}$ is differentiable at $\tilde{\bm{\pi}}$,
and its derivative at $\tilde{\bm{\pi}}$ is given by \cite{Griffiths1964}
\begin{align}
\frac{\partial \psi^\mathrm{gc}}{\partial \pi_{i}}(\tilde{\bm{\pi}})
= \lim_{N\to\infty} \bm{x}_N^\mathrm{gc}(\tilde{\bm{\pi}}).
\end{align}
Therefore, using Eq.~\eqref{eq:tdforce-woLaplace_0},
Eq.~\eqref{eq:tdforce-formula} can be rephrased as
\begin{align}
\lim_{N\to\infty} \left( \tilde{\pi}_{(j)} - \frac{\partial\eta}{\partial x_{(j)}}\left(\bm{x}_N^\mathrm{gc}(\tilde{\bm{\pi}})\right) \right) = 0. \label{eq:tdforce-woLaplace}
\end{align}
We derive Eq.~\eqref{eq:tdforce-woLaplace}
under reasonable conditions.
\subsection{Assumptions}
Let us measure the distance between the density matrices by the relative entropy
\begin{align}
D\infdivx*{\hat{\rho}}{\hat{\sigma}}
\equiv \mathrm{Tr} \left[ \hat{\rho} \left\{ \log \hat{\rho} - \log \hat{\sigma} \right\} \right].
\end{align}
Since the ``distance'' between $\hat{\rho}_N^\eta$ and $\hat{\rho}_N^\mathrm{gc}(\bm{\pi})$ is expected to by minimized at $\bm{\pi}=\tilde{\bm{\pi}}$,
it is reasonable to assume that
\begin{enumerate}[label={(\roman*)}]
\item \hfill\vspace{-\abovedisplayskip}\vspace{-\baselineskip}
\begin{align}
\left.\frac{\partialD\infdivx*{\hat{\rho}_N^\mathrm{gc}(\bm{\pi})}{\hat{\rho}_N^\eta}}{\partial\pi_{(i)}}\right|_{\bm{\pi}=\tilde{\bm{\pi}}} = o(N). \label{eq:tdforce-woLaplace_1}
\end{align}
\label{cond:tdforce-woLaplace_1}
\end{enumerate}
Note that a similar condition
\begin{align}
\left.\frac{\partialD\infdivx*{\hat{\rho}_N^\eta}{\hat{\rho}_N^\mathrm{gc}(\bm{\pi})}}{\partial\pi_{(i)}}\right|_{\bm{\pi}=\tilde{\bm{\pi}}} = o(N)
\end{align}
is equivalent to Eq.~\eqref{eq:tdforce-woLaplace_0},
which is a necessary condition for $\hat{\rho}_N^\mathrm{gc}(\tilde{\bm{\pi}})$ and $\hat{\rho}_N^\eta$ to describe the same equilibrium state.
Therefore, it seems reasonable to assume Eq.~\eqref{eq:tdforce-woLaplace_1}.
In addition, we assume
\begin{enumerate}[label={(\roman*)}]
\setcounter{enumi}{1}
\item $\hat{\rho}_N^\mathrm{gc}(\tilde{\bm{\pi}})$ has normal fluctuations \cite{Jaksic2010,Goderis1989}:
\begin{align}
\braket{\delta\hat{x}_{N(i_1)} \cdots \delta\hat{x}_{N(i_{2n})}}_N^\mathrm{gc}(\tilde{\bm{\pi}})
&= O(N^{-n}),
\end{align}
\label{cond:tdforce-woLaplace_2}
\end{enumerate}
where $\braket{\bullet}_N^\eta \equiv \mathrm{Tr} \left[ \bullet \rho_N^\eta \right]$.
Here, for simplicity of notation,
we abbreviate
$\braket{\cdots (\bullet-\braket{\bullet}_N^\eta) \cdots}_N^\eta$
to $\braket{\cdots \delta\bullet \cdots}_N^\eta$.
Finally, we assume
\begin{enumerate}[label={(\roman*)}]
\setcounter{enumi}{2}
\item the $m \times m$ positive-semidefinite matrix
\begin{align}
\Delta_{ij} \equiv \lim_{N\to\infty} N \Braket{\delta\hat{x}_{N(i)};\delta\hat{x}_{N(j)}}_N^\mathrm{gc}(\tilde{\bm{\pi}}) \label{eq:tdforce-woLaplace_3}
\end{align}
is invertible,
\label{cond:tdforce-woLaplace_3}
\end{enumerate}
where $\braket{\bullet;\bullet}_N^\eta$ denotes
the generalized Bogoliubov-Duhamel inner product:
\begin{align}
\braket{\hat{A};\hat{B}}_N^\eta
&\equiv \int_0^1 d\nu\
\braket{
e^{+ \nu N \eta(\hat{\bm{x}}_N)} \hat{A}^\dagger
e^{- \nu N \eta(\hat{\bm{x}}_N)} \hat{B}
}_N^\eta.
\end{align}
The validity of this assumption is explained as follows.
Since $\bm{x}^\mathrm{gc}(\tilde{\bm{\pi}})$ is assumed to be outside the phase transition region,
$\bm{x}^\mathrm{gc}(\bm{\pi})$ is analytic and one-to-one in a neighborhood of $\tilde{\bm{\pi}}$.
Therefore, $\displaystyle \frac{\partial x_{(j)}^\mathrm{gc}}{\partial \pi_{(i)}}(\tilde{\bm{\pi}})$ is an invertible matrix.
On the other hand, we have
\begin{align}
\Delta_{ij} = \lim_{N\to\infty} \frac{\partial x_{N(j)}^\mathrm{gc}}{\partial \pi_{(i)}}(\tilde{\bm{\pi}}).
\end{align}
It is expected that we can interchange the limit and differentiation except at phase transition points (see Lemma~\ref{lemma:AA_cor}) as
\begin{align}
\Delta_{ij} = \lim_{N\to\infty} \frac{\partial x_{N(j)}^\mathrm{gc}}{\partial \pi_{(i)}}(\tilde{\bm{\pi}})
= \frac{\partial x_{(j)}^\mathrm{gc}}{\partial \pi_{(i)}}(\tilde{\bm{\pi}}).
\end{align}
Therefore, $\Delta_{ij}$ should be invertible.
Note that assumptions~\ref{cond:tdforce-woLaplace_2}-\ref{cond:tdforce-woLaplace_3}
are assumptions on the GCE, not on the SE.
\subsection{Proof}
To prove Eq.~\eqref{eq:tdforce-woLaplace},
we first prove the following lemma.
\begin{lemma} \label{lemma:bound_Bogoliubov-Duhamel}
If
\begin{align}
\braket{\delta\hat{x}_{N(i_1)} \cdots \delta\hat{x}_{N(i_{2n})}}_N^\eta = O(N^{-n}),
\end{align}
then
\begin{align}
\braket{
\delta\hat{x}_{N(i_1)} \cdots \delta\hat{x}_{N(i_m)};
\delta\hat{x}_{N(j_1)} \cdots \delta\hat{x}_{N(j_n)}
}_N^\eta
= O(N^{-\frac{m+n}{2}}). \label{eq:bound_Bogoliubov-Duhamel}
\end{align}
\end{lemma}
\begin{proof}
Applying the Cauchy-Schwarz inequality
\begin{align}
\left| \braket{\hat{A};\hat{B}}_N^\eta \right|^2
&\leq \braket{\hat{A};\hat{A}}_N^\eta \braket{\hat{B};\hat{B}}_N^\eta,
\end{align}
and Brooks Harris inequality \cite{Mermin1966,BrooksHarris1967}
\begin{align}
\braket{\hat{A};\hat{A}}_N^\eta
\leq \frac{1}{2} \braket{\{\hat{A}^\dagger,\hat{A}\}}_N^\eta,
\end{align}
we have
\begin{align}
&\left| \braket{
\delta\hat{x}_{N(i_1)} \cdots \delta\hat{x}_{N(i_m)};
\delta\hat{x}_{N(j_1)} \cdots \delta\hat{x}_{N(j_n)}
}_N^\eta \right|^2\\
&\leq \frac{1}{4} \Braket{ \left\{
\delta\hat{x}_{N(i_m)} \cdots \delta\hat{x}_{N(i_1)},
\delta\hat{x}_{N(i_1)} \cdots \delta\hat{x}_{N(i_m)}
\right\} }_N^\eta
\Braket{ \left\{
\delta\hat{x}_{N(j_n)} \cdots \delta\hat{x}_{N(j_1)},
\delta\hat{x}_{N(j_1)} \cdots \delta\hat{x}_{N(j_n)}
\right\} }_N^\eta.
\end{align}
Therefore, by assumption,
we obtain Eq.~\eqref{eq:bound_Bogoliubov-Duhamel}.
\end{proof}
\begin{proof}[Proof of Eq.~\eqref{eq:tdforce-woLaplace}]
By assumption~\ref{cond:tdforce-woLaplace_2}
and Lemma~\ref{lemma:bound_Bogoliubov-Duhamel},
we have
\begin{align}
\braket{
\delta\hat{x}_{N(i)};
\delta\hat{x}_{N(j_1)} \cdots \delta\hat{x}_{N(j_n)}
}_N^\mathrm{gc}
= O(N^{-\frac{1+n}{2}}).
\end{align}
In addition, since $\eta$ is a polynomial,
expanding in power series around $\bm{x}_N^\mathrm{gc}(\bm{\pi})$,
$\eta(\hat{\bm{x}}_N)$ can be written as a finite sum as
\begin{align}
\eta(\hat{\bm{x}}_N)
&= \eta(\bm{x}_N^\mathrm{gc}(\bm{\pi}))
+ \sum_j \frac{\partial\eta}{\partial x_{(j)}}(\bm{x}_N^\mathrm{gc}(\bm{\pi})) \delta\hat{x}_{N(j)}
+ \cdots.
\end{align}
Therefore, using
\begin{align}
\frac{\partialD\infdivx*{\hat{\rho}_N^\mathrm{gc}(\bm{\pi})}{\hat{\rho}_N^\eta}}{\partial\pi_{(i)}}
&= N^2 \Braket{
\delta \left[ \partial_{\pi_{(i)}}\mathrm{gc}(\bm{\pi};\hat{\bm{x}}_N) \right];
\mathrm{gc}(\bm{\pi};\hat{\bm{x}}_N) - \eta(\hat{\bm{x}}_N)
}_N^\mathrm{gc}(\bm{\pi}),
\end{align}
we have
\begin{align}
\frac{\partialD\infdivx*{\hat{\rho}_N^\mathrm{gc}(\bm{\pi})}{\hat{\rho}_N^\eta}}{\partial\pi_{(i)}}
&= \sum_j N^2 \left\{ \Braket{
\delta\hat{x}_{N(i)};
\delta\hat{x}_{N(j)}
}_N^\mathrm{gc}(\bm{\pi}) \left( \pi_{(j)} - \frac{\partial\eta}{\partial x_{(j)}}(\bm{x}_N^\mathrm{gc}(\bm{\pi})) \right)
+ O(N^{-\frac{1+2}{2}}) \right\}.
\end{align}
Therefore, assumption~\ref{cond:tdforce-woLaplace_1} can be rewritten as
\begin{align}
\sum_j \Delta_{ij} \lim_{N\to\infty} \left( \tilde{\pi}_{(j)} - \frac{\partial\eta}{\partial x_{(j)}}\left(\bm{x}_N^\mathrm{gc}(\tilde{\bm{\pi}})\right) \right)
= 0.
\end{align}
Hence, by assumption~\ref{cond:tdforce-woLaplace_3},
we obtain Eq.~\eqref{eq:tdforce-woLaplace}.
\end{proof}
\section{squeezed ensemble} \label{sec:ensemble}
\subsection{Setup} \label{sec:setup}
We consider a sequence of quantum spin systems
indexed by the number of lattice sites $N$.
We assume that the equilibrium state can be specified uniquely
by an $m$-tuple of additive quantities $(X_{(0)}=E,X_{(1)},\cdots,X_{(m-1)})$ for each $N$.
Let $(\hat{X}_{N(0)}=\hat{H}_N,\hat{X}_{N(1)},\cdots,\hat{X}_{N(m-1)})$ be a tuple of the corresponding observables.
To take the thermodynamic limit,
we introduce an additive quantity per site
\begin{align}
x_{(i)} \equiv X_{(i)}/N,
\end{align}
and the corresponding additive observable per site
\begin{align}
\hat{x}_{N(i)} \equiv \hat{X}_{N(i)}/N.
\end{align}
For simplicity of notation,
tuples are denoted by bold symbols like
\begin{align}
\bm{x} = (x_{(0)},x_{(1)},\cdots,x_{(m-1)}).
\end{align}
The thermodynamic state space $\Gamma$
is an open subset of $\mathbb{R}^m$, spanned by $\bm{x}$.
Hereafter,
we assume that all quantities are nondimensionalized with an appropriate scale.
Let $s$ be the thermodynamic entropy per site,
which is a function of $\bm{x}$; $s=s(\bm{x})$.
According to thermodynamics,
$s$ is concave and differentiable everywhere in $\Gamma$
\cite{Callen1985,Lieb1999}.
In particular, $s$ is {\em strictly} concave
except in first-order phase transition regions.
On the other hand, in first-order phase transition regions
$s$ becomes linear in a certain direction(s) and therefore {\em not} strictly concave \cite{Shimizu2021,Callen1985}.
This is the reason why the GCE cannot give a phase coexistence state \cite{Yoneta2019}.
We call $s$ and its derivatives
the `genuine thermodynamic quantities.'
In particular, the first derivatives are the thermodynamic forces.
We denote by
$\Pi_{(i)}$ the thermodynamic force conjugate to $X_{(i)}$, i.e.,
\begin{align}
\Pi_{(i)}(\bm{x}) \equiv \frac{\partial s}{\partial x_{(i)}}(\bm{x}).
\end{align}
Particularly,
\begin{align}
\beta(\bm{x}) = \Pi_{(0)}(\bm{x})
\end{align}
is the inverse temperature
of the equilibrium state specified by $\bm{x}$.
\subsection{Definition} \label{sec:ensemble_def}
We introduce a general class of ensembles
which give equilibrium values of the local observables
and the genuine thermodynamic quantities
in the thermodynamic limit.
Let $\eta$ be a polynomial,
independent of $N$,
with real coefficients in $m$ noncommutative indeterminates.
We impose the following conditions on $\eta$.
Firstly, we require
\begin{enumerate}[label={(\Alph*)}]
\setcounter{enumi}{0}
\item $\eta(\hat{\bm{x}}_N)$ is self-adjoint for all $N$. \label{cond:A}
\end{enumerate}
Since $\eta$ is defined for noncommutative indeterminates,
it is possible to substitute the commutative quantities
into the indeterminates as a special case.
Therefore, the polynomial $\eta$ defines a real function on $\mathbb{R}^m$ uniquely.
When there is no danger of confusion,
we will use the same symbol $\eta$ for the function.
We also require
\begin{enumerate}[label={(\Alph*)}]
\setcounter{enumi}{1}
\item $s-\eta$ has the unique maximum point $\bm{x}_\mathrm{max}^\eta$ in $\Gamma$.
\label{cond:B}
\end{enumerate}
Since $s$ is concave,
this condition is always satisfied
when $\eta$ is taken as a strictly convex function.
However, this does {\em not} mean that $\eta$
should always be taken as strictly convex,
as will be exemplified in Eq.~(\ref{eq:eta_gc}).
Using $\eta$ that meets the above conditions,
we define the {\em squeezed ensemble (SE) associated with $\eta$} by
\begin{align}
\hat{\rho}_N^\eta &\equiv \frac{\displaystyle e^{-N\eta(\hat{\bm{x}}_N)}}{\displaystyle \mathrm{Tr}\left[e^{-N\eta(\hat{\bm{x}}_N)}\right]},\\
\psi_N^\eta &\equiv - \frac{1}{N} \log \mathrm{Tr} \left[ e^{-N\eta(\hat{\bm{x}}_N)} \right].
\end{align}
The condition~\ref{cond:A} ensures
that $\hat{\rho}_N^\eta$ is a density matrix.
Note that simultaneous eigenstates of $\hat{\bm{X}}_N$
do not appear in the definition of the SE.
Therefore,
the SE is well-defined
even in the case where $\hat{\bm{X}}_N$ cannot be diagonalized simultaneously.
Various ensembles are included in the class of SEs.
For instance,
$s$ is {\em strictly} concave
except in first-order phase transition region,
as mentioned in Section~\ref{sec:setup},
and therefore we can take $\eta$ as a linear function,
\begin{align}
\eta(\bm{x}) = \mathrm{gc}(\bm{x}) \equiv \sum_i \pi_{(i)} x_{(i)}. \label{eq:eta_gc}
\end{align}
Here, $\bm{\pi}$ is a tuple of positive constants
and is equal to $\bm{\Pi}$,
as will be shown later by Eq.~\eqref{eq:tdforce-formula_GCE}).
The SE associated with this $\eta$ is just the grand canonical ensemble (GCE),
whose density matrix $\hat{\rho}_N^\mathrm{gc}$ is
the grand canonical Gibbs state.
For example,
when $m=2$ and $X_{(1)}$ is the total magnetization,
the GCE describes the equilibrium state
with the inverse temperature $\pi_{(0)}$
and the thermodynamic magnetic field $-\pi_{(1)}/\pi_{(0)}$.
\subsection{Properties of Density Matrix of the SE}
At the first-order phase transition point,
the GCE gives a statistical mixture of macroscopically distinct states
in many cases \cite{Penrose1971,Binder1980,Yoneta2019},
as shown in examples in Section~\ref{sec:d2Ising}.
The condition~\ref{cond:B} makes the SEs free from such deficiency.
In fact, as shown in Appendix~\ref{sec:derivation},
the SE gives the state
specified by $\bm{x}_\mathrm{max}^\eta$
in the thermodynamic limit:
\begin{align}
x_{N(i)}^\eta \equiv \mathrm{Tr} \left[ \hat{x}_{N(i)} \hat{\rho}_N^\eta \right] \to \bm{x}_{\mathrm{max}(i)}^\eta. \label{eq:x-N-eta}
\end{align}
In addition, as discussed in Appendix~\ref{sec:derivation},
the condition~\ref{cond:B} ensures that
$\hat{\rho}_N^\eta$ has the vanishingly small variance of $\hat{x}_{N(i)}$
even in the first-order phase transition region, i.e.,
\begin{align}
\lim_{N\to\infty} \mathrm{Tr} \left[ \left( \hat{x}_{N(i)} - x_{N(i)}^\eta \right)^2 \hat{\rho}_N^\eta \right] = 0.
\end{align}
This is in contrast to the case of the grand canonical Gibbs state.
Moreover, $\hat{\rho}_N^\eta$ can be characterized by the principle of equal probability.
Under the constraint of
\begin{align}
\mathrm{Tr} \left[ \eta(\hat{\bm{x}}_N) \hat{\rho} \right]
= \mathrm{Tr} \left[ \eta(\hat{\bm{x}}_N) \hat{\rho}_N^\eta \right],
\end{align}
we can show that the
density matrix $\hat{\rho}$ that maximizes the von Neumann entropy,
\begin{align}
S^\mathrm{vN}(\hat{\rho}) \equiv - \mathrm{Tr} \left[ \hat{\rho} \log \hat{\rho} \right],
\end{align}
exists uniquely and is equal to $\hat{\rho}_N^\eta$.
To summarize, $\hat{\rho}_N^\eta$ gives the properties of a typical state
in which $\hat{\bm{x}}$ has a macroscopically definite value.
Therefore, by using the SE,
one can investigate microscopic structures of any equilibrium state in $\Gamma$,
including the first-order phase transition region.
It is worth mentioning that,
in general, $\hat{\rho}_N^\eta$ is not strictly invariant
under the time evolution described by the Hamiltonian which keeps the GCE invariant.
At first glance, this might seem to imply
that $\hat{\rho}_N^\eta$ would describe a nonequilibrium state.
However, as shown in Appendix~\ref{sec:invariance},
we can show under reasonable assumptions
that the SE is invariant in the thermodynamic limit.
\subsection{Genuine thermodynamic quantities} \label{sec:gtdq}
We can also obtain genuine thermodynamic quantities,
such as the entropy and the thermodynamic force, easily from the SE.
In fact,
as shown in Appendix~\ref{sec:derivation}-\ref{sec:derivation_tdforce},
we obtain
\begin{align}
s_N^\eta
&\equiv \eta\left(\bm{x}_N^\eta\right) - \psi_N^{\eta}
\to s(\bm{x}^\eta), \label{eq:entropy-formula}\\
\Pi_{N(i)}^\eta
&\equiv \frac{\partial\eta}{\partial x_{(i)}}\left(\bm{x}_N^\eta\right)
\to \Pi_{(i)}(\bm{x}^\eta) \label{eq:tdforce-formula}
\end{align}
in the thermodynamic limit,
where
\begin{align}
x_{(i)}^\eta \equiv \lim_{N\to\infty} x_{N(i)}^\eta.
\end{align}
Since higher-order moments of $\hat{\bm{x}}_N$ vanish
as shown in Appendix~\ref{sec:derivation}, we have
\begin{align}
\mathrm{Tr} \left[ \eta(\hat{\bm{x}}_N) \hat{\rho}_N^\eta \right]
= \eta(\hat{\bm{x}}_N^\eta) + o(N^0).
\end{align}
Therefore, we can rephrase Eq.~\eqref{eq:entropy-formula} as
\begin{align}
S^\mathrm{vN}(\hat{\rho}_N^\eta)/N \to s(\bm{x}^\eta).
\end{align}
Using Eq.~\eqref{eq:tdforce-formula},
one can obtain the thermodynamic forces of the equilibrium state
specified by $\bm{x}^\eta$
just by calculating $\bm{x}^\eta$
because $\eta$ is a known function.
This is a great advantage of the SE over the MCE
because, even in the case
where $\hat{\bm{X}}_N$ can be diagonalized simultaneously,
in order to calculate the thermodynamic force using the MCE,
one needs to differentiate the entropy,
and it gives very noisy results in numerical calculation.
\section{Free Spin} \label{sec:freeSpin}
Before studying a system which undergoes a first-order phase transition,
we confirm the validity of our formulation by applying it to the free spins:
\begin{align}
\hat{H}_N &= \hat{X}_{N(0)} = 0,\\
\hat{X}_{N(1)} &= \sum_i \hat{\sigma}_i^x,\\
\hat{X}_{N(2)} &= \sum_i \hat{\sigma}_i^y.
\end{align}
Note that $\hat{X}_{N(1)}$ and $\hat{X}_{N(2)}$ do not commute,
so they cannot be diagonalized simultaneously.
We choose the parameter space of $\bm{\kappa}$ as $K\equiv\mathbb{R}^2$,
and $\eta$ as
\begin{align}
\eta(\bm{\kappa};\bm{x}) = \kappa_{(1)} x_{(1)} + \frac{1}{2} \lambda (x_{(2)}-\kappa_{(2)})^2, \label{eq:freeSpin_eta}
\end{align}
where $\lambda$ is a positive constant.
Since this $\eta$ satisfies all conditions~\ref{cond:A}-\ref{cond:D},
this is an SE.
On the other hand,
the GCE for this system is obtained
by taking $\eta$ as Eq.~\eqref{eq:GCE_gc}, i.e.,
\begin{align}
\mathrm{gc}(\bm{\pi};\bm{x}) &= \pi_{(0)} x_{(0)} + \pi_{(1)} x_{(1)} + \pi_{(2)} x_{(2)}.
\end{align}
Since this simple model does not undergo the first-order phase transition,
this $\eta$ for the GCE also satisfies conditions~\ref{cond:A}-\ref{cond:D}.
Therefore, the SE associated with $\eta$ given by Eq.~\eqref{eq:freeSpin_eta}
should be equivalent, in the thermodynamic limit, to the GCE
not only for the thermodynamic function
but also for the expectation values of the local observables.
We will confirm this numerically.
For the SE the statistical-mechanical quantities are calculated
using the thermal pure quantum formulation \cite{Sugiura2012,Sugiura2013},
whereas those for the GCE can easily be calculated analytically.
\subsection{Local Observable}
First, we confirm that
the density matrix $\hat{\rho}_N^\eta$
describes the equilibrium state
which has the thermodynamic forces given by Eq.~\eqref{eq:tdforce-formula}.
Since this model does not undergo the first-order phase transition,
the equilibrium state is uniquely specified
by the tuple of thermodynamic forces $\bm{\Pi}$.
Therefore, in order for the SE and the GCE
to describe the same equilibrium state
in the thermodynamic limit,
we choose the parameters $\bm{\pi}$ and $\bm{\kappa}$ so that
\begin{align}
\lim_{N\to\infty} \bm{\Pi}_N^\mathrm{gc}(\bm{\pi}) = \lim_{N\to\infty} \bm{\Pi}_N^\eta(\bm{\kappa}).
\end{align}
Now we fix $\bm{\kappa}$
and calculate statistical-mechanical quantities in the SE,
including $\bm{\Pi}_N^\eta(\bm{\kappa})$.
Then we set $\bm{\pi}$ in such a way
that $\bm{\pi}=\bm{\Pi}_N^\eta(\bm{\kappa})$.
In this situation, indeed, we have
\begin{align}
\bm{\Pi}_N^\mathrm{gc}(\bm{\pi}=\bm{\Pi}_N^\eta(\bm{\kappa}))
= \bm{\Pi}_N^\eta(\bm{\kappa})
\end{align}
from Eq.~\eqref{eq:tdforce-formula_GCE}.
To confirm the equivalence for the expectation values of the local observables,
we have plotted in Fig.~\ref{fig:freeSpin_N-z} the $N$ dependence of
\begin{align}
\left| x_{N(2)}^\eta(\bm{\kappa})
- x_{N(2)}^\mathrm{gc}(\bm{\Pi}_N^\eta(\bm{\kappa})) \right|.
\end{align}
We confirm that the difference decreases with increasing $N$,
proportionally to $N^{-1}$.
This is consistent with the results obtained by Laplace's method (see Appendix~\ref{sec:derivation_Laplace}).
\begin{figure}
\centering
\includegraphics[keepaspectratio, width=\linewidth]{freeSpin_N-z.pdf}
\caption{$N$ dependence of the difference between $x_{N(2)}^\eta$ and $x_{N(2)}^\mathrm{gc}(\bm{\Pi}_N^\eta)$ for $\bm{\kappa}=(1,1)$.}
\label{fig:freeSpin_N-z}
\end{figure}
\subsection{Thermodynamic function}
Next, we confirm that the thermodynamic function $\psi^\eta$
is equivalent to the thermodynamic entropy via Eqs.~\eqref{eq:s-psi}-\eqref{eq:psi-s}.
Since the system under consideration has short-range interactions,
the grand canonical thermodynamic function $\psi^\mathrm{gc}$
is equivalent to the thermodynamic entropy:
\begin{align}
s(\bm{x}) &= \inf_{\bm{\pi} \in \mathbb{R}^3} \left\{ \mathrm{gc}(\bm{\pi};\bm{x}) - \psi^\mathrm{gc}(\bm{\pi}) \right\}.
\label{eq:s-psigc}
\end{align}
On the other hand,
according to Eq.~\eqref{eq:s-psi},
the thermodynamic function $\psi^\eta$ is equivalent to this $s$:
\begin{align}
\psi^\eta(\bm{\kappa}) = \inf_{\bm{x} \in \Gamma} \left\{ \eta(\bm{\kappa};\bm{x}) - s(\bm{x}) \right\}.
\end{align}
To confirm this equivalence for the thermodynamic function,
we have plotted in Fig.~\ref{fig:freeSpin_N-psi} the $N$ dependence of
\begin{align}
\left| \psi_N^\eta(\bm{\kappa}) - \inf_{\bm{x} \in \Gamma} \left\{ \eta(\bm{\kappa};\bm{x}) - s(\bm{x}) \right\} \right|,
\end{align}
where $s(\bm{x})$ is calculated from
the grand canonical thermodynamic function using Eq.~\eqref{eq:s-psigc}.
We confirm that the difference decreases with increasing $N$,
in almost proportion to $N^{-1}$.
This is consistent with the asymptotic behavior $O(N^{-1} \log N)$
obtained by Laplace's method (see Appendix~\ref{sec:derivation_Laplace}).
\begin{figure}
\centering
\includegraphics[keepaspectratio, width=\linewidth]{freeSpin_N-psi.pdf}
\caption{$N$ dependence of the difference between $\psi_N^\eta$ and the generalized Legendre-Fenchel transformation (Eq.~\eqref{eq:s-psi}) of the thermodynamic entropy $s$ for $\bm{\kappa}=(1,1)$.}
\label{fig:freeSpin_N-psi}
\end{figure}
\section{Introduction}
The statistical ensemble is a fundamental concept in equilibrium statistical mechanics \cite{Gibbs1902,Landau1980,Toda1983},
and various types of ensembles have been devised
\cite{Hetherington1987,Challa1988_PRL,Challa1988_PRA,Johal2003,Ray1991,Gerling1993,Tsallis1988,Beck2003,Cohen2004,Costeniuc2005,Costeniuc2006,Toral2006,Penrose1971,Ellis2000,Touchette2010,Yoneta2019}.
The thermodynamic functions of the macroscopic systems with short-range interactions are independent of the ensemble used in the calculation \cite{Ruelle1999}.
This fact is called the equivalence of ensembles.
Therefore, to obtain thermodynamic functions of the macroscopic system,
one can employ any ensemble.
However, to investigate other properties, such as microscopic structures of the phase interfaces or thermodynamics of the systems
with finite size effects,
one must choose an appropriate ensemble.
The appropriate choice of ensemble is particularly important
in the first-order phase transition region,
where several phases can coexist in various proportions \cite{Gibbs1876,Gibbs1878}.
Such an equilibrium state is called a `phase coexistence state.'
The coexisting phases in a phase coexistence state
cannot be distinguished by any thermodynamic forces,
such as temperature,
because every thermodynamic force
takes the same value between the coexisting phases.
For this reason, the `canonical' ensembles
that include thermodynamic forces
as parameters have some limitations
in the first-order phase transition regions \cite{Penrose1971,Binder1980}.
For example,
when the periodic boundary conditions are imposed,
the grand canonical ensemble (GCE)
gives either a single phase state or a statistical mixture of single phases,
and it cannot give a phase coexistence state
(see \cite{Yoneta2019};
we will also give examples in Sec.~\ref{sec:d2Ising}).
This problem of the GCE can be solved for some specific models by imposing clever boundary conditions \cite{Dobrushin1973,vanBeijeren1975}.
However, for general systems such as systems with a disordered phase,
it is not clear whether such boundary conditions exist
and what boundary conditions should be imposed.
The coexisting phases in a phase coexistence state can be distinguished
not by a thermodynamic force but
by a proper additive quantity,
called the `order parameter' (in a broad sense).
Therefore, the microcanonical ensemble (MCE)
can give phase coexistence states for general systems
because it is specified by a set of additive quantities
including the order parameter.
This makes it possible to investigate microscopic structures,
such as those of phase interfaces,
of the phase coexistence state \cite{Harris1985}.
The characteristics of different ensembles are also pronounced
when investigating finite systems
because the equivalence of ensembles does not hold for finite systems \cite{Harris1985,Stump1987,Gross2001}.
For example,
in finite systems which undergo first-order phase transitions
(in the thermodynamic limit)
the concavity of the Boltzmann entropy is broken generally,
even with short-range interactions (see Appendix~C of Ref.~\cite{Yoneta2019}).
The concavity breaking leads to thermodynamic anomalies
such as negative specific heat \cite{Bixon1989,Labastie1990,Gross1990,Gross1997,Litz1992,Nielsen1994,Reyes-Nava2003},
which was indeed observed experimentally \cite{DAgostino2000,Schmidt2001,Gobet2002}.
Nevertheless,
the GCE fails to give such thermodynamic anomalies of finite systems,
while the MCE correctly gives them directly and quantitatively \cite{Harris1985,Stump1987,Gross2001,Junghans2006,Junghans2008,Chen2009,Penrose1971,Binder1980,Troster2012}.
Unfortunately, however,
the MCE has a fundamental problem in quantum systems
when the additive observables
that specify the MCE do not commute with each other.
Since such observables cannot be diagonalized simultaneously,
it is generally impossible to construct the MCE
using their simultaneous eigenstates.
The same problem is also present in the restricted ensemble,
which is obtained by projecting the GCE onto the subspace spanned by the eigenstates of an additive observable
\cite{Forster1975,Goldenfeld1992,Penrose1971,Ellis2000}.
Such a projection often generates
superposition of macroscopically distinct states \cite{Tatsuta2018},
which cannot be a thermal equilibrium state.
To resolve the problem of noncommutative additive observables,
von Neumann proposed to construct the MCE using a commutative set of observables
that approximate the original set of additive observables \cite{Neumann1929}.
However, such a set of commutative observables are hard to construct,
and their explicit forms are obtained
only in limited cases \cite{Davidson1985,Lin1997,Ogata2013,Hastings2009}.
That is, so far, no method has been established
for constructing the MCE for general quantum systems.
In this paper, we propose a new class of ensembles
which are free from the above problems of the conventional ensembles,
by generalizing the squeezed ensemble (SE) \cite{Yoneta2019}.
The SE was introduced as a class of ensembles
with macroscopically well-defined energy.
We here generalize it to the case
where the equilibrium state is specified by several additive quantities
which can be noncommutative.
We derive various formulas for this novel class of ensembles,
including the one by which thermodynamic forces are obtained directly
from the expectation values of additive observables without knowing entropy.
Unlike the MCE,
the generalized SE does not require simultaneous eigenstates in its definition.
Hence, it is well-defined and can be constructed in any system.
Therefore,
the generalized SE can give phase coexistence states for general systems
(and thermodynamic anomalies when the systems are finite).
Moreover, the generalized SE is convenient for practical calculations
because of good analytic properties.
It can be numerically constructed more easily than the MCE
even when the additive observables commute with each other.
Furthermore, efficient numerical methods
are applicable in almost the same manner as in the GCE.
The validity of the SEs are shown under reasonable assumptions,
and confirmed by numerical calculations.
As an illustration,
we apply the generalized SE to the two-dimensional transverse field Ising model
at finite temperature.
In this model,
coexisting phases are distinguished
by the order parameter
which does not commute with the Hamiltonian.
Phase coexistence states
of such a quantum system at finite temperature are obtained
for the first time (to the author's knowledge).
We conclude that our formulation provides a concrete method
to construct phase coexistence states of general quantum systems.
\section{Time invariance of the SE} \label{sec:invariance}
In general, the density matrix given by the SE
is not invariant
even under the time evolution described by the Hamiltonian
which keeps the GCE (describing the same equilibrium state) invariant.
We here investigate the time evolution of the SE
and show that the SE is invariant
in the thermodynamic limit.
We consider a quantum spin system
on the $\nu$-dimensional hypercubic lattice $\Lambda$
with side length $L$.
We assume that the additive observable $\hat{X}_{N(i)}$ is expressed as
\begin{align}
\hat{X}_{N(i)} = \sum_j \gamma_j(\hat{o}_{(i)}),
\end{align}
where $\gamma_j$ is the $j$-lattice translation
and $\hat{o}_{(i)}$ is an $N$-independent local observable.
We consider the time evolution of the SE
under the mechanical forces
which keep the grand canonical Gibbs state
(with the same thermodynamic forces as the SE)
invariant.
Such a dynamics is generated by the Hamiltonian
\begin{align}
\hat{G}_N \equiv \hat{H}_N + \frac{\Pi_{(1)}^\eta}{\Pi_{(0)}^\eta} \hat{X}_{N(1)} + \cdots + \frac{\Pi_{(m-1)}^\eta}{\Pi_{(0)}^\eta} \hat{X}_{N(m-1)},
\end{align}
where
\begin{align}
\bm{\Pi}^\eta \equiv \lim_{N\to\infty}\bm{\Pi}_N^\eta.
\end{align}
Note that $\hat{G}_N= N \mathrm{gc}\left(\bm{\pi}=\bm{\Pi}^\eta;\hat{\bm{x}}_N\right) / \Pi_{(0)}^\eta$.
Then we rescale the time variable and the Hamiltonian as
\begin{align}
\tau &\equiv t / \Pi_{(0)}^\eta,\\
\hat{\Gamma}_N &\equiv \Pi_{(0)}^\eta \hat{G}_N = N \mathrm{gc}\left(\bm{\pi}=\bm{\Pi}^\eta;\hat{\bm{x}}_N\right).
\end{align}
We define the time-dependence of operators by the Heisenberg picture:
\begin{align}
\hat{A}(\tau) &\equiv e^{+ i \hat{\Gamma}_N \tau} \hat{A} e^{- i \hat{\Gamma}_N \tau}.
\end{align}
In our arguments, the following lemma plays a key role.
\begin{lemma}[Lieb-Robinson Bound \cite{Lieb1972,Nachtergaele2006,Nachtergaele2010}] \label{lemma:Lieb-Robinson}
There exist positive constants $C$, $\mu$ and $v$,
such that for any observables $\hat{A}$ and $\hat{B}$
with finite supports $U \subset \Lambda$ and $V \subset \Lambda$, respectively,
and for any $\tau \in \mathbb{R}$,
\begin{align}
\left\|\left[\hat{A}(\tau), \hat{B}\right]\right\|
&\leq C \left\|\hat{A}\right\| \left\|\hat{B}\right\|
\min \left[\left|U\right|,\left|V\right|\right]
e^{-\mu\left(d(U,V)-v|\tau|\right)},
\end{align}
where $d(x,y)$ is the distance
which is defined to be the shortest path length
that one needs to connect $x$ to $y$.
\end{lemma}
\begin{proof}
It follows from Eq.~(2.15) in \cite{Nachtergaele2010}.
\end{proof}
The time invariance of the SE is now stated as follows.
\begin{theorem} \label{theorem:invariance}
If the following conditions are fullfiled:
\begin{enumerate}[label={(\roman*)}]
\item for any $i$, the variance of $\hat{x}_{N(i)}$ in the SE is $O(N^{-1})$
\begin{align}
\mathrm{Tr} \left[ \left( \hat{x}_{N(i)} - x_{N(i)}^\eta \right)^2 \hat{\rho}_N^\eta \right] = O(N^{-1}),
\end{align}
\label{cond:invariance_variance}
\item for any $i$, the finite size effects on $x_{N(i)}^\eta$ is $O(N^{-1/2})$
\begin{align}
x_{N(i)}^\eta = x_{(i)}^\eta + O(N^{-1/2}),
\end{align}
\label{cond:invariance_finite-size-effect}
\end{enumerate}
then
\begin{align}
\frac{d}{d\tau} \mathrm{Tr} \left[ \hat{A}(\tau) \hat{\rho}_N^\eta \right]
&= \begin{cases}
\displaystyle O \left( \left(L^{-1/2} \log L \right)^\nu\right) & (v|\tau| \lesssim \log L)\\ \\
\displaystyle O \left( \left(L^{-1/2} |\tau| \right)^\nu\right) & (\log L \ll v|\tau| \ll L)
\end{cases}
\label{eq:invariance_hat-tilde}
\end{align}
for any observables $\hat{A}$ with finite supports $U \subset \Lambda$.
\end{theorem}
Assumptions~\ref{cond:invariance_variance}-\ref{cond:invariance_finite-size-effect} is implied by the analysis using Laplace's method
when $s$ is twice continuously differentiable
and is also confirmed numerically
(see Section~\ref{sec:freeSpin}-\ref{sec:d2Ising}).
\begin{proof}
First, we construct a local observable which approximates $\hat{A}(\tau)$.
Take a real number $l$ such that $1 \ll l \ll L$.
Let $\Lambda^\mathrm{in} \equiv \left\{ j \in \Lambda \mid d(U,j)<l \right\}$,
then $\left|\partial \Lambda^\mathrm{in}\right|=O\left(l^{\nu-1}\right)$.
Since $\hat{\Gamma}_N$ is additive,
there exist observables $\hat{\Gamma}_N^\mathrm{in}$, $\hat{\Gamma}_N^\mathrm{out}$ and $\hat{\Gamma}_N^\mathrm{int}$
such that
$\mathrm{supp\ } \hat{\Gamma}_N^\mathrm{in} = \Lambda^\mathrm{in}$,
$\mathrm{supp\ } \hat{\Gamma}_N^\mathrm{out} = \Lambda \setminus \Lambda^\mathrm{in}$,
$\left\| \hat{\Gamma}_N^\mathrm{int} \right\|=O\left(l^{\nu-1}\right)$ and
$\hat{\Gamma}_N = \hat{\Gamma}_N^\mathrm{in}+\hat{\Gamma}_N^\mathrm{out}+\hat{\Gamma}_N^\mathrm{int}$.
Now, we defined an observable on $\Lambda^\mathrm{in}$ as
\begin{align}
\tilde{A}(\tau)
&\equiv e^{+ i (\hat{\Gamma}_N-\hat{\Gamma}_N^\mathrm{int}) \tau} \hat{A} e^{- i (\hat{\Gamma}_N-\hat{\Gamma}_N^\mathrm{int}) \tau}
= e^{+ i \hat{\Gamma}_N^\mathrm{in} \tau} \hat{A} e^{- i \hat{\Gamma}_N^\mathrm{in} \tau}.
\end{align}
Applying Lemma~\ref{lemma:Lieb-Robinson}, we have
\begin{align}
&\left\|\hat{A}(\tau) - \tilde{A}(\tau)\right\|\\
&= \left\|
e^{+i \hat{\Gamma}_N \tau} \hat{A} e^{-i \hat{\Gamma}_N \tau}
- e^{+i \left(\hat{\Gamma}_N-\hat{\Gamma}_N^\mathrm{int}\right) \tau} \hat{A} e^{-i \left(\hat{\Gamma}_N-\hat{\Gamma}_N^\mathrm{int}\right) \tau}
\right\|\\
&= \left\|\int_{0}^{\tau} \frac{d}{d\sigma} \left(
e^{+i \left(\hat{\Gamma}_N-\hat{\Gamma}_N^\mathrm{int}\right) (\tau-\sigma)}
e^{+i \hat{\Gamma}_N \sigma}
\hat{A}
e^{-i \hat{\Gamma}_N \sigma}
e^{-i \left(\hat{\Gamma}_N-\hat{\Gamma}_N^\mathrm{int}\right) (\tau-\sigma)}
\right) d\sigma \right\|\\
&= \left\|\int_{0}^{\tau}
e^{+i \left(\hat{\Gamma}_N-\hat{\Gamma}_N^\mathrm{int}\right) \sigma}
\left[
e^{+i \hat{\Gamma}_N \sigma}
\hat{A}
e^{-i \hat{\Gamma}_N \sigma},
\hat{\Gamma}_N^\mathrm{int}
\right]
e^{-i \left(\hat{\Gamma}_N-\hat{\Gamma}_N^\mathrm{int}\right) \sigma}
d\sigma \right\|\\
&\leq \int_{0}^{|\tau|} \left\|\left[ \hat{A}(\sigma),\hat{\Gamma}_N^\mathrm{int} \right]\right\| d\sigma\\
&\leq \frac{C}{\mu v} \left\|\hat{A}\right\| \left\|\hat{\Gamma}_N^\mathrm{int}\right\| \left|U\right| e^{-\mu\left(l-v|\tau|\right)}\\
&= O\left( l^{\nu-1} e^{-\mu\left(l-v|\tau|\right)} \right).
\label{eq:invariance}
\end{align}
Thus, we see that $\tilde{A}(\tau)$ is an observable on $\Lambda^\mathrm{in}$
which approximates $\hat{A}(\tau)$ with an error of $O\left( l^{\nu-1} e^{-\mu\left(l-v|\tau|\right)} \right)$.
Since $\eta$ is a polynomial,
expanding in power series around $\bm{x}^\eta$,
$\eta(\hat{\bm{x}}_N)$ can be written as a finite sum as
\begin{align}
\eta(\hat{\bm{x}}_N)
= \mathrm{gc}\left(\bm{\pi}=\bm{\Pi}^\eta;\hat{\bm{x}}_N\right)
+ \sum_{K,\{i_k\}} c_{i_1 \cdots i_K} \prod_{k=1}^{K} \left(\hat{x}_{N(i_k)}-x_{(i_k)}^\eta\right)
+ \mathrm{const.}.
\end{align}
Note that $K$ is greater than or equal to $2$.
Substituting this into
\begin{align}
\frac{d}{d\tau} \mathrm{Tr} \left[ \hat{A}(\tau) \hat{\rho}_N^\eta \right]
&= - i \mathrm{Tr} \left[ \left[ \hat{A}(\tau), \hat{\Gamma}_N \right] \hat{\rho}_N^\eta \right],
\end{align}
we have
\begin{align}
\frac{d}{d\tau} \mathrm{Tr} \left[ \hat{A}(\tau) \hat{\rho}_N^\eta \right]
&= \sum_{K,\{i_k\}} i c_{i_1 \cdots i_K} \underbrace{N \mathrm{Tr} \left[ \left[ \hat{A}(\tau), \prod_{k=1}^{K} \left(\hat{x}_{N(i_k)}-x_{(i_k)}^\eta\right) \right] \hat{\rho}_N^\eta \right]}_{\equiv (\ast)}.
\end{align}
Decomposing $\hat{A}(\tau)$ into
$\tilde{A}(\tau) + (\hat{A}(\tau)-\tilde{A}(\tau))$,
we have
\begin{align}
(\ast) &= \underbrace{\mathrm{Tr} \left[ \left[ \tilde{A}(\tau), \prod_{k=1}^{K} \left(\hat{x}_{N(i_k)}-x_{(i_k)}^\eta\right) \right] \hat{\rho}_N^\eta \right]}_{\equiv (\ast 1)}
+ \underbrace{\mathrm{Tr} \left[ \left[ \hat{A}(\tau)-\tilde{A}(\tau), \prod_{k=1}^{K} \left(\hat{x}_{N(i_k)}-x_{(i_k)}^\eta\right) \right] \hat{\rho}_N^\eta \right]}_{\equiv (\ast 2)}.
\end{align}
Then
\begin{align}
(\ast 1)
&= \sum_{k'=1}^K \mathrm{Tr} \left[
\prod_{k=1}^{k'-1} \left(\hat{x}_{N(i_k)}-x_{(i_k)}^\eta\right)
\hat{q}_{N(i_{k'})}
\prod_{k=k'+1}^K \left(\hat{x}_{N(i_k)}-x_{(i_k)}^\eta\right)
\hat{\rho}_N^\eta \right],
\end{align}
where $\displaystyle \hat{q}_{N(i_{k})} \equiv \left[ \tilde{A}(\tau), \hat{X}_{N(i_{k})} \right]$.
Since $K \geq 2$, applying the Cauchy-Schwartz inequality
\begin{align}
\left| \mathrm{Tr} \left[ \hat{A} \hat{B}^\dagger\hat{\rho}_N^\eta \right] \right|^2
\leq \mathrm{Tr} \left[ \hat{A} \hat{A}^\dagger\hat{\rho}_N^\eta \right] \mathrm{Tr} \left[ \hat{B} \hat{B}^\dagger\hat{\rho}_N^\eta \right],
\end{align}
we have
\begin{align}
|(\ast1)|
&\leq \left| \mathrm{Tr} \left[
\hat{q}_{N(i_1)}
\prod_{k=2}^K \left(\hat{x}_{N(i_k)}-x_{(i_k)}^\eta\right)
\hat{\rho}_N^\eta \right] \right|
+ \cdots
+ \left| \mathrm{Tr} \left[
\prod_{k=1}^{K-1} \left(\hat{x}_{N(i_k)}-x_{(i_k)}^\eta\right)
\hat{q}_{N(i_K)}
\hat{\rho}_N^\eta \right] \right|\\
&\leq \sqrt{\mathrm{Tr} \left[
\hat{q}_{N(i_1)}
\prod_{k=2}^{K-1} \left(\hat{x}_{N(i_k)}-x_{(i_k)}^\eta\right)
\prod_{k=K-1}^{2} \left(\hat{x}_{N(i_k)}-x_{(i_k)}^\eta\right)
\hat{q}_{N(i_1)}^\dagger
\hat{\rho}_N^\eta \right]}
\sqrt{\mathrm{Tr} \left[
\left(\hat{x}_{N(i_K)}-x_{(i_K)}^\eta\right)^2
\hat{\rho}_N^\eta \right]}\\
&+ \cdots\\
&+ \sqrt{\mathrm{Tr} \left[
\left(\hat{x}_{N(i_1)}-x_{(i_1)}^\eta\right)^2
\hat{\rho}_N^\eta \right]}
\sqrt{\mathrm{Tr} \left[
\hat{q}_{N(i_K)}^\dagger
\prod_{k=K-1}^{2} \left(\hat{x}_{N(i_k)}-x_{(i_k)}^\eta\right)
\prod_{k=2}^{K-1} \left(\hat{x}_{N(i_k)}-x_{(i_k)}^\eta\right)
\hat{q}_{N(i_K)}
\hat{\rho}_N^\eta \right]}.
\end{align}
Usnig assumption~\ref{cond:invariance_variance}-\ref{cond:invariance_finite-size-effect},
we have
\begin{align}
\mathrm{Tr} \left[
\left(\hat{x}_{N(i)}-x_{(i)}^\eta\right)^2
\hat{\rho}_N^\eta \right]
= \mathrm{Tr} \left[
\left(\hat{x}_{N(i)}-x_{N(i)}^\eta\right)^2
\hat{\rho}_N^\eta \right]
+ \left(x_{N(i)}^\eta-x_{(i)}^\eta\right)^2
= O(N^{-1}).
\end{align}
In addition, $\displaystyle \left\|\hat{q}_{N(i)}\right\| = O(l^\nu)$ and $\displaystyle \left\|\hat{x}_{N(i)}-x_{(i)}^\eta\right\| = O(N^0)$.
Therefore, we obtain
\begin{align}
|(\ast1)|
&= O(N^{-1/2} l^\nu).
\end{align}
On the other hand, using H\"{o}lder's inequality and Eq.~\eqref{eq:invariance_hat-tilde}, we have
\begin{align}
|(\ast2)|
&\leq N \left\| \left[\hat{A}(\tau)-\tilde{A}(\tau), \prod_{k=1}^{K} \left(\hat{x}_{N(i_k)}-x_{(i_k)}^\eta\right)\right] \hat{\rho}_N^\eta \right\|_1\\
&\leq 2 N \left\| \hat{\rho}_N^\eta \right\|_1 \left\| \hat{A}(\tau)-\tilde{A}(\tau) \right\| \prod_{k=1}^{K} \left\| \hat{x}_{N(i_k)}-x_{(i_k)}^\eta \right\|\\
&= O\left(N l^{\nu-1} e^{-\mu\left(l-v|\tau|\right)}\right).
\end{align}
Therefore, we have
\begin{align}
\frac{d}{d\tau} \mathrm{Tr} \left[ \hat{A}(\tau) \hat{\rho}_N^\eta \right]
&= O(N^{-1/2} l^\nu) + O\left(N l^{\nu-1} e^{-\mu\left(l-v|\tau|\right)}\right).
\end{align}
Then, letting $l = v|\tau| + \frac{2\nu}{\mu} \log L$,
we obtain Eq.~\eqref{eq:invariance}.
\end{proof}
\section{Noncommutative $\hat{\bm{X}}_N$} \label{sec:noncommutative}
When $\hat{\bm{X}}_N$ commute with each other,
they can be diagonalized simultaneously
and analytic techniques such as Laplace's method can be applied.
Then it can be shown that the SE gives a correct equilibrium state under the conditions~\ref{cond:A}-\ref{cond:B}.
Furthermore, useful formulas for genuine thermodynamic quantities are derived.
On the other hand, when $\hat{\bm{X}}_N$ do not commute with each other,
such powerful techniques are not applicable.
This is a serious obstacle in the formulation of the statistical mechanics.
To resolve this problem,
it was proposed to use a commutative set of observables $\check{\bm{X}}_N$
that approximate the original set of additive observables $\hat{\bm{X}}_N$.
Although such a set of commutative observables are hard to construct,
it is possible to perform the analysis formally
assuming the existence of $\check{\bm{X}}_N$.
We shall show below that the statistical-mechanical properties of the SE are identical
for $\hat{\bm{X}}_N$ and for $\check{\bm{X}}_N$ in the thermodynamic limit.
Then to study the asymptotic behaviors of the expectation values and variances of $\hat{\bm{X}}_N$,
and the thermodynamic function,
we can perform the analysis by treating $\hat{\bm{X}}_N$
as if they were commutative.
This makes it possible to formulate statistical mechanics
for systems with noncommutative $\hat{\bm{X}}_N$ using the SE.
\subsection{Setup}
We consider a quantum spin system on the $\nu$-dimensional hypercubic lattice.
We assume that the additive observable $\hat{X}_{N(i)}$ is expressed as
\begin{align}
\hat{X}_{N(i)} = \sum_j \gamma_j(\hat{o}_{(i)}),
\end{align}
where $\gamma_j$ is the $j$-lattice translation
and $\hat{o}_{(i)}$ is an $N$-independent local observable.
Then there exists an $m$-tuple of observables $\bm{\check{X}}_N=(\check{X}_{N(0)},\check{X}_{N(1)},\cdots,\check{X}_{N(m-1)})$
such that \cite{Ogata2013}
\begin{align}
\lim_{N\to\infty} \left\| \hat{X}_{N(i)}-\check{X}_{N(i)} \right\| &= 0, \label{eq:hat-check}\\
\left[ \check{X}_{N(i)}, \check{X}_{N(j)} \right] &= 0.
\end{align}
The SE for $\dot{\bm{x}}_N\equiv\dot{\bm{X}}_N/N (\dot{\bullet}=\hat{\bullet},\check{\bullet})$
is defined as
\begin{align}
\psi_N^{\dot{\eta}}
&\equiv - \frac{1}{N} \log \mathrm{Tr} \left[ e^{-N\eta(\dot{\bm{x}}_N)} \right],\\
\rho_N^{\dot{\eta}}
&\equiv \frac{\displaystyle
e^{-N\eta(\dot{\bm{x}}_N)}
}{\displaystyle
\mathrm{Tr} \left[ e^{-N\eta(\dot{\bm{x}}_N)} \right]
},\\
\braket{\bullet}_N^{\dot{\eta}}
&\equiv \mathrm{Tr} \left[ \bullet \rho_N^{\dot{\eta}} \right].
\end{align}
\subsection{Thermodynamic function}
\begin{theorem} \label{theorem:psi}
If $\psi_N^{\hat{\eta}}$ and $\psi_N^{\check{\eta}}$ converge in the thermodynamic limit, then
\begin{align}
\lim_{N\to\infty} \psi_N^{\hat{\eta}}
= \lim_{N\to\infty} \psi_N^{\check{\eta}}.
\end{align}
\end{theorem}
This implies that, even when $\hat{\bm{X}}_N$ do not commute,
the thermodynamic function of the SE for $\hat{\bm{x}}_N$,
$\displaystyle\lim_{N\to\infty}\psi_N^{\hat{\eta}}$,
is equivalent to the grand canonical entropy for $\hat{\bm{x}}_N$,
$s^{\hat{\mathrm{GC}}}$,
via Eq.~\eqref{eq:s-psi}-\eqref{eq:psi-s}
(Fig.~\ref{fig:noncommutative_genLeg}).
Here, the grand canonical entropy for $\dot{\bm{x}}_N$,
$s^{\dot{\mathrm{GC}}}$,
is defined by the Legendre-Fenchel conjugate
of the grand canonical thermodynamic function for $\dot{\bm{x}}_N$,
$\displaystyle\lim_{N\to\infty}\psi_N^{\dot{\mathrm{gc}}}$.
\begin{figure}[H]
\centering
\begin{tikzpicture}
\node[draw] at (+3,+3) {$\hat{\bm{X}}_N$};
\node (hat_s) at (0,+3) {$s^{\hat{\mathrm{GC}}}$};
\node (hat_max) at (+2.598,+1.5) {$\displaystyle\lim_{N\to\infty}\psi_N^{\hat{\mathrm{gc}}}$};
\node (hat_eta) at (-2.598,+1.5) {$\displaystyle\lim_{N\to\infty}\psi_N^{\hat{\eta}}$};
\draw[dotted,thick] (-4,0) to (+4,0);
\node[draw] at (+3,-3) {$\check{\bm{X}}_N$};
\node (check_s) at (0,-3) {$s^{\check{\mathrm{GC}}}$};
\node (check_max) at (+2.598,-1.5) {$\displaystyle\lim_{N\to\infty}\psi_N^{\check{\mathrm{gc}}}$};
\node (check_eta) at (-2.598,-1.5) {$\displaystyle\lim_{N\to\infty}\psi_N^{\check{\eta}}$};
\draw[<->] (hat_s) to node[sloped,above=0.3ex] {\small Legendre} (hat_max);
\draw[<->,dotted,color=red] (hat_s) to node[sloped,above=0.3ex] {\small Eq.~\eqref{eq:s-psi}-\eqref{eq:psi-s}} (hat_eta);
\node[color=blue,rotate=90] at (+2.598,0) {\Huge $=$};
\node[color=blue,rotate=90] at (-2.598,0) {\Huge $=$};
\draw[<->] (check_s) to node[sloped,above=0.3ex] {\small Legendre} (check_max);
\draw[<->] (check_s) to node[sloped,above=0.3ex] {\small Eq.~\eqref{eq:s-psi}-\eqref{eq:psi-s}} (check_eta);
\draw[thick, <->] (0,1.5) arc (90:-210:1.5);
\end{tikzpicture}
\caption{Schematic diagram showing the equivalence
between the thermodynamic function of the SE for $\hat{\bm{x}}_N$
and the grand canonical entropy for $\hat{\bm{x}}_N$ (red dashed line).
By Theorem~\ref{theorem:psi} (right blue equal sign),
we have $\displaystyle\lim_{N\to\infty}\psi_N^{\hat{\mathrm{gc}}}=\lim_{N\to\infty}\psi_N^{\check{\mathrm{gc}}}$.
Therefore, the grand canonical entropies,
which are defined from the grand canonical thermodynamic functions,
are equal for $\hat{\bm{x}}_N$ and for $\check{\bm{x}}_N$, $s^{\hat{\mathrm{GC}}}=s^{\check{\mathrm{GC}}}$.
Since $\check{\bm{X}}_N$ commute with each other,
we can show the equivalence between $\displaystyle\lim_{N\to\infty}\psi_N^{\check{\eta}}$ and $s^{\check{\mathrm{GC}}}$ as in Appendix~\ref{sec:derivation}.
On the other hand, again by Theorem~\ref{theorem:psi} (left blue equal sign),
we have $\displaystyle\lim_{N\to\infty}\psi_N^{\hat{\eta}}=\lim_{N\to\infty}\psi_N^{\check{\eta}}$.
Therefore, $\displaystyle\lim_{N\to\infty}\psi_N^{\hat{\eta}}$ is equivalent to $s^{\hat{\mathrm{GC}}}$.}
\label{fig:noncommutative_genLeg}
\end{figure}
\begin{proof}[Proof of Theorem~\ref{theorem:psi}]
Using Eq.~\eqref{eq:hat-check} and $\|\hat{x}_{N(i)}\|\leq\|\hat{o}_{(i)}\|(<\infty)$,
we have
\begin{align}
\lim_{N\to\infty}\left\| \eta(\hat{\bm{x}}_N) - \eta(\check{\bm{x}}_N) \right\|=0. \label{eq:hat-check_eta}
\end{align}
That is, for any $\epsilon > 0$, there exists $N_0\in\mathbb{N}$ such that
$\left\| \eta(\hat{\bm{x}}_N) - \eta(\check{\bm{x}}_N) \right\|<\epsilon$ for any $N>N_0$.
Then it follows that
\begin{align}
- \eta(\check{\bm{x}}_N) - \epsilon < - \eta(\hat{\bm{x}}_N) < - \eta(\check{\bm{x}}_N) + \epsilon.
\end{align}
Therefore, by using Weyl's inequality \cite{Bhatia1997}, we have
\begin{align}
\mathrm{Tr} \left[ e^{- N \eta(\check{\bm{x}}_N) - N \epsilon} \right]
< \mathrm{Tr} \left[ e^{- N \eta(\hat{\bm{x}}_N)} \right]
< \mathrm{Tr} \left[ e^{- N \eta(\check{\bm{x}}_N) + N \epsilon} \right].
\end{align}
By definition of $\psi_N^{\dot{\eta}}$, this implies that
\begin{align}
\psi_N^{\check{\eta}} - \epsilon < \psi_N^{\hat{\eta}} < \psi_N^{\check{\eta}} + \epsilon.
\end{align}
\end{proof}
\subsection{Observable}
\begin{theorem} \label{theorem:exp}
Let $\theta$ be a polynomial with real coefficients
in $m$ noncommutative indeterminates.
We assume that $\theta$ is independent of $N$,
and that $\theta(\hat{\bm{x}}_N)$ is self-adjoint for all $N$.
If the following conditions are fulfilled:
\begin{enumerate}[label={(\roman*)}]
\item $\left\{\braket{\theta(\hat{\bm{x}}_N)}_N^{\hat{\eta}}\right\}_{N\in\mathbb{N}}$ and $\left\{\braket{\theta(\check{\bm{x}}_N)}_N^{\check{\eta}}\right\}_{N\in\mathbb{N}}$ converge in the thermodynamic limit,
\label{cond:exp_converge}
\item $\psi_N^{\check{\eta}}$ converges in the thermodynamic limit, \label{cond:psi_converge}
\item There exist a closed and bounded interval $I$ and $M>0$ such that
\begin{itemize}
\item $0 \in I$
\item for any $\lambda \in I$ and $N\in\mathbb{N}$,
the variance of $\theta(\check{\bm{x}}_N)$
in the SE associated with $\eta+\lambda\theta$ for $\check{\bm{x}}_N$ satisfies
\begin{align}
\mathrm{Tr} \left[ \left( \theta(\check{\bm{x}}_N) - \braket{\theta(\check{\bm{x}}_N)}_N^{\check{\eta},\check{\theta}}(\lambda) \right)^2 \rho_N^{\check{\eta},\check{\theta}}(\lambda) \right]
< \frac{M}{N},
\end{align}
where
\begin{align}
\rho_N^{\dot{\eta},\dot{\theta}}(\lambda)
&\equiv \frac{\displaystyle
e^{-N\left(\eta(\dot{\bm{x}}_N)+\lambda\theta(\dot{\bm{x}}_N)\right)}
}{\displaystyle
\mathrm{Tr} \left[ e^{-N\left(\eta(\dot{\bm{x}}_N)+\lambda\theta(\dot{\bm{x}}_N)\right)} \right]
},\\
\braket{\bullet}_N^{\dot{\eta},\dot{\theta}}(\lambda)
&\equiv \mathrm{Tr} \left[ \bullet \rho_N^{\dot{\eta},\dot{\theta}}(\lambda) \right],
\end{align}
\end{itemize}
\label{cond:variance}
\end{enumerate}
then we have
\begin{align}
\lim_{N\to\infty} \braket{\theta(\hat{\bm{x}}_N)}_N^{\hat{\eta}}
= \lim_{N\to\infty} \braket{\theta(\check{\bm{x}}_N)}_N^{\check{\eta}}.
\end{align}
\end{theorem}
For example, suppose $\theta(\bm{x})=(x_{(i)})^2$.
Let $I=[0,\delta]$ for some small positive constant $\delta$.
Then, for all $\lambda \in I$,
the convexity of $\eta+\lambda\theta$ is stronger than or equal to the convexity of $\eta$.
Therefore, as show in Appendix~\ref{sec:derivation},
the variance of $(\check{x}_{N(i)})^2$ in the SE associated with $\eta+\lambda\theta$ for $\check{\bm{x}}_N$
is $O(N^{-2})$.
Hence, assumption~\ref{cond:variance} is fulfilled.
By Theorem~\ref{theorem:exp}, we have
\begin{align}
\lim_{N\to\infty} \braket{(\hat{x}_{N(i)})^2}_N^{\hat{\eta}}
= \lim_{N\to\infty} \braket{(\check{x}_{N(i)})^2}_N^{\check{\eta}}.
\end{align}
That is,
the second order moment of $x_{(i)}$ in the SE for $\hat{\bm{x}}_N$
is equal to that in the SE for $\check{\bm{x}}_N$
in the thermodynamic limit.
In the same way,
it can be shown that the first order moment of $x_{(i)}$ in the SE for $\hat{\bm{x}}_N$
is equal to that in the SE for $\check{\bm{x}}_N$
in the thermodynamic limit.
To prove Theorem~\ref{theorem:exp}, we first prove the following lemma.
\begin{lemma} \label{lemma:AA_cor}
Let $\{f_n\}_{n \in \mathbb{N}}$ be a sequence of real-valued differentiable functions on $I$
such that $\{f_n(x_0)\}_{n \in \mathbb{N}}$ is bounded for some $x_0 \in I$
and $\{f_n'\}_{n \in \mathbb{N}}$ is uniformly bounded.
Then there exists a subsequence $\{f_{n_k}\}_{k \in \mathbb{N}}$ that converges uniformly on $I$.
\end{lemma}
\begin{proof}
By the Arzel\`{a}-Ascoli theorem,
it is sufficient to show
the uniform boundedness and equicontinuity
of $\{f_n\}_{n \in \mathbb{N}}$.
(uniform boundedness)
Uniform boundedness of $\{f_n'\}_{n \in \mathbb{N}}$ implies by the mean value theorem that for all $n \in \mathbb{N}$ and $x,y \in I$,
\begin{align}
\left| f_n(x)-f_n(y) \right| \leq M \left| x-y \right|, \label{eq:AA_Cor_MeanValueThm}
\end{align}
where $\displaystyle M \equiv \sup_{n \in \mathbb{N}, x \in I}|f_n'(x)|$.
In addition, since $\{f_n(x_0)\}_{n \in \mathbb{N}}$ is bounded,
we have, for all $n \in \mathbb{N}$ and $x \in I = [a,b]$,
\begin{align}
\left| f_n(x) \right| \leq F + M |b-a|,
\end{align}
where $\displaystyle F \equiv \sup_{n \in \mathbb{N}}|f_n(x_0)|$.
Therefore, $\{f_n\}_{n \in \mathbb{N}}$ is uniformly bounded.
(equicontinuity)
Given $\epsilon > 0$, let $\displaystyle \delta \equiv \frac{\epsilon}{M}$.
Then, using Eq.~(\ref{eq:AA_Cor_MeanValueThm}),
we have
\begin{align}
\left| x-y \right| < \delta
\Rightarrow \left| f_n(x)-f_n(y) \right| < \epsilon
\end{align}
for all $x, y \in I$.
Therefore, $\{f_n\}_{n \in \mathbb{N}}$ is equicontinuous.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theorem:exp}]
Let us introduce the generating functions
\begin{align}
\phi_N^{\dot{\eta},\dot{\theta}}(\lambda)
&\equiv - \frac{1}{N} \log \mathrm{Tr} \left[ e^{-N\left(\eta(\dot{\bm{x}}_N)+\lambda\theta(\dot{\bm{x}}_N)\right)} \right].
\end{align}
Taking the derivative of $\phi_N^{\dot{\eta},\dot{\theta}}$,
we obtain the expectation value of $\theta(\dot{\bm{x}}_N)$ in $\rho_N^{\dot{\eta},\dot{\theta}}(\lambda)$ as
\begin{align}
\frac{d\phi_N^{\dot{\eta},\dot{\theta}}}{d\lambda}(\lambda)
&= \braket{\theta(\dot{\bm{x}}_N)}_N^{\dot{\eta},\dot{\theta}}(\lambda).
\end{align}
In particular, at $\lambda=0$, it coincides with the expectation value in $\rho_N^{\dot{\eta}}$.
Since $\phi_N^{\check{\eta},\check{\theta}}(0) = \psi_N^{\check{\eta}}$,
using assumption~\ref{cond:psi_converge},
we observe that $\left\{ \phi_N^{\check{\eta},\check{\theta}} (0) \right\}_{N\in\mathbb{N}}$ is bounded.
In addition, since $\displaystyle\frac{d\phi_N^{\check{\eta},\check{\theta}}}{d\lambda}(\lambda)=\braket{\theta(\check{\bm{x}}_N)}_N^{\check{\eta},\check{\theta}}(\lambda)\leq\left\|\theta(\check{\bm{x}}_N)\right\|$,
$\displaystyle\left\{\frac{d\phi_N^{\check{\eta},\check{\theta}}}{d\lambda}\right\}_{N\in\mathbb{N}}$ is uniformly bounded on any closed and bounded interval.
Therefore, by Lemma~\ref{lemma:AA_cor},
there exists a subsequence $\displaystyle\left\{\phi_{N_k}^{\check{\eta},\check{\theta}}\right\}_{k\in\mathbb{N}}$
that converges (uniformly on compacts in $\mathbb{R}$).
For this limit, in the same way as in the proof of Theorem~\ref{theorem:psi},
it can be shown that
\begin{align}
\phi^{\eta,\theta}(\lambda)
\equiv \lim_{k\to\infty}\phi_{N_k}^{\hat{\eta},\hat{\theta}}(\lambda)
= \lim_{k\to\infty}\phi_{N_k}^{\check{\eta},\check{\theta}}(\lambda)
\end{align}
for all $\lambda \in \mathbb{R}$.
Since $\theta(\check{\bm{x}}_N)$ commutes with $\eta(\check{\bm{x}}_N)$,
we have
\begin{align}
\frac{d^2\phi_N^{\check{\eta},\check{\theta}}}{d\lambda^2}(\lambda)
&= - N \mathrm{Tr} \left[ \left( \theta(\check{\bm{x}}_N) - \braket{\theta(\check{\bm{x}}_N)}_N^{\check{\eta},\check{\theta}}(\lambda) \right)^2 \rho_N^{\check{\eta},\check{\theta}}(\lambda) \right].
\end{align}
Thus, using assumption~\ref{cond:variance},
we find that $\displaystyle\left\{\frac{d^2\phi_N^{\check{\eta},\check{\theta}}}{d\lambda^2}\right\}_{N\in\mathbb{N}}$ is uniformly bounded on $I$.
Hence, by Lemma~\ref{lemma:AA_cor},
there exists a subsequence $\displaystyle\left\{\frac{d\phi_{N_{k_l}}^{\check{\eta},\check{\theta}}}{d\lambda}\right\}_{l\in\mathbb{N}}$ that converges uniformly on $I$.
Therefore, $\phi^{\eta,\theta}$ is continuously differentiable on $I$
and its derivative is given by
\begin{align}
\frac{d\phi^{\eta,\theta}}{d\lambda}(\lambda)
= \lim_{l\to\infty}\frac{d\phi_{N_{k_l}}^{\check{\eta},\check{\theta}}}{d\lambda}(\lambda)
= \lim_{l\to\infty}\braket{\theta(\check{\bm{x}}_{N_{k_l}})}_{N_{k_l}}^{\check{\eta},\check{\theta}}(\lambda)
\end{align}
for all $\lambda \in I$ \cite{Rudin1976}.
Since $\phi_{N_k}^{\hat{\eta},\hat{\theta}}$ is concave,
using Griffiths's lemma\cite{Griffiths1964}, we obtain
\begin{align}
\frac{d\phi^{\eta,\theta}}{d\lambda}(\lambda)
= \lim_{k\to\infty}\frac{d\phi_{N_k}^{\hat{\eta},\hat{\theta}}}{d\lambda}(\lambda)
= \lim_{k\to\infty}\braket{\theta(\hat{\bm{x}}_{N_k})}_{N_k}^{\hat{\eta},\hat{\theta}}(\lambda).
\end{align}
for all $\lambda \in I$.
From the above, together with assumption~\ref{cond:exp_converge},
we have
\begin{align}
\lim_{N\to\infty}\braket{\theta(\check{\bm{x}}_N)}_N^{\check{\eta},\check{\theta}}(0)
= \lim_{l\to\infty}\braket{\theta(\check{\bm{x}}_{N_{k_l}})}_{N_{k_l}}^{\check{\eta},\check{\theta}}(0)
= \lim_{k\to\infty}\braket{\theta(\hat{\bm{x}}_{N_k})}_{N_k}^{\hat{\eta},\hat{\theta}}(0)
= \lim_{N\to\infty}\braket{\theta(\hat{\bm{x}}_N)}_N^{\hat{\eta},\hat{\theta}}(0).
\end{align}
\end{proof}
\section{Parameter of squeezed ensemble} \label{sec:parameter}
Since the equilibrium state described by the SE depends on $\eta$,
one can obtain equilibrium states of various $\bm{x}$
by taking various forms of $\eta$.
Hence, it is convenient for practical calculations
if $\eta$ depends on certain parameters.
Suppose that $\eta$ is parametrized on a parameter space $K$.
That is, each $\bm{\kappa} \in K$ corresponds to a polynomial $\eta(\bm{\kappa};\cdot)$
which satisfies conditions~\ref{cond:A}-\ref{cond:B}.
Then each $\bm{\kappa}$ corresponds to a SE
associated with $\eta(\bm{\kappa};\cdot)$.
Thus, a statistical-mechanical quantity in the SE can be regarded as a function of $\bm{\kappa}$.
\subsection{GCE}
As mentioned in Section~\ref{sec:ensemble_def},
the GCE can be regarded as a kind of the SE
except at phase transition points.
To distinguish the parameters of the GCE
from those of the SE associated with general $\eta$,
we use the notation $\bm{\pi} = (\pi_{(0)}, \cdots, \pi_{(m-1)})$
for the parameters $\bm{\kappa}$ of the GCE.
That is, $\eta$ for the GCE is denoted as
\begin{align}
\mathrm{gc}(\bm{\pi};\bm{x}) &= \sum_i \pi_{(i)} x_{(i)}. \label{eq:GCE_gc}
\end{align}
Then it is seen from Eq.~\eqref{eq:tdforce-formula} that
\begin{align}
\pi_{(i)} = \Pi_{(i)}(\bm{x}^\mathrm{gc}(\bm{\pi})), \label{eq:tdforce-formula_GCE}
\end{align}
as is expected.
In this particular case, the parameters of the SE coincide with thermodynamic quantities.
For general SEs, however, the parameters are not directly related to thermodynamic quantities.
Nevertheless, one can calculate thermodynamic quantities from the SEs
via Eqs.~\eqref{eq:s-psi} and \eqref{eq:psi-s} as shown below,
and via Eqs.~\eqref{eq:entropy-formula} and \eqref{eq:tdforce-formula}.
\subsection{Parameter dependence of the density matrix of the SE}
On the $\bm{\kappa}$ dependence of $\eta$,
we impose the following conditions~\ref{cond:C} and \ref{cond:D}.
Hence, together with the two conditions in Section~\ref{sec:ensemble_def},
four conditions~\ref{cond:A}-\ref{cond:D} are imposed on $\eta$.
The third condition is
\begin{enumerate}[label={(\Alph*)}]
\setcounter{enumi}{2}
\item $\eta$ is continuous as a function of $\bm{\kappa}$. \label{cond:C}
\end{enumerate}
Then, for the maximum point $\bm{x}_\mathrm{max}^\eta$ of $s-\eta$,
which is uniquely determined by
\begin{align}
\frac{\partial s}{\partial x_{(i)}}(\bm{x}_\mathrm{max}^\eta)
- \frac{\partial \eta}{\partial x_{(i)}}(\bm{\kappa};\bm{x}_\mathrm{max}^\eta)
= 0,
\end{align}
we can prove,
applying an implicit function theorem for continuous maps \cite{Kumagai1980},
that it moves continuously in $\bm{\kappa}$.
This should be contrasted with the GCE
at a first-order phase transition point.
In this case, since condition~\ref{cond:B} is not satisfied,
the GCE is not an SE and $\bm{x}_\mathrm{max}^\mathrm{gc}$ is not unique.
Then, before and after passing through the first-order phase transition point,
$\bm{x}_\mathrm{max}^\mathrm{gc}$ switches discontinuously from one to the other.
Consequently, $\bm{x}^\mathrm{gc}$ changes discontinuously in $\bm{\pi}$,
and the GCE is unable to describe the equilibrium states
in the first-order phase transition region.
\subsection{Thermodynamic function}
The fourth condition imposed on $\eta$ is
\begin{enumerate}[label={(\Alph*)}]
\setcounter{enumi}{3}
\item $\bm{x}_\mathrm{max}^\eta$ is surjective onto $\Gamma$.
\label{cond:D}
\end{enumerate}
This condition plays a crucial role for obtaining {\em all} thermodynamic properties
from the thermodynamic function of the SE, as follows.
Let us define the {\em thermodynamic function associated with $\eta$} as the thermodynamic limit of $\psi_N^\eta$:
\begin{align}
\psi^\eta \equiv \lim_{N \to \infty} \psi_N^\eta.
\label{eq:psi_eta}
\end{align}
Using Eq.~\eqref{eq:x-N-eta}, Eq.~\eqref{eq:entropy-formula} can be rephrased as
\begin{align}
\psi^\eta(\bm{\kappa}) &= \inf_{\bm{x} \in \Gamma} \left\{ \eta(\bm{\kappa};\bm{x})-s(\bm{x}) \right\}. \label{eq:s-psi}
\end{align}
Then, owing to condition \ref{cond:D},
we can invert Eq.~\eqref{eq:s-psi} (see Appendix~\ref{sec:genLeg_proof}) as
\begin{align}
s(\bm{x}) &= \inf_{\bm{\kappa} \in K} \left\{ \eta(\bm{\kappa};\bm{x})-\psi^\eta(\bm{\kappa}) \right\}. \label{eq:psi-s}
\end{align}
Using this relation,
one can obtain the entropy only from $\psi^\eta$ without knowing $\bm{x}^\eta$.
In this sense, $\psi^\eta$ is equivalent to the thermodynamic entropy.
Hence, all thermodynamic functions are obtained from $\psi^\eta$.
From a physical point of view,
relation~\eqref{eq:s-psi} is a generalization of the equivalence of the entropy and the grand canonical thermodynamic function.
From a mathematical point of view, the relation is a generalization of the Legendre-Fenchel transformation.
\section{Summary and Conclusion}
We have proposed a new class of ensembles,
which we call the (generalized) squeezed ensembles (SEs).
Since it is well-defined even when the order parameter(s)
does not commute with other additive observables
such as the Hamiltonian,
it solves the fundamental problems
faced by conventional ensembles
when dealing with quantum systems
with first-order phase transitions.
The validity of the SEs are shown
under reasonable assumptions,
and confirmed by numerical calculations.
In contrast to the case of the grand canonical ensemble,
additive quantities
which specify the equilibrium state
always have macroscopically definite values in the SE.
Furthermore, unlike the microcanonical ensemble,
the SE can be defined and constructed
even when the additive observables
do not commute with each other.
Utilizing these advantages of the SEs,
we have provided, for the first time,
a concrete method for constructing phase coexistence states
of general quantum systems.
Various ensembles are included in the class of SEs.
One can choose an appropriate SE depending on the purpose,
such as one that is easy to construct.
In addition, good analytic properties of the SEs
yield practical formulas for thermodynamic quantities
such as the entropy and the thermodynamic force.
For these reasons,
the SEs are convenient for practical calculations.
As an illustration, we have applied our formulation
to the two-dimensional transverse field Ising model
at finite temperature,
whose order parameter does not commute with the Hamiltonian.
We have confirmed that
the SE is composed of the thermodynamically identical states with large domains.
In addition, we have obtained thermodynamic anomalies
peculiar to phase coexistence states separated by phase interfaces.
These results evidence that the SE successfully gives a phase coexistence state.
We expect that the SEs enable the incorporation of quantum effects
into the analysis and design of materials with phase interfaces.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,453
|
{"url":"https:\/\/mammothmemory.net\/physics\/lenses\/convex-lenses\/principal-focal-length-and-centre-of-curvature.html","text":"# Principal focal length and centre of curvature\n\nIf you have two convex lenses of identical shape but made of different materials, a parallel ray of light going through each lens will have a different principal focal length.\n\nHowever, two convex lenses made out of the same material (same refractive index (n)) will have different principal focal lengths if they have different centres of curvature (radius of spheres). Another way of saying this is, if two lenses of the same material have different shapes - one a flat surface and the other a rounded one, then they will have different principal focal lengths.\n\nSlightly curved lens = Long focal length\nVery curved lens = Short focal length\n\nThe greater the curvature of a lens the greater the refraction of the ray of light. Which obviously means that the less the curvature of a lens the less the refraction will be.","date":"2021-09-28 04:25:14","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\": 0.8103837966918945, \"perplexity\": 633.734650960502}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780060201.9\/warc\/CC-MAIN-20210928032425-20210928062425-00380.warc.gz\"}"}
| null | null |
Why I Hike
Outerwear Guide
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Waterproof Rainwear
Wrinkle-Free Shirts & Pants
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Hiking & Travel
Fishing & Field
Wrinkle-Free
Plus Shop
Trenches & Coats
Casual Comfort Shoes
Home & Bedding
Guides & Athletes
Big City Mountaineers
American Hiking Society
The Heroes Project
American Forests
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Eddie Bauer |
Global Labor Practices
Pro Deals
Global Labor Practice
Eddie Bauer is committed to providing customers with the highest quality, innovation and value in our products. We believe this commitment is best met through strong relationships with our associates and by selecting business partners who share our commitment to ethical practices and agree to our standards of business conduct.
While we do not own or manage any factories, we do have a comprehensive Global Labor Practices Program to monitor for compliance and encourage best practices in all facilities that are producing for us. This program began in 1995 with the establishment of Eddie Bauer's Standards for Business Partnerships. OurFactory Workplace Code of Conduct includes standards of conduct to which we expect our manufacturing partners to adhere, including the prohibition of forced and child labor, effective policies against harassment, abuse and discrimination, compliance with health, safety and environmental standards, guaranteeing freedom of association and collective bargaining, paying appropriate wages and benefits, regulating hours of work and ensuring the payment of overtime compensation. All of our vendors must agree to comply with these standards in writing, and pass an initial audit of their facilities as to the various elements of the Code, before we will do business with them. company.info.global.labor.practice.msg4=Our code is enforced in numerous ways: corporate education, vendor training and certification, announced and surprise factory audits by an independent third party auditor, worker interviews, corrective action plans for any identified violations, termination for serious/repeated noncompliance, outreach to civil society groups knowledgeable about local working conditions, risk assessments and benchmarking with other companies to adopt best practices.
???company.info.global.labor.practice.msg4???
California Transparency in Supply Chains Act of 2010 Disclosure
You will need Adobe® Reader® to view PDF documents.
Download a free copy from the Adobe website.
Eddie Bauer's Factory Workplace Code of Conduct
Eddie Bauer is committed to providing customers with the highest quality and value in our products. We believe this commitment is met in part through strong relationships with our associates and by selecting business partners who share our commitment to ethical practices and agree to our standards of business conduct. The following conditions are required for factories producing our products:
There shall not be any use of forced labor, whether in the form of prison labor, indentured labor, bonded labor or otherwise.
No person shall be employed at an age younger than 15 (or 14 where the law of the country of manufacture allows) or younger than the age for completing compulsory education in the country of manufacture where such age is higher than 15.
Harassment or Abuse
Every employee shall be treated with respect and dignity. No employee shall be subject to any physical, sexual, psychological or verbal harassment or abuse.
No person shall be subject to any discrimination in employment, including hiring, salary, benefits, advancement, discipline, termination or retirement, on the basis of gender, race, religion, age, disability, sexual orientation, nationality, political opinion, or social or ethnic origin.
Employers shall provide a safe and healthy working environment to prevent accidents and injury to health arising out of, linked with, or occurring in the course of work or as a result of the operation of employer facilities.
Freedom of Association and Collective Bargaining
Employers shall recognize and respect the right of employees to freedom of association and collective bargaining.
Wages and Benefits
Employers recognize that wages are essential to meeting employees' basic needs. Employers shall pay employees, as a floor, at least the minimum wage required by local law or the prevailing industry wage, whichever is higher, and shall provide legally mandated benefits.
Except in extraordinary business circumstances, employees shall (i) not be required to work more than the lesser of (a) 48 hours per week and 12 hours overtime or (b) the limits on regular and overtime hours allowed by the law of the country of manufacture or, where the laws of such country do not limit the hours of work, the regular work week in such country plus 12 hours overtime and (ii) be entitled to at least one day off in every seven day period.
Overtime Compensation
In addition to their compensation for regular hours of work, employees shall be compensated for overtime hours at such premium rate as is legally required in the country of manufacture or, in those countries where such laws do not exist, at a rate at least equal to their regular hourly compensation rate.
Business partners shall comply with the environmental laws and regulations of thelocal region or country in which they operate
Compliance with applicable local laws and with this Code are conditions for becoming, and remaining, a business partner of Eddie Bauer and will be agreed to, in writing, as a term of engagement. Eddie Bauer will take appropriate action, including termination of our relationship, with any business partner in violation of our standards.
To facilitate effective monitoring and enforcement, our business partners are expected to provide full access to their production facilities and to relevant records relating to employment practices. We will undertake affirmative measures, such as unannounced, on-site inspections of facilities, to implement and monitor these standards.
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,601
|
package com.artbeatte.exercises.tests;
import com.artbeatte.exercises.strings.RunLengthEncoding;
import com.artbeatte.exercises.strings.RunLengthEncodingTestCase;
import com.artbeatte.testrunner.MethodParameterTestCase;
import com.artbeatte.testrunner.MethodTestCase;
import com.artbeatte.testrunner.SystemTestRunner;
import com.artbeatte.testrunner.TestRunner;
/**
* @author art.beatte
* @version 11/17/15
*/
public class RunLengthEncodingTest {
public static void main(String[] args) {
TestRunner testRunner = new SystemTestRunner();
for (String test : RunLengthEncodingTestCase.TESTS.keySet()) {
RunLengthEncoding rle = new RunLengthEncoding(test);
testRunner.addTestCase(new MethodTestCase<>(rle, "encode", RunLengthEncodingTestCase.TESTS.get(test)));
testRunner.addTestCase(
new MethodParameterTestCase<>(rle, "decode", String.class, rle.encode(), test));
}
testRunner.runTests();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,794
|
Ember Google Charts [](https://travis-ci.org/sir-dunxalot/ember-google-charts) [](https://www.npmjs.com/package/ember-google-charts)
======
Ember Google Charts makes it very easy to implement [Google Charts](https://developers.google.com/chart/) in Ember CLI apps.
All dependencies are lazy loaded using the [Google JS API Loader](https://developers.google.com/loader/?hl=en), which intelligently caches requests between a user's sessions.
## Installation
```sh
ember install ember-google-charts
```
See the [demo app](http://sir-dunxalot.github.io/ember-google-charts/) here.
- [Charts](#charts)
- [Default options](#default-options)
- [Locales](#locales)
- [Resize](#resize)
- [Actions](#actions)
- [chartDidRender()](#chartdidrender)
- [packagesDidLoad()](#packagesdidload)
- [Events](#events)
- [Custom Charts](#custom-charts)
- [Content Security Policy](#content-security-policy)
- [Testing](#testing)
- [renderChart()](#renderchart)
- [assertChart()](#assertchart)
- [Development](#development)
## Usage
### Charts
There are six types of chart supported out of the box:
- Area Charts (`{{area-chart}}`)
- Bar Charts (`{{bar-chart}}`)
- Geo Charts (`{{geo-chart}}`)
- Line Charts (`{{line-chart}}`)
- Pie Charts (`{{pie-chart}}`)
- Scatter Charts (`{{scatter-chart}}`)
To add a chart to any route, simply add the relevant component:
```hbs
<AreaChart @data={{this.model}} @options={{this.options}} />
```
Or if you're using an old template syntax:
```hbs
{{area-chart data=data options=options}}
```
Data and options should be in the format expected by the given chart, as detailed in the [Google Charts documentation](https://developers.google.com/chart/interactive/docs/).
For example:
```js
/* stats/route.js */
import Route from '@ember/routing/route';
export default Route.extend({
model() {
return [
['Task', 'Hours per Day'],
['Work', 11],
['Eat', 2],
['Commute', 2],
['Watch TV', 2],
['Sleep', 7],
];
},
});
```
```js
/* stats/controller.js */
import Controller from '@ember/controller';
export default Controller.extend({
options: {
title: 'How I spend my days',
height: 300,
width: 400,
animation: {
startup: true,
easing: 'inAndOut',
},
},
});
```
```hbs
{{!-- stats/template.hbs --}}
<PieChart @data={{this.model}} @options={{this.options}} />
```
You can pass data as an array (see above example) or as a [Google Data Table](https://developers.google.com/chart/interactive/docs/datatables_dataviews):
```js
/* stats/route.js */
import Route from '@ember/routing/route';
export default Route.extend({
model() {
return google.visualization.arrayToDataTable([
['Year', 'Sales', 'Expenses'],
['2004', 1000, 400],
['2005', 1170, 460],
], false);
},
});
```
```hbs
{{!-- stats/template.hbs --}}
<PieChart @data={{this.model}} @options={{this.options}} />
```
For more information about data tables and how to create them, see the [Google Charts guides](https://developers.google.com/chart/interactive/docs/datatables_dataviews).
Where possible, this addon default to using Material Charts over Google's 'classic' design.
It's very easy to add non-default charts (e.g. table charts or gauge charts) - [see the custom charts docs here](#custom-charts)
#### Design
Indicate which design you want: `classic` or `material`.
```hbs
<BarChart @data={{this.model}} @options={{this.options}} @design="classic" />
```
Only some chart types support Material Charts. See the [Google Charts documentation](https://developers.google.com/chart/interactive/docs) Chart Types to learn more.
#### Default Options
Default options for all charts can be set in the `GoogleChartsService`.
**Default options are always merged with the options you pass into a component.** Passed in options will only override specific options properties, not the whole options object.
```js
/* services/google-charts.js */
import GoogleChartsService from 'ember-google-charts/services/google-charts';
export default GoogleChartsService.extend({
defaultOptions: {
backgroundColor: '#389fcc',
annotations: {
alwaysOutside: true,
},
},
});
```
You can also set default options for individual components by overriding `defaultOptions` for the component. For example, if you want a [custom chart component](#custom-charts) to use different default options:
```js
/* components/gantt-chart.js */
import GoogleChart from 'ember-google-charts/components/google-chart';
import renderMaterialChart from 'ember-google-charts/utils/render-material-chart';
export default GoogleChart.extend({
type: 'gantt',
defaultOptions: {
backgroundColor: '#389fcc',
annotations: {
alwaysOutside: true,
},
},
renderChart: renderMaterialChart,
});
```
```
```
#### Locales
You can set the language of the charts you render by specifying the language code in the `google-charts` service:
```js
/* services/google-charts.js */
import GoogleChartsService from 'ember-google-charts/services/google-charts';
export GoogleChartsService.extend({
language: 'fr',
});
```
For more information on locales, see the [Google Charts documentation](https://developers.google.com/chart/interactive/docs/basic_load_libs#loadwithlocale).
Please note, Google Charts dependencies can only be loaded for a single language. This is a [limitation](https://developers.google.com/chart/interactive/docs/basic_load_libs#basic-library-loading) of the Google API loader.
#### Resize
By default charts will rerender when the window size changes. You can opt out of this by setting `responsiveResize` to false:
```hbs
{{pie-chart data=data responsiveResize=false}}
```
### Actions
Two actions are available for you to hook on to:
#### chartDidRender()
This fires when the Google chart has rendered and is ready for interaction via Google Charts public methods.
This action receives the `chart` object of the rendered chart.
```js
/* stats/controller.js */
import Controller from '@ember/controller';
export default Controller.extend({
actions: {
selectCountry(chart) {
chart.setSelection('someValue');
},
},
});
```
```hbs
{{!-- stats/template.hbs --}}
<GeoChart
@data={{this.model}}
@options={{this.options}}
@chartDidRender=(action 'selectCountry')
/>
```
#### packagesDidLoad()
This fires when the Google chart has finished loading the required Google packages for a specific chart.
This action receives no params.
```js
/* stats/controller.js */
import Controller from '@ember/controller';
export default Controller.extend({
actions: {
checkGoogleExists() {
if (window.google) {
// Do something...
}
},
},
});
```
```hbs
{{!-- stats/template.hbs --}}
<LineChart
@data={{this.model}}
@options={{this.options}}
@packagesDidLoad=(action 'checkGoogleExists')
/>
```
### Events
It's easy to listen to events emitted by a chart:
```js
/* stats/controller.js */
import Controller from '@ember/controller';
export default Controller.extend({
actions: {
addChartEventListeners(chart) {
const { google: { visualization } } = window;
visualization.events.addListener(chart, 'onmouseover', function(event) {
/* Do something here... */;
});
}
},
});
```
```hbs
{{!-- stats/template.hbs --}}
<LineChart
@data={{this.model}}
@options={{this.options}}
@chartDidRender=(action 'addChartEventListeners')
/>
```
For more information on events, see the [Google Charts event documentation](https://developers.google.com/chart/interactive/docs/events).
### Custom Charts
All chart components in this addon extend from a single core component: the `GoogleChartComponent`.
1. Find the type of chart in [the Google guides](https://developers.google.com/chart/interactive/docs/) and see what Google Charts package it requires
2. Update the [Google Chart service](https://github.com/sir-dunxalot/ember-google-charts/blob/master/addon/services/google-charts.js) `packages` property with the new Google Charts package you require (if applicable)
3. Specify whether to use `'material'` or `'classic'` design (depending on what Google Charts documentation says the chart type supports)
```js
/* components/gantt-chart.js */
import GoogleChart from 'ember-google-charts/components/google-chart';
export default GoogleChart.extend({
design: 'classic',
type: 'gantt',
});
```
```js
/* services/google-charts.js */
import GoogleChartsService from 'ember-google-charts/services/google-charts';
export GoogleChartsService.extend({
googlePackages: ['corechart', 'bar', 'line', 'scatter', 'gantt'], // Added gantt to defaults
});
```
If preferred, you can write your own `renderChart` method. Use the [`renderChart` util as your guide](https://github.com/sir-dunxalot/ember-google-charts/blob/master/addon/utils/render-chart.js).
`renderChart()` receives the DOM Element in which to render the chart followed by the chart properties, which is an object that should include the following properties:
- `data` (see usage instructions)
- `design` (`'material'` or `'classic'`)
- `options` (see usage instructions)
- `type` (e.g. `'bar'`, `'line'`, etc)
`renderChart()` must return a promise that resolves with the chart object (`resolve(chart)`).
### Content Security Policy
You will need to add the following to your app's [content security policy](https://github.com/rwjblue/ember-cli-content-security-policy) to mitigate CSP errors:
```js
contentSecurityPolicy: {
'script-src': "'self' 'unsafe-eval' *.google.com *.gstatic.com",
'style-src': "'self' 'unsafe-inline' *.google.com *.googleapis.com *.gstatic.com",
'font-src': "'self' *.gstatic.com *.googleapis.com",
}
```
## Testing
This addon makes two test helpers available that you can use in your app's test suite:
- `renderChart()`
- `assertChart()`
### renderChart()
`renderChart()` is an async helper that renders a chart using `@ember/test-helpers`'s `render()` method.
You must pass in an ES6 tagged template string, as is expected by `render()`, [documented here](https://github.com/emberjs/ember-test-helpers/blob/master/API.md#render), so this helper is designed for use in integration tests.
For convenience, `renderChart()` returns the chart's DOM element.
For example:
```js
/* tests/integration/some-test */
import { module } from 'qunit';
import { setupRenderingTest } from 'ember-qunit';
import { renderChart } from 'ember-google-charts/test-support';
module('Integration | Component | pretty color', function(hooks) {
setupRenderingTest(hooks);
test('Rendering the expenses chart', async function(assert) {
this.set('data', [
['Year', 'Sales', 'Expenses'],
['2004', 1000, 400],
['2005', 1170, 460],
['2006', 660, 1120],
['2007', 1030, 540],
]);
const chart = await renderChart(hbs`{{area-chart data=data}}`);
/* Now run some assertions... */
assert.ok(chart.textContent.indexOf('2007') > -1,
'Should contain 2007 data');
});
});
```
`renderChart()` adds a delay to your test suite that can be removed if you desire (but this may fail test suites in remote environments, like Travis):
```js
const chart = await renderChart(hbs`{{area-chart data=data}}`, {
delay: 0, // Or some number of milliseconds
});
```
### assertChart()
`assertChart()` runs a series of predefined assertions on any chart element to assert that the chart has been rendered correctly.
`assertChart()` expects several params to be passed:
- `assert`, which is available in all 'ember-qunit' tests
- `chart`, which is the chart's element and is returned by the [`renderChart()` test helper](#renderchart)
- `properties`, which is an object that should include the properties passed into the chart component:
- `data`
- `design` (`'material'` or `'classic'`)
- `options`
- `type` (e.g. `'bar'`, `'line'`, etc)
Here is an example, which also uses `renderChart()`:
```js
/* tests/integration/some-test */
import { module } from 'qunit';
import { setupRenderingTest } from 'ember-qunit';
import { assertChart, renderChart } from 'ember-google-charts/test-support';
module('Integration | Component | pretty color', function(hooks) {
setupRenderingTest(hooks);
const data = [
['Year', 'Sales', 'Expenses'],
['2004', 1000, 400],
['2005', 1170, 460],
['2006', 660, 1120],
['2007', 1030, 540],
];
const options = {
title: 'Yearly expenses',
animation: {
startup: true,
easing: 'inAndOut',
},
};
test('Rendering the expenses chart', async function(assert) {
this.setProperties({
data,
options,
});
const chart = await renderChart(hbs`{{area-chart data=data options=options}}`);
assertChart(assert, chart, {
data,
design: 'classic', // Because it's not a Material Chart
options,
type: 'area',
});
});
});
```
## Development
All PRs and issues are welcome.
- `git clone https://github.com/sir-dunxalot/ember-google-charts.git`
- `cd ember-tooltips`
- `npm install && bower install`
- `ember s`
- `ember test`, `ember try:testall`, or the `/tests` route
Please include tests and documentation updates with any new features.
You do not need to bump the version when you have a PR.
To release an update to the demo app:
```sh
git checkout master # make sure you're on master branch
ember github-pages:commit --message "Some commit message" # Builds the app
git push origin gh-pages:gh-pages # Deploys the app
```
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,691
|
\section{Introduction}
Superstring propagation on manifolds of $SU(n)$ holonomy requires extended
world-sheet superconformal symmetry \cite{wsscs}. Gepner has shown how to
describe such vacua using
tensor products of $N=2$ superconformal minimal models \cite{gep}. The
underlying Calabi-Yau (CY) manifolds are complete intersections in
weighted projective space \cite{wps}.
In principle, mirror symmetry allows the computation of world-sheet
instanton corrections \cite{wsic} to the low energy effective lagrangian.
More recently, second-quantized mirror
symmetry \cite{fhsv} has opened the way to computations of
non-perturbative string corrections for $N=2$ heterotic - type II dual
pairs \cite{kv}.
The extension of these results to $N=1$ models calls for a better understanding
of the relation between heterotic and type I vacua. The conjectured 10D
duality between the $SO(32)$ heterotic string and the type I superstring
\cite{dyn} has passed several tests \cite{test}, and
is expected to persist, in some non-naive form, after
compactification \cite{vwdual}. In this respect it is worth noticing that
the relation between type I and heterotic dilatons depends on the
space-time dimension $D$ according to \cite{wroma2}
\begin{equation}
\phi_{H} = {6-D \over 4} \ \phi_{I} \ - \ {D-2 \over 16} \ \log \det G_I \quad ,
\end{equation}
where $G_I$ is the internal metric in the type I string-frame. For instance,
in six dimensions the heterotic dilaton is related to
internal gravitational modes of the type I string, rather than
to the type I dilaton \cite{blpssw}.
In the past, type I models have been studied to a lesser extent than
models of oriented closed strings.
The initial proposal \cite{cargese} of identifying open-string
theories as {\it parameter-space orbifolds} of left-right symmetric theories
of oriented closed strings has been
brought to a consistent systematization \cite{bs,bpstor}, realizing
Chan-Paton (CP) symmetry breaking at the microscopic level.
The first consistent 6D
$N=1$ chiral open-string models \cite{bs} differ markedly from
perturbative heterotic K3 compactifications \cite{orbifold,gep,eoty},
since they include different numbers of tensor multiplets
that take part in a generalized Green-Schwarz (GS) mechanism \cite{tensor}.
Recently, additional instances of $N=1$ type I 6D models
have been constructed as toroidal
orbifolds \cite{gpdp,gjdp}, along the lines of \cite{ps}. A nice geometrical
setting for all these 6D models is provided by the F-theory proposal
of \cite{vafaf},
where non-trivial scalar backgrounds allow for an effective 12D dynamics.
The variety of 6D models with different numbers of tensor multiplets may
then be related to a corresponding variety of compactifications on
elliptically-fibered CY threefolds \cite{morvaf}.
In this letter we discuss descendants of
type IIB models based on rational superconformal field theories.
We mainly concentrate on 6D models, where a precise
comparison with orbifold compactifications can be drawn. This
allows a complementary view of several known results, while introducing
a novelty, models with {\it no} tensor multiplets at all.
Here we discuss only the simplest Gepner models, related to free conformal
field theories. We also relate the models without an open sector
of \cite{gjdp} to the ``crosscap constraint'' of \cite{fps,pss}.
This consistency condition
for conformal field theories on open and unoriented surfaces allows,
in general,
multiple choices for the Klein-bottle projection of the closed spectrum.
For minimal and $SU(2)$ WZW models, this was
discussed in \cite{pss}. For six-dimensional models, it retrieves the
unique anomaly-free $N=1$ spectrum with 9 tensor multiplets and 12
hypermultiplets found in \cite{gjdp}.
More complicated models, including $N=1$ type I models in four dimensions,
can be analyzed in a similar fashion and are full of interesting surprises.
For instance,
starting from the $Z$ orbifold, one can build chiral $N=1$ type I vacua in D=4
\cite{wroma2}, where the moduli space of
untwisted closed-string scalars, $Sp(8)/SU(4)\times U(1) $, is strongly
suggestive of a twelve-dimensional interpretation \cite{bfpss}.
A more complete analysis of rational superconformal models and of some
of their deformations will be presented elsewhere \cite{abpss}.
\section{Open Descendants of Gepner Models}
In addition to the Virasoro generators $L_n$, the $N=2$ superconformal algebra
includes two supercurrents $G^{\pm}_r$ and a $U(1)$ current $J_n$. It
admits an automorphism, known as spectral flow, under which
\begin{equation}
L_n\rightarrow L_n +\eta J_n + {\eta^2 c\over 6} \delta_{n,0} \ ,\qquad
J_n\rightarrow J_n +{\eta c\over 3}\delta_{n,0} \ ,\qquad
G^{\pm}_r \rightarrow G^{\pm}_{r\pm\eta} \quad ,
\end{equation}
where $0\le \eta <1$ and $c$ is the central charge, that connects different
sectors of the spectrum.
In models describing superstring propagation, the coupling to the $N=1$
world-sheet gravitino induces a restriction to the Ramond (R)
($G$-periodic) and Neveu-Schwarz (NS) ($G$-anti-periodic) sectors.
The primary fields of the $N=2$ superconformal algebra
are labeled by the conformal weight $h$ and the $U(1)$ charge $q$.
In the NS sector, the massless vertex operators involve
chiral primary fields with $h=q/2$. Their supersymmetric partners in
the R sector, with $h=c/24$, arise from the spectral flow with
$\eta =1/2$. The unique R descendant of the identity ($h=0$) is
identified with the target-space supercharge.
Gepner has shown how to construct string vacua with spacetime supersymmetry
tensoring $N=2$ superconformal minimal models \cite{gep}, that
form a discrete series with central charge $c_k = 3k/(k+2)$. For $d=(D-2)$
non compact (transverse) dimensions, the
total internal central charge is
$c_{I} = \sum_i c_{k_i} = 12-3d/2$, where $c_{k_i}$ are the central charges
of the various factors. The correct periodicity of the
world-sheet supercurrent is ensured if characters from the NS
and R sectors are combined separately. Target-space supersymmetry
follows if the supercharge has a local operator product expansion with all
vertex operators. The required truncation ($G$-projection), induced by
the restriction to integral $U(1)$ charges, is consistent with
modular invariance if $c_{I} = 3n$, and
the ``orbifoldized'' tensor product of $N=2$ minimal models
then describes superstring propagation on a manifold of $SU(n)$ holonomy.
The general procedure for constructing type IIB vacua, described in \cite{eoty}, starts with the identification
of the graviton orbit in the NS sector, $N^{+}_{0}$.
The other massless ($m$) and massive ($M$) NS orbits, $N^{+}_{\alpha}$,
$\alpha=1,\ldots,n_m+n_M$, are exposed by
a modular $S$ transformation on $N^{+}_{0}$. Their projections
$N^{-}_{\alpha}$
are obtained by a $T$ transformation, while the R orbits $R^{+}_{\alpha}$ are
obtained by an $S$ transformation from $N_{\alpha}^{-}$. Finally, a spectral
flow with $\eta =1$ determines the projected R orbits, $R^{-}_{\alpha}$.
Introducing the supersymmetric characters
\begin{equation}
X_\alpha =
{1\over 2} \biggl( \
V_d (N^{+}_{\alpha} + N^{-}_{\alpha}) +
O_d (N^{+}_{\alpha} - N^{-}_{\alpha}) -
C_d (R^{+}_{\alpha} - R^{-}_{\alpha}) -
S_d (R^{+}_{\alpha} + R^{-}_{\alpha}) \
\biggr) \quad ,
\end{equation}
where $\{ V_d , O_d , C_d , S_d\}$ are level-one $SO(d)$ characters, a
modular invariant partition function with space-time supersymmetry
can be written as
\begin{equation}
{\cal T}_{susy} = \sum_{\alpha=0}^{n_m+n_M}
\ell_\alpha |X_\alpha |^2 \quad ,
\label{tsusy}
\end{equation}
where $\ell_\alpha = S_{0\alpha} / S_{\alpha 0}$ are the multiplicities
of each orbit.
In order to construct open descendants of L-R symmetric closed models
as in \cite{bs},
one has to resolve the fixed-point ambiguity in the definition of the
characters.
The different modular invariant torus amplitudes may
be related to the propagation on different complex manifolds, consistently with
the ADE classification of simple singularities \cite{wps}.
\section{D=8 and Other Toroidal Compactifications}
Gepner models with $c=3$ describe
compactifications on rational tori. For instance, the orbifoldized
tensor products $(k=1)^3$ and $(k=1)\times (k=4)$ correspond to the $SU(3)$
torus, while $(k=2)^2$ corresponds to the
$SU(2)\otimes SU(2)$ torus \cite{gep}.
The resulting open descendants
are among the rational toroidal compactifications discussed in \cite{bpstor}.
There it was shown how
a quantized background for the NS-NS antisymmetric tensor
reduces the size of the CP group and how CP
symmetry breaking may proceed
via Wilson lines. This setting plays a crucial role in establishing
a correspondence between F-theory on K3 and heterotic string on $T_2$
\cite{sen}.
In general, the Klein bottle amplitude allows
for the introduction of signs $\epsilon_i = \pm 1$ in the projection of the closed-string
spectrum. This was discussed in detail for $SU(2)$ WZW models in \cite{pss}.
In a generic $r$-dimensional toroidal compactification at radii $R_i$, with
$i=1,\ldots,r$, it is possible to choose a Klein bottle amplitude
(neglecting irrelevant factors)
\begin{equation}
{\cal K} =\prod_{i=1}^{r}
\left(\sum_{m_i\ {\rm even}} q^{
({m_i\over R_i})^2}
+ \epsilon_i \sum_{m_i\ {\rm odd}}
q^{({m_i\over R_i})^2} \right) \quad ,
\label{exotic}
\end{equation}
where the conventional choice is $\epsilon_i = 1$.
When at least one of the $\epsilon_i$ equals $-1$,
there are no massless tadpoles, and thus one can not introduce
boundary states and open strings. From a microscopic viewpoint, the
consistency of this choice may be justified from the crosscap constraint
of \cite{fps,pss}, that indeed, for a one-dimensional torus, leaves
only one relative sign between even and odd momentum sums.
Notice that all choices of signs in (\ref{exotic}), that determine the
(anti)symmetrization of the states, respect the fusion rules.
This procedure may be generalized to
any rational model with $Z_2$ automorphisms, where the
Klein bottle projections that forbid the introduction of
the open sector do not involve massless characters
in the transverse channel. In the next section, we shall exploit this
possibility to recover the results of \cite{gjdp}, but we should mention
the simplest instance, toroidal compactification to four dimensions,
that gives a string setting to $N=4$ supergravity coupled to
six vector multiplets.
\section{Six-Dimensional Models}
In six dimensions there are several possible types of Gepner models \cite{gep}.
They correspond to toroidal
compactifications, to orbifolds of rational tori and
to some interacting rational $N=2$ superconformal field
theories\footnote{More precisely, in this case the $N=2$ superalgebra extends to
an $N=4$ superalgebra, that includes an $SU(2)$ subalgebra.}.
The starting point is a ``parent'' type IIB theory, whose chiral spectrum is
uniquely fixed by target-space $N=(2,0)$ supersymmetry.
Indeed, aside from (non-chiral) models with extended supersymmetry, that
correspond to rational points of toroidal compactifications,
any modular invariant torus amplitude
results in a massless spectrum including the $N=(2,0)$ supergravity
multiplet (the graviton, 5 self-dual tensors
and 2 right-handed ($R$) gravitini) and 21 tensor multiplets
(one anti-self-dual tensor, 5 scalars and 2 left-handed
($L$) tensorini). The scalar fields of the resulting low-energy
supergravity parametrize the coset $SO(5,21)/SO(5) \times
SO(21)$.
In the open descendants supersymmetry is reduced to $N=(1,0)$, and
the unoriented closed spectrum consists of the $N=(1,0)$
supergravity multiplet (the graviton, a self-dual tensor and a
$R$ gravitino) coupled to
$n^{c}_{T}$ tensor multiplets (an anti-self-dual tensor, a $L$ tensorino and
a scalar) and
$n^{c}_{H}$ hypermultiplets (four scalars and a $L$ hyperino). The open
unoriented sector contains
$n^{o}_{V}$ vector multiplets (a vector and a $R$ gaugino) and $n^{o}_{H}$
charged hypermultiplets.
It should be appreciated that $n_{T}^{c} + n^{c}_{H}$
is fixed to be 21 since the Klein-bottle projection simply halves the fermi
degrees of freedom.
The different models that we describe give rise to different values of both
$n^{c}_{T}$ and $n^{c}_{H}$, while the presence of a self-dual
tensor in the $N=(1,0)$ supergravity multiplet leaves a net number of
$n^{c}_{T} -1$
anti-self-dual tensors. The tensor fields that flow in the
transverse channel and correspond to unphysical R-R scalars take part in a
generalized GS mechanism \cite{tensor}.
\subsection{The $(k=2)^4$ Models with $Z_2$ Symmetry}
The first class of models that we discuss is
obtained tensoring four copies of the $k=2$ superconformal model.
It corresponds to the $Z_2$ orbifold of the maximal torus of
$SO(8)$ \cite{eoty}. Each $k=2$ minimal model with $c=3/2$ is
equivalent to the direct product of the Ising model with $c=1/2$ and a free
boson ($c=1$) theory at $R=\sqrt{8}$. Denoting by $\{ o,\psi ,\sigma\}$ the
three characters of the Ising model and by $\rho_m$, $m=-3,\ldots ,4$, the
eight characters of the $c=1$ theory one finds that
the $N=2$ characters in the
NS sector are given by $(o,\psi ) \times (\rho_{2p})$
and $\sigma \times (\rho_{2p +1})$ while those in the
R sector are given by $(o,\psi ) \times
(\rho_{2p+1})$
and $\sigma \times (\rho_{2p})$.
The graviton orbit, that gives rise to the identity character, reads
\begin{equation}
N^{+}_{0} = (o \rho_0 + \psi \rho_{4} )^4 + (o \rho_2 + \psi \rho_{-2} )^4
+ (o \rho_{-2} + \psi \rho_{2} )^4 + (o \rho_4 + \psi \rho_{0} )^4 \quad .
\end{equation}
A proper definition of the characters requires the resolution of a fixed-point
ambiguity. There are 5 more massless ($h=1/2$) and 10 massive ($h=1$) characters
in one-to-one correspondence with those of the first class of models in
\cite{bs}. In this case, the $P= T^{1/2} S T^2 S T^{1/2}$ modular transformation
that relates direct and transverse M\"obius channels \cite{bs}
is diagonal. In particular, $P_{11}=-1$, and thus only
symplectic CP groups are present in the open-string sector. This is in marked
contrast with the results of \cite{gpdp,gjdp}.
In order to switch to unitary or orthogonal CP groups, one has to introduce
discrete Wilson lines \cite{bs}, relative phases between crosscap and boundary
operators, that break in part the internal symmetry \cite{abpss}.
The CP symmetry is reduced, since a non-vanishing NS-NS
antisymmetric tensor is present in the $SO(8)$ torus \cite{bpstor}.
An exhaustive analysis of all modular invariant torus amplitudes gives
three type IIB parent theories. The
spectra of the corresponding open descendants are listed in table 1 for
simple choices of CP multiplicities consistent with tadpole cancellation.
Non standard Klein projections,
with only massive characters in the transverse channel, are allowed in all
these cases, and give rise to the anomaly free spectrum with
$n^{c}_{T}=9$ and $n^{c}_{H}=12$ with no open-string states. This should
be contrasted with conventional $B_{16}$ models, where massless tadpoles
require that the same closed spectrum be accompanied by anomaly-free open
sectors.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Mod. & $n^{c}_{T}$ & $n^{c}_{H}$ & CP Gauge Group &
$n^{o}_{V}$ & Charged Hypermultiplets & $n^{o}_{H}$
\\
\hline
\hline
${\bf A}_{16}$ & 7 & 14 & $Sp(4)^{\otimes 4}$
& 40 & $({\bf 4},{\bf 1},{\bf 4},{\bf 1})
\oplus ({\bf 1},{\bf 4},{\bf 1},{\bf 4})$ & 96
\\
& & & & & $\oplus 2 ({\bf 4},{\bf 1},{\bf 1},{\bf 4}) \oplus
2 ({\bf 1},{\bf 4},{\bf 4},{\bf 1})$ &
\\
\hline
${\bf B}_{16}$ & 9 & 12 & $Sp(4) \otimes Sp(4)$
& 20 & $({\bf 10},{\bf 1})\oplus ({\bf 1},{\bf 10})$ & 20
\\
\hline
${\bf D}_{16}$ & 5 & 16 & $Sp(8)^{\otimes 4}$
& 144 & $({\bf 8},{\bf 8},{\bf 1},{\bf 1})
\oplus ({\bf 1},{\bf 1},{\bf 8},{\bf 8})$ & 256
\\
& & & & & $\oplus ({\bf 8},{\bf 1},{\bf 8},{\bf 1})\oplus
({\bf 1},{\bf 8},{\bf 1},{\bf 8})$ &
\\
\hline
\end{tabular}
\end{center}
\caption{Open descendants of $(k=2)^4$ $Z_2$ Gepner model.}
\end{table}
\subsection{The $(k=2)^4$ Models with $Z_4$ Symmetry}
Another class of rational models arises from a $Z_4$ orbifold of the
$SU(2)^{\otimes 4}$ torus\footnote{At the closed-string level, this
rational model is equivalent to the
$Z_2$ orbifold of the $SO(8)$ torus
\cite{eoty}, but for the absence of a NS-NS antisymmetric tensor
background.}.
The type IIB partition function may be written in terms of 64
characters, and corresponds to the second class of models in
\cite{bs}. In addition to the identity character, there is another massless
self-conjugate character,
16 complex massless characters, 2 massive self-conjugate characters with
$h=3/2$, two sets of 8 massive characters each with
$h=3/4$ and $h=5/4$, respectively, and, finally, 12 self-conjugate and 16
complex massive characters with $h=1$.
An exhaustive analysis of all modular invariant torus amplitudes gives
rise to seven $N=(2,0)$ type IIB parent theories. The
spectra of the corresponding open descendants are listed in table 2 for
simple choices of CP multiplicities consistent with tadpole cancellation.
The diagonal (${\bf D}_{64}$) and charge-conjugation
(${\bf C}_{64}$) modular invariants give rise to quite different descendants.
The ${\bf D}_{64}$ model with standard Klein bottle projection
yields a closed-string spectrum with one tensor multiplet
and 20 hypermultiplets, while the ${\bf C}_{64}$ model leads to
9 tensor multiplets and 12 hypermultiplets. Similar results have been
found in
\cite{gpdp,gjdp}. In the ${\bf C}_{64}$ case, although the
closed spectrum
is anomaly free, one has to introduce open strings to cancel unphysical
tadpoles in the transverse channel. The resulting CP group is at most
$Sp(16)\otimes Sp(16)$ for the ${\bf D}_{64}$
case and $U(8)\otimes U(8)$ for the
${\bf C}_{64}$ case. The introduction of discrete Wilson
lines \cite{bs} leads to CP group enhancement
to $U(16)\otimes U(16)$ for the ${\bf D}_{64}$ model, with the
anomaly-free spectrum also found in \cite{gpdp}.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Mod. & $n^{c}_{T}$ & $n^{c}_{H}$ & CP Gauge Group &
$n^{o}_{V}$ & Charged Hypermultiplets & $n^{o}_{H}$
\\
\hline
\hline
${\bf A}_{64}$ & 5 & 16 & $Sp (8) \otimes Sp(8)$ & 72 & $({\bf 28},{\bf 1})
\oplus ({\bf 1},{\bf 28}) \oplus 2 ({\bf 8},{\bf 8})$ & 184
\\
\hline
${\bf B}_{64}$ & 7 & 14 & $Sp (8) \otimes Sp(8)$ & 72 & $2 ({\bf 8},{\bf 8})$ &
128
\\
\hline
${\bf C}_{64}$ & 9 & 12 & $U(8) \otimes U(8)$ & 128 & $({\bf 8},{\bf 8}^*)
\oplus ({\bf 8}^*,{\bf 8})$ & 128
\\
\hline
${\bf D}_{64}$ & 1 & 20 & $Sp (16) \otimes Sp(16)$ & 272 & $({\bf 120},{\bf 1})
\oplus ({\bf 1},{\bf 120}) \oplus ({\bf 16},{\bf 16})$ & 496
\\
\hline
${\bf E}_{64}$ & 9 & 12 & $U(8)$ & 64 & ${\bf 64}$ & 64
\\
\hline
${\bf F}_{64}$ & 9 & 12 & $U(4) \otimes U(4)$ & 32 & $({\bf 16},{\bf 1}) \oplus
({\bf 1},{\bf 16})$ & 32
\\
\hline
${\bf G}_{64}$ & 9 & 12 & $U(4)$ & 16 & ${\bf 16}$ & 16
\\
\hline
\end{tabular}
\end{center}
\caption{Open descendants of $(k=2)^4$ $Z_4$ Gepner model.}
\end{table}
\subsection{The $(k=1)^6$ Models}
The last class of models that we would like to discuss descends from a $Z_3$
orbifold of the maximal torus of $SU(3)\otimes SU(3)$,
or equivalently \cite{eoty} from the tensor product of
six copies of the $k=1$ minimal $N=2$ superconformal theory with $c=1$.
The latter is equivalent to a free boson at radius $R=\sqrt{12}$.
The twelve primary fields have conformal weights
$h_m=m^2/24$ with $m=-5,...,6$, and the corresponding characters will be
denoted by $\xi_m$.
The graviton orbit \cite{eoty} that gives rise to the identity
character
\begin{equation}
N^{+}_{0} = (\xi_0 + \xi_6 )^6 + (\xi_2 +\xi_{-4})^6 + (\xi_4 + \xi_{-2})^6
\end{equation}
is the only self-conjugate one. Resolving the fixed point ambiguity, one finds
20 more massless characters, and two sets of 30 characters with $h=5/6$
and $h=13/6$.
For the diagonal model (${\bf D}_{81}$), the unoriented closed
spectrum resulting from a standard Klein bottle projection
includes the supergravity multiplet and
one hypermultiplet from the identity character.
The other 20 massless characters give half hypermultiplet each. The resulting
spectrum contains {\it no tensor multiplets}, and
the dilaton must thus lie in a hypermultiplet.
This closed spectrum is anomalous and requires the introduction of
open strings. Tadpole cancellation in the transverse
channel selects the CP gauge group $SO(8)$ and,
aside from the vector multiplet, there are 10 hypermultiplets
in the adjoint representation.
This representation is anomaly free and the chiral
fermion content of the open-string spectrum exactly cancels the gravitational
anomaly in the closed-string spectrum.
Upon reduction on $T_2$ to $D=4$, the theory would not
be asymptotically free. It would be very interesting to find a model with no
tensor multiplets with an asymptotically free CP group.
The analytic prepotential of the
resulting effective field theory would not receive spacetime instanton
corrections.
The charge conjugation modular invariant (${\bf C}_{81}$)
leads to an open descendant with 10 hypermultiplets and
10 tensor multiplets arising from massless characters different from the identity,
and one hypermultiplet and the supergravity multiplet arising from the identity.
The closed spectrum is anomalous and tadpole cancellation requires the
introduction of open strings. The gauge group is once again
$SO(8)$, but the spectrum contains only the vector multiplet. The open spectrum
differs from the one found in \cite{gpdp,gjdp} for the $Z_3$ orbifold, since the
Gepner model involves a rank 6 NS-NS antisymmetric tensor background.
An exhaustive analysis of all modular invariant torus amplitudes results in six
equivalent type IIB parent theories. The type I spectra are summarized in table
3 for some simple choices of the CP multiplicities. Although it is not evident
from the table, $SO(8)$ is always allowed. This is precisely
the subgroup of $SO(32)$ preserving a generic configuration of 24 instantons on
K3 as required by anomaly cancellation, and suggests an F-theory interpretation
for these models. For an elliptic CY 3-fold $X$ fibered over a base $B$, the
number of tensor multiplets is $n_T=h_{11}(B)-1$, the rank of
the gauge group is
$r_V=h_{11}(X)-h_{11}(B)-1$ and the number of neutral hypermultiplets is
$n^0_H=h_{12}(X)+1$ \cite{morvaf}.
A large class of elliptic CY 3-folds, studied by
Voisin and Borcea and classified by Nikulin in terms of three invariants
$(r,a,\delta )$, have Hodge numbers $h_{11} (X) = 5 + 3r -a$ and
$h_{12} (X) = 65 - 3r - 2a$. F-theory compactification on these spaces leads
to an $SO(8)^{k+1}$ gauge group, with
$k=(r-a)/2$, and to $g=(22-r-a)/2$ adjoint hypermultiples. It is
tempting to conjecture that the type I models with $G=SO(8)$
correspond to the choice $r=a=n_{T}^{c} + 1$.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Mod. & $n^{c}_{T}$ & $n^{c}_{H}$ & CP Gauge Group &
$n^{o}_{ V}$ & Charged Hypermultiplets & $n^{o}_{H}$
\\
\hline
\hline
${\bf A}_{81}$ & 6 & 15 & $U(4)$ & 16 & $4\times {\bf 16} \oplus 3\times ({\bf 6}
\oplus {\bf 6}^* )$ & 100
\\
\hline
${\bf B}_{81}$ & 8 & 13 & $U(4)$ & 16 & $2\times {\bf 16} \oplus {\bf 6}
\oplus {\bf 6}^* $ & 44
\\
\hline
${\bf C}_{81}$ & 10 & 11 & $SO(8)$ & 28 & --- & 0
\\
\hline
${\bf D}_{81}$ & 0 & 21 & $SO(8)$ & 28 & $10\times {\bf 28}$ & 280
\\
\hline
${\bf E}_{81}$ & 10 & 11 & $SO(8)$ & 28 & --- & 0
\\
\hline
${\bf F}_{81}$ & 10 & 11 & $SO(8)$ & 28 & --- & 0
\\
\hline
\end{tabular}
\end{center}
\caption{Open descendants of $(k=1)^6$ Gepner model.}
\end{table}
\section{Anomaly Cancellation and Final Remarks}
Aside from $U(1)$ anomalies, tadpole cancellations guarantee the absence
of gauge and gravitational anomalies for all the models discussed so far.
Indeed, the antisymmetric tensor fields that flow in the transverse
channel take part in a generalized GS mechanism \cite{tensor} that allows the
cancellation of not necessarily factorized 6D chiral anomalies.
In table 4 we report the anomaly polynomials for the models discussed
in the previous section. These expressions determine the kinetic
terms of the gauge fields and their singularities at finite
coupling \cite{tensor}, that have been associated to phase
transitions \cite{dmw} with tensionless
strings \cite{blpssw}. Antisymmetric tensor fields, however,
can not cancel chiral anomalies
for abelian gauge groups with charged hypermultiplets.
It has been proposed \cite{fms,blpssw} that a mechanism
similar to the one taking place in D=4 be at work in this case. In six
dimensions the relevant field
is a 4-form potential, dual to a pseudo-scalar. The massless
type I spectra that we have discussed include several R-R scalars.
These admit exact Peccei-Quinn symmetries
and may be dualized into 4-form potentials, that can couple to abelian fields
via $A_4\wedge dA_1$. In the presence
of chiral anomalies, these R-R fields could then provide the
longitudinal degrees of freedom needed to exclude the (massive)
abelian vector fields from the low-energy spectrum, much in the same way as in
$D=4$ \cite{dsw,wroma2}.
\begin{table}
\begin{center}
\begin{tabular}{|c|l|}
\hline
Model & Anomaly Polynomial
\\
\hline
\hline
${\bf A}_{16}$ & $- {1\over 16} ({1\over 2} {\rm tr}R^2
- {\rm tr} F_{1}^{2} - {\rm tr} F_{2}^{2} - {\rm tr} F_{3}^{2} -
{\rm tr} F_{4}^{2} )^2 $
\\
& $+ {1\over 8} ({\rm tr} F_{1}^{2} + {\rm tr} F_{2}^{2} - {\rm tr}
F_{3}^{2} - {\rm tr} F_{4}^{2} )^2 +
{1\over 16} ({\rm tr} F_{1}^{2} - {\rm tr} F_{2}^{2} +
{\rm tr} F_{3}^{2} - {\rm tr} F_{4}^{2} )^2 $
\\
\hline
${\bf D}_{16}$ & $- {1\over 32} ({\rm tr} F_{1}^{2} + {\rm tr} F_{2}^{2} +
{\rm tr} F_{3}^{2} + {\rm tr} F_{4}^{2} + {\rm tr} R^2 )^2
+ {3 \over 32} ({\rm tr} F_{1}^{2} - {\rm tr} F_{2}^{2} -
{\rm tr} F_{3}^{2} + {\rm tr} F_{4}^{2} )^2$
\\
& $+ {1\over 32} ({\rm tr} F_{1}^{2} + {\rm tr} F_{2}^{2} -
{\rm tr} F_{3}^{2} - {\rm tr} F_{4}^{2} )^2
+ {1\over 32} ({\rm tr} F_{1}^{2} - {\rm tr} F_{2}^{2} +
{\rm tr} F_{3}^{2} - {\rm tr} F_{4}^{2} )^2$
\\
\hline
${\bf A}_{64}$ & $-{1\over 8} ({1\over 2} {\rm tr} R^2 - {\rm tr} F_{1}^{2}
- {\rm tr} F_{2}^{2} )^2 + {1\over 8} ({\rm tr} F_{1}^{2}
- {\rm tr} F_{2}^{2} )^2$
\\
\hline
${\bf B}_{64}$ & $-{1\over 16} ({1\over 2} {\rm tr} R^2 - {\rm tr} F_{1}^{2}
- {\rm tr} F_{2}^{2} )^2 + {3\over 16} ({\rm tr} F_{1}^{2}
- {\rm tr} F_{2}^{2} )^2$
\\
\hline
${\bf C}_{64}$ & ${1\over 4} ({\rm tr} F_{1}^{2} - {\rm tr} F_{2}^{2} )^2
+ {1\over 3} ({\rm tr} F_{1} - {\rm tr} F_{2}) [
({\rm tr} F_{1}^{3} - {\rm tr} F_{2}^{3}) - {1\over 16} {\rm tr} R^2
({\rm tr} F_{1} - {\rm tr} F_{2}) ]$
\\
\hline
${\bf D}_{64}$ & $- {1\over 16} ({\rm tr} R^2 - {\rm tr} F_{1}^{2} -
{\rm tr} F_{2}^{2} )^2 + {1\over 16} ({\rm tr} F_{1}^{2} -
{\rm tr} F_{2}^{2} )^2$
\\
\hline
${\bf A}_{81}$ & $- {3\over 8} ( {1\over 4} {\rm tr} R^2 + 2 {\rm tr} F^2 )^2
+ {1\over 8} ({\rm tr} R^2 {\rm tr} F - 16 {\rm tr} F^3 ) {\rm tr} F$
\\
\hline
${\bf B}_{81}$ & $- {1\over 8} ( {1\over 4} {\rm tr} R^2 - 2 {\rm tr} F^2 )^2
+ {1\over 24} ({\rm tr} R^2 {\rm tr} F - 16 {\rm tr} F^3 ) {\rm tr} F$
\\
\hline
${\bf C}_{81}$ & $ {1\over 8} ({1\over 4} {\rm tr} R^2 - {\rm tr} F^{2})^2$
\\
\hline
${\bf D}_{81}$ & $-{9\over 8} ({1\over 4} {\rm tr} R^2 - {\rm tr} F^2 )^2$
\\
\hline
${\bf E}_{81}$ & ${1\over 8} ({1\over 4} {\rm tr} R^2 - {\rm tr} F^2 )^2$
\\
\hline
${\bf F}_{81}$ & ${1\over 8} ({1\over 4} {\rm tr} R^2 - {\rm tr} F^2 )^2$
\\
\hline
\end{tabular}
\end{center}
\caption{Non-vanishing anomaly polynomials for the models in tables 1,2,3,
with $F_i$
the field strength of the $i$-th factor in the CP group. $R$ is the
curvature 2-form, and tr denotes the trace in the fundamental representation.}
\end{table}
The list of 6D Gepner models is clearly not exhausted by these
simple cases, and a more complete analysis will be reported in \cite{abpss}.
We have also constructed the simplest 4D Gepner model with $N=1$
supersymmetry,
$(k=1)^9$. Since it is related to the $Z_3$ orbifold of the maximal torus of
$SU(3)^{\otimes 3}$, the CP multiplicities are reduced \cite{bpstor} by
the non-vanishing (quantized) NS-NS antisymmetric tensor background.
The spectrum is encoded in 2187 characters.
Apart from the identity, only 168 of them are massless.
There are many torus amplitudes that give rise to different descendants.
In particular, the type IIB theory based on the charge
conjugation invariant may be related to the compactification on a CY threefold
with Hodge numbers $h_{11}=84$ and $h_{21}=0$. The open descendant has
$84+1$ chiral multiplets in the unoriented closed spectrum and an $Sp(4)$ CP group
with 84 chiral multiplets in the adjoint representation in the open
spectrum. In the diagonal case one finds the same closed
spectrum and CP group without charged matter, while the
type IIB spectrum, with 84 $N=2$ vector multiplets and one universal
hypermultiplet, may be related to the mirror of the above threefold.
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HomeCommunityartsKDT's Block Party Offers Limited Run of "To T or Not to T?"
KDT's Block Party Offers Limited Run of "To T or Not to T?"
July 1, 2022 arts 0
In the spring of 2017, the Center Theatre Group launched the Block Party series at the Kirk Douglas Theatre. Each year, CTG highlights a couple of productions from local "intimate" (formerly 99-seat) theatre companies in Los Angeles as a way to bring attention to the talent showcased in the more than 250 local theatre companies in town. Another thing it does – provides a larger stage for a multitude of voices that are not normally heard from in the larger theatres in town.
To T or Not to T?, the current Block Party entry on stage at KDT, is a perfect example of this. A one-man show written and performed by the talented D'Lo is as the subtitle suggests, "a comedic trans journey through (T)estosterone and masculinity." Originally produced at the Los Angeles LGBT Center, the show is a very personal look at not only queer and trans issues, but also how these cause conflict with his Tamal immigrant family.
Using his commitment ceremony to his partner as a framing device, D'Lo relates his journey from his childhood in Lancaster, CA, where he struggled with gender identity and his family's expectations, through coming out first as bisexual through his decision to transition via the use of "T." D'Lo is a talented comedian and storyteller, weaving a poignant tale of how he found peace with his identity and his family (including the young death of his sister) through his feminist perspective and his cultural and ancestral roots. One highlight of the show is the recreation of his father's speech at D'Lo's commitment ceremony, which he returns to again and again to punctuate his journey.
D'Lo has been ably guided in this journey by director and dramaturge Adelina Anthony, who makes sure the pace never dulls, as well as Tanya Orellana's vividly colored and playful set; and Rose Malone and Edwin Peraza's creative lighting and sound design, respectively.
D'Lo has a story worth hearing, one that helps bridge the gaps between us as humans. If we had more large theatres committed to presenting shows like this to the broader community, we might have more understanding and less judgement.
To T or Not to T? at the Kirk Douglas Theatre, only through July 10. Tuesday-Friday 8pm, Saturday 2 & 8 pm, Sunday 1 & 6:30 pm. For tickets, go to CenterTheatreGroup.org
Brenna Guthrie
Sustainability Subcommittee Focuses on Power; Clean Energy Gets Even Less Expensive
Police and Fire Both Warn Against Illegal Fireworks
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« Telluride '18: An Introduction | Main | Telluride '18: The Old Man & the Gun »
Telluride '18: First Man
starring Ryan Gosling, Claire Foy, Pablo Schreiber, Christopher Abbott
screenplay by Josh Singer
directed by Damien Chazelle
by Walter Chaw Damien Chazelle's First Man is the Super 8 shrine for Terrence Malick that Oscar voters never knew they needed. It's a mutant clumping-together of The Tree of Life (all the sad Texas scenes) and Philip Kaufman's The Right Stuff (all the astronaut stuff), mixed in with a few scenes that are gritty and true (most of them involving a frankly extraordinary Claire Foy), even if Chazelle remains overly fond of snap zooms and the handheld aesthetic in long shots. It's best, even exceptional, when it's not hagiography and passing fine when it's doing what it "ought" to be doing. Like playing a classical music waltz when stoic-to-the-point-of-deranged astronaut/engineer Neil Armstrong (Ryan Gosling) initiates the first-ever orbital docking manoeuvre, because 2001: A Space Odyssey; or doing a little riff on Bill Conti's amazing score for The Right Stuff right before the first closed-cabin testing. Could be homage. Could be the movie just doing what seems right as a shorthand for emotional engagement. If that's the case, more's the pity, as Chazelle proves in the first thirty minutes or so of his film--which revolve around an orbital "bounce" for a test plane and the death of Armstrong's toddler daughter to cancer--that he's capable of evoking real emotion, and employing smart contrasts in style and action, if he would only let go of the desire to impress.
First Man is essentially the study of a man who is completely closed-off to his family in the belief that there might be relief somewhere in the great elsewhere. Its closest analogue in that way is Close Encounters of the Third Kind, where another half-crazed pilgrim abandons his family to his extraterrestrial calling. Like that film, it's ever in danger of feeling like a movie about grief and loss from a privileged vantage, and like that film, though not as successfully, it manages to hit the emotional notes regardless. Armstrong gets a job with NASA's Gemini program and he and wife Janet (Foy) agree that it's a "fresh start"--though it becomes clear that this reboot is Neil's and not Janet's, nor is it one for their marriage. Janet is constantly in danger of being just another wife character from these flyboy/military melodramas, waiting behind a screen door for the letter to come, but there's a scene before Armstrong leaves to go to the moon where she tells him that she's done serving as his emotional surrogate and spokesperson with the two sons he doesn't seem to ever connect with. I appreciate another, quieter moment where a recently-widowed neighbour, Pat (Olivia Hamilton), stands frozen before an open car trunk until Janet, hugging her cardigan around herself, walks out to help her remember how to go into her house. The film is littered with great moments. There's a heartbreaking bit where the eldest Armstrong boy (Luke Winters) reaches out a hand solemnly for his father to shake in lieu of a hug. Not heartbreaking for the gesture, but for the strong decision to have Neil solemnly shake it. I also liked the sentimentality of showing how the world responded to the moon landing and the pang of real pain when a French woman in broken English declares that she always trusted the United States to lead in these areas.
So many good choices in First Man--and so many that are either obvious or unfortunate. Most of the flight sequences are shot from the point-of-view of the pilots or in such tight closeup that it's impossible to know what's happening spatially. The near-fatal orbital docking manoeuvre ends with the capsule holding Armstrong and Dave Scott (Christopher Abbott) spinning around at revolutions a second and is resolved basically with Armstrong grimacing and flipping switches on and off for a period long past useful or, crucially, interesting. Better is earlier in that sequence when Armstrong refuses to answer Ground Control because he's busy doing math longhand. It's character-developing and establishes some parameters. Then the picture immediately goes into noises and shaking. I understand that this is a choice, putting the viewers in the same situation as the astronauts who were blind but for their tiny portals, sort of searching out the universe from inside a steel can. But all these (many) scenes really do is make one nauseated. Still, it can be forgiven as a stylistic decision because at least the style is Chazelle's. Less easy to overlook is the open cribbing of Malick's nostalgic elegies; ditto Kaufman's iconic depiction of the sometimes-perverse rigours of astronaut training.
First Man is the first Damien Chazelle movie I've liked and there are times I liked it quite a lot. Gosling essentially plays his wounded masculinity thing from Drive, with the saving grace being that he's really good at silent and taciturn. And of course there's Foy, who will win the Best Supporting Actress Oscar next year and deserve it. At the movie's core is a conversation about depression--the moon, everything else is a MacGuffin I wish First Man hadn't spent quite as much time on. Chazelle and Gosling's Armstrong is going through something terrible and personal--hollowing out from the inside, he distracts himself with aspiration. He looks for solace, and the film implies that what he finds instead is another kind of eternal blackness, standing against the universe at the edge of a crater he can't see the bottom of. He lets something drop there. I don't know if he did in real life, but it works superbly in the film. Foy plays Janet as frustrated. She walks around Neil on eggshells until she doesn't; she knows not to ask until she does. I love how a wake ends when Neil can't deal with being around people anymore and so leaves Janet at the party without telling her, forcing her to later suffer the humiliation of friends driving her home. The way she holds onto her dignity in the backseat, the way she tries to defend his, it's spectacular. She sees him standing in the backyard looking into the sky, and instead of romanticizing the mariner gazing at the gulf, First Man has her turning out the light to leave him to his dark. The last shot of the picture suggests both that maybe he's going to break through his glassed-in despair and that maybe he never will. But First Man's best moment is its quietest: Neil and Janet are talking about a pleasant nothing, there's a pause, and then they laugh together. In a flash, you understand what's really at stake here. It's everything, and it's lovely.
Posted in 2018, Authors: Walter Chaw, Biopic, Drama, Film Festivals, Telluride, Telluride 2018 | Permalink
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\section{Introduction}
One of the basic integrals that arises in many parts in physics is
\begin{equation}
\label{gaussint}
\int_{-\infty}^{\infty}\prod_{k=1}^n {{dx_k}\over{(2\pi)^{1\over 2}}}
e^{-{1\over 2}xA x}
= (Det A)^{-{1\over 2}}
\end{equation}
where $A$ is a real, symmetric matrix with posive eigenvalues.
For instance, let (\ref{gaussint}) describe the integration
of fluctuations around a classical solution in quantum mechanics, where
the Lagrangian has been expanded up to second order.
The dimension of $A$ is then infinite, and (\ref{gaussint}) is
divergent on
both sides. The equation is therefore undefined as it stands.
As a basic example take
a one--dimensional
harmonic potential, and the fluctuations
in the time interval $0<\tau<L$. Then
\begin{equation}
xAx = \int_{0}^{L} d\tau (\partial_{\tau}x)^2 + \omega^2 x^2,
\end{equation}
and we have Dirichlet
boundary conditions for $x$
at $\tau=0$ and $\tau=L$.
The most straightforward way, is then to
go back to the Gaussian integral (\ref{gaussint}),
reintroduce a cut-off $\epsilon$
in space,
and modify the integration
measure depending on the cut-off so that the limit when $\epsilon$
goes to zero is finite.
In quantum mechanics this is feasible:
changing $(2\pi)^{-{1\over 2}}$ to $(2\pi\epsilon)^{-{1\over 2}}$,
and including one more factor $(2\pi\epsilon)^{-{1\over 2}}$,
turns (\ref{gaussint}) to a discrete approximation to
Feynman's sum over paths, which in the limit
gives
\begin{equation}
\label{sinh}
(\hbox{Regularized[$Det A$]})^{-{1\over 2}} = ({{2\pi\sinh (\omega L)}\over
L})^{-{1\over 2}},
\end{equation}
and this is the correct expression in the Greens function.
We may also observe that
the eigenvalues of $A$ are
${{\pi^2 n^2}\over{L^2}} + \omega^2$, and
the determinant is then formally
\begin{eqnarray}
\label{divergdet}
\det A \sim \prod_{n=1}^{\infty} ({{\pi^2 n^2}\over{ L^2}} + \omega^2)
\end{eqnarray}
One way to regularize the determinant is to introduce a
cut-off $\Lambda$ in the product (\ref{divergdet}),
check that in the limit of large $\Lambda$ the result separates
into one finite factor and one factor divergent with
$\Lambda$, and keep the finite factor
as the regularized result.
For (\ref{divergdet}), this gives the same result as
(\ref{sinh})\cite{Parisi}.
A regularization can also be found from
zeta function of the operator;
\begin{equation}
\label{zetadef}
Z_A(s) = \sum \lambda_k^{-s}
\end{equation}
which converges when the real part of $s$ is large enough. When
this function can be analytically continued to
be regular in a neighbourhood
of the origin, then
\begin{equation}
\label{zetaprimedef}
\hbox{Regularized[$Det A$]} = e^{-Z_A'(0)}
\end{equation}
For (\ref{divergdet}), this again gives the same result as
(\ref{sinh}).
The zeta function method
was first introduced in the context of regularizing
expressions like (\ref{gaussint}) by Hawking\cite{Hawking},
to study fluctuating fields in a
background of curved space.
In dimension higher than one, it is not evident that all
regularizations of the determinant give the same result.
In this paper we will compute $Z_A'(0)$, with $A$ the
Laplace operator with Dirichlet boundary conditions in two--dimensional
simplicial domains.
In the language of field theory this is
the gaussian model, a free massless theory, albeit with unusual and
obstructing boundary conditions. Even so, the direct limit
is far from trivial, and the most relevant result we are aware of,
works only for lattice Laplacians, discretized
on ectangular ($M\times N$) domains\cite{DD}:
\begin{eqnarray}
\label{DDasympt}
\det A \sim 2^{9\over 4} e^{ {GMN}\over{\pi} }
(1+\sqrt 2 )^{ -{{(M+N)}\over 2} } (MN)^{-{1\over 4}}\eta (q)
({M\over N})^{1\over 4}
\end{eqnarray}
Here, in lattice units, $MN$ is the area, $2(M+N)$ is the length
of the boundary, $q=e^{-2\pi {N\over M}}$ is the modular parameter,
G is Catalan's constant and $\eta(q)$ is the modular form of
Dedekind. There are now no less than three terms separately
diverging with the size of the lattice.
If, with hindsight, we use that
for rectangular domains
$Z_A(0) = {1\over 4}$, we can rewrite (\ref{DDasympt}) in terms of
an explicit lattice spacing $a$:
\begin{eqnarray}
\label{DDsimple}
\det A \sim \mu_{A}^{{Area}\over{a^2}}\mu_{L}^{{Length}\over{a}}
(a^2)^{Z_A(0)} e^{-Z_A(0)\log Area - B}
\end{eqnarray}
where $e^{-B}$ are the various remaining terms in
(\ref{DDasympt}) which
agree\footnote{up
to a constant factor $2^{{11}\over{4}}$} with
$e^{-Z_A'(0)}$\cite{ItzZub2torus}({\it see} Appendix 11).
If nothing else, it seems likely that a discretization on a
rectangular grid, of a domain which is not itself of rectangular
shape, will give rise to oscillating terms in the cut--off.
If (\ref{DDsimple}) is to be generally valid in two
dimensions, it can probably only be of a smoothened discretized
determinant, where
the smoothening goes over cut--off scales.
Assuming that this can be done, and
considering that the area, the length of the
boundary, and $Z_A(0)$ are all integrals of local distributions
({\it see} section 2), it is possible to introduce local
cut--off dependent counter--terms, such that the
the finite remaining piece is $e^{-Z_A'(0)}$.
It therefore at least makes sense to define the regularized
determinant to be $e^{-Z_A'(0)}$, and this is the view we
take in the rest of this paper. We will use the notation
$Z_D'(0)$ for our generic case: the zeta function of the Laplacian with
Dirichlet boundary conditions in a simplicial domain $D$,
with the topology of a disc. We will freely change the index of
the zeta function to denote various special cases, and even
contributions to the regularized determinant from some parts
of the domain.
If we look for physical relevance, we must have fluctuating
geometry, as otherwise all would dissappear in an overall normalization.
It is a quite old idea that the elementary excitations of
non--Abelian gauge theories are string--like
objects\cite{Wilson,Polyakov1,Alvarez}.
In lattice gauge theories, the statistical weight of a Wilson
loop, when a quark and an anti--quark are taken apart for
some time, is the area of the smallest area delimited by
the loop. A phenomenological model
of the excitations was proposed\cite{Polyakov1}, where the
statistical weight of a surface is its area, and
the path integral goes over imbeddings in $d$--dimensional
external space (``$x^{\mu}$''), and over internal two--dimensional
geometry (``$g^{ab}$''):
\begin{equation}
\label{Polyakovint}
Z \sim \int D[g^{ab}] D[x^{\mu}] \,
e^{-{1\over 2}[\int \sqrt{g} g^{ab} \partial_{a} x^{\mu}
\partial_{b} x^{\mu}]}
\end{equation}
Our computations are relevant to a part of the investigations
of (\ref{Polyakovint}). With $g^{ab}$ fixed, the integration over
$x^{\mu}$ is just a quadratic integral
with Dirichlet boundary conditions like (\ref{gaussint}).
Our calculations hence
gives the finite piece of this determinant.
As is well known, reparametrization
invariance of the action in (\ref{Polyakovint}), gives rise
to Faddeev--Popov determinants, which for smooth surfaces turn
out to be determinants of Laplacians acting on vector fields,
with modified Dirichlet boundary conditions. We have not
investigated these determinants, and we are not quite sure they
are relevant when
we consider piece--wise flat
surfaces with sharp corners.
At least in a class of simplices with fixed number of corners,
such a surface is its own model, and we would not have any more
reparametrization invariance.
In this respect, an approach closer to the simplicial
discretization of (\ref{Polyakovint})
would seem to be more appropriate\cite{KKM}.
Let us now see why it may be interesting to investigate
the determinants on simplicial disc--like domains.
{\it Smooth} disc--like manifolds with {\it smooth boundary}
can always be mapped conformally onto one another. If we
denote the conformal factor $\sigma(x)$, the base metric and
curvature by $\hat g$ and $\hat R$, and the base geodetic
curvature of the boundary by $\hat k$, a celebrated
result\cite{Polyakov1,Alvarez} says that
\begin{eqnarray}
\label{Liouvilleaction}
Z'_{D'}(0) =
Z'_{D}(0) - {1\over{4\pi}}
\int_{\partial D} d\hat s \hat n\cdot\partial\sigma
+ {1\over{6\pi}}
\int_{\partial D} d\hat s \hat k\sigma \nonumber \\
+{1\over{12\pi}}
\int_{D} d^2z
\sqrt{\hat g} [\hat g^{ab} \partial_{a} \sigma
\partial_{b} \sigma + \hat R\sigma]
\end{eqnarray}
The integration constant can be computed from the upper
half sphere\cite{Weisberger}. If we
disregard the boundary terms, the equations of motion
of (\ref{Liouvilleaction}) are Liouville's equation,
and we refer to the combined action in (\ref{Liouvilleaction})
as the Liouville action.
In ordinary Feynman path integrals (\ref{gaussint}) the
typical path is quite rough, i.e. a nowhere smooth random walk.
It seems likely that the typical surface
that enters in \ref{Polyakovint} is also quite rough.
On two--dimensional smooth surfaces, one
can open up a corner with angle $2\pi\alpha$
(or $\pi\alpha$ at the boundary) with a coordinate transformation
which is conformal and regular everywhere but at the corner,
where it instead has a logarithmic singularity.
The kinetic energy term
in (\ref{Liouvilleaction}) will then be logarithmically
divergent at the corner.
In other words, the action for smooth (and conformal) coordinate
changes is infinite for these transformations.
One possible procedure is then to
introduce a cut-off $r_0$ at a corner,
and simply remove the quantities diverging
when $r_0$ goes to zero\cite{HP}. This is obviously dangerous;
if one is not careful one easily gets a spurious finite
piece from a logarithmic divergence, and
$Z_D'(0)$ is a well-defined mathematical object, which
has a value, the surface being smooth or not.
The correct interpretation is that a conformal transformation
does not change $Z_D(0)$, but a non--conformal one
does. In fact, for piece--wise flat surfaces, $Z_D(0)$ can
be written as a sum over a rational functions of the
opening angles of the corners ({\it see} section 2, Appendix 6).
A typical non--conformal transformation will change the
opening angles by a shear, so it will change $Z_D(0)$.
The infinity of (\ref{Liouvilleaction}) under non--conformal
transformations is therefore a mirror of actually $Z_D(0)$ changing.
Our computation give some more explicit results on determinants,
to which one does not have access from smooth models. This
can have some mathematical interest by itself. More speculatively,
it is possible that a definition of determinants by a precise
calculation of $Z'_D(0)$, may yield a better regularization
of (\ref{Polyakovint})
than does (\ref{Liouvilleaction}) and its Faddeev--Popov
ghosts.
Certainly, such a result would go far beyond what is actually done
here: we have not begun to address a computation of a sum
over surfaces as in
(\ref{Polyakovint}).
The conclusions of this paper can now be stated as the following
propositions:
\\\\
{\bf Proposition 1.} {\it Domains with disc--like topology may
be mapped to the upper half complex plane, the boundary being
mapped to the real axis, and the corners being mapped to
branch--points $\omega_j$. If $z$ is the coordinate in the domain,
the transformation satisfies
\begin{eqnarray}
{{d\omega}\over{dz}}
= \phi_D\prod_j (\omega-\omega_j)^{1-\alpha_j},
\end{eqnarray}
and the representation is determined by the angles
and the lengths of the sides,
up to a rational fractional transformation of the upper half
plane. Choosing one parametrization
the normalized area of the simplex is:
\begin{eqnarray}
\hbox{Area}_D = \int_{\Im\omega > 0} {{d\bar\omega\wedge d\omega}\over{2i}}
\prod_j |\omega-\omega_j|^{2\alpha_j-2},
\end{eqnarray}
and the variation
of $Z_D'(0)$ under a general shear and dilatation
can be written;
\begin{eqnarray}
\delta[Z_D'(0)|_{\hbox{Area} = A}]
= \delta[Z_D'(0)] + \delta[Z_D(0)\log A] - \delta[Z_D(0)\log\hbox{Area}_D]
\end{eqnarray}
where then value of the zeta function at the origin is;
\begin{eqnarray}
Z_D(0) = \sum_{\hbox{interior corners}}{1\over{12}}({1\over{\alpha_j}}
-\alpha_j)
+\sum_{\hbox{boundary corners}}{1\over{24}}({1\over{\alpha_j}} -\alpha_j)
\end{eqnarray}
and the opening angles are written $2\pi\alpha_j$ in the
interior, and $\pi\alpha_j$ on the boundary.
}
\\\\
{\bf Proposition 2.} {\it
When the area is chosen $\hbox{Area}_D$, the
difference of $Z_D'(0)$ between two
simplices, that differ by an infinitessimal
transformation, which is regular and conformal everywhere except
at the corners, can be written as a sum over quantities
with point mass support at the corners, and a line density on
the boundary. The integral over of the boundary term is identically
zero
for polygons, but gives
\begin{equation}
-4\pi\sum_{\hbox{interior corners}}
\delta\alpha_i,
\end{equation}
if there are corners in the interior.
}
\\\\
{\bf Proposition 3.} {\it The contribution to $\delta Z_D'(0)$ from
the point mass in one
corner $c$, is a sum
\begin{equation}
\delta Z_D'(0)|_{c} = \delta Z'_{\alpha_c}(0) +
Z_{\alpha_c}(0)\cdot\delta[\hbox{other corners}]
\end{equation}
where $\delta Z'_{\alpha_c}(0)$ is exclusively determined locally
at the corner, $Z_{\alpha_c}(0)$
is the contribution to $Z_{D}(0)$ from the corner,
and the influence from the
other corners depend
on the opening angles of these, and on the lengths of the sides:
\begin{equation}
Z_{\alpha_c}(0)\cdot\delta[\hbox{other corners}] = Z_{\alpha_c}(0)
[\sum_{c'\neq c}
\delta\alpha_{c'}\log |\omega_{c'}-\omega_c|^{2}
-(1-\alpha_{c'})
[{{\delta(\omega_{c'}-\omega_c)}\over{\omega_{c'}-\omega_c}} + \hbox{c.c}]]
\end{equation}
}
\\\\
{\bf Proposition 4.} {\it The strictly local contributions
may be given in integrated form, and are for a corner
on the boundary:
\begin{eqnarray}
Z'_{\alpha}(0) = {1\over{12}}({1\over{\alpha}} -\alpha)(\gamma -\log 2)
-{1\over{12}}({1\over{\alpha}} + 3 +\alpha)\log\alpha +\tilde J(\alpha) ,
\end{eqnarray}
and for a corner in the interior;
\begin{eqnarray}
Z'_{\alpha}(0) = {1\over{6}}({1\over{\alpha}} -\alpha)(\gamma -\log 2)
-{1\over{6}}({1\over{\alpha}} + \alpha)\log\alpha +2\tilde J(\alpha) ,
\end{eqnarray}
where the term $\tilde J$ has the integral representation;
\begin{eqnarray}
\label{Jdefmain}
\tilde J(\alpha) =
\int_{0}^{\infty}
{1\over{e^{x} -1}}
[{1\over {2x}}(\coth({x\over{2\alpha}}) -
\alpha \coth({x\over{2}}))
-{1\over {12}}({1\over{\alpha}}-\alpha)] dx.
\end{eqnarray}
We have here included an integration constant in our definition
of $\tilde J$.
}
\\\\
Special situations are quite important to us, as they provide
necessary checks. We therefore list the them separately:
\\\\
{\bf Proposition 5.} {\it
For triangles one may choose a parametrization where the
branchpoints lie fixed in $0$, $1$ and $\infty$. The normal
area of a triangle with
angles $\pi\alpha_1,\pi\alpha_2,\pi\alpha_{3}$ is then
\begin{equation}
Area(\alpha_1,\alpha_2,\alpha_{3}) = {{\pi}\over 2}
{{\Gamma(\alpha_1)\Gamma(\alpha_2)\Gamma(\alpha_{3})}\over
{\Gamma(1-\alpha_1)\Gamma(1-\alpha_2)\Gamma(1-\alpha_{3})}}
\end{equation}
and all terms but the strictly local in the variation of $Z'_D(0)$
vanish.
The determinant of a triangular domain is thus
\begin{equation}
Z_T'(0) = \sum_{p=1,2,3} Z'_{\alpha_p}(0)
\end{equation}
with $Z'_{\alpha}(0)$ as in Proposition 4.}
\\\\
{\bf Proposition 6.} {\it For regular polygons with $n$ corners
one can choose a parametrization by mapping to the unit disc,
where the corners are regularly spaced on the unit circle.
The radius of the circumscribed circle is with this parametrization:
\begin{equation}
R_n = {{\Gamma(1+{1\over n})\Gamma(1-{2\over n})}
\over{\Gamma(1-{1\over n})}}
\end{equation}
and the determinant in a regular polygon with radius
of circumscribed circle $R$ is;
\begin{eqnarray}
Z_{P_n}'(0) = Z_{P_n}(0)\log{{R^2}\over{R_n^2}} -
{1\over{3(n-2)}}\log n + n Z_{1 - {2\over n}}'(0)
\end{eqnarray}
where $Z_{P_n}(0) = {{n-1}\over{6(n-2)}}$.
Specializing to $n=2$ we obtain the determinant in a square:
\begin{eqnarray}
Z_{P_4}'(0) = {1\over 4}\log\hbox{Area}
+{1\over 2}\log{{\Gamma({3\over 4})}\over{\Gamma({1\over 4})}}
+ {1\over 4}\log\pi + {5\over 4}\log 2
\end{eqnarray}
Taking the limit as $n\to\infty$ we obtain the determinant
in a disc:
\begin{eqnarray}
Z_{P_{\infty}}'(0) =
{1\over 3}\log R + {5\over 12} + {1\over 2}\log\pi + {1\over 6}\log 2
+ 2\zeta'(-1)
\end{eqnarray}
}
\\\\
The paper is organized as follows: standard results on
heat kernels and zeta functions
are summarized in section 2. In section 3 a representation is found
for $\delta Z'(0)$ which treats the corners and the rest in different
ways; this is the main idea. In both cases the representation
is in terms of the short--times asymptotics of the heat kernel.
In the interior this representation is standard, and gives
the variational form of (\ref{Liouvilleaction}). That the
underlying space is flat, and with disc--like topology,
actually simplifies things so much that this contribution
gives only the simple sum in Proposition 1.
In the corners the appropriate asymptotics is the
Sommerfeldt
heat kernel in an infinite sector.
Most of the derivations are by calculation, and most
are elementary (although sometimes
cumbersome)
We have chosen to present those in appendices, largely in a self--contained
way, but without any reference to physical
arguments. The appendices can therefore be read
more or less independently
from the rest of the paper.
Section 4 summarizes
the results.
\section{Heat kernel and Zeta function phenomenology}
The diagonal elements of the heat kernel on smooth manifolds
admit an asymptotic expansion for short
times. In two dimensions the expansion goes as
\begin{equation}
\label{asympt}
K_D(x,x,t) \sim {{c_{-1}(x)}\over t} +
{{c_{-{1\over 2}}(x)}\over {t^{1\over 2}}} + c_0(x) + \ldots
\end{equation}
McKean and Singer\cite{McKean} proved the existence of this
expansion for smooth manifolds up to terms $t$, and for manifolds
with smooth boundary up to $t^{1\over 2}$. In the interior
the coefficients $c_i$
are polynomials in the curvature tensor,
and on the boundary they are line distributions, with weight
depending on the curvature of the boundary.
For polygonal domains in the plane, Kac\cite{Kac} proved the expansion up
to $c_0(x)$, which in this case is made up of point masses at the corners.
Since
the curvature of a polygon can be said to be concentrated at the corners,
the result is in some sense natural,
and $c_0(x)$ for a smooth boundary can indeed
be found as the limiting case of
polygons with angles coming closer and closer to $\pi$.
However, the converse is
not true, that is, if one wants to compute $Z_D(0)$ for a manifold with
tips and corners, it is not possible to use a smoothened
approximation.
The integrated form of (\ref{asympt}) is
\begin{equation}
\label{asymptint}
\int_D dx^2 K_D(x,x,t) = \Theta_D (t) \sim {{c_{-1}}\over t} +
{{c_{-{1\over 2}}}\over {t^{1\over 2}}} + c_0 + \ldots
\end{equation}
The zeta function is defined by a Mellin transform of the trace of the
heat kernel as
\begin{eqnarray}
Z_D(s) = {1\over{\Gamma(s)}}
\int_0^{\infty} dt t^{s-1} \Theta_D (t)
\end{eqnarray}
The integral only converges in the lower limit if $\Re s>-\alpha_i$ for
all terms $c_{\alpha_i}t^{\alpha_i}$ in the heat kernel, i.e.
$\Re s>1$ in two dimensions.
One can get around the pole by a partial integration, where the
boundary terms vanish for $\Re s>1$:
\begin{eqnarray}
{1\over{\Gamma(s)}}
\int_0^{\infty} dt t^{s-1} \Theta_D (t) =\nonumber \\
{1\over{\Gamma(s)}}
[({{t^{s-1}}\over{s-1}} t \Theta_D (t))]_0^{\infty}
- {1\over{\Gamma(s)}}
\int_0^{\infty} dt {{t^{s-1}}\over{s-1}}{{\partial}\over{\partial t}}
[t \Theta_D (t)]
\end{eqnarray}
The second integral now gives an analytic continuation down to
$\Re s>{1\over 2}$.
Further on the zeta function has poles at
$s=-\alpha_i$, with residue
${{c_{\alpha_i}}\over{\Gamma(\alpha_i)}}$, except at zero and
the negative integers, where the inverse gamma function has zeros,
which gives finite values $Z(-n) = (-)^nc_{-n}n!$\cite{BSV}.
For positive $n$ greater than 1, a method that goes back to
Euler\cite{Watson}
gives a representation
of the zeta function as a convolution of electrostatic
Greens functions:
\begin{eqnarray}
Z_D(n+1) = Tr_x ({1\over{-\Delta}})^{n+1} = \int dx\int dy_1 \ldots \int dy_n
G_D(x,y_1)G_D(y_1,y_2)\ldots G_D(y_n,x)
\end{eqnarray}
This method can be extended to calculate the finite part of $Z_D(s)$ at
$s=1$\cite{IML}. Hence, if the expansion (\ref{asymptint}) goes
on indefinitely and one knows the Greens function,
one may in principle calculate the
zeta function at the integers, and the residues at all the poles.
But if this is true, and the zeta function does not grow
too fast at infinity, then the zeta function is completely
determined, which (somewhat indirectly) determines the
eigenvalues. Hence all the information about the
eigenvalues, and on all quantities depending on them,
is then contained in the Green's function and the
asymptotic expansion of the heat kernel for short times.
It is natural to consider what we will call
a {\it zeta function density}:
\begin{eqnarray}
\label{zetalocal}
Z_D(x,x,s) = {1\over{\Gamma(s)}}
\int_0^{\infty} dt t^{s-1} K_D(x,x,t)
\end{eqnarray}
For $\Re s<1$ we define the zeta function density by analytic
continuation from (\ref{zetalocal}).
At zero, the zeta function density is determined by the
asymptotics of the heat kernel for short times as
in (\ref{asympt}):
\begin{eqnarray}
\label{zeta0finite}
Z_D(x,x,0) = c_0(x)
\end{eqnarray}
We may just as well consider the density associated
to the derivative of the zeta function at the origin,
but this quantity will depend also on
the heat kernel for long times. These questions are addressed
in more detail in Appendix 1.
One could also look at
off--diagonal elements in (\ref{zetalocal}), or
equivalently, the operator $({1\over{-\Delta}})^s$.
Since the off--diagonal Mellin transforms are finite
for all $s$, the analytical continuation brings
no off--diagonal subtractions, and effectively the diagonal and
off--diagonal elements are treated differently for $\Re s<1$.
It would be interesting to have a proper regularization
of off--diagonal terms, so that one might consider
$Z_D'(0)$ as the trace of a (regularized)
operator $Q\sim -\log(-\Delta)$,
and in terms of which a regularized Laplace operator
would be $e^{-Q}$ (in the operator sense).
In this paper we will only consider the diagonal
elements, and we therefore refrain from calling
the zeta function density the matrix elements of an operator
when $\Re s < 1$.
Let us consider a dilation
$x\to\lambda x$ of the domain. That changes the eigenvalues
in a simple way: $E_n\to\lambda^{-2}E_n$, and the zeta function
accordingly also changes simply: $Z_{\lambda D}(s)\to\lambda^{2s}Z_{D}(s)$.
For the quantities arond the origin this implies:
\begin{eqnarray}
\label{zetadil}
Z_{\lambda D}(0)=Z_{D}(0)\qquad Z_{\lambda D}'(0)=Z_{D'}(0)+\log\lambda^2
Z_{D}(0)
\end{eqnarray}
The invariance of $Z_{D}(0)$ under dilations is a special case
of invariance under regular conformal transformations. The
change in $Z_{D}'(0)$ logarithmically proportional to the variation
of the area, is contained in Proposition~1,
and was already noticeable in the asymptotic expansion
of the discretized determinant in rectangular domains
(\ref{DDsimple}).
\section{A variational formula}
In this section we will write down a variational formula.
It can be formulated either in terms of the asymptotics of
the heat kernel for short times, or in terms of the zeta function
density from section 2.
Let us first motivate why we give two computations, that essentially
only differ in that they work on opposite sides of the
Mellin transform: the approach using the heat kernel is more
traditional, but requires more delicate treatment to separate
out the finite piece. Note that our formulae also differ by
(a fairly simple term) from a definition of the regularized
determinant often used in field theory\cite{Alvarez,Polyakov1}.
The reason for this discrepancy is that we also consider
transformations that are not conformal, and here the
additional term matters. The approach using the zeta function is
computationally somewhat simpler, once the notion of a zeta
function density is admitted.
Let now $D$ and $D'=D+\delta D$ be two domains that differ infinitesimally.
Then, for $\Re s>1$,
\begin{equation}
\label{varyT}
Z_{D'}(s)-Z_D(s) = \sum_{n=0}^{\infty}[E_n'^{-s} - E_n^{-s}] =
-s\sum_{n=0}^{\infty}({{\delta E_n}\over{E_n}})E_n^{-s}
\end{equation}
and this definition is analytically continued to lower values of
$s$.
We can consider $\{E_n'\}$ to be the eigenvalues of a
modified operator
$(-\Delta-\delta\Delta)$ in the domain $D$, so that we write
\begin{equation}
\label{smallvaryT}
Z_{D'}(s)-Z_D(s) = s Tr_x [(\delta\Delta) Z_D(x,x,s+1)]
\end{equation}
The variations we will consider map one simplex on another. They
therefore leave the following sum invariant:
\begin{equation}
\label{invariant}
{\cal S}(\alpha_j) = + \sum_{\hbox{boundary corners}} (1-\alpha_j) +
\sum_{\hbox{interior corners}} (2-2\alpha_j)
=2
\end{equation}
where the opening angle of a corner on the boundary
(in the interior) is $\pi\alpha$ ($2\pi\alpha$).
The analytical expression for our variation
will therefore only be determined up to the variation of
a ($s$--dependent)
function of ${\cal S}(\alpha_i)$,
that is, up to terms linear in the variations of the
angles.
If we want to evaluate the variation of the derivative
at zero we can do it as
\begin{equation}
{d\over {ds}}\delta Z_{D}(s)|_{s=0} =
\lim_{s\to 0}
{1\over s}Tr_x [ (Z_{D'}(x,x,s) - Z_{D}(x,x,s))
- (Z_{D'}(x,x,0) - Z_{D}(x,x,0))]
\end{equation}
which can be rewritten as
\begin{equation}
\label{finitevary}
{d\over {ds}}\delta Z_{D}(s)|_{s=0} =
\hbox{Finite}_{s\to 0}
{1\over s}Tr_x [ Z_{D'}(x,x,s) - Z_{D}(x,x,s)]
\end{equation}
In general there will be a pole at the origin in $s$
in (\ref{finitevary}), with a residue equal to the
variation of $Z_D(0)$. Combining (\ref{finitevary})
and (\ref{smallvaryT}) we have
\begin{equation}
\label{smallfinitevary}
{d\over {ds}}\delta Z_{D}(s)|_{s=0} =
\hbox{Finite}_{s\to 0}
Tr_x [(\delta\Delta) Z_D(x,x,s+1)]
\end{equation}
It is now convenient to choose the variation
in a particular way.
Think first
of the Laplace--Beltrami operator in
curved space, with metric tensor $g^{ab}$:
$\Delta = g^{-{1\over 2}}
\partial_a g^{ab}g^{{1\over 2}}\partial_b$.
In two dimensions we can choose the particular form
$g_{ab} = e^{2\sigma(x)} \hat g_{ab}$, and for a
smooth surface with disc
topology we can take $\hat g^{ab}$ to be the standard
flat metric. In this coordinate system, the
Laplacian has the following form
$\Delta = e^{-2\sigma}\partial^2_{aa}$.
A conformal variation of the Laplace--Bertrami operator
is then $\delta\Delta = (-2\delta\sigma)\Delta$, and
the final variational formula is
\begin{equation}
\label{varyPERS}
{d\over {ds}}\delta Z_{D}(s)|_{s=0} =
\hbox{Finite}_{s\to 0}
Tr_x [2\delta\sigma(x) Z_D(x,x,s)]
\end{equation}
The formula (\ref{varyPERS}) holds also
when we include
corners with varying opening angles,
the only new effect then being that prefactor
weight $\delta\sigma$ has a logarithmic
singularity at the corner.
We will show in Appendix 3 that in flat domains far
from the boundary, the density $Z_D(x,x,s)$ vanishes
at the origin. Hence the interior of flat domains,
far from the corners, give no contribution at all
to $Z_D'(0)$. This is in agreement with
(\ref{Polyakovint}), since our base space is flat (hence
$\hat R$ is zero), and the kinetic energy term of a regular
conformal variation can only give a boundary term.
At straight boundaries, there is
an undetermined
contribution to $Z_D'(0)$, proportional to the boundary length.
The contribution to the variation, $\delta Z_D'(0)$, is however
fully determined following (\ref{varyPERS}), and picks out the normal
derivative of the variation $\delta\sigma(x)$. Here
again we have exact agreement with (\ref{Liouvilleaction}).
For simplices, the boundary term integrates trivially,
and we are left with only the corner contributions as the
important parts.
Considering that the zeta function density
$Z_D(x,x,0)$, has components with point mass support
at the corners, it may be surmised that integrated
against the logarithm of the distance from the tip,
it gives an infinite contribution to
(\ref{varyPERS}). This is indeed the case, that
infinity is precisely the variation of $Z_D(0)$,
which we subtract by taking the finite part as
$s$ goes to zero. It is now important to realise
that we can choose small areas around the corners,
with some radius $r_0$, which we can let tend to zero
at the end of the calculation. It therefore does not matter
if we compute the answer up to terms proportional to
$r_0$ or $r_0^2$, since these will eventually drop out.
This means that we can actually substitute the true (unknown)
$Z_D(x,x,s)$, with a sufficiently good approximation,
computed from the Mellin transform of an approximation
to the true heat kernel valid for short times.
The technique for doing this is explained in Appendix~5,
and the approximation to the heat kernel -- the
Sommerfeldt heat kernel in a sector -- is described
in Appendix~4.
The calculations of the contributions
to $Z_D'(0)$ from the corner is then
done in Appendix~6.
Let us now derive an alternative formula to
(\ref{varyPERS}) using only the heat kernel for
short times. We begin with the following
equality (derived in Appendix 1):
\begin{equation}
\label{AlvarezcorrR}
Z_D'(0) =
\gamma Tr_x Z_D(x,x,0)
+ \hbox{Finite}_{\epsilon\to 0} Tr_x
\int_{\epsilon}^{\infty} {{dt}\over t} K_D(x,x,t)
\end{equation}
The heat kernel can be expanded in a complete set of states:
\begin{equation}
K_D(x,x,t) = \sum_v |\psi_v(x)|^2 e^{-\lambda_v t}
\end{equation}
and the
variation of (\ref{AlvarezcorrR}) will be
\begin{eqnarray}
\label{AlvarezcorrR2}
\delta Z_D'(0) =
\gamma \delta Tr_x Z_D(x,x,0)
-\hbox{Finite}_{\epsilon\to 0}
\int_{\epsilon}^{\infty} dt
\sum_v <v|(2\delta\sigma\Delta) |v> e^{-\lambda_v t} \nonumber \\
= \gamma \delta Tr_x Z_D(x,x,0)
+\hbox{Finite}_{\epsilon\to 0}
Tr_x [2\delta\sigma(x) K_D(x,x,\epsilon)]
\end{eqnarray}
When we are in the interior, or at a straight boundary,
the zeta function density $Z_D(x,x,0)$ is zero.
The remaining trace over the heat kernel in (\ref{AlvarezcorrR2})
then agrees with a well--known definition of the regularized
determinant in field theory, and
the short--time expansion of the heat kernel can be done
by standard means\cite{Polyakov1,Alvarez}.
We then find the
variational form of (\ref{Liouvilleaction}).
In piece--wise flat simplices there are further simplifications,
such that the only non--zero term is the integral over
the boundary of the normal derivative of $\delta\sigma$ in
(\ref{Liouvilleaction}).
Close to the corners, we cannot
however use the same short--time expansion of the heat kernel.
Again we have to turn to the Sommerfeldt heat kernel,
and carefully extract the constant piece of the trace
of the heat kernel
for short times. Finally, we must
take into consideration that the variation of $Z_D(0)$ is not zero
at the corner, and add the first term on the right--hand side of
(\ref{AlvarezcorrR2}). The computation following these lines
is done in Appendix~7.
\section{Calculations \& Conclusions}
In this section we describe the computations
in the appendices, and discuss the results.
Appendix~1 is an expansion on Section~2. We establish that
the zeta function density of the derivative at zero is well--defined,
but depends on the heat kernel for long times. An
alternative definition of the variation of the regularized
determinant in terms on the heat kernel for short times
is derived. Appendix~2 presents a general
parametrization of variation of simplices, and
computes the contribution from corners in the
interior as in Proposition~2. In Appendix~3 we treat in some
detail the case of a straight boundary. We derive that it
cannot give a a term in $Z_D'(x,x,0)$ more singular than
a line density, and hence a contribution to $Z_D'(0)$
proportional to boundary length. We also derive the
contribution from the line integral on the boundary to
the variation of $Z_D'(0)$. Appendix~4 presents
the Sommerfeldt heat kernel in a sector, and Appendix~5
states an elementary integral, that is convenient when
one wants to compute the Mellin transform using the Sommerfeldt
heat kernel.
In Appendix~6 we compute the contribution to $Z_D(0)$, and
the strictly local contribution to
$Z_D'(0)$ from a corner with opening angle $\alpha$,
using the approach of the zeta function density.
We call these quantities $Z_{\alpha}(0)$ and $Z_{\alpha}'(0)$.
We perform the computations for both corners on the boundary
and corners in the interior. In Appendix~7 we do the same
thing, using the slightly more involved approach of the
analysis of the short--time properties of the Sommerfeldt heat
kernel. In Appendix~8 we investigate $Z_{\alpha}'(0)$ in
the special cases,
where the opening angle is ${1\over n}$. In Appendix~9
we investigate the asymptotic behaviour of $Z_{\alpha}'(0)$
as $\alpha$ is large, small or close to one.
In Appendix~10 we treat a few special cases using our general
variational formula. The additional computations done here
essentially boil
down to taking care of the expression in Proposition~3,
that has support at one corner, but depends on the
angles of all the corners, and on the lengths of the
sides. In the very special case of triangles, this
additional term is absent, but in polygons it is important.
In Appendix~11 we state summarily the cases we know to
compare with,
where the $Z_D'(0)$ are known, either because the eigenvalues
are known, or because these domains are conformal images
of some domain where the eigenvalues are known. This
is important information: it fixes an integration
constant that we cannot reach; and it provides
much needed analytical checks.
In conclusion we have shown that the determinants on piece--wise
smooth surfaces are different from the determinants on
smooth surfaces. The formula that describe variations
of determinants on smooth surfaces (\ref{Liouvilleaction})
give infinite results for variations that change the
opening angles of the corners. This infinity is not real,
it is only an imprint of actually $Z_D(0)$ changing.
We have parametrized the variations of the domains
such they are
conformal everywhere outside the corners,
and separated the contributions to the variation
of $Z_D'(0)$ into four parts:
one (Proposition~1), that expresses the variation of
$Z_D(0)$ times the logarithm of the area;
one (Proposition~2), which
is an integral over the
boundary of the normal derivative of the
conformal parameter; one (Proposition~3)
which has local support at the corners, but depend
on the opening angles of all the corners and on the lengths
of the sides; and one (Proposition~4),
which is determined strictly locally at the corners.
The first part amounts to determining how the normalized
area changes with the angles and the sides.
We can trivially integrate the second part, which is
identical to one
of the terms in the variation of determinants on smooth surfaces
(\ref{Liouvilleaction}).
We have integrated the fourth part, both for corners
on the boundary and in the interior.
For general opening angles this part has an integral
representation (\ref{tibc}, \ref{Jdef2}), which for the
special opening angles $\alpha={1\over n}$, can
be resolved into a finite sum (\ref{finalfixed}).
We have been able to parametrize the area and integrate
the third part, for the special cases of triangles
(where the third part vanishes) and a class of polygons,
which includes certain interpolations between pairs of
regular polygons (Appendix~10).
In general, we have not been able to integrate the third part,
which therefore has to be left in variational
form (Proposition~3).
The wider applicability of our approach evidently
depends on better handling of the parts we have
not integrated.
We do not expect that one will in general
be able even to express the normalized area in closed
form, as a function of the angles and the sides.
Perhaps one may however find a limiting form, which
is valid for small opening angles.
Considered as an action for simplicial surfaces,
it is possible that the full expression of $Z_D'(0)$
strongly damps out corners with small opening
angles.
As is well known, most investigations of sums
over random surfaces in physical dimensions
lead to very rough surfaces, dominated by long
sharp spikes.
If the full expression of $Z_D'(0)$ as a function
of one opening angle has a singularity at the
origin -- with the right sign of the prefactor --
then surfaces with spikes will be more strongly
damped by the action $Z_D'(0)$, then by any finite
expansion in local curvature.
We hope to return to questions in this direction
in the future.
\section{Acknowledgements}
E.A. wishes to thank
Claude~Itzykson for initially encouraging
him to study these problems, and
Service de Physique Th\'eorique at Saclay
for hospitality in 1986--87. We thank Jean--Marc Luck for communicating
numerical results on the regularized determinants in triangles,
that proved a valuable check.
This work was supported by the
Swedish Natural Science Research Council under contracts
U--FR--1778--101, S--FO~1778--302 and F-FU 8230-306,
and by the G\"oran Gustavsson Foundation.
\newpage
\section* {A1. Local and nonlocal quantities}
\label{s:app1}
\renewcommand{\theequation}{A1.\arabic{equation}}
\setcounter{equation}{0}
In this appendix we collect some results on the zeta
function density and compare it to other approaches in
the literature. An important aspect is locality: a Mellin
transform of a heat kernel, at a point, integrated over some
subdomain, or over the whole domain, is said to be local,
if it is determined only by the expansion of the heat kernel
for short times.
We start with the definition (\ref{zetalocal})
\begin{equation}
Z_D(x,x,s) = {1\over{\Gamma(s)}}
\int_0^{\infty} dt t^{s-1} K_D(x,x,t)
\end{equation}
It is useful to explicitly analytically continue the integral beyond
the poles at $s=1$, $s={1\over 2}$ and also beyond $s=0$. We will
then get out a pole in $s$ at $s=0$ from the integral,
which will combine with the gamma--function
in front to give a regular function. We thus have a representation of
the zeta function density around $s=0$ in terms of a regular prefactor
and a convergent integral:
\begin{equation}
Z_D(x,x,s) = {1\over{(s-1)(s-{1\over 2})s\Gamma(s)}}
\int_0^{\infty} dt t^{s}
{{\partial}\over{\partial t}}[
t^{1\over 2}{{\partial}\over{\partial t}}[t^{1\over 2}
{{\partial}\over{\partial t}} t K_D(x,x,t)]]
\end{equation}
Using the asymptotic expansions as $t$ is close to zero:
\begin{eqnarray}
K_D(x,x,t) \sim {{c_1(x) }\over t} +
{{c_{1\over 2}(x) }\over {t^{1\over 2}}} +
c_0(x) + c_{-{1\over 2}}(x) t^{1\over 2} + \ldots \nonumber \\
{{\partial}\over{\partial t}} t K_D(x,x,t) \sim
{1\over 2}{{c_{1\over 2}(x) }\over {t^{1\over 2}}} +
c_0(x) + {3\over 2} c_{-{1\over 2}}(x) t^{1\over 2} + \ldots \nonumber \\
t^{1\over 2}[{{\partial}\over{\partial t}}[ t^{1\over
2}{{\partial}\over{\partial t}} t K_D(x,x,t)] \sim
{1\over 2}c_0(x) +
{3\over 2} c_{-{1\over 2}}(x) t^{1\over 2}+ \ldots \nonumber
\end{eqnarray}
we may evaluate as follows
\begin{eqnarray}
\hbox{Res} Z(x,x,s) |_{s=1} =& c_1(x) \nonumber \\
\hbox{Res} Z(x,x,s) |_{s={1\over 2}} =& \pi^{1\over 2}c_{1\over 2}(x)
\nonumber \\
Z(x,x,0) =& c_0(x) \nonumber
\end{eqnarray}
We may also evaluate the derivative with respect to $s$ at
the origin:
\begin{equation}
\label{zetaprimdef}
Z_D'(x,x,0) =
(3+\gamma)c_0(x) +
2\int_0^{\infty} dt \log t
{{\partial}\over{\partial t}}[
t^{1\over 2}{{\partial}\over{\partial t}}[t^{1\over 2}
{{\partial}\over{\partial t}} t K_D(x,x,t)]]
\end{equation}
It is quite clear that $Z_D'(x,x,0)$ is a quantity
which depends also on the heat kernel at large times.
An alternative formula is obtained by taking
the integral from
$\epsilon$ to infinity, where $\epsilon$ is some small positive
number.
As the integrand in \ref{zetaprimdef}
behaves as $t^{-{1\over 2}}\log t $
for small $t$, the integral is convergent, and the limit
as $\epsilon$ goes to zero is harmless. Partial integrations
will however bring in divergent terms from the lower
boundary. A short calculation gives
\begin{eqnarray}
Z_D'(x,x,0) =
(3+\gamma)c_0(x) + \nonumber \\
(c_0(x)\log\epsilon - 2c_{1\over 2}(x) \epsilon^{-{1\over 2}}
-c_1(x)\epsilon^{-1} - 3 c_0(x))
+\int_{\epsilon}^{\infty} {{dt}\over t} K_D(x,x,t)
\end{eqnarray}
which can be simplified to
\begin{equation}
\label{Alvarezcorr}
Z_D'(x,x,0) =
\gamma c_0(x)
+ \hbox{Finite}_{\epsilon\to 0}
\int_{\epsilon}^{\infty} {{dt}\over t} K_D(x,x,t)
\end{equation}
Let $\tilde K_D$ be an asymptotic approximation to $K_D$, valid
for short times. We may then write the integral in
\ref{Alvarezcorr} as
\begin{eqnarray}
\label{intdivided}
\hbox{Finite}_{\epsilon\to 0}\int_{\epsilon}^{1} {{dt}\over t} \tilde
K_D(x,x,t)
+ \int_{0}^{1} {{dt}\over t} [K_D(x,x,t) - \tilde K_D(x,x,t)]
+ \int_{1}^{\infty} {{dt}\over t} K_D(x,x,t)
\end{eqnarray}
The second and third terms are finite, if the asymptotic
approximation is sufficiently good. Hence they
can only contribute to $Z_D'(x,x,0)$ as a smooth density.
The only term which can give a distribution contributions,
the space integral of which contains a finite piece even if
taken over a vanishingly small area,
is the first in \ref{intdivided}, and the term $\gamma c_0(x)$. Both
only depend on the heat kernel for short times.
When we integrate \ref{Alvarezcorr} over the whole domain,
we obtain
\begin{equation}
Z_D'(0) =
\gamma Z_D(0)
+ \hbox{Finite}_{\epsilon\to 0} Tr_{x}
\int_{\epsilon}^{\infty} {{dt}\over t} K_D(x,x,t)
\end{equation}
\section* {A2. Variation and parametrization}
\label{s:app2}
\renewcommand{\theequation}{A2.\arabic{equation}}
\setcounter{equation}{0}
We normalise the areas by choosing the simplest form of the
Schwarz--Cristoffel transformations that map the disc--like
domains to
the upper half complex plane. Let $z$ be the variable in the domain,
and $\omega$ the variable in the upper half plane. Then we
have:
\begin{equation}
\label{SC}
{{d\omega}\over{dz}}
= \phi_D\prod_j (\omega-\omega_j)^{1-\alpha_j}
\end{equation}
The vertices of the domain map to
the branch points $\omega_j$, which are determined up to a rational
fractional transformation of the upper half plane.
This redundancy may be eliminated (up to permutations)
by fixing the positions
of the images of three vertices.
An overall
phase factor $\phi_D$ is determined by the orientation of
the domain $D$ in the $z$--plane.
For the simplest case, triangles, the three branch points may
be put in $0$, $1$ and $\infty$ in the $\omega$--plane, and the
side of the triangle
between the vertices that are mapped to $0$ and $1$ can be
placed parallel to the real axis in the $z$--plane.
We then have the familiar formula:
\begin{equation}
\label{triangleSC}
{{d\omega}\over{dz}}
= \omega^{1-\alpha_0}(1-\omega)^{1-\alpha_1}
\end{equation}
A variation of the shape of the domain,
brings a simultaneous variation of all the
angles, and of the positions of the branch points:
the expression for the variation hence concretely depends
on the parametrization chosen of the branch points in
terms of the opening angles and the side lengths.
Let us first consider the behaviour close to a branch point, say
$\omega_0$. By a linear change of variables,
$\tilde\omega = \omega -\omega_0$ before the variation,
and
$\tilde\omega = \omega -\omega_0'$ after the variation,
we can consider
the branch point $\omega_0$ to lie at the origin, and
not to move under the variation.
Taking a small corner around the inverse image of the branch point
(in the $z$--plane), and ignoring terms that vary slowly over the
corner, we have
\begin{eqnarray}
\label{varycorner}
2\delta\sigma_{\hbox{corner}}
= \log |{{dz'}\over{dz}}|^2 = \log |{{dz'}\over{d\tilde\omega}}|^2
- \log |{{dz}\over{d\tilde\omega}}|^2 = \nonumber \\
\delta\alpha_0\log |\alpha_0 z|^{2\over{\alpha_0}}
+ \sum_{j\neq 0} \delta\alpha_j\log |\omega_j-\omega_0|^2
- \sum_{j\neq 0} (1-\alpha_j)
[{{\delta(\omega_j-\omega_0)}\over{\omega_j-\omega_0}} + \hbox{c.c}]
\end{eqnarray}
In the bulk (i.e. away from the corners), we do not shift the
$\omega$--coordinates, and have instead:
\begin{eqnarray}
\label{varybulk}
2\delta\sigma_{\hbox{bulk}}
= \log |{{dz'}\over{dz}}|^2 = \log |{{dz'}\over{d\omega}}|^2
- \log |{{dz}\over{d\omega}}|^2 = \nonumber \\
\sum_{j} \delta\alpha_j\log |\omega-\omega_j|^2
- \sum_{j} (1-\alpha_j)
[{{\delta\omega_j}\over{\omega-\omega_j}} + \hbox{c.c}]
\end{eqnarray}
However, by Appendix 3, in piece--wise flat domains, the only
bulk contribution will arise from the normal derivative
at the boundary of \ref{varybulk}, so let us compute that
(in the $\omega$-plane, that is, at the real axis):
\begin{eqnarray}
\label{normderiv}
2\hat n\cdot\nabla \delta\sigma_{\hbox{bulk}}=
-2\sum_{j} \delta\alpha_j{{\Im\omega_j}\over{ |\omega-\omega_j|^2}}
+2 \sum_{j} (1-\alpha_j)
\Im[{{\delta\omega_j}\over{(\omega-\omega_j)^2}}]
\end{eqnarray}
If $\omega_j$ is on the real axis, \ref{normderiv} vanishes
identically. If $\omega_j$ lies in the upper complex plane
(an interior corner) we are to integrate the normal
derivative along the boundary. The second sum in
\ref{normderiv} then clearly gives zero, while the first
integrates to
\begin{eqnarray}
\label{normintegrate}
\int_R 2\hat n\cdot\nabla \delta\sigma_{\hbox{bulk}} ds=
-4\pi\sum_{\Im\omega_j > 0} \delta\alpha_j
\end{eqnarray}
For simplices with disc topology, all the contributions
except the rather simple result \ref{normintegrate}
therefore
come from the corners, and
for polygons, there are no interior corners, and
\ref{varycorner} gives the only relevant terms.
We have not been able to put \ref{varycorner} in a form
which is manifestly a total variation, but in all the
special cases where we checked it, it turns out to
be so. In some cases
it is possible to choose the parametrization of a family
of polygons
such that the corners on the boundary do not move: in this
case all the contributions are from the first two terms
in \ref{varycorner}.
For triangles we have the final simplification in that we can
use the parametrization
(\ref{triangleSC}), and then the only contributions are from the
first term in \ref{varycorner}.
\section* {A3. Contribution of a straight boundary}
\label{s:app3}
\renewcommand{\theequation}{A3.\arabic{equation}}
\setcounter{equation}{0}
On piece--wise flat surfaces, away from the corners,
we can always approximate the heat kernel with the
free space heat kernel, or the heat kernel close to a
straight boundary. The error term ($K_D - \tilde K_D$
in \ref{intdivided}) is exponentially small in ${1\over t}$.
In free space the diagonal elemeent of the heat kernel
is $K_D(x,x,t) = {1\over{4\pi t}}$. That leads to a contribution
to $Z(x,x,s)$ from $\tilde K_D$
in \ref{intdivided}, which has
a singularity at $s=1$ with residue ${1\over{4\pi}}$,
but which is regular at lower $s$.
Inside the piece--wise flat surfaces
there are therefore no contributions to $Z(0)$,
or to the variation of $Z'(0)$.
The heat kernel in the half plane with Dirichlet conditions
on the $x$--axis is expressed as the difference between a
``direct'' and a ``reflected'' term:
\begin{equation}
K((x_1,y_1),(x_2,y_2),t) = {1\over{4\pi t}}
(\exp(-{{(x_1-x_2)^2+(y_1-y_2)^2}\over{4 t} })-
\exp(-{{(x_1-x_2)^2+(y_1+y_2)^2}\over {4t} }))
\end{equation}
The diagonal elements are then
\begin{equation}
K((x,y),(x,y),t) = {1\over{4\pi t}}
(1-\exp(-{{y^2}\over t }))
\end{equation}
We compute the Mellin transform multiplied
with a convergence factor $\exp (-\mu t)$, and $s$
between ${1\over 2}$ and $1$:
\begin{equation}
{1\over {\Gamma(s) 4\pi}}\int_0^{\infty} dt t^{s-1}
{{ 1 - \exp(-{{y^2}\over t }})\over{t}} e^{-\mu t}
= - {{\Gamma(1-s)}\over{4\pi\Gamma(s)}} ({{y^2}})^{s-1} +
{\cal O} (\mu^{2-s},\mu)
\end{equation}
To investigate the contributions to $Z(s)$ we may integrate $y$
over a strip along the boundary with width $\delta$ and length $L$.
We then obtain:
\begin{eqnarray}
- L{{\Gamma(1-s)}\over{4\pi\Gamma(s)}} {{\delta^{2s-1}}\over{2s-1}}
\end{eqnarray}
When we continue to lower values of $s$ we find
a pole at $s={1\over 2}$ with residue
$-{L\over{8\pi}}$
which implies that the short term asymptotics of the heat kernel
has one term which is a line density at
the boundary, with coefficient $-t^{-{1\over 2}}{1\over{8\pi^{1\over 2}}}$.
At the origin, the contribution goes as $s$, hence there is no
contribution to $Z_D(0)$ from a straight edge. We obtain contributions
to $Z_D'(0)$ as
\begin{eqnarray}
\label{boundary}
{L\over{4\pi\delta}}
\end{eqnarray}
The interpretation of \ref{boundary} is, that apart from the
finite and regular terms in $Z'(x,x,0)$, there is a distribution,
which is defined by the analytic continuation of $y^{2s-2}$.
If we integrate over strips parallel to the boundary, the
terms as in \ref{boundary} would cancel pairwise, and we would
be left with the outer boundary term. Hence we conclude that
a straight boundary gives a contribution to $Z_D'(x,x,0)$
which is no more singular than a line density, and
hence gives a contribution proportional to boundary length.
In general
one could integrate $Z_D((x,y),(x,y),s)$ against a smooth
test--function $f(x,y)$,
with $s$ in the interval $[{1\over 2}, 1]$,
and then analytically continue to smaller $s$. Such a test--function
is the conformal variation $\delta\sigma(x,y)$. Assuming an expansion
normal to the boundary at $x$ as
$2\delta\sigma(x,y) = a_0(x) + y\cdot a_1(x) + \ldots$ we find
\begin{eqnarray}
\label{distr}
\int_0^L dx\int_0^{\delta} dy 2\delta\sigma(x,y) Z((x,y),(x,y),s) \nonumber \\
= \int_0^L dx\int_0^{\delta} dy
[ - {{\Gamma(1-s)}\over{4\pi\Gamma(s)}} ({{y^2}})^{s-1} ]
[ a_0(x) + y\cdot a_1(x) + \ldots] \nonumber \\
= - [{{\Gamma(1-s)}\over{4\pi\Gamma(s)}} {{{\delta}^{2s-1}}\over{2s-1}}
\int_0^L dx a_1(x)] -
[{{\Gamma(1-s)}\over{4\pi\Gamma(s)}} {{{\delta}^{2s}}\over{2s}}
\int_0^L dx a_1(x)] + \ldots
\end{eqnarray}
The contribution to the variation of $Z'_D(0)$ is the limit of
\ref{distr} as $s$ tend to zero, which may clearly be expressed in
the normal derivative at the boundary of the conformal variation:
\begin{eqnarray}
\delta Z'_{\hbox{straight boundary}}(0) = -{1\over{4\pi}}
\int_0^L \hat n \cdot \nabla(\delta\sigma) ds
\end{eqnarray}
\section* {A4. The Sommerfeldt kernel}
\label{s:app4}
\renewcommand{\theequation}{A4.\arabic{equation}}
\setcounter{equation}{0}
A. Sommerfeldt in 1896 solved the problem of diffraction of light
by a perfectly conducting half--plane\cite{SOpt}. The solution
takes the form of a kernel periodic in the angle variable with periodicity
$4\pi$; the difference of one ``direct'' and one ``reflected'' wave
vanishes at $0$ and $2\pi$.
We will need the solution to the diffusion problem in a sector with
opening angle $\pi\alpha$, which is quite analogous, but for completeness
given here. (The solution of diffusion problem at an interior
corner with total angle $2\pi\alpha$ is obtained in the same way,
by keeping only the ``direct'' term.)
If the opening angle is of the form ${{\pi}\over n}$ the sector
can be refleced in its side $2n$ times to precisely cover $2\pi$, and the
solutions to both the diffusion problem and the diffraction problem are
expressed by the method of images. Sommerfeldts solution is a substitute
when the reflections do not make up a full turn, and is given by
a certain finite number of image charges, and a correction term.
It has the following integral representation:
\begin{equation}
K_S(r,\phi ; r',\phi' ; t) = {1\over {4\pi t}}
\exp ( - { {r^2+{r'}^2}\over{4t} })
[ \nu_{\alpha}({{rr'}\over{2t}},\phi-\phi')
- \nu_{\alpha}({{rr'}\over{2t}},\phi+\phi')]
\end{equation}
where the important part is
\begin{equation}
\nu_{\alpha}(a,\phi) = {1\over {2\pi\alpha}}
\int_{A+B} \exp ( a\cos \delta ) {{d\delta}\over
{1-e^{-{{i(\delta +\phi)}\over\alpha}}}}
\end{equation}
$A$ and $B$ are paths in the plane of complex $\delta$ that go asymptotically
from $\pi + i\cdot\infty$ to $-\pi + i\cdot\infty$, and $-\pi - i\cdot\infty$
to $\pi - i\cdot\infty$, respectively.
Essentially this is a superposition of free heat kernels between $ x$ and
$y'$, where $|y'| = r'$ and
$| x-y'|^2 = r^2+r'^2 - 2rr'\cos(\delta )$. In the bands
${{(4n+1)\pi}\over 2} < |Re(\delta )| < {{(4n+3)\pi}\over 2}$
the contour integral can be taken to infinity since
$\Re(| x-y'|^2 ) \to \infty$,
but not in between.
$K_S$ satisfies the heat equation because it is superposition of free heat
kernels, it is symmetric in $(x,y)$
and periodic in $\phi$ and $\phi'$ with period $2\pi\alpha$
by construction of $\nu_{\alpha}$,
and it vanishes at the boundaries of the sector because it is the
difference between a direct and a reflected term.
Furthermore, away from the imaginary axis it is analytic in $\alpha$.
By deforming $A$ and $B$ into the straight lines $\pi + iy$, $-\pi+iy$ and
$[-\pi,\pi]$, $\nu_{\alpha}$ can be written as
\begin{eqnarray}
\nu_{\alpha}(a,\phi) = \sum_{k: -1<2\alpha k - \phi/\pi < 1}
\exp(a\cos(2\pi\alpha k - \phi )\nonumber \\
- {{\sin(\pi/\alpha )}\over {2\pi\alpha}}
\int_{-\infty}^{\infty}{{\exp(-a\cosh y) }\over{\cosh (y/\alpha - i\phi/\alpha)
-\cos \pi/\alpha }} dy
\end{eqnarray}
{}From the image charges of the ``direct'' term, it
follows that the normalization of the kernel is correct.
If for some $k$ an image charge wanders
through the line $\pm \pi$,
we ought to take half the residue and the principal value of the integral
in the usual fashion, but then
$\alpha$ equals ${{\pi}\over n}$,
and the prefactor of the integral is zero: that is;
we have just a solution by images.
\section* {A5. A useful integral}
\label{s:app5}
\renewcommand{\theequation}{A5.\arabic{equation}}
\setcounter{equation}{0}
In this appendix we discuss an elementary but useful integral. We would
like to compute a contribution to $Z_D(0)$ or $Z_D'(0)$ by
substituting the true (unknown) heat kernel in a compact
domain, with essentially something
like a heat kernel in the full plane. This includes the case with
a flat boundary, where the heat kernel can be written as the difference
between a free heat kernel and an image charge reflected over the
boundary, and also the case in a sector, since the
Sommerfeldt heat kernel can be written as a linear
superposition of free heat kernels.
The first problem is that the eigenvalues in an infinite domain
accumulate to zero; the Mellin transforms are therefore not
convergent for any parameter $s$. We are however interested
in Mellin transforms of heat kernels in compact domains, which are
asymptotic to free heat kernels for short times, but fall of
exponentially for large times. Furthermore, we are only
interested in the singular contribution, so the problem can be
resolved by limiting the integral in the Mellin transform to the interval
$[0,1]$. In practice such an integral is not analytically tractable. Instead
we can introduce a convergence factor $e^{-\mu t}$, including
which the Mellin transform converges for $Re(s) > 1$,
and look for the singular part in
the $\mu$--independent part of the result.
We therefore consider
\begin{equation}
\int_0^{\infty} t^{s-1} e^{-{a\over t}} e^{-\mu t} dt
= 2({a\over{\mu}})^{s\over 2} K_s((4a\mu )^{1\over 2})
= \Gamma (s) \mu^{-s} + \Gamma(-s) ({a})^{s} + {\cal O} (\mu^{1-s},\mu)
\end{equation}
by the power series expansion of the modified Bessel function.
Two variants, with $s$ around the origin, are
\begin{equation}
{1\over{\Gamma(s)}}\int_0^{\infty} t^{s-1}(1 - e^{-{a\over t}}) e^{-\mu t} dt
= -{{\Gamma(-s)}\over {\Gamma(s)}} ({a})^{s} + {\cal O} (\mu^{1-s},\mu)
\end{equation}
and
\begin{equation}
{1\over{\Gamma(s)}}\int_0^{\infty} t^{s-1}{{e^{-{a\over t}}}\over t} e^{-\mu t}
dt
= {1\over{(s-1)}}\mu^{1-s} + {{\Gamma(1-s)}\over {\Gamma(s)}}({a})^{s-1} +
{\cal O} (\mu^{2-s},\mu)
\end{equation}
As $\mu$ tends to zero, in the first case, the value at zero is 1,
and the derivative is
$2\gamma + \log a $ by the expansion
$\Gamma (s) \sim {1\over s} - \gamma + \ldots$
In the second case the value at zero vanishes, but the derivative is
${1\over a}$.
\section* {A6. The singular corner contribution}
\label{s:app6}
\renewcommand{\theequation}{A6.\arabic{equation}}
\setcounter{equation}{0}
We will here compute the contributions to $Z_D(0)$
and $\delta Z_D'(0)$ from a corner.
For $Z_D(0)$ we are by Appendix 2
and Appendix 4 looking
for the limit as $s$ tends to zero of
\begin{equation}
\label{z0contribution}
\int_{\hbox{corner}} d^2 x Z(x,x,s)
\end{equation}
where the zeta function density is computed from
the Sommerfeldt heat kernel, and the corner is delimited
by a cut-off radius $r_0$.
For $\delta Z_D'(0)$ we use the variational
formula (\ref{varyPERS}), and look
for the term finite in $s$ as $s$ tends
to zero of
\begin{equation}
\label{zprim0contribution}
\int_{\hbox{corner}} d^2 x [2\delta\sigma(x)Z(x,x,s)]
\end{equation}
An inspection of \ref{varycorner} shows that the conformal
factor close to the corner contains two types of terms, one
which is proportional to the logarithm of the distance to
the tip of the corner, and one which is constant over the corner.
We choose in the following to keep the constant factors
in first term in \ref{varycorner}, so that what we compute
will directly be the stricly local
contribution to $\delta Z_D'(0)$, which can then be given in
integrated form. The contributions of the last two terms
in \ref{varycorner} will be proportional to the
contribution to $Z_D(0)$, and are easily included by
comparison.
Our computations therefore boil down to \ref{z0contribution}
and
\begin{eqnarray}
\label{intonconstant}
\int_0^{\pi\alpha} \int_0^{r_0}r d\phi dr [{{\delta\alpha}\over{\alpha}}
2\log\alpha Z_{\alpha}(x,x,s) ] \\
\label{intonlog}
\int_0^{\pi\alpha} \int_0^{r_0}r d\phi dr [{{\delta\alpha}\over{\alpha}}
\log r^2 Z_\alpha (x,x,s) ]
\end{eqnarray}
where we can directly integrate over
$\phi$ since our test functions (a constant and $\log r^2$) do not
depend on $\phi$.
It will turn out that the reflected term in the Sommerfeldt
kernel only contributes to $\delta Z_D'(0)$, and
only by a simple term.
We can therefore in parallel compute the contributions to a corner
at the boundary with opening angle $\pi\alpha$, and a corner in the interior
with opening angle $2\pi\alpha$, which almost only differ by a factor
of two.
By analyticity in $\alpha$ we can choose the convenient range
${1\over 2} < \alpha < 1$.
\subsection*{The image charges}
The charge from the direct term, and the
image charges from the reflected
term give contributions to the Mellin
transform, integrated over the area of the corner, as
\begin{equation}
\label{refintegral}
{1\over {4\pi t}}
\int_0^{r_0} r dr \int_0^{\pi\alpha} d\phi [
1 -
\sum_{k: -1<2\alpha k - 2\phi/\pi < 1}
\exp(-{{r^2}\over{2t}}(1-\cos(2\pi\alpha k - 2\phi )))]
\end{equation}
The direct term charge can be neglected, as along a straight boundary.
With ${1\over 2} < \alpha < 1$ the only relevant values of $k$
in the reflected term are
$0$ and $1$.
For $k=0$ the integral over $\phi$ is limited between 0
and ${{\pi}\over 2}$, while
for $k=1$ it goes between $\pi\alpha - {{\pi}\over 2}$ and $\pi\alpha$. With
$y=r\sin(\phi )$ in the first case, and $y=r\sin(\phi - \pi\alpha )$
in the second, we find with $s$ between $1\over 2$ and $1$:
\begin{eqnarray}
\int_0^{{\pi}\over 2} d\phi r Z_{\alpha}((r,\phi),(r,\phi),s) &=&
- {{\Gamma(1-s)}\over{4\pi\Gamma(s)}}r^{2s-1}
2\int d\phi \sin^{2s-2}\phi \nonumber \\
&=&-{{\Gamma(1-s)r^{2s-1}
\Gamma({1\over 2})\Gamma(s-{1\over 2})}
\over
{4\pi\Gamma(s)\Gamma(s)}}
\end{eqnarray}
When we integrate up to a radius $r_0$
and continue to lower $s$ we find, as we should, a pole at $s={1\over 2}$
with residue $-{{2r_0}\over{8\pi}}$.
To investigate the contributions close to zero, we use
the formulae
\begin{eqnarray}
\label{rintegrations}
\int_0^{r_0} dr r^{2s-1} = {{r_0^{2s}}\over{2s}}
= {1\over {2s}} + {1\over 2}\log r_0^{2} + {\cal O}(s) \nonumber \\
\int_0^{r_0} dr r^{2s-1}\log r^2 =
{{\partial}\over{\partial s}} [{{r_0^{2s}}\over{2s}}]
= -{1\over {2s^2}} + {1\over 4}(\log r_0^{2})^2 + {\cal O}(s) \nonumber
\end{eqnarray}
where we understand that we first integrate over $r$, and then
analytically continue to the neighbourhood of the origin.
The contribution to $\delta Z'_{\alpha}(0)$ from the reflected image
charges (hence only for boundary corners) is thus
\begin{eqnarray}
\label{imagecontr}
- {{\delta\alpha}\over{4\alpha}}
\end{eqnarray}
\subsection*{The reflected term: the integral}
The integral part of the reflected term comes from
\begin{equation}
{{\sin \pi/\alpha}\over {8\pi^2\alpha t}}
\int_0^{\pi\alpha} r d\phi \int_{-\infty}^{\infty}
{{\exp (-{{r^2 (1 + \cosh y )}\over{2t}} )}\over
{\cosh (y/\alpha - 2i\phi/\alpha ) - \cos \pi/\alpha }} dy ,
\end{equation}
where we have integrated
over the angle variable $\phi$.
The $\phi$--integration forms part of closed contour from $-i\infty$ to
the origin, along the real axis to $\pi\alpha$, and down parallel to the
imaginary axis to $\pi\alpha - i\infty$. The integrals over the two lines
cancel. If $y>0$ there are two poles inside the contour, but their
residues cancel. The angle integral is thus zero.
\subsection*{The direct term: the integral}
We consider
\begin{eqnarray}
\label{direct1}
-{{\sin \pi/\alpha}\over {8\pi^2\alpha t}}
\int_0^{\pi\alpha} r d\phi\int_{-\infty}^{\infty}
{{\exp (-{{r^2 (1 + \cosh y )}\over{2t}} )}\over
{\cosh y/\alpha - \cos \pi/\alpha }} dy
\end{eqnarray}
Following Appendix 2, the
Mellin transform of \ref{direct1},
performed termwise inside the integral over $y$,
gives
\begin{equation}
-r^{2s-1} {{\Gamma (1-s)\sin (\pi/\alpha) }\over {\Gamma (s) 8\pi }}
\int_{-\infty}^{\infty}
{1 \over
{(\cosh y/\alpha - \cos \pi/\alpha )(\cosh^2 y/2)^{1-s}}} dy
\end{equation}
The integral is most tractable if considered as a correlation
integral between the two factors in the denominator, then
by Percival's formula\footnote{This transformation brings us closer to
the Lebedev--Kantorovich tranform of D. Ray, that was
used by McKean and Singer in \cite{McKean}.}
we have
\begin{equation}
r^{2s-1}{{\Gamma (1-s) \alpha }\over {\Gamma (s) 8\pi }}
2^{2-2s}\int_{-\infty}^{\infty} B (1-s +iy,1-s-iy) [
{{\sinh \pi y \cosh \pi\alpha y - \sinh \pi\alpha y \cosh \pi y}\over
{\sinh \alpha\pi y }}] dy
\end{equation}
This expression can be written more compactly as
\begin{equation}
\label{Zscompact}
r^{2s-1}{{\Gamma (1-s) \alpha 2^{2-2s} }\over {\Gamma (s) 8\pi }}
A(s)
\end{equation}
where the numerical values are determined by the
integral
\begin{equation}
\label{Asdef}
A(s) = \int_{-\infty}^{\infty} B (1-s +iy,1-s-iy) [
{{\sinh \pi y \cosh \pi\alpha y - \sinh \pi\alpha y \cosh \pi y}\over
{\sinh \alpha\pi y }}] dy
\end{equation}
Using the identity
\begin{equation}
B (1+iy,1-iy) = {{\pi y}\over
{\sinh \pi y }}
\end{equation}
we have
\begin{equation}
A(0) = \int_{-\infty}^{\infty} [\pi y \coth \alpha\pi y -
\pi y \coth \pi y] dy \\
\end{equation}
and
\begin{equation}
A'(0) = \int_{-\infty}^{\infty}(2\psi(2)-\psi(1+iy) -\psi(1-iy)) [\pi y \coth
\alpha\pi y -\pi y \coth \pi y] dy
\end{equation}
where $\psi(x+1) = -\gamma +\sum_{n=1}^{\infty} ({1\over n} - {1\over {n+x}})$.
\subsection*{Contribution to $Z_D(0)$}
We integrate
\ref{Zscompact}
over the radial variable:
\begin{equation}
{{r_0^{2s}}\over{2s}}[{{s\alpha}\over{2\pi}} A(0) + {\cal O} (s^2)]
\end{equation}
$A(0)$ can be computed by introducing a factor $e^{i\epsilon y}$, and closing
the integral over $y$ in the upper half plane.
The residues contribute
\begin{eqnarray}
-2\pi \sum_{n=1}^{\infty} n({1\over{\alpha^2}}e^{-n\epsilon/\alpha} -
e^{-n\epsilon}) \nonumber \\
= -{\pi\over 2} ( {1\over {\alpha^2\sinh^2 (\epsilon/2\alpha )}}
- {1\over {\sinh^2 (\epsilon/2)}})
\end{eqnarray}
which, in the limit as $\epsilon \to 0$, becomes ${\pi\over 6}(
{1\over {\alpha^2}}
- 1)$
The value of $Z_{\alpha}(0)$ is therefore
\begin{eqnarray}
\label{z0magic}
{1\over 24}({1\over {\alpha}} - \alpha)
\end{eqnarray}
a result first derived by
McKean and Singer\cite{McKean}.
\subsection*{Contribution to $\delta Z_D'(0)$}
Using again \ref{rintegrations} and the expansion of $A(s)$
and the prefactors
at the origin, we obtain:
\begin{eqnarray}
\label{sexpansion}
[{{\delta\alpha}\over{\alpha}}]
[s{{\alpha}\over{2\pi}} A(0) +
{{s^2\alpha}\over{2\pi}}(A'(0) + A(0)(2\gamma -2\log 2))]
[- {1\over {2s^2}} + { { \log \alpha }\over s}] + \hbox{h.o.t}
\end{eqnarray}
We have a pole in $s$:
\begin{eqnarray}
-{1\over {s}} {{\delta\alpha}\over{\alpha}}{\alpha\over{4\pi}} A(0)
\end{eqnarray}
Since ${{\alpha}\over{4\pi}} A(0)$ is the contribution to $Z_D(0)$
from the corner, we can write this variation as
\begin{eqnarray}
\label{modvaryZ}
{1\over s} {{\delta\alpha}\over{24}}(-{1\over{\alpha^2}}+1)
= {1\over s} \delta [{1\over{24}}({1\over{\alpha}}+\alpha)]
\end{eqnarray}
Let us work out that this variation is only seemingly
in contradiction
with the known expression of $Z_{\alpha}(0)$ in \ref{z0magic}.
The variation of $Z_D(x,x,0)$
is of course a point mass at the corner, with weight
${{\delta\alpha}\over{24}}(-{1\over{\alpha^2}}-1)$.
The variational forms (\ref{varyT}) and (\ref{smallvaryT})
are however for the variation of $Z_D(s)$, not for the
zeta function density. The variations we consider go between
different simplices with disc--like topology, and therefore leave
the sum $\sum_{\hbox{i.c.}}(2-2\alpha_j) + \sum_{\hbox{b.c.}}(1-\alpha_j)$
invariant ({\it see} equation~\ref{invariant}).
The analytical expressions of the variation are then
only determined up to terms linear in the variations of the angles,
which is the discrepancy we have in \ref{modvaryZ}.
The contribution finite in $s$ in \ref{sexpansion} is
\begin{eqnarray}
\delta\alpha
[{{ A(0)}\over{4\pi}}(2\log\alpha-2\gamma + 2\log 2)
-{{ A'(0)}\over{4\pi}}]
\end{eqnarray}
We compute $A'(0)$ by
introducing a convergence factor $e^{i\epsilon y}$.
Since $\psi(\bar z ) = \bar \psi (z)$, it is enough to consider
\begin{equation}
C_n = \int_{-\infty}^{\infty}({1\over n} - {1\over {n-iy}}) [\pi y \coth
\alpha\pi y -\pi y \coth \pi y] dy
\end{equation}
in terms of which $A'(0) = 2 A(0) - 2\Re\sum_{n=1}^{\infty}C_n$.
The two terms in $C_n$ have residues in the upper half plane, and sum to
\begin{eqnarray}
&\, &-{{\pi}\over {2n}} ( {1\over {\alpha^2\sinh^2 (\epsilon/2\alpha )}}
- {1\over {\sinh^2 (\epsilon/2)}}) \nonumber \\
&\, &+ 2\pi \sum_{m=1}^{\infty}
m({1\over{\alpha^2(n+m/\alpha)}}e^{-m\epsilon/\alpha} -
{1\over{(n+m)}}e^{-m\epsilon}) \nonumber \\
&=& - {{\pi}\over {6n}} ( {1\over {\alpha^2}} - 1)
+ \pi(1-{1\over{\alpha}}) \nonumber \\
&\, & -2\pi ne^{\epsilon n}\int_{\epsilon}^{\infty}
e^{-\mu n} [{1\over{\alpha(e^{\mu/\alpha} -1) }}- {1\over{(e^{\mu} -1) }}] d\mu
+ {\cal O} (\epsilon )
\end{eqnarray}
With some care the sum in $n$ is:
\begin{eqnarray}
\label{cruzint}
&\, & \sum_{n=1}^{\infty} C_n = -2\pi I(\alpha) = \nonumber \\
&\,&-2\pi
\int_{0}^{\infty}
{1\over{e^{\mu} -1}}
[{1\over {4\sinh^2 (\mu/2)}} - {1\over {4\alpha^2\sinh^2 (\mu/2\alpha )}}
-{1\over {12}}({1\over{\alpha^2}}-1)] d\mu
\end{eqnarray}
Now we can collect all the strictly local contributions
to $Z_D'(0)$ and write
\begin{eqnarray}
\label{totalvary}
\delta Z_{\alpha}'(0) = -{{\delta\alpha}\over{4\alpha}} +
\delta\alpha [{1\over{12\alpha}}({1\over{\alpha}} - \alpha)
(\log\alpha-\gamma + \log 2 -1) - I(\alpha)]
\end{eqnarray}
Here we have taken a boundary corner, and hence included the
contribution from the reflected image charges.
It is clearly of interest to give \ref{totalvary} in
integrated form, whence we introduce the primitive function
of $I(\alpha)$:
\begin{eqnarray}
\label{Jdef1}
J(\alpha) =
\int_{0}^{\infty}
{1\over{e^{\mu} -1}}
[{1\over {2\mu}}\coth({{\mu}\over{2\alpha}}) -
{{\alpha}\over {4\sinh^2 ({{\mu}\over 2})}}
-{1\over {12}}({1\over{\alpha}}+\alpha)] d\mu \\ \nonumber
I(\alpha) = -{d\over{d\alpha}} J(\alpha)
\end{eqnarray}
We know from Appendix 3 that the contribution to $Z_D'(0)$
({\it note:} not the variation) of a straight boundary
is proportional to the boundary length. The integrated form
of \ref{totalvary} should therefore be zero at $\alpha=1$.
As in the variation of $Z_D(0)$, we should in addition
expect undetermined linear terms in the angles.
A rather long derivation, to be given in Appendix~8 as we
investigate the special values of $\alpha={1\over n}$, shows that
we may integrate \ref{totalvary}, compare with one of
the integrable cases, include the integration constant in
the contribution from the corners, and simplify to:
\begin{eqnarray}
\label{tibc}
Z'_{\alpha}(0) = {1\over{12}}({1\over{\alpha}} -\alpha)(\gamma -\log 2)
-{1\over{12}}({1\over{\alpha}} + 3 +\alpha)\log\alpha +\tilde J(\alpha)
\end{eqnarray}
where the last term has the more symmetric integral representation
\begin{eqnarray}
\label{Jdef2}
\tilde J(\alpha) =
\int_{0}^{\infty}
{1\over{e^{x} -1}}
[{1\over {2x}}(\coth({x\over{2\alpha}}) -
\alpha \coth({x\over{2}}))
-{1\over {12}}({1\over{\alpha}}-\alpha)] dx
\end{eqnarray}
\subsection*{Difference at interior corners}
The only change is that the angle--independent direct term
is integrated over twice as wide an angle, and the contribution
from the reflected image charges are absent.
The contribution to $Z_D(0)$ is then
\begin{eqnarray}
Z_{\alpha}(0) = {1\over {12}}({1\over {\alpha}} - \alpha)
\end{eqnarray}
and the strictly local contributions to $Z_D'(0)$
\begin{eqnarray}
\label{tiic}
Z'_{\alpha}(0) = {1\over{6}}({1\over{\alpha}} -\alpha)(\gamma -\log 2)
-{1\over{6}}({1\over{\alpha}} + \alpha)\log\alpha +2\tilde J(\alpha)
\end{eqnarray}
\subsection*{Summary of corner contributions}
For simplices with the disc topology, the contributions to the
variation of the zeta function derivative are given by \ref{varycorner}
and a simple sum over the interior corners.
There are three terms in \ref{varycorner}, of which the first is
the only one that arises in triangular domains. It is written
in final variational form valid for corners
on the boundary in \ref{totalvary}, \ref{cruzint}, and
in integrated form in \ref{tibc}, \ref{Jdef2}.
The last two terms in \ref{varycorner} are of the same type
as \ref{intonconstant}, i.e. in variation form they will
give something proportional to the contribution to $Z_D(0)$
from the corner. We give those expressions here, under the
(slight) simplifying assumption that none of the branch point
in the $\omega$--plane lies at infinity:
\begin{eqnarray}
\label{varyextra}
\delta Z'_{\hbox{extra}} = \sum_{i,j}
Z_{\alpha_i}(0)[
\delta\alpha_j\log |\omega_j-\omega_i|^{-2}
-(1-\alpha_j)
[{{\delta(\omega_j-\omega_i)}\over{\omega_j-\omega_i}} + \hbox{c.c}]]
\end{eqnarray}
\section* {A7. Computation using the heat kernel}
\label{s:app7}
\renewcommand{\theequation}{A7.\arabic{equation}}
\setcounter{equation}{0}
This approach was used by Dowker\cite{Dowker} to compute the
contribution to $Z_D(0)$ from a corner.
We begin with the asymptotic formulae for the trace
of the heat kernel in a domain $D$:
\begin{eqnarray}
\label{tracec}
Tr_x [ K_D(x,x,\epsilon)] &\sim& {{c_1 }\over {\epsilon}} +
{{c_{1\over 2} }\over {\epsilon^{1\over 2}}}
+Z_D(0) +
{\cal O}(\epsilon^{1\over 2})\\
\label{varytracec}
Tr_x[2\delta\sigma(x) K_D(x,x,\epsilon)] &\sim&
{{\delta[c_1] }\over {\epsilon}} +
{{2\delta[c_{1\over 2}] }\over {\epsilon^{1\over 2}}}
-\log\epsilon\, \delta[Z_D(0)]
-\gamma\delta [Z_D(0)] \nonumber \\
&+& \delta[Z_D'(0)] +
{\cal O}(\epsilon^{1\over 2})
\end{eqnarray}
As in Appendix~6 we only need to consider the image charges
from the reflected term and the integral from the
direct term.
\subsection*{The direct term: the integral}
It is convenient to go back to the original form
of the Sommerfeldt kernel (including the line integrals
and the direct charge):
\begin{equation}
K_{\hbox{direct}}(r,\phi ; r,\phi ; t) = {1\over {8\pi^2\alpha t}}
\int_{A+B}
{{d\delta}\over
{1-e^{-{{i\delta}\over\alpha}}}} e^{-{{r^2}\over{2t}}(1-\cos \delta)}
\quad - (\hbox{the pole at $\delta=0$})
\end{equation}
We are to integrate the kernel, by itself or
multiplied by ${{\delta\alpha}\over{\alpha}}
\log \alpha^2r^2$, over the corner, and then
catch the finite piece as $t$ tends to zero.
The integration contour is chosen such that the prefactor
of $r^2$ in the exponent always has negative real part.
We can therefore extend
the integral over all $r$
(as in Appendix~5), without introducing significant errors.
For the contribution to $Z_D(0)$ we then find
\begin{eqnarray}
\int_0^{\pi\alpha} r d\phi
\int_0^{r_0} K_{\hbox{direct}}(x,x,t)
= -{1\over{8\pi}}\int_{A'+B'}
{{d\delta}\over
{1-e^{-{{i\delta}\over\alpha}}}} {1\over{1-\cos \delta}}
+ {\cal O}(e^{-C\cdot {{r_0^2}\over t}}
\end{eqnarray}
where we have afterwards moved over the integration contour to the
straight lines parallel to the imaginary axis at $\pi$
and $-\pi$.
Except for an error exponentially small in ${1\over t}$,
we have only left a term independent of $t$, that will
give us what we want. By a change of variables,
$\delta\to -\delta$, the integration
contour $A'$ goes into $-B'$, and vice versa, and the second
factor in the integrand (${1\over{1-\cos \delta}}$) is unchanged.
It is therefore sufficient to keep in the first factor,
the part odd under this reflection, and we have
\begin{eqnarray}
-{1\over{16\pi i}}\int_{A'+B'}
d\delta \cot ({{\delta}\over{2\alpha}})
{1\over{1-\cos \delta}}
\end{eqnarray}
This integral is now in fact convergent at infinity, so we can close
it, and evaluate it by computing the residue of the pole at
the origin. Hence
\begin{eqnarray}
\int_0^{\pi\alpha} r d\phi
\int_0^{r_0} dr K_{\hbox{direct}}(x,x,t)
&=& -{1\over{8\pi}}2\pi i \hbox{Res}[
\cot ({{\delta}\over{2\alpha}})
{1\over{1-\cos \delta}}]_{\delta=0}\nonumber \\
&=& {1\over{24}}({1\over{\alpha}} - \alpha)
\end{eqnarray}
which is, of course, the right answer.
To compute the variation of $Z_D'(0)$, it is again convenient to
extend the integral over
the radius to infinity. We then find
\begin{eqnarray}
\int_0^{\pi\alpha} d\phi r
\int_0^{\infty}
{1\over{\alpha}}\log (\alpha^2 r^2) e^{-{{r^2}\over{2t}}(1-\cos\delta)}
= {{\pi t}\over{1-\cos\delta}}(-\gamma+\log {{2\alpha^2 t}
\over{1-\cos\delta}})
\end{eqnarray}
(we do not write out the variational factor $\delta\alpha$,
except where necessary),
and hence
\begin{eqnarray}
\label{smalltexpr}
\int_0^{\pi\alpha} r d\phi
\int_0^{r_0}dr [2\delta\sigma(x) K_{\hbox{direct}}(x,x,t)] =\nonumber \\
-{1\over{8\pi\alpha}}\int_{A'+B'}
{{d\delta}\over
{1-e^{-{{i\delta}\over\alpha}}}} {1\over{1-\cos \delta}}
(-\gamma+\log {{2\alpha^2 t}\over{1-\cos\delta}})
\end{eqnarray}
Pulling out the term divergent as $\log t$ we have
\begin{eqnarray}
\log t {{\delta\alpha}\over{\alpha}}
{1\over{24}}({1\over{\alpha}} - \alpha)
\end{eqnarray}
which is equal to $-\log t\, \delta[Z_{\alpha}(0)]$, up to the
term linear in the the angle.
Taking into account that the finite part of \ref{smalltexpr}
is $\delta[Z_{\alpha}'(0)] - \gamma \delta[Z_{\alpha}(0)]$ we have
\begin{eqnarray}
\delta[Z_{\alpha}'(0)] = \delta[Z_{\alpha}(0)](-\log (2\alpha^2) +2\gamma)
\nonumber \\
- {{\delta\alpha}\over{8\pi\alpha}}\int_{A'+B'}
{{d\delta}\over
{1-e^{-{{i\delta}\over\alpha}}}} {1\over{1-\cos \delta}}
\log {1\over{1-\cos\delta}}
\end{eqnarray}
We choose the branch of $\log {1\over{1-\cos\delta}}$ symmetric under
$\delta\to -\delta$. Then we can again substitute for
${1\over{1-e^{-{{i\delta}\over\alpha}}}}$ its anti-symmetric
part,
and the integral is convergent at infinity.
Essentially $\log {1\over{1-\cos\delta}}$ is a function of
$\delta^2$, so it can be chosen to have a branch cut along the
negative real axis in the $\delta^2$--plane. That translates
to two branch cuts in the $\delta$--plane, along the positive
and negative real axis. Evenness under $\delta\to -\delta$
then determines the phase choice in the left half plane.
The best we can do now is to
pull both integration
contours in to the imaginary axis and integrate
over the branch cuts. In addition we have a singularity
at the origin. We separate out a small circle around the origin
with radius $\epsilon$, and an integral along the branch cuts from
$\pm i\epsilon$ to $\pm i\infty$. The integrals along the branch cuts
give
\begin{eqnarray}
{1\over{4\alpha}}
\int_{i\epsilon}^{i\infty}
d\delta \cot({{\delta}\over{2\alpha}}) {1\over{1-\cos \delta}} \nonumber \\
= {1\over{4\alpha}}
-{1\over{\epsilon^2}} - {1\over{12}}({1\over{\alpha}} - \alpha)
+ {1\over{8\alpha^2}} \int_{\epsilon}^{\infty}
dy \coth({y\over{2}}) {1\over{\sinh^2({y\over{2\alpha}})}}
\end{eqnarray}
Close to the origin we estimate ($\delta = \epsilon e^{i\theta}$):
\begin{eqnarray}
\log {1\over{1-\cos\delta}} &=& \log 2 - 2\log\epsilon
-2i\theta + {1\over{12}}\delta^2 + \ldots
\qquad\hbox{$\Re\delta>0$} \nonumber \\
&=& \log 2 - 2\log\epsilon -2i\theta +2i\pi + {1\over{12}}\delta^2
+ \ldots
\qquad\hbox{$\Re\delta<0$} \nonumber
\end{eqnarray}
which eventually gives
\begin{eqnarray}
{1\over{24}}({1\over{\alpha^2}} -1) (\log 2 -2\log\epsilon)
- {1\over{24}} + {1\over{2\epsilon^2}}
\end{eqnarray}
Collecting all terms from the direct integral we have
\begin{eqnarray}
\delta [Z_{\alpha}'(0)]_{\hbox{direct integral}}
&=& \delta\alpha[ {1\over{24}}({1\over{\alpha^2}} -1)
(-2\gamma+2\log(2\alpha) -2\log\epsilon)\nonumber \\
&\, & - {1\over{2\epsilon^2}} +{1\over{4\alpha}}
-{1\over{12}}({1\over{\alpha^2}} + 1) \nonumber \\
&\, & + {1\over{8\alpha^2}} \int_{\epsilon}^{\infty}
dy \coth({y\over{2}}) {1\over{\sinh^2({y\over{2\alpha}})}}]
\end{eqnarray}
\subsection*{The image charges}
We should compute the integral
\begin{equation}
\delta [Z_{\alpha}'(0)]_{\hbox{image charges}}
= -\delta\alpha[ \int_0^{{\pi}\over 2} r d\phi
\int_0^{r_0} dr {1\over{\pi\alpha t}}
\log(\alpha r) e^{-{{r^2}\over{2t}}(1-\cos 2\phi)}]
\end{equation}
The problem is that we have to separate out a $r_0$-dependent piece
diverging as $t^{-{1\over 2}}$, before we can have the finite value
at the origin, and as it stands it is not possible to extend
the integral over $r$ to infinity. If the integral would have been
over a small square, we could have separated in $x$ and $y$,
and then the integral would be much easier. Let us first see that the
segment of the square outside the
quadrant is unimportant, and then compute the integral over
the square.
The difference is
\begin{equation}
- \int_0^{r_0} dy \int_{\sqrt{r_0^2-y^2}}^{r_0} dx {1\over{2\pi\alpha t}}
\log(\alpha^2(x^2+y^2) ) e^{-{{y^2}\over t}}
\end{equation}
which is essentially
\begin{equation}
- \int_0^{r_0} dy {{y^2}\over{4\pi\alpha t r_0}}
[\log(\alpha^2 r_0^2) +{{y^2}\over{r_0^2}} + \ldots] e^{-{{y^2}\over t}}
\end{equation}
which only gives a contribution of order $t^{1\over 2}$.
The integral over the square is
\begin{equation}
- \int_0^{r_0} dx \int_0^{r_0} dy {1\over{2\pi\alpha t}}
\log(\alpha^2(x^2+y^2) e^{-{{y^2}\over t}}
\end{equation}
After first integrating over $x$, we can extend the integral over $y$
to infinity and find:
\begin{equation}
- \int_0^{\infty} dx {1\over{2\pi\alpha t}}
(\log\alpha^2 +2r_0(\log r_0 -1) +\pi y + \ldots)e^{-{{y^2}\over t}}
\nonumber \\
= \hbox{Const.}\cdot t^{-{1\over 2}} - {1\over{4\alpha}}
\end{equation}
\subsection*{Summary}
The terms diverging with the cutoff $\epsilon$ can be brought
into the integral, and then give, after some algebra,
\begin{equation}
\delta [Z_{\alpha}'(0)]
= \delta\alpha[ -{1\over{12}}({1\over{\alpha^2}} -1)
(1+\gamma-\log 2\alpha) - {1\over{4\alpha}} + J'(\alpha)]
\end{equation}
which is the same result as in \ref{totalvary}.
\section* {A8. Corners with opening angle ${1\over n}$}
\label{s:app8}
\renewcommand{\theequation}{A8.\arabic{equation}}
\setcounter{equation}{0}
We proceed to simplify the the general expressions
of the contributions to the
regularized determinants, when the angles are of the form ${{\pi}\over n}$.
We recall that the contribution to $Z_D'(0)$ from a
(boundary) corner
is
\begin{eqnarray}
A + B\alpha -{1\over 4}\log\alpha +
{1\over{12\alpha}}(\gamma -\log 2 -\log\alpha)
-{1\over{12}}\alpha\log\alpha+J(\alpha) \nonumber
\end{eqnarray}
where the last term has the integral representation
\begin{eqnarray}
\label{Jdefhere}
J(\alpha) =
\int_{0}^{\infty}
{1\over{e^{\mu} -1}}
[{1\over {2\mu}}\coth({x\over{2\alpha}}) -
{{\alpha}\over {4\sinh^2 ({{\mu}\over 2})}}
-{1\over {12}}({1\over{\alpha}}+\alpha)] d\mu
\end{eqnarray}
and have included an undetermined term linear in the
angles ($B\alpha$), and an integration constant, that we
will not need explicitly ($A$).
The analysis proceeds by computing the integral for
$\alpha$ being ${1\over n}$. Let $y$ stand for $e^{\mu}$.
Decomposing various rational functions of $y$:
\begin{eqnarray}
\log (y^n-1) &= &\sum_{v=0}^{n-1} \log (y-\lambda_v)
\qquad\qquad \lambda_v = e^{{2\pi i v}\over n}\nonumber \\
{n\over{y^n-1}} &= &\sum_{v=0}^{n-1} {1\over{\lambda_vy-1}}\nonumber \\
{1\over {y-1}}({n\over{y^n-1}} +{n\over 2}) &=
&{1\over{(y-1)^2}} + {1\over{2(y-1)}} -
\sum_{v=1}^{n-1} {{\lambda_v}\over{1-\lambda_v}}
{1\over{y-\lambda_v}}\nonumber \\
\sum_{v=1}^{n-1} {{\lambda_v}\over{1-\lambda_v}}
&= &{{1-n}\over{2}} \nonumber
\end{eqnarray}
we can absorb the terms proportional to ${1\over n}$ into $B$, and
write the modified integral:
\begin{eqnarray}
\label{tildeJdefn}
\tilde J({1\over n}) =
{1\over n}\int_{\epsilon}^{\infty}
[-{1\over {x}}
\sum_{v=1}^{n-1} {{\lambda_v}\over{1-\lambda_v}}
{1\over{y-\lambda_v}} - {{n^2-1}\over{12(y-1)}}]
dx
\end{eqnarray}
The cut--off $\epsilon$ is introduced for later convenience.
In fact, we can go backwards and express \ref{tildeJdefn} as
\begin{eqnarray}
\label{tildeJdefc}
\tilde J(\alpha ={1\over n}) =
\int_{0}^{\infty}d\mu
{1\over{e^{\mu}-1}}
[({1\over {2\mu}})(\coth({{\mu}\over{2\alpha}})
- \alpha\coth({{\mu}\over{2}})) -
{1\over{12}}({1\over{\alpha}}-\alpha)]
\end{eqnarray}
Since up to simple terms in $\alpha$ that
we keep, \ref{tildeJdefc} and
\ref{Jdefhere} agree on the integers, and both expressions are
analytic in ${1\over{\alpha}}$, then they must agree overall.
We are therefore allowed to substitute for \ref{Jdefhere}
the more symmetric expression \ref{tildeJdefc}.
Continuing the analysis of \ref{tildeJdefn} we use
\begin{eqnarray}
\int_{\epsilon}^{\infty} {{dx}\over x}
{{\lambda_v}\over{y-\lambda_v}} &=&
\int_{n\epsilon}^{\infty} {{dx}\over x} \sum_{p=1}^n \lambda_{vp}
[{{e^{(1-{p\over n})x}}\over{e^x-1}}] \nonumber \\
\int_{\epsilon}^{\infty} {{dx}\over x}
{{e^{(1-a)x}}\over{e^x-1}} &=& {1\over{\epsilon}}
-({1\over 2} -a)(\gamma + \log\epsilon) + \log\Gamma(a)
-{1\over 2}\log (2\pi) + \hbox{small as $\epsilon$}\nonumber \\
\sum_{v=1}^{n-1} [{{\lambda_{vp}}\over{1-\lambda_v}}]
&=& p - {{1+n}\over 2} \nonumber
\end{eqnarray}
and express the integral $\tilde J$ as
\begin{eqnarray}
\label{tildeJdef2}
\tilde J({1\over n}) =
-{1\over n}\sum_{v=1}^{n-1}\sum_{p}^{n}
{{\lambda_{vp}}\over{1-\lambda_v}}
\int_{n\epsilon}^{\infty}
{{dx}\over x}
{{e^{(1-{p\over n})x}}\over{e^x-1}} + {{n^2-1}\over{12n}}\log\epsilon
\end{eqnarray}
which may finally be simplified to
\begin{eqnarray}
\label{tildeJdef3}
\tilde J({1\over n}) =
{{1-n^2}\over{12n}}(\gamma+\log n) -{1\over{4n}}\log n +
{1\over 4}(1-{1\over n})\log(2\pi) + \sum_{p=1}^{n-1}
({1\over 2} - {p\over n})\log\Gamma({p\over n})
\end{eqnarray}
Absorbing further terms proportional to $1\over n$ into $B$,
we have finally
\begin{eqnarray}
\label{finalexpr}
Z'_{1\over n}(0) = \hat A + \hat B{1\over n} +
{{1-n}\over{12}}\log 2 + ({1\over 4} - {1\over{12n}})\log n +
\sum_{p=1}^{n-1}
({1\over 2} - {p\over n})\log\Gamma({p\over n})
\end{eqnarray}
\subsection*{The integration constant}
We need one exact value to fix an integration constant.
For the equlateral triangle, the regularized determinant
is ({\it see} Appendix~11):
\begin{eqnarray}
Z'_{\hbox{equilateral}}(0)
= {1\over 2}\log\pi -{1\over 6}\log 2 + {2\over 3}\log 3 + {1\over 2}
\log {{\Gamma({1\over 3})}\over {\Gamma({2\over 3})}}
\end{eqnarray}
If we on the other hand use \ref{finalexpr}, we find
\begin{eqnarray}
Z'_{\hbox{equilateral}}(0) = 3Z_{1\over 3}'(0) =
-{1\over 2}\log 2 + {2\over 3}\log 3 + {1\over 2}
\log {{\Gamma({1\over 3})}\over {\Gamma({2\over 3})}}
+ 3\hat A + \hat B
\end{eqnarray}
so we have the integration constant
\begin{eqnarray}
3\hat A + \hat B = {1\over 3}\log 2 + {1\over 2}\log\pi
\end{eqnarray}
\subsection*{Simplified general expression}
We choose to include the integration constant in the
stricly local contribution to $Z_D'(0)$, by adding
\begin{eqnarray}
(1-\alpha)[{1\over 6}\log 2 + {1\over 4}\log\pi]
\end{eqnarray}
to a boundary corner (and twice this quantity to an interior
corner).
The final expression with an opening angle ${1\over n}$ then reads
\begin{eqnarray}
\label{finalfixed}
Z'_{1\over n}(0) = (1-{1\over n})({1\over 6}\log 2 + {1\over 4}\log\pi)
+
{{1-n}\over{12}}\log 2 + ({1\over 4} - {1\over{12n}})\log n
\nonumber \\
+ \sum_{p=1}^{n-1}
({1\over 2} - {p\over n})\log\Gamma({p\over n})
\end{eqnarray}
For a general opening angle we go back to
\ref{tildeJdefc}, keep track of the various terms proportional to
${1\over n}$ that had been absorbed into $B$, and arrive at
\begin{eqnarray}
\label{totalintegratedbc}
Z'_{\alpha}(0) = {1\over{12}}({1\over{\alpha}} -\alpha)(\gamma -\log 2)
-{1\over{12}}({1\over{\alpha}} + 3 +\alpha)\log\alpha +\tilde J(\alpha)
\end{eqnarray}
Throughout this appendix, we have simplified on the integral $J$, which
is the same for corners on the boundary and in the interior. The
strictly local contribution to $Z_D'(0)$ from a corner in the
interior is thus
\begin{eqnarray}
\label{totalintegratedic}
Z'_{\alpha}(0) = {1\over{6}}({1\over{\alpha}} -\alpha)(\gamma -\log 2)
-{1\over{6}}({1\over{\alpha}} + \alpha)\log\alpha +2\tilde J(\alpha)
\end{eqnarray}
\section* {A9. Asymptotics of $Z_{\alpha}'(0)$}
\label{s:app9}
\renewcommand{\theequation}{A9.\arabic{equation}}
\setcounter{equation}{0}
In this section we investigate $Z_{\alpha}'(0)$ when $\alpha$ is
large or small or close to one.
\subsection*{Large and small $\alpha$}
It is convenient to divide up the terms in $Z_{\alpha}'(0)$
according to whether they are
symmetric or antisymmetric under the transformation
$\alpha\to{1\over{\alpha}}$.
If we take the integral $\tilde J$, it makes sense
to first introduce a finite symmetric alternative
expression
\begin{eqnarray}
\label{JSdef}
J_S(\alpha) =
\int_{\epsilon}^{\infty}
{{dy}\over y} {1\over{e^{y\sqrt{\alpha}}-1}}
{1\over{e^{y\over{\sqrt{\alpha}}}-1}}
-{1\over{2\epsilon^2}} +
({1\over{\sqrt{\alpha}}}+ \sqrt{\alpha}){1\over{2\epsilon}}
+{1\over {12}}(\alpha+3+{1\over{\alpha}})\log\epsilon
\end{eqnarray}
where the limit as $\epsilon\to 0 $ is understood.
Using the elementary integrals
\begin{eqnarray}
\int_{\epsilon}^{\infty} {{dx}\over{e^x-1}}
&=& -\log\epsilon + {\cal O}(\epsilon)\nonumber \\
\int_{\epsilon}^{\infty} {{dx}\over{e^x-1}} {1\over x}
&=& {1\over{\epsilon}} + {1\over 2}\log\epsilon +
{{\gamma}\over 2} - {1\over 2}\log 2\pi + {\cal O}(\epsilon)\nonumber \\
\int_{\epsilon}^{\infty} {{dx}\over{e^x-1}} {1\over {x^2}}
&=& {1\over{2\epsilon^2}} - {1\over{2\epsilon}} -
{1\over {12}}\log\epsilon +
+ {1\over{12}} - {{\gamma}\over{12}} - \zeta'(-1) +
{\cal O}(\epsilon) \nonumber \\
\int_{\epsilon}^{\infty} {{dx}\over x} {{e^x}\over {(e^x-1)^2}}
&=& {1\over{2\epsilon^2}}
{1\over {12}}\log\epsilon +
+ {1\over{12}} + {{\gamma}\over{12}} + \zeta'(-1)
+ {\cal O}(\epsilon) \nonumber \\
\end{eqnarray}
we can express the difference between $\tilde J$ and $J_S$
as
\begin{eqnarray}
\label{Jdiff}
\tilde J(\alpha) = J_S(\alpha)
+ {1\over{24}}(\alpha+3+{1\over{\alpha}})\log\alpha
+{{1+\alpha}\over 4}(\gamma-\log 2\pi) -
\alpha({{\gamma}\over{12}}+\zeta'(-1))
\end{eqnarray}
and divide up $Z_{\alpha}'(0)$ in symmetric, antisymmetric,
and linear parts:
\begin{eqnarray}
\label{zprimsep}
Z_{\alpha}'(0) &=& [J_S(\alpha) + {{\gamma}\over{12}}({1\over{\alpha}}
+3+\alpha) - {1\over 4}\log 2\pi] \nonumber \\
&\, & - [{1\over{12}}({1\over{\alpha}}-\alpha) \log 2
+{1\over{24}}({1\over{\alpha}} +3+\alpha)\log\alpha] \nonumber \\
&\, & -\alpha[{1\over 4}\log 2\pi +\zeta'(-1)]
\end{eqnarray}
The behaviour of $J_S$ as $\alpha$ turns to zero is found
expanding the integral in \ref{JSdef}, and subtracting the
terms in the expansion divergent with $\epsilon$:
\begin{eqnarray}
\label{JSexpansion}
J_S(\alpha) &\sim_{\alpha\to 0}&
{1\over{24}}({1\over{\alpha}}
+3+\alpha)\log\alpha
+ ({{1-\gamma}\over{12}}-\zeta'(-1)){1\over{\alpha}} \nonumber \\
&\, & + {1\over 4}(\log 2\pi -\gamma) +
\sum_{n=3}^{\infty} {{\zeta(n)B_{n+1}}\over{n(n+1)}}\alpha^n
\end{eqnarray}
and from this we find
\begin{eqnarray}
\label{zprimexpansion}
Z_{\alpha}'(0) &\sim_{\alpha\to 0}&
{1\over{\alpha}}({1\over{12}}(1-\log 2)-\zeta'(-1))
+ \alpha ({{\gamma+\log 2}\over{12}}-{1\over 4}\log 2\pi-\zeta'(-1))
\nonumber \\
&\, & +\sum_{n=3}^{\infty} {{\zeta(n)B_{n+1}}\over{n(n+1)}}\alpha^n
\end{eqnarray}
Using the symmetry under $\alpha\to{1\over{\alpha}}$ we
have the asymptotic expansion for large $\alpha$:
\begin{eqnarray}
\label{zprimexpansion2}
Z_{\alpha}'(0) &\sim_{\alpha\to \infty}&
-{1\over{12}}({1\over{\alpha}} +3+\alpha)\log\alpha +
\alpha ({1\over{12}}(1-\log 2)-{1\over 4}\log 2\pi-2\zeta'(-1))
\nonumber \\
&\, & + {1\over{\alpha}} ({{\gamma-\log 2}\over{12}})
+\sum_{n=3}^{\infty} {{\zeta(n)B_{n+1}}\over{n(n+1)}}\alpha^{-n}
\end{eqnarray}
\subsection*{Development of $Z_{1+\epsilon}'(0)$ }
It is convenient to go over to the first definition
of the integral $J(\alpha)$ and write
\begin{eqnarray}
Z_{\alpha}'(0) =
{1\over{12}}({1\over{\alpha}} -\alpha)(\gamma -\log 2)
-{1\over{12}}({1\over{\alpha}} + 3 +\alpha)\log\alpha +J(\alpha)
-\alpha \Delta J
\nonumber
\end{eqnarray}
where $J$ has the integral representation
\begin{eqnarray}
J(\alpha) =
\int_{0}^{\infty}
{1\over{e^{\mu} -1}}
[{1\over {2\mu}}\coth({{\mu}\over{2\alpha}}) -
{{\alpha}\over {4\sinh^2 ({{\mu}\over 2})}}
-{1\over {12}}({1\over{\alpha}}+\alpha)] d\mu \nonumber
\end{eqnarray}
and the difference has the integral representation
\begin{eqnarray}
\Delta J &=&
\int_{0}^{\infty} dx
[ {1\over x}({1\over{(e^x-1)^2}} + {1\over{2(e^x-1)}})
-{{e^x}\over{(e^x-1)^3}} - {1\over{6(e^x-1)}}] \nonumber \\
&=& -{1\over 6}\gamma - {5\over{24}} +{1\over 4}\log(2\pi) + \zeta'(-1)
\end{eqnarray}
The derivative of $J(\alpha)$ can be expanded around $\alpha=1$, and
the successive terms evaluated in Mathematica, which gives
\begin{eqnarray}
J'(1+\epsilon) =
-{1\over 36} \epsilon + {1\over 16} \epsilon^2 + {\cal O}(\epsilon^3)
\end{eqnarray}
and putting the various terms together we have for a corner
on the boundary:
\begin{eqnarray}
\label{zprimexpb}
Z_{1+\epsilon}'(0) &=&
({1\over 6}\log 2 - {5\over{24}} -
{1\over 4}\log(2\pi) - \zeta'(-1))\epsilon \nonumber \\
&\quad & + ({14\over 72} +{{\gamma-\log 2}\over{12}})\epsilon^2
+ (-{29\over 144} - {{\gamma-\log 2}\over{12}})\epsilon^3
+ {\cal O}(\epsilon^4)
\end{eqnarray}
The expansion for a corner in the interior is easily obtained
by adding the expansion of ${1\over 4}\log(1+\epsilon)$, and
doubling that result.
\section* {A10. Special cases}
\label{s:app10}
\renewcommand{\theequation}{A10.\arabic{equation}}
\setcounter{equation}{0}
\subsection*{Triangles}
For triangles one can choose a convenient parametrization
in terms of the Schwarz--Christoffel transformation that maps
a point ($z$) in the triangle to a point ($\omega$) in
the upper complex plane:
\begin{equation}
{{d\omega}\over{dz}}
= \omega^{1-\alpha_0}(1-\omega)^{1-\alpha_1}
\end{equation}
This fixes the area in terms of the angles to be
\begin{equation}
\label{normareaa10}
Area(\alpha_1,\alpha_2,\alpha_{3}) = {{\pi}\over 2}
{{\Gamma(\alpha_1)\Gamma(\alpha_2)\Gamma(\alpha_{3})}\over
{\Gamma(1-\alpha_1)\Gamma(1-\alpha_2)\Gamma(1-\alpha_{3})}}
\end{equation}
One corner (with opening angle $\pi\alpha_0$) will
map to the origin in the $\omega$--plane, one corner
($\pi\alpha_1$) to one, and the last corner maps to infinity.
The parametrization is uniform over the space of triangles,
in the sense that the branchpoints do not move.
Hence the third term in \ref{varycorner} is identically
zero.
The logarithm of the distance between the origin and one is
zero, hence the second terms in \ref{varycorner} are also
zero. All we have left is then the strictly local contribution,
so for this choice of the normalized areas, we have
for triangles:
\begin{equation}
Z_T'(0) = \sum_{p=1,2,3} Z'_{\alpha_p}(0)
\end{equation}
An integration constant was included in the definition of
$Z'_{\alpha}(0)$ in Appendix~8.
By numerically solving for about the first one thousand
eigenvalues of the Laplacian in isosceles
triangles, and then estimating the analytic continuation
of the zeta functions, Luck found that the quotient
\begin{equation}
\zeta_T = {{Z_T'(0)}\over{Z_T(0)}}
\end{equation}
varies surprisingly little over the triangles (still
taking the area \ref{normareaa10})\cite{JMLprive}.
We can express this $\zeta_T$ as
\begin{equation}
\zeta_T = {{\sum_{p=1,2,3} Z'_{\alpha_p}(0)}\over{\sum_{p=1,2,3}
Z_{\alpha_p}(0)}}
\end{equation}
It has a maximum for the equilateral triangle, where
it equals $4.591151\ldots$
The minimum of $\zeta_T$ is obtained as an angle tends to zero,
and the value follows from the asymptotic expansion \ref{zprimexpansion}:
\begin{eqnarray}
\label{zprimelimit}
\hbox{Limit}_{\alpha\to 0} {{Z_{\alpha}'(0)}
\over{Z_{\alpha}(0)}} = 2(1-\log 2)-24\zeta'(-1)
= 4.583813\ldots
\end{eqnarray}
\subsection*{Polygons}
In this case it is convenient to take a parametrization
where the interior of the polygon ($z$) is mapped onto
the interor of a circle ($u$) by a transformation that satisfies
\begin{equation}
{{du}\over{dz}}
= \prod_v (u-e^{i\phi_v})^{1-\alpha_v}
\end{equation}
where $e^{i\phi_v}$ are the branchpoints, and the interior
angles are $\pi\alpha_v$.
The important special case is a {\it regular} $n$--polygon,
$P_n$, that
can be mapped to the unit circle by a transformation that
satisfies
\begin{equation}
{{du}\over{dz}}
= \prod_{v=0}^n (u-e^{{2\pi i v}\over n})^{{2\over n}}
= (u^n-1)^{{2\over n}}
\end{equation}
This parametrization fixes the radius of the circumscribed
circle:
\begin{equation}
\label{normalradius}
R_n = {{\Gamma(1+{1\over n})\Gamma(1-{2\over n})}
\over{\Gamma(1-{1\over n})}}
\end{equation}
and the area, which is proportional to $R_n^2$.
We can now choose a family of polygons that smoothly
interpolate between a regular $m$--polygon (at $a=0$),
and a regular $mn$--polygon (at $a=1$):
\begin{equation}
{{du}\over{dz}}
= (u^m-1)^{{2\over n}(1-a)}(u^{mn}-1)^{{2\over {mn}}a}
\end{equation}
which is convenient, since the parametrization
is then uniform in
the family. The variation of $Z_D'(0)$ will therefore
be determined by the strictly local terms, and by the second
term in \ref{varycorner}. Let us express $Z_D'(0)$
as a function of the parameter $a$ in the family,
and write out these
two terms in \ref{varycorner} in more detail:
they give the following contribution
to ${d\over{da}}Z_{D(a)}'(0)$ from the corner $v$, which
is mapped to the branchpoint $e^{{2\pi i v}\over{mn}}$;
\begin{eqnarray}
{d\over{da}}Z_{D(a)}'(0)]_{\hbox{from corner $v$}}
= {d\over{da}}Z_{\alpha_v(a)}'(0) +
Z_{\alpha_v(a)}(0) {{d\lambda_v(a)}\over{da}} \nonumber \\
{{d\lambda_v(a)}\over{da}} = - \sum_{v'\neq v}
{{d\alpha_{v'}(a)}\over{da}}\log |u_{v'}-u_v|^2 \nonumber
\end{eqnarray}
which can be rewritten
\begin{eqnarray}
\label{varynotmanifest}
{d\over{da}}Z_{D(a)}'(0)]_{\hbox{from corner $v$}}
= {d\over{da}}[Z_{\alpha_v(a)}'(0) +
{1\over{12}}({1\over{\alpha_v(a)}} - \alpha_v(a))\lambda_v(a)]
\nonumber \\
+{{\lambda_v(a)}\over 6}{{d\alpha_v(a)}\over{da}}
\end{eqnarray}
We have to differ between whether $n$ does or does not divide
$v$. For the first case
\begin{eqnarray}
\lambda_v(a) &=& -{2\over m}(1-a)\log{{u^m-1}\over{u-1}}|_{u=1}
-{2\over{mn}} a\log{{u^{mn}-1}\over{u-1}}|_{u=1} \nonumber \\
&=&-{2\over m}(1-a + {a\over n})\log m - {{2a}\over{mn}}\log n
\qquad n \mid v
\end{eqnarray}
while for the second
\begin{eqnarray}
\lambda_v(a) = -{2\over m}(1-a)\log |e^{{2\pi i v}\over n} -1| -
{{2a}\over{mn}}\log mn
\qquad n \not\mid v
\end{eqnarray}
The expression \ref{varynotmanifest} contains one part
which is a total variation, and an extra term. Let us first
do the second, summed over the corners $v$:
\begin{eqnarray}
{1\over 6}\sum_v \lambda_v(a) {{d\alpha_v(a)}\over{da}} &=&
{1\over 6} \sum_{n\mid v}
(-{2\over m}(1-a)\log m - {2\over{mn}}\log mn)({2\over m}(1-{1\over n})
\nonumber \\
&+& {1\over 6} \sum_{n\not\mid v}
(-{2\over m}(1-a)\log |e^{{2\pi i v}\over n} -1| - {{2a}\over{mn}}\log mn)
(-{2\over{mn}}) \nonumber
\end{eqnarray}
which can eventually be simplified to
\begin{eqnarray}
-{2\over 3}{{1-a}\over m}\log m +
{2\over 3}{{1-a}\over {mn}}\log mn \nonumber
\end{eqnarray}
from which we have
\begin{eqnarray}
\int_0^1 da {1\over 6}\sum_v \lambda_v(a) {{d\alpha_v(a)}\over{da}}
= {1\over 3}({1\over{mn}}\log mn - {1\over m}\log m)
\end{eqnarray}
The total variation in \ref{varynotmanifest} is summed over
the corners, and gives in integrated form:
\begin{eqnarray}
nZ_{1 - {2\over n}}'(0) - {{2(n-1)}\over{3n(n-2)}} \log n
+ \hbox{Const.}
\end{eqnarray}
We are now ready to write down $Z_{P_n}'(0)$ for
a regular polygon with $n$ corners and radius of circumscribed
circle $R$:
\begin{eqnarray}
Z_{P_n}'(0) &=& Z_{P_n}(0)\log{{R^2}\over{R_n^2}} -
{1\over{3(n-2)}}\log n + n Z_{1 - {2\over n}}'(0)
\end{eqnarray}
where $Z_{P_n}(0) = {{n-1}\over{6(n-2)}}$ and the normal
radius is as in \ref{normalradius}. The integration constant
turns out to be zero, since it has already been
incorporated in the definition of $Z_{\alpha}'(0)$.
\subsection*{Square and disc}
For the square we obtain
\begin{eqnarray}
Z_{P_4}'(0) &=&
{1\over 2}\log({{\Gamma({3\over 4})4R}\over{\sqrt{\pi}\Gamma({1\over 4})}})
- {1\over 6}\log 4 + 4({1\over 8}\log\pi + {5\over{24}}\log 2) \nonumber \\
&=& {1\over 2}\log{{\Gamma({3\over 4})}\over{\Gamma({1\over 4})}}
+ {1\over 4}\log\hbox{Area} + {1\over 4}\log\pi + {5\over 4}\log 2
\end{eqnarray}
which agrees with the exact result from \ref{squareexact}.
The disc is obtained as the limit when $n$ tends to infinity.
We then find:
\begin{eqnarray}
Z_{P_{\infty}}'(0) &=&
{1\over 3}\log R - 2{d\over{d\alpha}}Z_{\alpha}'(0)|_{\alpha=1} \nonumber \\
&=& {1\over 3}\log R + {5\over 12} + {1\over 2}\log\pi + {1\over 6}\log 2
+ 2\zeta'(-1)
\end{eqnarray}
which agrees with \ref{discexact}.
\section* {A11. Integrable domains.}
\label{s:app11}
\renewcommand{\theequation}{A11.\arabic{equation}}
\setcounter{equation}{0}
In this appendix we collect the cases known to us,
where one can directly deduce the derivative of the
zeta function at zero.
In a rectangle with side lengths $A$ and $B$, the eigenvalues
of the Laplacian with Dirichlet boundary conditions are
\begin{equation}
\label{eigenvaluesrectangle}
\lambda_{mn} = \pi^2({{m^2}\over{A^2}} + {{m^2}\over{A^2}})
\end{equation}
The zeta function around the origin is
\begin{eqnarray}
Z_{\hbox{rectangle}}(0) = &{1\over 4} \nonumber \\
Z_{\hbox{rectangle}}'(0) =
& {1\over 4}\log (AB) - \log [2^{-{1\over 2}} ({B\over A})^{1\over 4}
\eta(q)] \nonumber
\end{eqnarray}
where $\eta$ is the modular form of Dedekind:
\begin{eqnarray}
\eta(q) = q^{1\over{24}}\prod_{m=1}^{\infty} (1-q^m)
\qquad q = e^{-2\pi\sqrt{{B\over A}}}
\end{eqnarray}
For the square, we have the simpler expression
\begin{eqnarray}
\label{squareexact}
Z_{\hbox{square}}'(0) =
{1\over 4}\log A^2 + {1\over 4}\log
[\pi 2^5{{\Gamma^2({3\over 4})}\over{\Gamma^2({1\over 4})}}]
\end{eqnarray}
Three triangles tile the plane by reflections in the side:
the equilateral (${1\over 3},{1\over 3},{1\over 3}$);
the bisected equilateral
${1\over 2},{1\over 3},{1\over 6}$, and
the right angle isosceles (${1\over 2},{1\over 4},{1\over 4}$).
In these domains one can solve for the eigenmodes of the
Laplacian by superposition of plane waves \cite{Itz,IL},
We normalize the areas in terms of the side lengths ($a$) of the
equilateral, the sidelengths of the legs in the right angle
isosceles,
and the sidelength of the longest side in the
bisected equilateral:
\begin{eqnarray}
\label{eigenvaluestriangles}
E_{({1\over 3},{1\over 3},{1\over 3}),n,m} &=&
({{4\pi}\over{3a}})^2(n^2+m^2-nm)\qquad n>m>0\nonumber \\
E_{({1\over 2},{1\over 4},{1\over 4}),n,m} &=&
({{\pi}\over{a}})^2(n^2+m^2)\qquad n>m>0\nonumber \\
E_{({1\over 2},{1\over 3},{1\over 6}),n,m} &=&
({{4\pi}\over{3a}})^2(n^2+m^2+nm)\qquad n>m>0 \nonumber
\end{eqnarray}
The corresponding zeta functions can be written in terms
of the Riemann zeta-function and
Dirichlet $L$-series\cite{IL},
which may in turn be resolved into sums of zeta
functions of Hurwitz $H(x,s)$ with different arguments $x$.
Using
\begin{eqnarray}
\label{Hurwitzzeta}
H(x,s) = \sum_{k=0}^{\infty} {1\over{(x+k)^s}} \nonumber \\
H(x,0) = {1\over 2} -x \qquad\qquad {d\over{ds}}H(x,s)|_{s=0}
= \log {{\Gamma(x)}\over{\sqrt{2\pi}}} \nonumber
\end{eqnarray}
and the normal areas
determined
by the representation of the Schwarz--Christoffel transformation;
\begin{eqnarray}
\label{intareas}
A(\alpha_0,\alpha_1,\alpha_{\infty}) &= {{\pi}\over{2}}
{{\Gamma(\alpha_0)\Gamma(\alpha_1)\Gamma(\alpha_{\infty})}
\over {\Gamma(1-\alpha_0)\Gamma(1-\alpha_1)\Gamma(1-\alpha_{\infty})}}
\end{eqnarray}
one finds after some
algebra the following expressions for the determinants:
\begin{eqnarray}
\label{intzetas1}
Z_{({1\over 3},{1\over 3},{1\over 3})}'(0) &=&
{1\over 3}\log {{\hbox{Area}}\over{A_{({1\over 3}, {1\over 3},
{1\over 3})}}} +
\log {{\Gamma^{1\over 2}({1\over 3})\pi^{1\over 2}
3^{2\over 3} 2^{-{1\over 6}}}\over{\Gamma^{1\over 2}({2\over 3})}} \\
Z_{({1\over 2},{1\over 4},{1\over 4})}'(0) &=&
{3\over 8}\log {{\hbox{Area}}\over{A_{({1\over 2}, {1\over 4},
{1\over 4})}}} +
\log {{\Gamma^{1\over 2}({1\over 4})\pi^{1\over 2}2^{7\over 8}}
\over{\Gamma^{1\over 2}({3\over 4})}} \\
Z_{({1\over 2},{1\over 3},{1\over 6})}'(0) &=&
{5\over 12}\log {{\hbox{Area}}\over{A_{({1\over 2}, {1\over 3},
{1\over 6})}}} +
\log {{\Gamma({1\over 3})3^{11\over 24} 2^{2\over 9}\pi^{1\over 2}}
\over{\Gamma({2\over 3})}}
\end{eqnarray}
The eigenvalues of the Laplacian in on the upper half sphere
of radius one
are $l(l+1)$. If one imposes Dirichlet
boundary conditions at the equator, one selects out the
eigenmodes odd under reflection in a plane through the
equator. Each eigenvalue will then be $l$ times degenerate.
The zeta function of this spectrum is
\begin{equation}
Z_{\hbox{hemisphere}}(s) = \sum_{l=1}^{\infty} {l\over{(l(l+1))^s}}
\nonumber
\end{equation}
from which one can derive \cite{Weisberger}
\begin{equation}
\label{hemispehereexact}
Z_{\hbox{hemisphere}}'(0) = 2\zeta'(-1) + {1\over 2}\log 2\pi - 4
\end{equation}
The hemisphere can be mapped conformally to a disc with radius
$R$. The difference of the regularized determinants on
the hemisphere and on the disc can then be evaluated by
computing the Liouville action (\ref{Liouvilleaction})
of the conformal factor\cite{Weisberger}:
\begin{equation}
\label{discexact}
Z_{\hbox{disc}}'(0) = {1\over 6}\log 2 + {1\over 2}\log\pi +
{1\over 3}\log R + 2\zeta'(-1) + {5\over{12}}
\end{equation}
Equation \ref{discexact} has also been checked numerically
to great accuracy by computing the eigenvalues from the
zeros of Bessel functions, and directly investigating the
analytical continuation of the zeta function\cite{JMLprive}.
\newpage
|
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{"url":"https:\/\/testbook.com\/question-answer\/directions-identify-the-meaning-of-the-following--5fa23ceaec4b9b976bc1c045","text":"# Directions: Identify the meaning of the following:It was all Greek to me.\n\nThis question was previously asked in\nNIMCET 2020 Official Paper\nView all NIMCET Papers >\n1. Difficult to speak\n2. Difficult to write\n3. Difficult to arrange\n4. Difficult to understand\n\n## Answer (Detailed Solution Below)\n\nOption 4 : Difficult to understand\nFree\nCell\n314011\n10 Questions 10 Marks 7 Mins\n\n## Detailed Solution\n\nThe correct answer is 'Difficult to understand'.\n\nKey Points\n\n\u2022 Given Idiom:\u00a0It was all Greek to me\u00a0is used for\u00a0expressing that something is difficult to understand.\n\u2022 Example:\u00a0Can you make sense of these instructions? It's all Greek to me!\n\u2022 From the given options, the\u00a0fourth option\u00a0is\u00a0the\u00a0most appropriate meaning of the given idiom.\n\u2022 Hence, the correct answer is\u00a0option 4.\n\nAdditional Information\n\n\u2022 The idiom\/phrase\u00a0originates from:\u00a0This phrase was increasingly used by monk scribes in the Middle Ages, as knowledge of the Greek alphabet and language was dwindling among those who were copying manuscripts in monastic libraries.","date":"2021-10-26 05:29:24","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\": 0.8327391147613525, \"perplexity\": 6361.721101157396}, \"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-2021-43\/segments\/1634323587799.46\/warc\/CC-MAIN-20211026042101-20211026072101-00055.warc.gz\"}"}
| null | null |
Is there anything quite as beautiful as walking into a home with new wood floors? There is something that appeals in an organic way when we add wood flooring to our home. It is an opportunity to bring some of the "outdoors-in" by integrating natural elements into our interior space and since no two natural wood grains are the same your floor will have a unique look.
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Looking for more advice on hardwood flooring? Contact our customer service department at 855-433-2587 for free advice on wood floors and other options for your home.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,039
|
require 'swagger/blocks'
require 'zermelo/records/redis'
require 'flapjack/data/extensions/short_name'
require 'flapjack/data/validators/id_validator'
require 'flapjack/data/extensions/associations'
require 'flapjack/gateways/jsonapi/data/join_descriptor'
require 'flapjack/gateways/jsonapi/data/method_descriptor'
module Flapjack
module Data
class UnscheduledMaintenance
include Zermelo::Records::RedisSortedSet
include ActiveModel::Serializers::JSON
self.include_root_in_json = false
include Swagger::Blocks
include Flapjack::Data::Extensions::Associations
include Flapjack::Data::Extensions::ShortName
define_attributes :start_time => :timestamp,
:end_time => :timestamp,
:summary => :string
define_sort_attribute :start_time
belongs_to :check, :class_name => 'Flapjack::Data::Check',
:inverse_of => :unscheduled_maintenances
# TODO :check before_set -- should fail if already set
range_index_by :start_time, :end_time
before_validation :ensure_start_time
validates :start_time, :presence => true
validates :end_time, :presence => true
validates_with Flapjack::Data::Validators::IdValidator
def duration
self.end_time - self.start_time
end
swagger_schema :UnscheduledMaintenance do
key :required, [:id, :type, :start_time, :end_time]
property :id do
key :type, :string
key :format, :uuid
end
property :type do
key :type, :string
key :enum, [Flapjack::Data::UnscheduledMaintenance.short_model_name.singular]
end
property :start_time do
key :type, :string
key :format, :"date-time"
end
property :end_time do
key :type, :string
key :format, :"date-time"
end
property :relationships do
key :"$ref", :UnscheduledMaintenanceLinks
end
end
swagger_schema :UnscheduledMaintenanceLinks do
key :required, [:check]
property :check do
key :"$ref", :CheckLinkage
end
end
swagger_schema :UnscheduledMaintenanceUpdate do
key :required, [:id, :type]
property :id do
key :type, :string
key :format, :uuid
end
property :type do
key :type, :string
key :enum, [Flapjack::Data::UnscheduledMaintenance.short_model_name.singular]
end
property :end_time do
key :type, :string
key :format, :"date-time"
end
end
def self.swagger_included_classes
# hack -- hardcoding for now
[
Flapjack::Data::Check,
Flapjack::Data::Contact,
Flapjack::Data::Medium,
Flapjack::Data::Rule,
Flapjack::Data::ScheduledMaintenance,
Flapjack::Data::State,
Flapjack::Data::Tag,
Flapjack::Data::UnscheduledMaintenance
]
end
def self.jsonapi_methods
@jsonapi_methods ||= {
:get => Flapjack::Gateways::JSONAPI::Data::MethodDescriptor.new(
:attributes => [:start_time, :end_time, :summary],
:descriptions => {
:singular => "Get data for an un scheduled maintenance period.",
:multiple => "Get data for multiple unscheduled maintenance periods."
}
),
:patch => Flapjack::Gateways::JSONAPI::Data::MethodDescriptor.new(
:attributes => [:start_time, :end_time, :summary],
:descriptions => {
:singular => "Update data for an unscheduled maintenance period.",
:multiple => "Update data for unscheduled maintenance periods.",
}
),
:delete => Flapjack::Gateways::JSONAPI::Data::MethodDescriptor.new(
:descriptions => {
:singular => "Delete an scheduled maintenance period.",
:multiple => "Delete unscheduled maintenance periods."
}
)
}
end
def self.jsonapi_associations
unless instance_variable_defined?('@jsonapi_associations')
@jsonapi_associations ||= {
:check => Flapjack::Gateways::JSONAPI::Data::JoinDescriptor.new(
:get => true,
:number => :singular, :link => true, :includable => true,
:descriptions => {
:get => "Returns the check an unscheduled maintenance period applies to."
}
)
}
populate_association_data(@jsonapi_associations)
end
@jsonapi_associations
end
private
def ensure_start_time
self.start_time ||= Time.now
end
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,076
|
\section{\@startsection {section}{1}{\z@}%
{-3.5ex \@plus -1ex \@minus -.2ex}%
{2.3ex \@plus.2ex}%
{\normalfont\large\bfseries}}
\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
{-3.25ex\@plus -1ex \@minus -.2ex}%
{1.5ex \@plus .2ex}%
{\normalfont\bfseries}}
\begin{document}
\title{ \LARGE {\textsc {Central charges for AdS black holes}}}
\author{
{\large Malcolm Perry}$^{1,2,3}$\footnote{malcolm@damtp.cam.ac.uk.}, {\large Maria J. Rodriguez}$^{4,5}$\footnote{majo.rodriguez.b@gmail.com}, \\
\\
$^{1}${\small Department of Physics, Queen Mary University of London,}\\{\small Mile End Road, London E1 4NS, UK}\\
\\
$^{2}${\small DAMTP, Cambridge University, Centre for Mathematical Sciences,}\\ {\small Wilberforce Road, Cambridge CB3 0WA, UK}\\
\\
$^{3}${\small Trinity College, Cambridge, CB2 1TQ, UK}\\
\\
$^{4}${\small Department of Physics, Utah State University,}\\ {\small 4415 Old Main Hill Road, UT 84322, USA}\\
\\
$^{5}${\small Instituto de F\' isica Te\' orica UAM/CSIC, Universidad Aut\' onoma de Madrid,}\\ {\small 13-15 Calle Nicol\' as Cabrera, 28049 Madrid, Spain}\\
\\
\\
}
\maketitle
\abstract
Nontrivial diffeomorphisms act on the horizon of a generic 4D black holes and create distinguishing features referred to as soft hair. Amongst these are a left-right pair of Virasoro algebras with associated charges that reproduce the Bekenstein-Hawking entropy for Kerr black holes. In this paper we show that if one adds a negative cosmological constant, there is a similar set of infinitesimal diffeomorphisms that act non-trivially on the horizon.
The algebra of these diffeomorphisms gives rise to a central charge. Adding a boundary counterterm, justified to achieve integrability, leads to well-defined central charges with $c_L = c_R$. The macroscopic area law for Kerr-AdS black holes follows from the assumption of a Cardy formula governing the black hole microstates.
\newpage
\tableofcontents
\section{Introduction}
\label{sec:Intro}
The entropy of black hole can be understood by a type of 'soft hair' associated to a general class of nonabelian Vir$_L$ $\times$ Vir$_R$ diffeomorphisms. In fact, only recently, a general class of Vir$_L$ $\times$ Vir$_R$ diffeos of a generic spin $J$ Kerr black hole were considered in \cite{Haco:2018ske} to determine its entropy in a manner similar to their stringy black hole counterparts.
\indent Strikingly, using the horizon itself as the surface permitting the analysis allow one to infer that there are no independent interior black hole microstates. In the case of black hole solutions in General Relativity, the black hole Hilbert space must be contained within the Hilbert space of states on or outside the black hole horizon. The observations seem to apply equally to the real-world Kerr black hole case \cite{Haco:2018ske}, the Kerr-Newman black hole solution \cite{Haco:2019ggi} and to the stringy black holes with near-AdS$_3$ regions.
\indent An interesting property of many non-supersymmetric black holes, such as Kerr and Kerr-AdS, is that its entropy remains finite for the extreme configuration where the Hawking temperature vanishes. This suggests that for extremal black holes the entropy can be reproduced as the statistical entropy of the dual CFT using the Cardy formula.
Previous attempts to determine the microscopic entropy of the black hole and reproduce the macroscopic area law relied on the existence of a near horizon extremal-AdS$_2$ region. The so called Kerr/CFT correspondence \cite{Guica:2008mu}, employed the near horizon extremal (zero-temperature) Kerr to show that the microscopic entropy of the CFT, calculated using the Cardy formula, coincides with the macroscopic the extremal Kerr black hole entropy. \cite{Castro:2010fd} generalized these ideas, only assuming that the left and right CFTs have identical central charges, and argued that even away from extremality Kerr black hole's entropy was reproduced. Similar results were presented for the five-dimensional Myers-Perry black holes in \cite{Krishnan:2010pv} and non-vanishing cosmological constant in \cite{Mao:2016pwq,Grumiller:2016kcp}.
\indent In this paper we follow the soft-hair approach in a slightly different direction. In an effort to test the applicability of the soft-hair conjecture we will focus on the Kerr-AdS black holes. The twofold goal is to uniquely define the conformal phase space formalism relevant to the approach and find a boundary counterterm to achieve integrability for the existence of a well-defined charges. Assuming the existence of a quantum Hilbert space on which these charges generate the symmetries, as well as the applicability of the Cardy formula, the concrete example of Kerr-AdS that we explore gives identical central charges $c_L = c_R$ providing evidence for an actual explanation of the macroscopic entropy.
\indent The existence of a generic near-AdS$_3$ regions is not ordinarily sufficient to determine the conformal coordinates in which the Virasoro action takes a simple form and exhibits the conformal structure of the black hole geometry. To supplement the approach, we will argue that the Noether charge interpretation of the entropy first presented in \cite{Wald:1993nt} gives the necessary input to uniquely fix the relevant conformal coordinates. We will only assume that the theory admits stationary black hole solutions with a bifurcate Killing horizon.
\indent Having established the appropriate conformal coordinates, we will construct an explicit set of Vir$_L$ $\times$ Vir$_R$ vector fields which generate the hidden conformal symmetry in the near-horizon region. These act non-trivially on the horizon in the sense that their boundary charges are non-vanishing. The covariant phase space formalism provides a formula for the Virasoro charges as surface integrals on the horizon. Using the covariant phase space formalism we find a boundary counterterm -- that reduces to the Wald-Zoupas boundary counterterm for Kerr -- justified to achieve integrability for well-defined charge and gives $c_L = c_R$.
The plan of the paper is as follows. In Section 2 we use the entropy as Noether charge interpretation \cite{Wald:1993nt} to find expressions for the generators of these $R,L$ sectors in a {\it conformal coordinate} frame with a direct CFT interpretation. Section 3 presents the identification of the AdS$_3$ near horizon in conformal coordinates for generic Kerr-AdS black holes in which the Virasoro action takes the simple form. The application of the blended AdS$_3$ near horizon identification and $R,L$ entropy killing field generators relevant for its Noether construction is presented in Section 4. This section includes the explicit identification of the microscopic temperatures $T_L$ and $T_R$ in Kerr-AdS. In Section 5 we compute the covariant right-left moving Virasoro charges and find a counterterm for an integrable action. We then indicate how to do a similar calculation for the left-moving charges. In section 6 we discuss the case of extreme black holes. In Section 7 we briefly discuss the first law. Section 8 repeats the same exercise but for the inner black hole horizon. The discussion of our findings are provided in Section 9. We illustrate this black hole conformal structure method with the detailed Kerr black hole example in Appendix A.\\
Throughout this paper we use units such that $c=\hbar=k=G=1$.
\section{Entropy and $(L,R)$\,-\,generators}
\label{sec:Scoords}
Consider a stationary black hole solution with a bifurcate Killing horizon. According to Wald's computation \cite{Wald:1993nt}, the black hole entropy $S_{\pm}$ is simply $2\pi$ times the integral over the Noether charge associated with the Killing field
\begin{eqnarray} \label{killing}
\bar{\zeta}_{\pm}=\kappa_{\pm}\,{\zeta}_{\pm}\,,
\end{eqnarray}
Here $\kappa_{\pm}$ is the surface gravity \footnote{The Hawking temperature is given by $T_{\pm}=\kappa_{\pm}/(2\pi)$.} and ${\zeta}_{\pm}=\partial_t+\sum_i \, \Omega_{\pm}\partial_{\phi}$ are the horizon Killing fields (normalized so as to have unit surface gravity) vanishing on the bifurcation $2$-surfaces $\Sigma_{\pm}$. With the aim to find a frame with a direct CFT interpretation for the outer/inner event horizons, we proposed to consider new variables $t_{L,R}$ where the Killing field (\ref{killing}) generating the entropy is
\begin{eqnarray} \label{prop}
\bar{\zeta}_{\pm}\propto (\partial_{t_L}\pm\partial_{t_R})\,.
\end{eqnarray}
and vanishes respectively either on the inner or outer event horizon. This is possible, for $4$-dimensional black holes via the transformation
%
\begin{eqnarray}\label{trtl}
t_R&=& \alpha \, \phi+\beta\, t \,, \\
t_L&=& \gamma \,\phi+\delta \,t \,.\nonumber
\end{eqnarray}
for the following choice of parameters
\begin{eqnarray} \label{trtl1}
&&\beta=-\frac{1}{2} \left[\gamma \left(\Omega_+ - \Omega_-\right) + \alpha \left(\Omega_+ + \Omega_-\right)\right]\,\\
&&\delta=-\frac{1}{2} \left[\alpha \left(\Omega_+ - \Omega_-\right) + \gamma \left(\Omega_+ + \Omega_-\right)\right]\,.\nonumber
\end{eqnarray}
This choice, gives the Killing fields (\ref{killing}) in a frame where the right (R) and left (L) sectors of the CFT is straightforward
\begin{eqnarray} \label{zeta}
\bar{\zeta}_{\pm}=\pm \kappa_{\pm} K_{\pm} (\partial_{t_L}\mp \partial_{t_R})\,.
\end{eqnarray}
where
\begin{eqnarray}
K_{\pm}=\frac{1}{2} \left(\gamma \mp \alpha \right) (\Omega_- -\Omega_+)\,.
\end{eqnarray}
We can also now invert the relations to find expressions for the generators of these $R,L$ sectors \footnote{In turn, we can also rewrite the above formulae in the form
\begin{eqnarray}
\partial_{t_L}=\frac{1}{2} \left[\left(\frac{1}{K_+}-\frac{1}{K_-}\right)\partial_t+\sum_i \left(\frac{\Omega_+^{(\phi_i)}}{K_+}-\frac{\Omega_-^{(\phi_i)}}{K_-}\right)\partial \phi_i\right]\,,\\
\partial_{t_R}=\frac{1}{2} \left[\left(\frac{1}{K_+}+\frac{1}{K_-}\right)\partial_t+\sum_i \left(\frac{\Omega_+^{(\phi_i)}}{K_+}+\frac{\Omega_-^{(\phi_i)}}{K_-}\right)\partial \phi_i\right]\,,
\end{eqnarray}
}.
The choice of $t_R,t_L$ coordinates that we have derived is very suggestive of the fact that there maybe a universal correspondence of a 2d CFT for any 4-dimensional black hole solution. To realize this conjecture we derive, in the following section, a method to identify the microscopic temperatures $T_L$ and $T_R$ (or equivalently the parameters $\alpha, \gamma$). While the conformal coordinates are not uniquely defined, the choice made here is unique in the sense that it would later give the correct thermodynamic relations that verify of the first law of thermodynamics for black holes. As we will now show the coordinate choice $(t_L,t_R)$ also matches those found by computing the monodromies around the inner and outer horizon in \cite{Castro:2013kea}.
\section{Conformal coordinates}
\label{sec:ConfCoords}
In this section we propose a method for the identification of the microscopical temperatures of 2d CFT that relies on the properties of the black hole geometry close to the bifurcation surfaces. This work is a logical outgrowth of the observation in \cite{Haco:2018ske} that the existence of a generic near-AdS$_3$ regions constrains the define the so-called conformal coordinates $(w^\pm,y)$ and possibly the structure of the black hole geometry.
There is a large ambiguity in the choice of the conformal coordinates in which the Virasoro action takes the simple form. The procedure does not fix uniquely the 2d CFT temperatures $T_{L,R}$. For the purpose of CFT identification, one can take Wald's construction to define the black hole entropy as a Noether charge associated with the Killing fields in a frame where the right (R) and left (L) are distinguishable (see Section \ref{sec:Scoords} for details). This suggests a natural modification of the warped AdS space-times geometry identification. We propose the coordinate choice to be consistent with (\ref{trtl})-(\ref{trtl1}) where we have decomposed the coordinates, which separate the corresponding Killing fields in a (R,L) frame.
We now conjecture that the conformal coordinates $(w^\pm,y)$ clearly exhibiting the conformal structure are
\begin{eqnarray}\label{CFT}
w^+&=&R(r)\,e^{t_R}\,,\nonumber\\
w^- &=&R(r)\,e^{t_L}\,\\
y&=&Q(r)\,e^{(t_L+t_R)/2}\,\nonumber
\end{eqnarray}
where $(t_L,t_R)$ are defined in (\ref{trtl}) with (\ref{trtl1}), $R(r)^2+Q(r)^2=1$ and close to the event horizon $r=r_+$ the function $R^2(r) \sim c \,(r- r_+)$ with constant $c$. The bifurcation surface in the conformal coordinates at the outer event horizon is at $w_{\pm}=0$ and $w_{\pm}=\infty$ at the inner event horizon.
It follows from (\ref{CFT}) that a black hole solution (in Boyer-Lindquist type coordinates) under the transformation
\begin{eqnarray}\label{eq:coordtransf}
t&=& \frac{1}{2(\gamma \beta- \alpha \delta)}\left[(\alpha+\gamma) \ln\frac{w^+}{w^-}-(\alpha-\gamma) \ln{(w^+w^-+y^2)}\right]\,,\nonumber\\
\left(\frac{R(r)}{Q(r)}\right)^2&=&\frac{w^+ w^-}{y^2}\,\\
\phi&=&\frac{1}{2\alpha} \, \ln \frac{w^+ (w^+w^-+y^2)}{w^-} -\frac{\beta}{\alpha} \, t\,,\nonumber
\end{eqnarray}
becomes at the leading order around the bifurcation surface
\begin{eqnarray}
ds^2= \frac{4\,\rho_+(\theta)^2}{y^2}dw^+ dw^-+\frac{F(\theta)\sin^2\theta}{y^2 \rho_+(\theta)^2}dy^2+\rho_+(\theta)^2d\theta^2+...\,,
\end{eqnarray}
for specific values of $\alpha$ and $\gamma$. Here $\rho_+(\theta),F(\theta)$ are arbitrary functions of $\theta$. This is simply a warped AdS spacetime in conformal coordinates. Hence these coordinates are well-adapted to an analysis of 4D black holes mirroring that of the 3D BTZ black holes.
Physically this means that there is a unique 2d CFT dual description for generic types black hole geometries. We observe that under azimuthal identification $\phi\rightarrow\phi+2\pi$ of the conformal variables
\begin{eqnarray}
w^+\sim e^{2\pi \alpha} w^+\,,\qquad w^-\sim e^{2\pi \gamma} w^-\,,\qquad y \sim e^{\pi(\alpha+\gamma)} y\,.
\end{eqnarray}
This is the same as the identification employed in \cite{Maldacena:1998bw} that turns $AdS_3$ in Poincare coordinates into BTZ
with temperatures $(T_L, T_R)$
\begin{eqnarray}
w^+\sim e^{4\pi^2T_R} w^+\,,\qquad w^-\sim e^{4\pi^2 T_L} w^-\,,\qquad y \sim e^{2\pi^2(T_R+T_L)} y\,.
\end{eqnarray}
The periodicities analysis yields
\begin{eqnarray}
\alpha=2\pi T_R\,,\qquad \gamma=2\pi T_L\,.
\end{eqnarray}
or equivalently
\begin{eqnarray}\label{temRL}
T_R=\alpha/(2\pi)\,,\qquad T_L=\gamma/(2\pi)\,.
\end{eqnarray}
We argue, that with the proposed systematic approach one can identify a CFT dual for generic types black hole geometries, allowing the computation of
the left and right $2d$ CFT temperatures and the microscopic entropy using a Cardy formula. In the next section, we present results for the $2d$ CFT temperatures in explicit examples that include Kerr black hole, and Kerr-AdS black holes that are the main focus of the paper.
\section{Conformal Coordinates for Kerr-AdS Black Hole}
In the previous section we made a rather general proposal to identify the 2d CFT temperatures via the conformal coordinates definitions. In this section we put flesh on this proposal and show the explicit application of the proposed method to the Kerr-AdS black hole solution. In the flat space-time limit $L\rightarrow \infty$ our results reduce to the Kerr black hole identifications found in \cite{Haco:2018ske} (see the appendix).
The metric of the four dimensional Kerr-AdS black hole \cite{Carter:1968ks}, satisfying $R_{\mu\nu}=-3L^{-2}\,g_{\mu\nu}$ is given by
\begin{eqnarray}
ds^2&=&\rho^2\left(\frac{dr^2}{\Delta}+\frac{d\theta^2}{\Delta_{\theta}}\right)+\frac{\Delta_{\theta}\sin^2\theta}{\rho^2}\left(a\,dt-\frac{r^2+a^2}{\Xi}d\phi\right)^2-\frac{\Delta}{\rho^2}\left(dt-\frac{a\sin^2\theta}{\Xi} d\phi\right)^2\,,\nonumber\\
\rho^2&=&r^2+a^2\cos^2\theta\,\qquad \Delta=(r^2+a^2)(1+r^2L^{-2})-2Mr\,,\nonumber \\
&&\Delta_{\theta}=1-a^2L^{-2}\cos^2\theta\,,\qquad \Xi=1-a^2L^{-2}\,.
\end{eqnarray}
The metric is asymptotic to $AdS_4$ in a rotating frame, with angular velocity $\Omega_{\Lambda}=-a L^{-2}$. The outer and inner event horizons are located at $\Delta(r_{\pm})=0$. The physical parameters are
\begin{eqnarray}\label{thermodynamicsAdS}
E=\frac{M}{\Xi^2}\,,\qquad J&=&\frac{M a }{\Xi^2}\,,\\
T_{\pm}=\frac{\Delta'(r_{\pm})}{4\pi(r_{\pm}^2+a^2)}\,\qquad
\Omega_{\pm}&=&\frac{a\,\Xi}{r_\pm^2+a^2}\,,\qquad S_{\pm}=\frac{\pi (r_{\pm}^2+a^2)}{\Xi} \,.\nonumber
\end{eqnarray}
corresponding to the physical mass, angular momentum, Hawking's temperature, angular velocity of the horizon (as measured in the asymptotically rotating frame), and the entropy respectively. As shown in \cite{Gibbons:2004ai}, the angular velocity which is measured relative to a {\it non-rotating} frame at infinity is determined by
\begin{eqnarray}
\Omega^{\pm}_{\infty}=\Omega_{\pm}-\Omega_{\Lambda}
=\frac{a (1+r_\pm^2/L^2)}{r_\pm^2+a^2}\,.
\end{eqnarray}
To proceed with the interpretation, we first take the transformation (\ref{eq:coordtransf}) with (\ref{trtl1}), and set $c = 1/\Delta'(r_+)$ in the function $R^2(r) \sim c \,(r- r_+)$ close to the event horizon of the black hole, such that the black hole metric around the bifurcation surface is
\begin{eqnarray}
ds^2&=& \frac{4\,\rho_+(\theta)^2}{y^2}dw^+ dw^-+\frac{k^2\, \Delta_\theta(\theta) \sin^2\theta}{y^2 \rho_+(\theta)^2}dy^2+\frac{\rho_+(\theta)^2}{\Delta_\theta(\theta)}d\theta^2\nonumber \\
&+& \frac{4 w^{+}}{y^3} \left(\frac{a^2 \Delta_{\theta}(\theta)\sin ^2 \theta \,( 2\, \delta \, r_+ +1) }{\delta ^2 \rho_+(\theta)^2 }+\frac{\left(a^2+r_+^2\right) (r_- -r_+)} {r_+} \right) dw^{-} dy\\
&-& \frac{4 w^{-}}{ y^3 } \left(\frac{a^2 \Delta_{\theta}(\theta) \sin ^2(\theta ) \, (2 \,\delta \, r_+ -1) } {\delta ^2 \rho_+(\theta)^2}+\frac{(a^2+r_+^2) (r_- +r_+ )-2 a^2 r_+ \sin^2 \theta }{r_+} \right)dw^{+} dy\nonumber \\
&+&...\,,\nonumber
\end{eqnarray}
where
\begin{eqnarray}
\rho_+(\theta)^2=r_+^2+a^2\cos^2\theta\,,\qquad k= 2 a /\delta\,.
\end{eqnarray}
In this case, we find the parameters in the transformation become
\begin{eqnarray}
\alpha=\frac{\Delta'(r_+)}{2 \,a\Xi}\,,\qquad \beta=0\,,\qquad\gamma=\frac{(\Omega_+ +\Omega_-)}{(\Omega_- -\Omega_+)} \alpha
\,,\qquad \delta=-\Omega_+ (\alpha+\gamma)
\end{eqnarray}
and from (\ref{temRL}) one can identify the right and left-temperatures arising from the CFT giving
\begin{eqnarray}\label{AdsTemp}
T_R=\frac{\Delta'(r_+)}{4\pi \,a\Xi}\,,\qquad T_L=\frac{(\Omega_++\Omega_-)}{(\Omega_- -\Omega_+)}\frac{\Delta'(r_+)}{4\pi \,a\Xi }\,.
\end{eqnarray}
In the case of vanishing cosmological constant, $L\rightarrow \infty$, the results for the CFT temperatures reduce to those for Kerr black holes given in \cite{Haco:2018ske}. Further details about the conformal coordinates for Kerr are presented in Appendix \ref{app:Kerr}.
The general result (\ref{AdsTemp}) for the right and left-temperatures arising from the CFT for the AdS-Kerr black holes can be combined with the Cardy entropy formula
\begin{equation}
S_{\pm}=\frac{\pi^2}{3}\left(c_L T_{L}\pm c_R T_R \right)\,.
\end{equation}
Using (\ref{AdsTemp}) and the entropies (\ref{thermodynamicsAdS}), one finds that for Kerr-AdS black holes the left and right CFTs have identical central charges
\begin{eqnarray}\label{centralAdsKerr}
c_L=c_R=-\frac{6 a }{\delta }=\frac{ 6 a (r_+^2-r_-^2)}{\Delta'(r_+)}\,.
\end{eqnarray}
\subsection*{Inverse metric}
\begin{eqnarray}
&& g^{yy}\sim \frac{\delta ^2 y^2 \rho_+^2}{4 a^2 \sin^2\theta \Delta_{\theta} (\theta )}\,,\qquad
g^{\theta\theta}\sim\frac{\Delta_{\theta}}{\rho_+^2}\,,\qquad
g^{+ -} \sim \frac{y^2}{4 \rho_+^2}\,,\nonumber\\
&& g^{+ y} \sim- \frac{w_+ y}{4}\left(\frac{2 \, \delta \, r_+ +1}{\rho_+^2}- \frac{\delta ^2 \left(a^2+r_+^2\right) (r_+ - r_-)}{a^2 r_+ \sin^2\theta\, \Delta_{\theta}}\right)\,,\\
&& g^{- y} \sim \frac{ w^{-} y}{4 } \left(\frac{2 \,\delta \, r_+ -1 } { \rho_+^2}+\frac{\delta ^2 ((a^2+r_+^2) (r_- +r_+ )-2 a^2 r_+ \sin^2 \theta) }{a^2 r_+ \sin^2\theta\, \Delta_{\theta}} \right)\,.\nonumber
\end{eqnarray}
\subsection*{Volume element}
The volume element is
\begin{eqnarray}
\epsilon_{+- y \theta} =\frac{4 a \sin\theta \rho_+^2 }{\delta y^3}+ ... \,.
\end{eqnarray}
\subsection*{Conformal vectors}
We consider the conformal vector fields
\begin{eqnarray}\label{eq:vectorfields}
\zeta_{n}&=& \epsilon_n \partial_+ +\frac{1}{2} \partial_+ \epsilon_n y \partial_y \,,\\
\bar{\zeta}_{n}&=& \bar\epsilon_n \partial_- +\frac{1}{2} \partial_- \bar\epsilon_n y \partial_y \,.
\end{eqnarray}
and restrict the full set of functions $(\epsilon ,\bar\epsilon)$ so that $(\zeta, \bar\zeta)$ are invariant under $2\pi$ azimuthal rotations is
\begin{eqnarray}
\epsilon_n &=& \alpha \,(w^+)^{1+\frac{i n}{\alpha}}=2\pi T_R(w^+)^{1+\frac{i n}{2\pi T_R}}\,, \\
\bar\epsilon_n &=& \gamma \,(w^-)^{1+\frac{i n}{\gamma}}= 2\pi T_L(w^-)^{1+\frac{i n}{2\pi T_L}}\,.
\end{eqnarray}
Taking $\zeta_n\equiv \zeta(\epsilon_{n})$ and $\bar\zeta_n=\bar\zeta(\epsilon_{n})$, the vector fields (\ref{eq:vectorfields}) obey the Lie bracket algebra
\begin{eqnarray}
[ \zeta_m,\zeta_n ] = i (n-m)\zeta_{m+n} \,, \qquad
[ \bar\zeta_m, \bar \zeta_n ] = i (n-m) \bar\zeta_{m+n}\,.
\end{eqnarray}
and the two set commuting with another
\begin{eqnarray}
[\zeta_m,\bar\zeta_n]=0\,.
\end{eqnarray}
The zero modes in this case are
\begin{eqnarray}
\zeta_0&=& \alpha (w^+\partial_++\frac{1}{2} y\, \partial_y)=2\pi T_R(w^+\partial_++\frac{1}{2} y\, \partial_y)= \partial_{\phi}-\frac{2\pi T_L}{\delta}\, \partial_t \equiv - i \, \omega_R\,,\\
\bar \zeta_0 &=& \gamma (w^-\partial_-+\frac{1}{2} y \,\partial_y)= \frac{2\pi T_L}{\delta}\, \partial_t \equiv i \, \omega_L
\end{eqnarray}
\section{Covariant charges}
\label{sect:CovCharges}
In this section we construct the linearized covariant charges $\delta \mathcal{Q}_n=\delta \mathcal{Q}(\zeta,h;g)$ associated to the diffeos $\zeta_n$ acting on the horizon. We are interested in the central term $K_{m,n}$ in the Virasoro charge algebra (for the right movers)
\begin{eqnarray}
\{\mathcal{Q}_n,\mathcal{Q}_m\}=(m-n) \mathcal{Q}_{m+n}+K_{m,n}\,,
\end{eqnarray}
where the central term is given by
\begin{eqnarray}
K_{m,n}=\delta\mathcal{Q}(\zeta_n,\mathcal{L}_{\zeta_m} g;g) =\frac{c_R \,m^3}{12} \delta_{m+n}\,.
\end{eqnarray}
It turns out that the general form for the linearized charge associated to a diffeo $\zeta$ on a surface $\Sigma$ with boundary $\partial \Sigma$ is
\begin{eqnarray}
\delta \mathcal{Q} = \delta \mathcal{Q}_{IW}+\delta \mathcal{Q}_{ct}
\end{eqnarray}
On the one hand, there is a nonzero contribution from the Iyer-Wald charge
\begin{eqnarray}
\delta \mathcal{Q}_{IW}=\frac{1}{16\pi} \int_{\partial\Sigma} *F_{(IW)}
\end{eqnarray}
with
\begin{eqnarray}
{F_{(IW)}}_{ab}=\frac{1}{2} \nabla_{a} \zeta_b h+\nabla_{a}{h^c}_b\zeta_c+\nabla_c \zeta_a {h^c}_b+\nabla_c {h^c}_a \, \zeta_b-\nabla_a h\, \zeta_b - a \leftrightarrow b\,.
\end{eqnarray}
Here we follow the conventions of \cite{Haco:2018ske}. The variation $h^{ab}$ is defined as $g^{ab}\rightarrow g^{ab}+h^{ab}$. We take the metric perturbation $h^{ab}=\mathcal{L}_{\tilde{\zeta}_m}g^{ab}$ due to the second diffeomorphism $\tilde{\zeta}$ and $h=h^{ab}g_{ab}$.\\
On the other hand, one also needs to add a counterterm
\begin{eqnarray}
\delta \mathcal{Q}_{ct} =\frac{1}{16\pi}\int_{\partial\Sigma} {F_{(ct)}}_{ab}\, d \Sigma^{ab}\,
\end{eqnarray}
where $N$ is the volume two-form on the normal bundle to the $\Sigma_{bif}$.
\begin{eqnarray}
{F_{(ct)}}_{ab}=- 2 {N_{d}}^c\nabla_c(\zeta_a {h^d}_{b}) - a \leftrightarrow b\,,
\end{eqnarray}
Note that the addition of this counterterm is justified to achieve integrability. The nonzero contributions to $K_{n,m}$ come only from
\begin{eqnarray}
\delta\mathcal{Q}=\frac{1}{16 \pi} \int d\theta dw^+ \epsilon_{\theta + - y} ({F_{(IW)}}^{-y}+{F_{(ct)}}^{-y})
\end{eqnarray}
We find that
\begin{eqnarray}
{F_{(IW)}}^{-y}=4 h^{y-}\zeta^y\Gamma^-_{y-}\,,
\end{eqnarray}
Considering ${N_+}^+=1$, ${N_-}^-=-1$ then
\begin{eqnarray}
{F_{(ct)}}^{-y}&=&-2 \nabla_+(\zeta^- h ^{+y})+2 \nabla_+(\zeta^y h ^{+-})+2 \nabla_-(\zeta^- h ^{-y})-2 \nabla_-(\zeta^y h ^{--})\\
&=&2 \zeta^y(\nabla_+ h^{+-})+2 (\nabla_-\zeta^-) h^{-y}-2\zeta^y (\nabla_- h^{--})\\
&=&2 \zeta^y h^{-y}(\Gamma^+_{+y}+\Gamma^-_{-y}-2 \Gamma^-_{-y})\\
&=& 2 \zeta^y h^{-y} (\Gamma^+_{+y}-\Gamma^-_{-y})
\end{eqnarray}
Note that
\begin{eqnarray}
h^{+-}=0\,, \qquad h^{--}=0\,,\qquad \nabla_+ \zeta^-=0\,,\qquad \nabla_- \zeta^-=\Gamma^-_{-y}\zeta^y\,,
\end{eqnarray}
and also
\begin{eqnarray}
\nabla_+h^{+y}=0\,,\qquad \nabla_+h^{+-}=\Gamma^+_{+y} h^{-y}\,,\qquad \nabla_-h^{-y}=0\,,\qquad \nabla_-h^{--}=2\Gamma^-_{-y} h^{-y}\,.
\end{eqnarray}
Adding the terms together one finds
\begin{eqnarray}
\delta\mathcal{Q}&=&\frac{1}{16\pi}\int d\theta dw^+ \epsilon_{+- y \theta} (4 h^{y-}\zeta^y\Gamma^-_{y-}+ 2 \zeta^y h^{-y} (\Gamma^+_{+y}-\Gamma^-_{-y}))\,,\\
&=&\frac{1}{16\pi}\int d\theta dw^+ \frac{4 a \sin\theta \rho_+^2 }{\delta y^3} 2 h^{y-}\zeta^y(\Gamma^-_{y-}+\Gamma^+_{+y})\,,
\end{eqnarray}
By working at small $w^+$ and taking the $w^+$ limit (which amounts to approaching $\Sigma_{bif}$ along the future horizon) one finds
\begin{eqnarray}
&& h^{-y}=g^{+-}\partial_+ \zeta^y = \frac{y^3\tilde{\epsilon}''}{4\rho^2_+} \qquad \text{with} \qquad '=\partial_+\,,\\
&& \int d\theta \sin\theta =2, \,\, \text{and} \qquad \Gamma^+_{+y}+\Gamma^-_{-y}=-\frac{2}{y}.
\end{eqnarray}
Choosing $\zeta$ to be $\zeta_n$ and $\tilde\zeta$ to be $\zeta_m$, the variation becomes
\begin{eqnarray}
K_{m,n} = \delta\mathcal{Q}&=&\frac{1}{16\pi}\int d\theta dw^+ \frac{4 a \sin\theta \rho_+^2 }{\delta y^3} 2 \left(\frac{y^3\epsilon_m''}{4\rho^2_+}\right) \left(\frac{1}{2} y \epsilon_n' \right)\left(\frac{-2}{y}\right) \,\\
&=&-\frac{1}{16\pi}\int d\theta dw^+ \frac{2 a \sin\theta }{\delta } {\epsilon_m''} \epsilon_n' \\
&=&-\frac{1}{16\pi}\int \frac{dw^+ }{w^+ }\frac{ 4 a }{\delta } {\epsilon_m''} \epsilon_n' \,,\\
&=&-\frac{1}{16\pi}(4\pi^2 T_R)\frac{ 4 a }{\delta } \frac{i m^3}{2\pi T_R} \delta_{m+n,0}\,\\
&=&\frac{ a }{2\delta } i m^3 \delta_{m+n,0}\,.
\end{eqnarray}
Here we have computed the Dirac bracket of two charges. Passing to the commutator introduces a factor of $-i$
resulting in a central charge of
\begin{eqnarray}
c_R=\frac{6 a }{\delta }
\end{eqnarray}
in agreement with (\ref{centralAdsKerr}).
\subsection*{Left-moving charges}
The computation of the central charge in the left-moving sector exactly parallels the computation in the right moving sector. All one needs to do is to exchange $\epsilon$ for $\bar\epsilon$, interchange $w^+ \leftrightarrow w^-$ together with a similar exchange in the components of the tensors and Christoffel symbols and finally
replace $T_R$ by $T_L$. The result for the left-moving central charge $c_L$ is
\begin{equation} c_L = \frac{6a}{\delta},\end{equation}
which is exactly the same as $c_R$.
\section{Extremal AdS-Kerr black hole}
In the extreme limit (when $T_{\pm} =0 $), the inner and outer horizons degenerate to a single horizon at $r_0\equiv r_+=r_-$.
The extremality condition is $\Delta(r_0)=\Delta'(r_0)=0$ that implies
\begin{eqnarray}
a^2=\frac{r_0^2(1+3r_0^2/L^2)}{(1-r_0^2/L^2)}\,\nonumber \\
M=\frac{r_0(1+r_0^2/L^2)^2}{(1-r_0^2/L^2)}\,.
\end{eqnarray}
At extremality, the right temperature arising for the CFT vanishes. And, the left temperature in the extremal limit agrees exactly with previous results \cite{Hartman:2008pb,Lu:2008jk}
\begin{eqnarray}\label{tempextremal}
T_R=0\,,\qquad T_L=\frac{1+6r_0^2L^{-2}-3r_0^4L^{-4}}{2\pi(1-3r_0^2L^{-2})\sqrt{(1+3r_0^2L^{-2})(1-r_0^2L^{-2})}}\,.
\end{eqnarray}
where $r_0\le L/\sqrt{3}$ to restrict the values of $0<T_L$.
The central charge (\ref{centralAdsKerr}) becomes
\begin{eqnarray}\label{centralextremal}
c_L=\frac{12 r_0\sqrt{r_0^2(1+3 r_0^2/L^2)(1-r_0^2/L^2)}}{1+6 r_0^2/L^2-3r_0^4/L^4}\,.
\end{eqnarray}
For a unitary conformal field theory at temperature $T_L$, the microscopic entropy from the Cardy formula is given by
\begin{eqnarray}
S_+=\frac{\pi^2}{3} c_L T_L\,.
\end{eqnarray}
From (\ref{centralextremal}) and (\ref{tempextremal}), we therefore obtain the microscopic entropy
\begin{eqnarray}
S_+=\frac{2\pi r_0^2}{1-3r_0^2/L^2}\,.
\end{eqnarray}
This is in perfect agreement with the extreme black hole entropy given in (\ref{thermodynamicsAdS}). For vanishing cosmological constant $(L\rightarrow\infty)$ the results reduce to the asymptotically flat Kerr black hole.
\section{First Law of Thermodynamics}
In this section we discuss the first law of thermodynamics for the AdS-Kerr black holes \cite{Gibbons:2004ai}, that is given by
\begin{eqnarray}
\delta E = \pm T_{\pm} \delta S_{\pm} + \Omega^{\pm}_{\infty} \delta J\,.
\end{eqnarray}
Remarkably, the first law may also be written
\begin{eqnarray}
\delta S_{\pm} = \frac{\delta E_L}{T_L} \pm \frac{\delta E_R}{T_R}\,.
\end{eqnarray}
where \footnote{As described in \cite{Castro:2013kea}, we consider the change $\phi\rightarrow \hat{\phi}- a\, t/L^2$ in (\ref{trtl}).}
\begin{eqnarray}
\delta E_L&=&-\frac{\gamma}{\delta} \,\delta E+\frac{\gamma}{\delta} \frac{a }{ L^2} \, \delta J\,,\\
\delta E_R&=&-\frac{\gamma}{\delta} \, \delta E+ \left(-1+\frac{\gamma}{\delta} \frac{a }{ L^2}\right)\delta J\,,
\end{eqnarray}
and the right and left temperatures are defined in (\ref{AdsTemp}). We have also used here $\delta E_L=-\gamma\, \delta \mathcal{E}_L$ and $\delta E_R=\alpha\, \delta \mathcal{E}_R$ \footnote{Provided $\epsilon$ in the vector field $\zeta=\epsilon \partial_+ +\frac{1}{2} \partial_+ \epsilon y \partial_y$, and similarly $\bar{\epsilon}$ in $\bar{\zeta}=\bar{\epsilon} \partial_- +\frac{1}{2} \partial_- \bar{\epsilon} y \partial_y$, is invariant under $2 \pi$ azimuthal rotations \cite{Haco:2018ske} we define the normalized functions $\epsilon=2\pi T_{R} (w^+)^{1+\frac{i n}{2\pi T_R}}=\alpha \,(w^+)^{1+\frac{i n}{\alpha}}$ and $\bar{\epsilon}=2\pi T_{L} (w^-)^{1+\frac{i n}{2\pi T_L}}=\gamma \,(w^-)^{1+\frac{i n}{\gamma}}$.}.
\section{Inner black hole horizon}
We could choose the inner horizon $r_-$ instead to proceed with the approach. We first take the transformation (\ref{eq:coordtransf}) with (\ref{trtl1}), and set $\tilde{c}^2 = 1/\Delta'(r_-)$ in the function $R^2(r) \sim \tilde{c} \,(r- r_-)$ close to the Cauchy horizon of the black hole, such that the leading order of the black hole metric around the bifurcation surface is
\begin{eqnarray}
ds^2&=& \frac{4\,\rho_-(\theta)^2}{y^2}dw^+ dw^-+\frac{k^2\, \Delta_\theta(\theta) \sin^2\theta}{y^2 \rho_-(\theta)^2}dy^2+\frac{\rho_-(\theta)^2}{\Delta_\theta(\theta)}d\theta^2\nonumber +...\,,\nonumber
\end{eqnarray}
where
\begin{eqnarray}
\rho_-(\theta)^2=r_-^2+a^2\cos^2\theta\,,\qquad k= 2 a /\tilde{\delta}\,.
\end{eqnarray}
In this case we find the parameters in the transformation become
\begin{eqnarray}
\tilde{\alpha}=\frac{\Delta'(r_-)}{2 \,a\Xi}\,,\qquad \tilde{\beta}=0\,,\qquad\tilde{\gamma}=\frac{(\Omega_+ +\Omega_-)}{(\Omega_- -\Omega_+)}\tilde{ \alpha}
\,,\qquad \tilde{\delta}=-\Omega_- \, (\tilde{\alpha}+\tilde{\gamma})\,,
\end{eqnarray}
or
\begin{eqnarray}
\tilde{\alpha}={2\pi T_R}\,,\qquad \tilde{\beta}=0\,,\qquad \tilde{\gamma}=2\pi T_L
\,,\qquad \tilde{\delta}=-\frac{2\pi a\Xi \, (T_L+T_R)}{(a^2+r_-^2)}\,.
\end{eqnarray}
To proceed with the interpretation of the right and left-temperatures arising from the CFT we compute (\ref{temRL}) that brings them to the form
\begin{eqnarray}\label{AdsTempIn}
T_R=\frac{\Delta'(r_-)}{4\pi \,a\Xi}\,,\qquad T_L=\frac{(\Omega_++\Omega_-)}{(\Omega_- -\Omega_+)}\frac{\Delta'(r_-)}{4\pi \,a\Xi }\,.
\end{eqnarray}
The general result (\ref{AdsTempIn}) for the right and left-temperatures arising from the CFT for the AdS-Kerr black holes can be combined with the Cardy entropy formula
\begin{equation}
S_{\pm}=\frac{\pi^2}{3}\left(c_L T_{L}\pm c_R T_R \right)\,,
\end{equation}
using (\ref{AdsTempIn}) and the entropies (\ref{thermodynamicsAdS}), that for Kerr black holes the left and right CFTs have identical central charges
\begin{eqnarray}\label{centralAdsKerr}
c_L=c_R=-\frac{6 a }{\tilde{\delta}}=\frac{ 6 a (r_-^2-r_+^2)}{\Delta'(r_-)}\,.
\end{eqnarray}
\section{Discussion}
\label{sec:discussion}
The hidden conformal symmetries of black holes are of particular interest to explain the leading black hole microstate degeneracy.
Because many of the most intriguing results along these lines have been found in the context of asymptotically flat black holes, the extension of the soft hair approach to include spacetimes with cosmological constant is both natural and important.
Our investigations of the soft hair has allowed us to establish a set of infinitesimal diffeomorphisms that act non-trivially on the horizon of generic Kerr-AdS black holes. Amongst these are a left-right pair of Virasoro algebras with associated charges that reproduce the Bekenstein-Hawking entropy for AdS black holes.
The fact that there are generic near-AdS$_3$ regions exhibiting the conformal structure of the black hole geometry is in general not sufficient to determine the conformal coordinates relevant to the application of this formalism. We argued that the Noether charge interpretation of the entropy in \cite{Wald:1993nt} gives the necessary input to uniquely fix the relevant conformal coordinates. Adding a boundary counterterm, justified to achieve integrability, lead to well-defined central charges $c_L = c_R$.
We do not herein prove uniqueness of the counterterm or attempt to tackle other difficult problems related to the characterizing diffeomorphisms or charges.
It is worth emphasizing that the choice of counterterm considered in our paper - that generalizes the Wald-Zoupas counterterm for asymptotically AdS space-times - gives a result that is compatible with all other results in the literature for Kerr-AdS black holes, see e.g. \cite{Hartman:2008pb,Lu:2008jk}. In the case of asymptotically flat rotating black holes and electrically charged black holes, the contributions from the counterterm reduce to the same ones from the Wald-Zoupas counterterm \cite{Haco:2018ske,Haco:2019ggi}.
A connection between the generic Kerr-AdS black hole results in this paper and previous results involving their extremal counterparts indicates a close link between the different computational approaches. By providing a first law of thermodynamics we also made a step in giving further evidence of the robustness of the soft hair formalism in reproducing the Bekenstein-Hawking entropy.
\section*{Acknowledgements}
We are grateful to Andrew Strominger for useful conversations. We would like to thank the Mitchell Family Foundation for hospitality in 2018 at the Brinsop Court in England and 2019 Cook's Branch workshop and for continuing support. Part of this work was conducted at the `Fourth USU Strings and Black Holes Workshop', which was supported by the Department of Physics and the DGCAMP group at Utah State University. We would also like to thank the Centro de Ciencias de Benasque Pedro Pascual for the hospitality where some of the research was carried out. The work of MJR is partially supported through the NSF grant PHY-1707571, SEV-2016-0597 and PGC2018-095976-B-C21 from MCIU/AEI/FEDER, UE. MJP is supported by an STFC consolidated grant ST/L000415/1, String Theory, Gauge Theory and Duality.
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« Mysteries of Lisbon (2010) | Main | Barney's Version (2010) + No Strings Attached (2011) »
The Smurfs (2011)
½*/****
starring Neil Patrick Harris, Jayma Mays, Sofia Vergara, Hank Azaria
screenplay by J. David Stem & David N. Weiss and Jat Scherick & David Ronn
directed by Raja Gosnell
by Walter Chaw Between preaching its preach about not being pigeonholed and the importance of living life in the moment, Raja Gosnell's The Smurfs misses no opportunity to talk about the superficiality of Smurfette (voice of Katy Perry) discovering her secret shopping bug; Gargamel (Hank Azaria) turning an "old lady" into a balloon-chested hottie; and human hero Patrick (Neil Patrick Harris) helping his harridan cosmetics boss Odile (Sofia Vergara) sell gallons of snake oil to the Vanity Smurfs (voice of John Oliver) of the world. There's also a lot of pissing, puking, shitting, and farting; a disturbing running joke about putting heads on a pike; highly-imitable and often-disturbing cat violence; and a wave of overwhelming weariness that rolls off these Alvin and the Chipmunks/The Sorcerer's Apprentice pieces of shit that tend to flop but never hard enough to prevent the clockwork arrival of another something just like it. Fact is, the kid-movie market is too lucrative to not take homerun swings at it with '80s-nostalgic, high-concept falderal such as this; fact is, too, that The Smurfs, et al, come coated in critic-repellent asbestos, because no matter how deadening and odious something is, as long as your pliant and uncritical children enjoy it, it's fine. What were you expecting, Citizen Kane? Were that the same rationale applied to food made for children: what were you expecting, free of salmonella and rat turds?
Buy This at Allposters.com
Not helping is that any adaptation of Belgian acid-dropper Peyo's claim to immortality is most likely doomed to be deeply, fundamentally irritating. Three apples high, smurfs are little blue elf things that, when boiled, apparently give off a magic "essence" that baddie Gargamel wishes to gather, leading at the end to the second snapping of an all-powerful wand on the heels of Harry Potter and the Deathly Hallows Part 2. The Smurfs are named after their defining characteristic--like Disney's dwarfs, come to think of it--so time doesn't need to be spent on trying to figure out what the punchlines are once the Pez-dispenser screenplay-by-committee decides it's time to make another roundhouse lunge at funny. Failing that, The Smurfs falls on that old standby of a musical sequence when, during a session of "Guitar Hero", Brainy (voice of Fred Armisen), kilt-wearing Gutsy (not Mike Myers, shockingly), Clumsy (voice of Anton Yelchin), and Grouchy (voice of George Lopez) do their interpretation of an Aerosmith karaoke standard. Hard to watch? At least hard to watch. But, really, no harder to watch than Harris and beautiful ginger ostrich Jayma Mays as his pregnant opposite-sex mate Grace go through the motions of the workaholic-learns-to-be-a-loving-family-man masterplot.
Easy to take potshots at, The Smurfs ultimately isn't better or worse than every other movie in its genre. Things counting in its favour include the decision to cast Harris instead of Eddie Murphy or Kevin James; things counting against it are that it's terrible and, worse, terrible in exactly the same way as everything else exactly like it is terrible. Gosnell should be hung by his thumbs, but because his filmography is the fertile loam from which prefab franchises sprout, he's given the keys to the kingdom instead. The Smurfs, even as it opens in its "magical" parallel forest universe, never threatens to be interesting, never promises anything beyond broad slapstick and crowd-pleasing homilies. It identifies this Jacksonian ideal that runs deep into the bedrock of American culture of the valour of getting hands dirty and the importance of beating the holy hell out of Brainy whenever it can, physically and emotionally--and who cares, because he's for moderation in a go-go world. The worst part is that I didn't actually hate The Smurfs, since there's a cumulative effect of this stuff (see the Scooby-Doo flicks, for another instance) that tends to blunt its ability to inspire any kind of emotion whatsoever. Eventually, popular culture wins and you don't bother to fight; eventually, and it's coming for all of us, you stop realizing that there's a fight to be joined. Originally published: July 29, 2011.
Posted in 2011, Adventure, Authors: Walter Chaw, Comedy, Family, Theatrical | Permalink
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{"url":"https:\/\/ftp.aimsciences.org\/article\/doi\/10.3934\/cpaa.2009.8.2037","text":"# American Institute of Mathematical Sciences\n\nNovember\u00a0 2009,\u00a08(6):\u00a02037-2053. doi:\u00a010.3934\/cpaa.2009.8.2037\n\n## A logistic equation with refuge and nonlocal diffusion\n\n 1 Dpto. de An\u00e1lisis Matem\u00e1tico, Universidad de La Laguna, C\/. Astrof\u00edsico Francisco S\u00e1nchez s\/n, 38271 - La Laguna 2 Dpto. de Matem\u00e1ticas, FCEyN, Universidad de Buenos Aires, 1428 \u2013 Buenos Aires, Argentina\n\nReceived\u00a0 November 2008 Revised\u00a0 April 2009 Published\u00a0 August 2009\n\nIn this work we consider the nonlocal stationary nonlinear problem $(J* u)(x) - u(x)= -\\lambda u(x)+ a(x) u^p(x)$ in a domain $\\Omega$, with the Dirichlet boundary condition $u(x)=0$ in $\\mathbb{R}^N\\setminus \\Omega$ and $p>1$. The kernel $J$ involved in the convolution $(J*u) (x) = \\int_{\\mathbb{R}^N} J(x-y) u(y) dy$ is a smooth, compactly supported nonnegative function with unit integral, while the weight $a(x)$ is assumed to be nonnegative and is allowed to vanish in a smooth subdomain $\\Omega_0$ of $\\Omega$. Both when $a(x)$ is positive and when it vanishes in a subdomain, we completely discuss the issues of existence and uniqueness of positive solutions, as well as their behavior with respect to the parameter $\\lambda$.\nCitation: J. Garc\u00eda-Meli\u00e1n, Julio D. Rossi. A logistic equation with refuge and nonlocal diffusion. Communications on Pure & Applied Analysis, 2009, 8 (6) : 2037-2053. doi: 10.3934\/cpaa.2009.8.2037\n [1] Weiyi Zhang, Zuhan Liu, Ling Zhou. Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3767-3784. doi: 10.3934\/dcdsb.2020256 [2] Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2777-2808. doi: 10.3934\/dcds.2020385 [3] Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, , () : -. doi: 10.3934\/era.2021024 [4] Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete & Continuous Dynamical Systems - B, 2021\u00a0 doi: 10.3934\/dcdsb.2021129 [5] Claudianor O. Alves, C\u00e9sar T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934\/cpaa.2021058 [6] Meiqiao Ai, Zhimin Zhang, Wenguang Yu. First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits. Journal of Industrial & Management Optimization, 2021\u00a0 doi: 10.3934\/jimo.2021039 [7] Raffaele Folino, Ram\u00f3n G. Plaza, Marta Strani. Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3211-3240. doi: 10.3934\/dcds.2020403 [8] Rafael Lu\u00eds, Sandra Mendon\u00e7a. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934\/dcdsb.2020094 [9] Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934\/dcdsb.2020186 [10] Hai-Yang Jin, Zhi-An Wang. The Keller-Segel system with logistic growth and signal-dependent motility. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3023-3041. doi: 10.3934\/dcdsb.2020218 [11] Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems & Imaging, 2021, 15 (3) : 519-537. doi: 10.3934\/ipi.2021003 [12] Jos\u00e9 A. Carrillo, Bertram D\u00fcring, Lisa Maria Kreusser, Carola-Bibiane Sch\u00f6nlieb. Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3985-4012. doi: 10.3934\/dcds.2021025 [13] Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934\/dcdsb.2020194 [14] Lu Xu, Chunlai Mu, Qiao Xin. Global boundedness of solutions to the two-dimensional forager-exploiter model with logistic source. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3031-3043. doi: 10.3934\/dcds.2020396 [15] Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934\/dcdsb.2020042 [16] Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934\/dcdsb.2018035 [17] Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, 2021, 15 (3) : 475-498. doi: 10.3934\/ipi.2021001 [18] Annalisa Cesaroni, Valerio Pagliari. Convergence of nonlocal geometric flows to anisotropic mean curvature motion. Discrete & Continuous Dynamical Systems, 2021\u00a0 doi: 10.3934\/dcds.2021065 [19] Lara Abi Rizk, Jean-Baptiste Burie, Arnaud Ducrot. Asymptotic speed of spread for a nonlocal evolutionary-epidemic system. Discrete & Continuous Dynamical Systems, 2021\u00a0 doi: 10.3934\/dcds.2021064 [20] Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934\/dcdss.2016.9.109\n\n2019\u00a0Impact Factor:\u00a01.105","date":"2021-04-20 07:53:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.655800461769104, \"perplexity\": 7710.139629601926}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618039379601.74\/warc\/CC-MAIN-20210420060507-20210420090507-00354.warc.gz\"}"}
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<?php
//This file cannot be viewed, it must be included
defined('IN_EZRPG') or exit;
/*
Class: Module_Logout
This module clears the session data to logout the user.
*/
class Module_Logout extends Base_Module
{
/*
Function: start
Clears session data and redirects back to homepage.
*/
public function __construct(&$db, &$tpl, &$player=0)
{
unset($_SESSION['hash']);
unset($_SESSION['userid']);
session_unset();
session_destroy();
global $hooks;
$hooks->run_hooks('logout');
$msg = 'You have been logged out!';
header('Location: index.php?msg=' . urlencode($msg));
exit;
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"redpajama_set_name": "RedPajamaGithub"
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Anthony Depula (June 28, 1966 – April 4, 2009), known professionally as Tony D, was an American hip hop artist from Trenton, New Jersey.
Career
Although he was an MC and a DJ, he was most famous for being an influential producer in hip hop music. He was the producer behind Poor Righteous Teachers, YZ, Blaque Spurm, Baby Chill and The Funk Family. He has also produced tracks for Young Zee, Outsidaz, Pace Won, Blvd Mosse, King Sun, Scott Lark, Kwest tha Madd Lad and Shawn Lov.
Tony D was the first artist to have a record released on Mark Rae's burgeoning British Grand Central Records independent record label (then called Gone Clear Records). His other albums were released on his label Cha-Ching Records and 4th & B'way/Island/PolyGram Records, and he was a part of the group Crusaders for Real Hip-Hop, which released one album on Profile Records called Déjà Vu - It's '82.
Tony D's early records were sampled by several artists, including Naughty by Nature on the hit "O.P.P." and by Heavy D without permission. All cases were settled out of court.
Apart from Tony D, Depula was also known professionally under the names Harvee Wallbangar, Don Nots, Don Sluggo, the Goombata and, in his earliest credits, Grand Pubha Tony D.
Later life
Later in life, Depula hosted a weekly hip hop radio show on Princeton University's 103.3 FM WPRB, which aired in Central New Jersey.
Death
Depula died on April 4, 2009, after being involved in a car accident near his home in Hamilton Township, Mercer County, New Jersey.
Discography
Music Makes You Move (1988)
Euro-K (1990) (DJ mix)
Droppin' Funky Verses (1991) U.S. R&B #71
Deja Vu, It's '82 (with Crusaders for Real Hip-Hop) (1992)
Flav (Beats) from the Cave (1994)
Trenton Connection (1996) contains Come Back to You used on AIM Fabriclive.17
Pound for Pound (1997)
Master of the Moaning Beats (2001)
Revone - I'm Still Here (2002)
Da Goodfella (Cassette, EP) (2020)
References
External links
Discography of Tony D at Discogs
1966 births
2009 deaths
American hip hop DJs
Island Records artists
American hip hop record producers
Road incident deaths in New Jersey
Rappers from New Jersey
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Martin Tomczyk wasn't able to celebrate his maiden pole position for BMW for too long. After his fastest time in qualifying for the second DTM season round at Brands Hatch, it was found in scrutineering of his car that the 2011 champion's BMW M3 DTM was underweight. As a result, the stewards of the meeting excluded the 31-year old from qualifying, but he is allowed to start the race on Sunday (12.30 hrs local time/13.30 hrs CET) from last place on the grid.
Because of Tomczyk's exclusion, pole position went to Mike Rockenfeller with the Audi RS 5 DTM, who will be starting from first place on the grid for the second time in is DTM career, like in 2011 again at Brands Hatch. Augusto Farfus with the BMW M3 DTM will be lining up alongside Rockenfeller on the front row of the grid. With two further BMWs, Joey Hand and champion Bruno Spengler follow in third and fourth place respectively. As a result of Tomczyk's exclusion, the best-placed Mercedes-Benz driver, Gary Paffett, moves up to eleventh place.
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{"url":"http:\/\/t-news.cn\/Floc2018\/FLoC2018-pages\/SR-2018-07-08.html","text":"FLOC 2018: FEDERATED LOGIC CONFERENCE 2018\nSR ON SUNDAY, JULY 8TH\nDays:\nprevious day\nall days\n\nView: session overviewtalk overviewside by side with other conferences\n\n10:30-11:00Coffee Break\n12:30-14:00Lunch Break\n14:00-15:30 Session 40Q: Two-player games\n14:00\nInfinite-Duration Richman Bidding Games\nSPEAKER: Guy Avni\n\nABSTRACT. Two-player games on graphs are widely studied in formal methods as they model the interaction between a system and its environment. The game is played by moving a token throughout a graph to produce an infinite path. There are several common modes to determine how the players move the token through the graph; e.g., in turn-based games the players alternate turns in moving the token. We study the {\\em bidding} mode of moving the token, which, to the best of our knowledge, has never been studied in infinite-duration games. Both players have separate {\\em budgets}, which sum up to $1$. In each turn, a bidding takes place. Both players submit bids simultaneously, and a bid is legal if it does not exceed the available budget. The winner of the bidding pays his bid to the other player and moves the token. Reachability bidding games, called {\\em Richman games}, have been studied in \\cite{LLPU96,LLPSU99}. There, a central question is the existence and computation of {\\em threshold} budgets; namely, a value $t \\in [0,1]$ such that if \\PO's budget exceeds $t$, he can win the game, and if \\PT's budget exceeds $1-t$, he can win the game. We focus on parity and mean-payoff games. We show the existence of threshold budgets and show that the complexity of finding them coincides with the $NP \\cap coNP$ complexity of reachability bidding games. The solution for mean-payoff consists of our most technically challenging contribution, where we construct optimal strategies for the players while extending and generalizing the probabilistic connection that was known for reachability bidding games.\n\n14:40\nSolving Parity Games: Explicit vs Symbolic\n\nABSTRACT. In this paper we provide a broad investigation of the symbolic approach for solving Parity Games. Specifically, we implement in a fresh tool, called SymPGSolver, four symbolic algorithms to solve Parity Games and compare their performances to the corresponding explicit versions for different classes of games. By means of benchmarks, we show that for random games, even for constrained random games, explicit algorithms actually perform better than symbolic algorithms. The situation changes, however, for structured games, where symbolic algorithms seem to have the advantage. This suggests that when evaluating algorithms for parity-game solving, it would be useful to have real benchmarks and not only random benchmarks, as the common practice has been.\n\n15:30-16:00Coffee Break\n16:00-18:00 Session 42Q: Strategy Logic\n16:00\nQuantifying Bounds in Strategy Logic\n\nABSTRACT. Program synthesis automatically constructs programs from specifications. Strategy Logic is a powerful specification language whose goal is to give theoretical foundations for program synthesis in a multi-agent setting. One limitation of Strategy Logic is that it is purely qualitative. For instance it cannot specify quantitative properties of executions such as \"every request is quickly granted\", or quantitative properties of trees such as \"most executions of the system terminate\". In this work, we extend Strategy Logic to include quantitative aspects in a way that can express bounds on \"how quickly\" and \"how many\". We define Prompt Strategy Logic, which encompasses Prompt LTL (itself an extension of LTL with a prompt eventuality temporal operator), and we define Bounded-Outcome Strategy Logic which has a bounded quantifier on paths. We supply a general technique, based on the study of automata with counters, that solves the model-checking problems for both these logics.\n\n16:40\nStrategy Logic with Imperfect Information\n\nABSTRACT. We introduce an extension of Strategy logic for the imperfect-information setting, called SLii, and study its model-checking problem. As this logic naturally captures multi-player games with imperfect information, the problem turns out to be undecidable. We introduce a syntactical class of \"hierarchical instances\" for which, intuitively, as one goes down the syntactic tree of the formula, strategy quantifications are concerned with finer observations of the model. We prove that model-checking SLii restricted to hierarchical instances is decidable. This result, because it allows for complex patterns of existential and universal quantification on strategies, greatly generalises previous ones, such as decidability of multi-player games with imperfect information and hierarchical observations, and decidability of distributed synthesis for hierarchical systems.\n\n17:20\nDependences in Strategy Logic\n\nABSTRACT. Strategy Logic (SL) is a very expressive temporal logic for specifying and verifying properties of multi-agent systems: in SL, one can quantify over strategies, assign them to agents, and express LTL properties of the resulting plays. Such a powerful framework has two drawbacks: first, model checking SL has non-elementary complexity; second, the exact semantics of SL is rather intricate, and may not correspond to what is expected. In this paper, we focus on strategy dependences in SL, by tracking how existentially-quantified strategies in a formula may (or may not) depend on other strategies selected in the formula, revisiting the approach of [Mogavero et al., Reasoning about strategies: On the model-checking problem, 2014]. We explain why elementary dependences, as defined by Mogavero et al., do not exactly capture the intended concept of behavioral strategies. We address this discrepancy by introducing timeline dependences, and exhibit a large fragment of SL for which model checking can be performed in 2EXPTIME under this new semantics.","date":"2023-03-31 08:35:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.800502359867096, \"perplexity\": 1488.038154873158}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296949598.87\/warc\/CC-MAIN-20230331082653-20230331112653-00641.warc.gz\"}"}
| null | null |
Notes From Our Contributors
"What a treasure trove! The fact that these stories are all written by women makes this book even more intriguing. How wonderful to be part of this vibrant and beautiful anthology."
\-- _Susan Orlean, author of_ The Orchid Thief.
"This is the collection I wish I'd had when I was starting out as a writer. Back then, non-fiction was the purview of men; here's an unequivocal affirmation that it no longer is."
\-- _Elizabeth Kaye, author of_ Lifeboat No. 8.
"One of the many beauties of nonfiction as a genre is that it invites diversity in the stories it seeks to tell and, by extension, in the practitioners who tell those stories. At long last, the contributions of women as leaders in this field are being fully acknowledged."
\-- _Madeleine Blais, author of_ To the New Owners.
Here's the book you've been missing, the only one you'll need for the next week or two, brimming with tall tales, hairpin turns, and poignant moments, all of them true, and with deftly-captured personalities brought vividly to life in these pages.
\-- _Melissa Fay Greene, Author of_ Praying for Sheetrock.
This is an amazing collection of journalists who just happen to be female--it's a must read for any and all young writers, a much-needed road map for how to report, write, and think about stories.
\-- _Mimi Swartz, author, editor and two-time National Magazine Award winner_.
I'm so honored to be included in these pages with some of the true masters of our genre. I've been waiting for a book like this for longest time. _The Stories We Tell_ belongs in the permanent collection of anyone who loves reading and writing reported stories and essays.
\-- _Jeanne Marie Laskas, author of_ Concussion.
"I won't be here to witness it, but won't it be a fine day when anthology specifically focused on women journalists won't make any sense?"
\-- _Adrian LeBlanc, author of_ Random Family.
More Intriguing Text/Anthologies from The Sager Group
_Newswomen: Twenty-five Years of Front Page Journalism_
•
_Janet's World: The Inside Story of Washington Post Pulitzer Fabulist Janet Cooke_
•
_Next Wave: America's New Generation of Great Literary Journalists_
•
_Artful Journalism: Essays in the Craft and Magic of True Storytelling_
•
_Everyone Leaves Behind a Name: True Stories by Michael Brick_
•
_Danny Yukon and the Secrets of the Amazing Lamp_
**The Stories We Tell: Classic True Tales by America's Greatest Women Journalists**
Published at Smashwords
Copyright © 2017 by The Sager Group LLC
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.
Published in the United States of America.
Cataloging-in-Publication data for this book is available from the Library of Congress.
ISBN-13: 978-0-9980793-1-8
ISBN-10: 0-9980793-1-6
Cover Designed by Stravinski Pierre and Siori Kitajima, SF AppWorks LLC
www.sfappworks.com/
Formatted by Ovidiu Vlad
Published by The Sager Group LLC
www.TheSagerGroup.net
info@TheSagerGroup.net
**The Stories We Tell**
**CLASSIC TRUE TALES BY AMERICA'S GREATEST WOMEN JOURNALISTS**
**EDITED BY PATSY SIMS**
With Research By Jamie Ballard and Caitlynne Leary
Published by The Sager Group at Smashwords
_We tell ourselves stories in order to live._
—Joan Didion, _The White Album_
Table of Contents
Introduction by Patsy Sims
They Were Commandos by Madeleine Blais
After near-perfect seasons tarnished by losses in the state championships, the Amherst High School girls' basketball team might finally change its story.
The Cheerleaders by E. Jean Carroll
The tiny town of Dryden, New York, endures a strange, five-year string of murders, car accidents, and suicides—all of it tied to two popular high school cheerleaders.
Some Dreamers of the Golden Dream by Joan Didion
In this New Journalism classic, a suburban California woman is convicted of burning her husband to death in their family Volkswagen.
Wonder Dog by Melissa Fay Greene
How a family struggling with a child's serious cognitive and behavioral disabilities finds peace and reassurance with an assistance dog.
Holy Days: The World of a Hasidic Family by Lis Harris
A rare glimpse inside the Hasidic Jewish community of Brooklyn, New York—how an ancient and traditional culture is adapting to life in the modern age.
The Last Day by Robin Marantz Henig
A compelling profile of prominent psychologist and professor Sandy Bem, who voluntarily ended her life before succumbing to Alzheimer's disease.
On the Bus with B.B. King by Gerri Hirshey
On the road with the great blues man and his beloved guitar, Lucille. Classic music journalism by _Rolling Stone_ 's first female contributing editor.
Nureyev Dancing in His Own Shadow by Elizabeth Kaye
Celebrated and famously volatile ballet dancer Rudolf Nureyev, captured intimately at the end of a glorious career.
The New Face of Richard Norris by Jeanne Marie Laskas
Richard Norris undergoes complex transplant surgery, giving him a new face and a new life, but leaving him with many of the same problems.
The Troubled Life of Boys by Adrian Nicole LeBlanc
A look at high school boys as they struggle with the changing perceptions of masculinity in the feminist culture of the 1990s.
Prodigal Daughter by Jill Lepore
A writer researching a book about Ben Franklin's extraordinary sister Jane finds illuminating connections to her own spirited, artistic mother.
The Split by Suzannah Lessard
In Manhattan, the opulent Upper East Side, one of the nation's richest neighborhoods, gives way to Harlem, one of the poorest.
Yuja Wang and the Art of Performance by Janet Malcolm
The young pianist is known not only for the brilliance of her music, but also for her dramatic, often surprisingly sexy outfits.
La Matadora Revisa Su Maquillaje (The Bullfighter Checks Her Makeup) by Susan Orlean
Every time Cristina Sanchez kills a bull, she presents an unforgettable tableau—a self-possessed young woman elegantly and lethally playing out an ancient masculine ritual.
Takes on the Town by Lillian Ross
For decades, Lillian Ross's inspired short gems have set the gold standard for _The New Yorker_ 's iconic Talk of the Town section.
Kid, Twelve by Susan Sheehan
The Tomberlind family represents millions of America's working poor. They're not on welfare, don't live below the poverty line. And yet they struggle every day to get by.
Mrs. Kennedy at the Moment by Gloria Steinem
An intimate portrait of Jackie Kennedy as she confronts difficult decisions in the year following the assassination of her husband, President John F. Kennedy.
Mothers, Sisters, Daughters, Wives by Mimi Swartz
What happens to women in Texas as the state wages war on Planned Parenthood, cuts family planning funding, and passes new sonogram laws?
My Breast by Joyce Wadler
A heartfelt, engrossing, and unexpectedly humorous account of one woman's victorious battle with breast cancer.
Soul Survivor by Isabel Wilkerson
Civil rights activist Kwame Ture, formerly known as Stokely Carmichael, was a fiery leader of the Black Power movement during the 1960s.
Permissions
About the Editor
About the Publisher
Introduction
_The Stories We Tell_ celebrates the work of twenty women who have made major contributions to American longform journalism over the past half-century.
All counted, they have garnered at least four Pulitzer Prizes, seven Guggenheim fellowships, five Woodrow Wilson fellowships, three National Magazine Awards, four National Book Awards for nonfiction, three National Book Critics Awards—among many other honors. The National Book Foundation presented its Medal for Distinguished Contributions to American Letters to Joan Didion in 2007; Adrian Nicole LeBlanc was named a 2006 MacArthur Fellow.
While each has her own style, the women in these pages share the attributes of all good writers: meticulous research and reporting, careful attention to detail, a talent for choosing the perfect noun or verb. Above all, they are astute observers and sticklers for accuracy. Over the years, they have been both prolific and versatile, writing about a wide range of topics, as demonstrated by the selections here, which take us from Suzannah Lessard's look at the cultural divide in a New York neighborhood, where trendy shops and residences give way to housing projects; to Isabel Wilkerson's interview with civil rights activist Stokely Carmichael; to Janet Malcolm's profile of the brilliant and stylish young pianist Yuja Wang; to Robin Marantz Henig's poignant account of the determination of one Alzheimer's victim to end her life on her own terms.
Many of the writers spent long periods of time with their subjects. To profile Rudolf Nureyev, Elizabeth Kaye traveled with the famed ballet dancer for nearly a year; she stayed in the same hotels and spent time with him as he played the piano and watched movies into the wee hours. "I was there late, and I was there early, and I saw a lot," she recalls. "It was incredibly generous of him, and trusting."
In Lis Harris's case, it took more than a year just to find the ideal Hasidic family for her profile, a search made more difficult by her insistence that the family include a teenager. She spent another year observing their lives.
"I thought, where there are teenagers, there is rebellion," she explains. "I rejoiced when I finally found the family I wrote about because they had five teens in the clan. But surprise! There was no rebellion. It was one thing to rebel against your mom and dad, quite another to rebel against your parents, your entire community, the Rebbe—and God!"
Gerri Hirshey looks back on her month-long series of conversations aboard the private tour bus of the late B.B. King as "a music journalist's fever dream." She remembers King, who was seventy-three at the time, as a wise, compassionate man, "a combo of a funky Yoda and the Dalai Lama." Her story won that year's ASCAP-Deems Taylor/Virgil Thompson Award.
By contrast, Joan Didion didn't arrive in San Bernardino until a year after the trial ended in the tabloid-style murder case she chronicles in "Some Dreamers of the Golden Dream," considered one of the classics of literary journalism. Nevertheless, with the help of trial transcripts, news clips, interviews, her observations of the community, and even Chamber of Commerce handouts—the basic tools of the reporter's trade—she reconstructs a story that brings readers into real time.
Gloria Steinem's 1964 profile of Jacqueline Kennedy provides another example of a writer who, without benefit of extended access to her subject, combined reportorial ingenuity with literary skills to construct a look at the former First Lady one year after the assassination of her husband, President John F. Kennedy.
Gaining face time with celebrities often takes as much creativity as actually writing the stories. When Susan Orlean set out to profile Christina Sanchez, she was convinced the only way to get to the popular Spanish bullfighter was through her agent, a man who assured Orlean of a meeting. The man, however, turned out to be an imposter and was no longer answering his phone by the time she arrived in Madrid.
"I was so embarrassed and frustrated that I prepared to leave on the next flight, but my editor urged me to use some ingenuity and try to get to her on my own," Orlean recalls. "I managed to figure out where her mother lived and decided to go visit, in hopes she would plead my case to Christina. As luck would have it, when I arrived at her mother's, Christina was there, taking a short break from her travels, and she invited me to follow her for the next several days."
Orlean's lesson learned: persevere.
While a number of stories in this collection focus on well-known subjects, many more are about ordinary people: Madeleine Blais writes about a girls' high school basketball team battling to win the state championship; Adrian Nicole LeBlanc focuses on a group of teenage boys struggling to fit in at school; Melissa Fay Greene follows the training of service dogs for children with autism and other cognitive disabilities. Susan Sheehan's article brings readers into the lives of a twelve-year-old boy and his family trying to live on $21,723 a year. Jeanne Marie Laskas writes about the recipient of a face transplant. E. Jean Carroll offers a haunting account of a series of mysterious teenage deaths in a small upstate New York town. Mimi Swartz chronicles the long political battle over reproductive rights in Texas.
Only two of the anthology stories are intensely personal: Joyce Wadler's "My Breast: One Woman's Cancer Story," and Jill Lepore's "The Prodigal Daughter."
At the outset, Wadler says, she had no intention of writing about her fight against cancer. "I was focused on getting the best medical treatment I could and saving my life," she says. "But it was a story that wanted to be written and several months after surgery the first sentence popped into my mind: 'I have a scar on my breast.' After that, it felt like the story was writing itself."
Lepore sees herself as more of an historian than a personal essayist, and yet, in "Prodigal Daughter," she goes against her rule of not writing about herself, combining research for her book on Ben Franklin's sister with memories of her recently deceased mother to beget a powerful piece of nonfiction.
As a journalist who has worked for decades alongside these great women in the same male-dominated field, it gives me a great deal of satisfaction to bring them together under one cover. All of the writers collected herein had a hand in choosing the stories featured. In the emails and telephone calls we shared, all expressed their delight that such a volume was being compiled.
While male bylines still outnumber those of females in American magazines, the trend is changing, so much so that narrowing this anthology to twenty women has not been easy. As a result, The Sager Group is compiling a second collection of longform magazine writing by a younger generation of women—many of them inspired by the writers featured here.
_Patsy Sims_
_Madeleine Blais_
Madeleine H. Blais received her bachelor's degree from the College of New Rochelle in New Rochelle, New York, followed by a master's degree from the Columbia Graduate School of Journalism. She was also a Nieman Fellow at Harvard University, Class of 1986. She is a professor of Journalism at the University of Massachusetts, Amherst, where she teaches longform, memoir, and documentary film, and serves as Honors Director in Journalism. She is a faculty mentor in the Goucher College low-residency Masters of Fine Arts in Nonfiction Program.
From 1979-1987 she was on the staff of _Tropic Magazine of The Miami Herald_ where she won many awards, including a Pulitzer Prize for feature writing. She is the author of _The Heart Is an Instrument_ , a collection of journalism, and In _These Girls, Hope Is a Muscle_ , which was chosen as a finalist in general nonfiction by the National Book Critics Awards and was cited by _ESPN_ as one of the top one hundred sports books of the twentieth century. Her most recent book is _To the New Owners: A Martha's Vineyard Memoir_. She is currently at work on a biography of tennis great Alice Marble.
Her essays have appeared in _Superstition Review, Cogniscenti_ , and _The New Guard_ as well as in books such as _Bad Girls: 26 Writers Misbehave_ edited by Ellen Sussman, _Our Boston: Writers Celebrate the City They Love_ edited by Andrew Blauner, _A Story Larger than My Own_ edited by Janet Burroway, and _Double Take: Portraits Over Time_ by Maggie Evans Silverstein. She is on the editorial board of _River Teeth Magazine_.
They Were Commandos
After near-perfect seasons tarnished by losses in the state championships, the Amherst High School girls' basketball team might finally change its story.
The voice of the coach rises above the din of shuffling footsteps, loud greetings, the slamming of metal, the thud of books. "Listen up. I want you to check right now. Do you have your uniforms? Your shoes and your socks? Do you have any other items of clothing that might be needed?"
Coach Ron Moyer believes it's possible to pack abstractions along with one's gear, intangibles like "intensity" and "game face" and "consistency" and "defense." As the members of the Amherst Regional High School girls' basketball team prepare to board the Hoop Phi Express on their way to the Centrum in Worcester more than an hour away for the Massachusetts state championship, he tells them, "Today, I want you to pack your courage."
The team is 23–1 going into this game, losing only to Agawam, which, like the Haverhill team they are facing this evening, has some real height. Haverhill, known for aggressive ball, nothing dirty but just short of it, has two girls over six feet nicknamed the Twin Towers. Moyer has prepped his team with a couple of specialized plays, the Murphy and the Shoelace, and he tells them: "Expect to play a little football." Amherst girls have a reputation for being afraid to throw their elbows, but this year they have learned to take the words "finesse team" as an insult. Although Coach has been careful to avoid saying "state championship" to goad his team, last fall he did tell one aging gym rat in town: "I have the two best guards in the state Madeleine Blais 3 and probably the nation, but it all depends on the girls up front. There's an old saying— 'Guards win games, but forwards win championships.' We'll have to see."
At six foot six, Moyer looms over his players. With a thick cap of graying brown hair and bangs that flop down over his forehead, he resembles a grizzly bear on spindly legs. The girls are more like colts. For Moyer, turning them into a team has nothing to do with breaking their spirit and everything to do with harnessing it.
As Jen Pariseau listens to Coach before leaving for Worcester, her legs can't stop twitching. One of the six seniors on the team playing highschool hoop together for the last time, she has thick, dark eyebrows and long, lanky limbs. For her, tonight's game is the perfect revenge, not just against Haverhill but also against some of the rebuffs she suffered as an athlete on the way up. For three years, she played on one of Amherst's Little League teams, the Red Sox. She was pitcher, shortstop, and first baseman. When it was time to choose the all-star league, she was told her bunts were not up to par.
Jen's teammates are just as hyped up. Half of them are giving the other half piggybacks. There are lots of hand-slapping and nudges. They swirl around one another, everyone making a private point of touching Jamila Wideman, Jen's co-captain, as if one dark-haired, brown-eyed girl could transmit the power of her playing to all the others. Jamila is an all-America, recipient of more than 150 offers of athletic scholarships. On the court, the strong bones on her face are like a flag demanding to be heeded; she is a study in quickness and confidence, the ball becoming part of her body. Her nickname is Predator.
Jen Pariseau is two-time all-Western Mass, and together the two guards delighted fans all season with the way they delivered the ball to each other, sometimes in a dipsy doo behind the back or between the legs, often resulting in an open shot. JennyandJamila. In Amherst, it's one word.
Coach pauses. He looks as though he is about to rebuke the girls for all the squirming, but he shrugs and gives a big smile. "Let's go." Then, perhaps more to himself than to them: "While we're still young."
Shortly after five in the evening, the sky is thick and gray and hooded, the cloud cover a welcome hedge against what has been a bitter New England winter. The bus the girls board is different from the usual.
"Hooked up and smooth," says Jen Pariseau, admiring the special features, including upholstered seats, a toilet, four television sets, and a VCR mounted on the ceiling—a definite step up from the yellow tin cans they have taken to every other game. There are some cheerleaders on the bus as well as Tricia Lea, an assistant coach with her own high-school memories about what it was like to go up against those Hillies from Haverhill in their brown and yellow uniforms with the short shorts. "Haverhill. I don't know what they eat up there, but they can be slightly ruthless. Sportsmanship does not run very deep in that town."
A few years back, Coach had trouble convincing players and their families of the seriousness of the commitment to girls' basketball. JennyandJamila remember playing in varsity games five and six years ago when the gym would be empty of spectators except for their parents and maybe a few lost souls who had missed the late bus. Coach remembers girls who would cut practice to go to their boyfriends' games, and once during the playoffs, a team captain left to go on a school-sponsored cultural exchange for three weeks in the former Soviet Union. As far as he's concerned, the current policy could not be clearer: You want cultural exchange? You can have it with Hamp.
Tonight, Amherst is sending three "pep" buses to the game, unprecedented support for an athletic event, boys' or girls'. Amherst is a place that tends to prize thought over action, tofu over toughness. It prefers to honor the work of the individual dedicated to a life of monastic scholarship rather than some noisy group effort. But this season, there were sellout crowds. There was even that badge: a wary cop on the premises for the first time in the history of a girls' event.
Amherst is a college town, with the usual benign ineffectuality that makes most college towns as maddening as they are charming and livable. When the Chamber of Commerce sponsored a contest for town motto, Moyer submitted one that he still thinks should have won—"Amherst: Where sexuality is an option and reality is an alternative."
Amherst is, for the most part, smoke-free, nuclear-free, and eager to free Tibet. Ponchos with little projectiles of fleece have never gone out of style. Banners stretch across South Pleasant Street at the town common, including the vintage "Spay or Neuter Your Pet, Prevent Abandonment & Suffering." This is a town that saves spotted salamanders, Madeleine Blais 5 creating love tunnels (at taxpayers' eager expense) so that they can all descend from the hills in early spring and migrate to the marshy areas for sexual assignation without being squashed on Henry Street. There's a new band called Salamander Crossing; heavy metal it's not. A famous local headline: "Well-Dressed Man Robs Amherst Bank." Amherst is an achingly democratic sort of place in which tryouts for Little League, with their inevitable rejections, have caused people to suggest that more teams should be created so that no one is left out. There are people in Amherst who still think "politically correct" is a compliment. The program notes for the spring musical _Kiss Me, Kate_ pointed out politely that _The Taming of the Shrew_ , on which it is based, was "well, Shakespearean in its attitude toward the sexes."
The downtown area seems to support pizza joints, Chinese restaurants, ice-cream parlors, and bookstores and not much else. It's hard to find a needle and thread, but if you wish you can go to the Global Trader and purchase for $4 a dish towel with a rain-forest theme. The surrounding communities range from the hard and nasty inner-city poverty of Holyoke, the empty factories in Chicopee and the blue-collar scrappiness in Agawam to the cornfields and asparagus patches in Whately and Hatfield and Hadley and the shoppers' mecca that is Northampton. They tend to look on Amherst with eye-rolling puzzlement and occasional contempt as the town that fell to earth.
The girls on the Hurricanes know they live in a kindly, ruminative sort of place. Sometimes they joke about how if they weren't playing ball, they'd be "tipping cows"—a basically useless activity necessitated by the unfortunate tendency of cows to sleep standing up.
With the playoffs looming, the six senior girls—JennyandJamila, Kathleen Poe, Kristin Marvin, Patri Abad, and Kim Warner—were treated to a late lunch by Jamila's father, John Edgar Wideman, winner of two PEN/ Faulkners as well as numerous other awards, and author of the nonfiction meditation _Brothers and Keepers; Philadelphia Fire_ , a fictional visitation of the Move bombing in 1985; _The Homewood Trilogy_ , about growing up black in Pittsburgh.
It was at that lunch that the team's center, Kristin, in trying to sum up the peculiar, almost consoling, lack of outward drama in a town like Amherst, confessed that the night before she had a dream.
"My Mom and I, we went to Stop and Shop and while we were there, we went down, you know, all our usual aisles in the regular order, picking out all the things we usually buy, and after that we got in line to check out."
"That's it?" said the other girls.
Jamila's father thought maybe the dream had another layer and so he tried a gentle psychoanalytic probe. He had a quicksilver face, his expression changing in a flicker from stormy to melancholy to soft and forgiving. Now it was contemplative.
"Did you run into any unusual people?"
"No."
"How about money? Did you run out of money or anything?"
"No."
"Kristin," said her teammates, "that's so sad."
Kathleen, who is in the top ten academically in her class of 250, told Jamila's father that she tried reading a collection of his short stories, "the one called 'Jungle Fever.'"
"I'm not Spike Lee. It was just _Fever_."
"Mr. Wideman, I tried reading it," said Jen Pariseau, also in the top ten academically. "I found the shortest story I could, and you know what? I think I understood it. I can't guarantee it, but I think I did."
He looked at his guests at the table, a blur of happy faces and ponytails. Their teasing was a joy. He is a former basketball player for the University of Pennsylvania and a Rhodes scholar who played at Oxford, and his passion for the game is such that Jamila tells people she was born playing basketball. Girls' basketball is not boys' basketball being played by girls. It's a whole new game. There's no dunking. They can't jump as high. They can't play above the rim. But they can play with every bit as much style. And there's that added purity, that sense of excellence for its own sake. It's not a career option for girls; after college the game is over, so there is none of the desperate jockeying for professional favor.
As a black man, Wideman knows only too well the shallow triumph of token progress. He had told Kathleen's father, "This is just one team in one season." It alone cannot change the discrimination against girls and their bodies throughout history. But here in these girls, hope is a muscle.
"Here's to the senior girls," he said, looking at all of them.
They hoisted their ritual glasses of water.
"This is," he said, "as good as it gets."
To look at them, these six seniors on the team, who all appear to be lit from within, one would assume that their lives have been seamless journeys. In fact, as Jen Pariseau puts it, she does not come from a "Dan Quayle kind of family"—and neither do most of the others. Whatever sadness or disruption they've been dealt, an opposite force follows them onto the court. JennyandJamila have not gone it alone; they have had Kathleen's strong right hand, an almost irresistible force heading toward the basket. She never wastes a motion: The ball is in her hands one second, then quietly dropping through the hoop the next, without dramatics, almost like an afterthought. There's Kristin. Her flushed cheeks are not a sign of exhaustion but of some private fury. When the ball comes curling out of the basket, more often than not it is Kristin who has pushed and shoved her way to the prize.
The only underclass starter, Emily Shore, is so serious about her chance to play with the famous JennyandJamila that she spent the bulk of her summer lifting weights and battling in pickup games on Amherst's cracked and weather-ravaged outdoor courts with a succession of skeptical and then grudgingly appreciative young men.
They have become what every opponent fears most: a team with a mission.
As good as it gets. That is, of course, the exact sentiment the girls feel toward their fancy bus.
"Fasten your seat belts," says Coach. "Beverage service will commence shortly after takeoff. There'll be turbulence coming to Haverhill when the Hurricanes hit Worcester." Then he announces the people to whom he would like them to dedicate the entire season. "And that's to the 140 girls who are now playing youth basketball in Amherst for the first time this year."
Jen Pariseau says she wants to read a letter from Diane Stanton, the mother of Chris Stanton, the star of the boys' basketball team.
"Jenny and Jamila," the letter began. Diane Stanton said she was addressing them because she knew them the best, but the letter was for the whole team. "Your existence as a team represents a lot of things to a lot of women like me. . . . As a young girl I remember standing outside the Little League fence and watching the boys and knowing that I could hit and catch better than at least a third of them. When our high-school intramural field hockey team and softball team asked for leagues, we were told flatly—NO, because there was no money. . . . When this group of girl athletes got together to form an intramural basketball team, we were subjected to ridicule and anger from some of the student body. . . . I lost courage, I'm embarrassed to admit, in my junior year and would no longer play intramural sports. Part of it was a protest against the failure of my school. . . to recognize that we needed to play as much as boys. I know the struggle."
Coach gives the driver a signal and the vehicle starts to roll. A police car just ahead suddenly activates its lights and in a slow ceremony leads the vehicle to the corner of Main and Triangle Streets, where another officer has been summoned to stop all traffic. Coach is beaming and silently thanks his old pal, Captain Charlie Scherpa, over in the Police Department for coming through. In addition to being a guidance counselor, Moyer has been the girls' coach off and on since 1981, a task he enjoys because unlike with boys, whose arrogance and confidence often have to be eroded before he can get the team to work, this is all constructive. The way to build a girls' team is to build their individual self-confidence.
The bus heads down Main (a street that is most famous for being the site of the house where Emily Dickinson was born, where she lived, died, and wrote her poetry) to the corner of Northeast, where they get to run a red light, turning in front of Fort River Elementary School, then heading out to Route 9, where the escort lasts all the way to the town line. In an instant, the sign that says "Entering Pelham" appears, and in another instant a new one looms ahead that says "Entering Belchertown."
The girls watch the film they had chosen unanimously to pump them for the game—"A League of Their Own." The six seniors are lost in their own thoughts.
Kim Warner knows her mother, who works in personnel at the University of Massachusetts, will be at the game, plus her two sisters, plus her boyfriend's family. Her father lives in Florida, and although she sends him news accounts of all the games, he has never seen her play. She hasn't seen him since the tenth grade. She plans to go to Westfield State and major in early childhood education. On the way to the game, Kim writes a fantasy letter in her head: "Dear Dad, At long last a lot of hard work paid off."
Patri Abad's mother, a bilingual teacher, has to be at work, and although Patri will miss her, she knows she can count on a large cheering section of friends. She almost didn't get to play this year. During her junior year, she had moved to Chicago with her mother and her new stepfather. Patri, who is Cuban on her father's side and Puerto Rican on her mother's, prayed incessantly to the Virgin. She received constant mail from teammates like Lucia Maraniss, back when Lucia was a gushing eighth grader: "Patri, I will always remember you as one of the wisest, most caring and compassionate people I've ever met. I'm going to miss you very, very, very, very, very, very, very, very, very, very, very, very, very, very much."
Whether it was divine intercession or that fourteenth "very" from Lucia, the resolve of Patri's mother to stay in Chicago eventually vanished. They returned to the Happy Valley, as Amherst is called, and Patri could finish her senior year as a member of the Hurricanes. She has been accepted at Drew, Clark, and the University of Massachusetts, pre-med.
Kristin Marvin, also known as Jolly, Jolly Green, and Grace (her teammates have misinterpreted her tenacity as clumsiness), is going to Holy Cross College, pre-med. She likes medicine because it has a strong element of knowability. Her parents were divorced when she was young and she lived with a lot of uncertainty. Her mother has since married a builder whose first wife married Kristin's father, who works in Connecticut and often rushes to the games after work in his business suit. The marital realignment has created a circumstance in which the daughter of her stepfather and stepmother is Kristin's double stepsister.
Coach calls Kathleen Poe his silent assassin—the girl with two distinct personalities. The demure senior with the high grades, with applications at Williams, Haverford, Duke, and Dartmouth, is Kathleen; the girl on the court is her ferocious twin, Skippy. He concocted the dichotomy because when Kathleen first started playing she said "Excuse me" all the time and would pause to pick her opponents up off the floor. She wants to be like Jamila: someone you don't want to meet on the court but who will be a good friend off it.
Jamila plans to study law and African American studies at Stanford. Like her mother, Judy Wideman, who is in her second year of law school, she hopes to be a defense attorney. As a child of mixed races, she has told interviewers she identifies not with being black or white but with being herself. Still, her bedroom has pictures of Winnie Mandela, Jesse Jackson, and the children of Soweto. After the riot in Los Angeles, she wrote several poems that reflected her feelings.
In "Black," she wrote:
_I walk the tightrope between the fires_
_Does anyone know where I fall through?_
_Their forked daggers of rage reflect my eye_
_Their physical destruction passes me by_
_Why does the fire call me?_
Jen is known locally as the best thing that ever happened to Pelham, which is that little twinge on the highway on Route 9. Since Jen was two and her brother, Chris, was four, they have lived with their father, who is a manager of reservoirs and water treatment in Amherst. She is planning to play ball for Dartmouth and to major in engineering. She turned down Princeton, especially after the recruiter, who made a home visit, would not let her father, who had a stutter, talk.
The door to her room is plastered with Nike inspirational ads. She calls the wall above her bed her "strong women wall," and it is filled with pictures of her favorite role models, including Ann Richards and Toni Morrison. By her bedside, she keeps a clothbound book—given to her by her teammate Rita Powell—in which she writes favorite quotes, a customized Bartlett's.
Marilyn Monroe: "If I'm going to be alone, I'd rather be by myself."
Colette: "You will do foolish things, but do them with enthusiasm."
Zora Neale Hurston: "The dream is the truth."
The team bonding among these six seniors and the ten younger girls is one reason they have played so well: the sisterhood-is-powerful quest for unity. They have a team song, "Real Love," and they have team trinkets (beaded necklaces with their names and plastic rings and scrunchies with basketballs), team teddy bears, team towels. At team dinners, Jamila's mother carbo-loads them with slivered chicken cooked in garlic and oil and lemon and served on a bed of noodles. The meals often conclude with a dozen or so girls linking arms in a tight circle, swaying, singing, shouting, " _Hoop Phi!_ "
To witness adrenaline overload at its most frenetic, nothing beats the atmosphere on one of those yellow buses on the return home after a victory over Hamp. Northampton is a fine town, birthplace of Calvin Coolidge, home of Smith College. But, as Jen Pariseau says: "Something happens when we play Hamp. Both teams become brutes." Hamp fans are always
To witness adrenaline overload at its most frenetic, nothing beats the atmosphere on one of those yellow buses on the return home after a victory over Hamp. Northampton is a fine town, birthplace of Calvin Coolidge, home of Smith College. But, as Jen Pariseau says: "Something happens when we play Hamp. Both teams become brutes." Hamp fans are always trying to demoralize JennyandJamila with the scornful chant: "You're overrated; you're overrated."
A victory against Hamp, especially on their territory at Feiker gym, especially in front of at least one thousand people with several hundred more turned away at the door, was a great moment to whoop and cheer the whole way home, to sing Queen's famous anthem, "We Are the Champions," to slap the ceiling of the bus, to open the windows and to shout:
_Who'll rock the house?_
_The Hurricanes will rock the house._
_And when the Hurricanes rock the house_
_They rock it all the way down._
But even though they beat Hamp in the Western Mass Regional finals, they weren't really champions—not yet. Do they have what it takes, these sweet-looking girls reared in maple syrup country on land that includes the Robert Frost trail? Playing before a few thousand fans in what is almost your own backyard is nothing compared with a stadium that seats 13,800, where real pros play. Rocking Feiker is one thing, but the Centrum?
When the bus finally pulls in front of the Centrum and it is time to leap off, the girls have faces like masks. To the world, they are a bunch of teen-age girls; inside their heads, they are commandos. To the world, these teen-agers have pretty names: Patri, Kristin, Jen, Kathleen, Kim, Jamila, Sophie, Jade, Emily J., Emily S., Jan, Lucia, Carrie, Rita, Jessi, Julie. But as far as these girls are concerned, they are the codes that encapsulate their rare and superb skills, their specialty plays, their personal styles. They are Cloudy and Cougar and Jones-bones and Gumby and Grace and Skippy and Predator. They are warriors.
The girls crowd into a locker room. With much less commotion than usual, they dress in their baggy knee-length uniforms. They slap hands and stand tall. Meanwhile the arena is redolent of hot dogs, popcorn, sweat, and anticipation, one side of the bleachers filled with their people and the other side with the fans from Haverhill.
The girls walk out wordlessly. They look up.
You have to live in a small town for a while before you can read a crowd, especially in New England, where fences are deep in the soil. But if you've been in a town like Amherst for a while, you can go to an out-of-town game, even one in as imposing and cavernous a facility as the Centrum, and you can feel this sudden lurch of well-being that comes from the soothing familiarity of faces that are as much a part of your landscape as falling leaves, as forsythia in season, as rhubarb in June. You scan the rows, and for better, and sometimes for worse, you know who's who. You know whose parents don't talk to whom else and you know why. You know who has had troubles that never get discussed.
You see the lawyer that represented your folks or one of their friends in a land dispute or a custody case. You see the realtor who tried to sell a house next to the landfill to the new kids in town. You see the doctor who was no help for your asthma and the one who was. You see the teacher who declared your baby brother a complete mystery and the teacher who always stops to ask what your remarkable brother is up to now. You know which man is the beloved elementary-school principal, now retired. You recognize the plump-cheeked ladies from the cafeteria who specialize in homemade cinnamon buns for sixty-five cents. You see your family and you see the fathers and mothers and stepfathers and stepmothers of your teammates. You know whose brother flew in from Chicago for the game; whose step-grandparents came from Minnesota.
But what is most important about all this is how mute it is. The commonality is something that is understood, as tacit as the progression of the summer to fall to winter to spring, and just as comforting. Usually there is a buzz of cheering at the start of a game, but this time the Amherst crowd is nearly silent as the referee tosses the ball.
The Haverhill center taps the ball backward to her point guard. She comes down the court, swings the ball to the wing, who instantly dishes it inside to the center. Easy layup. Amherst blinks first. Two-nothing. In the Haverhill stands, the crowd cheers. It is the only pure cheer they will get.
Within a few seconds, the score is 6–4 Amherst, and something truly remarkable takes place. The Hurricanes enter into a zone where all of them are all-Americans. It's a kind of controlled frenzy that can overtake a group of athletes under only the most elusive of circumstances. It's not certain what triggers it, perhaps it's Jamila's gentle three-pointer from the wing, or more likely, when Jen drives the baseline and as she swoops beneath the basket like a bird of prey she releases the ball back over her head, placing it like an egg against the backboard and through the hoop. It may have been ten seconds later when Jamila steals the ball, pushing it down court in a three-on-one break, makes a no-look pass to Jen who just as quickly fires the ball across the lane to Kathleen for an uncontested layup. Whatever it is that started it, there is nothing Haverhill can do to stop it, and time-outs repeatedly called for by their hapless coach only fuel Amherst's frenzy further.
Even the sportscasters can't remember a 37 to 0 run in a state championship game. The halftime score is 51–6.
An astonished Amherst can hardly even cheer. One Amherst fan shouts: "Where's Dr. Kevorkian?" Another makes the very un-Amherst comment: "They should bring on the Haverhill boys for the second half."
Among the spectators is Kathleen's father, Donald Poe, an associate professor of psychology at Hampshire College, who saw how her defense, along with that of Kristin and Emily Shore, kept Haverhill's score so low. When his son, Chris, was an infant, Donald Poe tried to teach him to say "ball" as his first word, until he was told that "b" is a hard sound for a baby. He expected a son to be an athlete, and when Kathleen came along he didn't have that expectation. Yet whenever they go into the yard and she pitches a ball to him, it takes only five minutes before his hand hurts. She throws a heavy ball.
To him, what's important is not that Amherst win, but that the spirit of girls' sports endures. Next year, it doesn't have to be Amherst; it might be Westside in Springfield. Its junior varsity is undefeated. When he was in W. T. Woodson High School in Fairfax, Virginia, the girls were not allowed to use the boys' gym, which was fancy and varnished with a logo in the middle of the floor. The girls had a little back gym, without bleachers. After a game, whenever he saw the little kids asking his daughter for autographs, he was glad to see the girls, pleased that they now had models. But he was just as glad to see the boys asking; to him their respect for the girls' team was just as important.
The final score is 74–36.
After receiving the trophies and after collapsing in one huge hysterical teen-age heap, they all stand up. First they sing "Happy Birthday" to Kristin Marvin, who turns eighteen this day. Then they extend their arms toward their parents, teachers, brothers, sisters, even to some of those 140 little girls whose parents have allowed them a school night of unprecedented lateness, and in one final act as a team, these girls shout, in the perfect unison that has served them so well on the court, "Thank you."
Back in the locker room, Kristin Marvin sucks on orange slices and sloshes water on her face. She then stands on a back bench, raises her right fist, turns to her comrades and shouts: "Holy #@&*! We're the *@#&*@# champions!" And then she loses it. For the next half-hour, she throws herself into the arms of one teammate after another. She cries and hugs, and hugs and cries, and so do they.
Coach keeps knocking at the door, trying to roust the stragglers. Finally, he announces he is coming in and what greets him is a roomful of girls who return his level gaze with eyes that are rheumy and red as they sputter "last . . . final . . . never again."
He looks right at them and says: "You're wrong. This isn't the last. There will be more basketball." His tone is conversational, almost adult to adult.
"But. . ." they start to say.
"I promise you. There will be lots more basketball."
Still they regard him with disbelief. They can't decipher his real message, at least not at this moment. They can't fathom how the word "basketball" might have more than one meaning.
Over. The game was over. On the way home, they watched a videotape of the game. Jen was stunned at how it had all fallen into place: "We were so fluid it was scary." While they watched themselves, television viewers all over the state were witnessing recaps of the highlights and hearing the verdicts of professional commentators who claimed these girls had wandered into the wrong league: They shoulda been playing Calipari's men at U Mass; they coulda taught the Celtics a thing or two.
The girls would hear all that in the days to come, but at this moment they were mostly thinking about the present—when truth itself had become a dream. The bus was going backward, retracing its earlier path, down the Pike back through Palmer, where the only sense of abundance is in the fast-food stores, then through Bondsville with its gin mill and the sunken rusty playground with a metal fence, back through the center of Belchertown, a singularly flat stretch in a town with a singularly unfortunate name, and back in and out of Pelham—thanks to Jen, on the map at last.
Kathleen Poe wished that the whole team could sleep that night in the gym at the high school, the coziest, most homey, softest place she could now imagine, that they could all sink into its floor, become part of it forever. She kept trying out rhymes in her head, phrases popping into her mind like sudden rebounds: top and stop, pride and ride, forever and sever, heart, smart, true, you.
_Hoop Phi is one of an intangible, untouchable breed,_
_It satisfies the soul, and a life-long need._
_We represented our school, represented our sex,_
_Now maybe both will get some well-earned respect._
No one really wanted the ride to end. The bare trees, the velvety night air, the cocoon of the bus itself.
At the town line there awaited another police escort, this time back into town. The cruiser was once again full of proud, slow ceremony. At the corner of Main and Triangle, the cruiser seemed to lurch right to take the short-cut back to the school, but then as if that was only a feint, it continued to move forward, so that the girls would be brought through town the long way.
The bus, boisterous in its very bigness, moved past the red-bricked Dickinson homestead with its top-heavy trees, tall and thin with a crown of green: _We're somebody; who are you?_ Downtown was almost empty save for a couple of pizza eaters in the front window of Antonio's and a lone worker sweeping in the back shadows of Bart's Ice Cream. As the strobe lights from the cruiser bounced off the storefronts, the bus wheezed past St. Bridget's and the bagel place, turning right, then left, finally pulling into the school parking lot a few minutes shy of midnight.
All of a sudden one of the players shouted: "There are people there, waiting for us!" And, indeed, in the distance was a small crowd standing in the cold and in the dark, clapping.
When the bus came to a stop, Coach stood up. "I promise it won't be mushy. There's just one thing you should know. When you're the state champions, the season never ever ends. I love you. Great job. And now, I'd like everybody else on the bus to please wait so that the team can get off first."
Often the Hurricanes will bound off a bus in a joyous squealing clump. On this night, they rose from their seats, slowly, in silence. _State champs!_ For the final time this season, with great care bordering on tenderness, the teammates gathered their stuff, their uniforms, their shoes, their socks, their game faces, and their courage. And then in a decision that was never actually articulated but seemed to have evolved as naturally as the parabola of a perfect three-pointer, the Hurricanes waited for captain Jen Pariseau to lead the way, which she did, and one by one the rest of the women followed, with captain Jamila Wideman the last of the Hurricanes to step off the bus into the swirling sea of well-wishers and winter coats.
Overhead the sky was as low-hanging and as opaque as it had been earlier in the evening, but it didn't need stars to make it shine.
_E. Jean Carroll_
E. Jean Carroll is a journalist, advice columnist, and author whose "Ask E. Jean" column is the longest-running advice column in American publishing, having appeared in _Elle_ magazine since 1993. She has also been a writer for the television show _Saturday Night Live_ and a contributing editor for _Esquire_ , _Outside_ , and _Playboy_. Her stories have taken her around the globe, from her home state of Indiana to Papua New Guinea. "The Cheerleaders," which originally appeared in Spin, was selected as one of the best true crime reporting pieces in 2002. It also appeared in the 2002 edition of _Best American Crime Writing_.
Carroll is the author of four books: _Female Difficulties: Sorority Sisters_ , _Rodeo Queens, Frigid Women_ , _Smut Stars_ , and _Other Modern Girls_ ; _A Dog in Heat Is a Hot Dog_ and _Other Rules to Live By_ ; _Hunter_ : _The Strange and Savage Life of Hunter S. Thompson;_ and _Mr. Right, Right Now_.
In 2012 Carroll founded the matchmaking service Tawkify. She also developed the mobile app, Damn Love, a send-up of modern dating apps where players can ruin (virtual) relationships. A native of Fort Wayne, Indiana, she attended Indiana University in Bloomington. She currently lives in upstate New York.
The Cheerleaders
The tiny town of Dryden, New York, endures a strange, five-year string of murders, car accidents, and suicides—all of it tied to two popular high school cheerleaders.
Welcome to Dryden. It's rather gray and soppy. Not that Dryden doesn't look like the finest little town in the universe—with its pretty houses and its own personal George Bailey Agency at No. 5 South Street, it could have come right out of _It's a Wonderful Life_. (It's rumored the film's director, Frank Capra, was inspired by Dryden.) But the thriving, well-heeled hamlet is situated on the southern edge of New York's Finger Lakes region, under one of the highest cloud-cover ratios in America. This puts the nineteen hundred inhabitants into two philosophical camps: those who feel the town is rendered more beautiful by the "drama" and "poetry" of the clouds and those who say it's so "gloomy" it's like living in an old lady's underwear drawer.
If you live in Dryden, the kids from Ithaca, that cradle of metropolitan sophistication fifteen miles away, will say you live in a "cow town." ("There's a cow pasture right next to the school!" says one young Ithacan.) But Dryden High School, with its emerald lawns, running tracks, athletic fields, skating pond, pine trees, and 732 eager students, is actually a first-rate place to grow up. The glorious pile of salmon-colored bricks stands on a hill looking out on the town, the mountains, the ponds, and the honey-and-russet-colored fields stretching as far as the eye can see. In the summer, the Purple Lions of Dryden High ride out to the fields and the ponds and build bonfires that singe the boys' bare legs and blow cinders into the girls' hair.
In the summer of '96, many bonfires are built. The girls are practicing their cheerleading routines and the boys are developing great packs of muscles in the football team's weight room; everybody laughs and everybody roars and the fields around town look like they've been trampled by a pride of actual lions. In fact, the Dryden boys display such grit at the Preseason Invitational football game that fans begin to believe as the players do: that the upcoming season will bring them another division championship. This spirit lasts until about 6:30 p.m. on September 10, when Scott Pace, one of the most brilliant players ever to attend the school, the unofficial leader of the team, a popular, handsome, dark-haired senior, rushes out of football practice to meet his parents and is killed in a car crash.
It is strange. It is sad. But sadder still is the fact that Scott's older brother, Billy, a tall, dazzling Dryden athlete, as loved and admired as Scott, had been killed in a car crash almost exactly one year before. The town is shaken up very badly. But little does anyone dream that Scott Pace's death will be the beginning of one of the strangest high school tragedies of all time: how, in four years, a stouthearted cheerleader named Tiffany Starr will see three football players, three fellow cheerleaders, and the beloved football coach of her little country school all end up dead.
At a home football game, Friday evening, October 4, 1996, three weeks after the death of Scott Pace, townspeople keep talking about the team and the school "recovering" and "pulling together," but the truth is, nobody can deal. To the students of Dryden High, it just feels as if fate or something has messed up in a major way, and everybody seems as unhappy as can be.
The game tonight, in any case, is a change. Tiffany Starr, captain of the Dryden High cheerleaders, arrives. The short-skirted purple uniform looks charming on the well-built girl with the large, sad, blue eyes. Seventeen, a math whiz, way past button-cute, Tiffany is on the student council, is the point guard on the girls' basketball team, and has been voted "Best Actress" and "Class Flirt." She hails from the special Starr line of beautiful blonde cheerleaders; her twin sisters, Amber and Amy, graduated from Dryden two years before. Their locally famous father, Dryden High football coach Stephen Starr, has instilled in his daughters a credo that comes down to two words: "Be aggressive!"
And right now the school needs cheering. Though her heart is breaking for Scott, Tiffany wants to lead yells. But as she walks in, the cheerleading squad looks anxiously at her, and one of them says, "Jen and Sarah never showed up at school today."
"What?" says Tiffany.
Tiffany taught Jennifer Bolduc and Sarah Hajney to cheer, and her first thought is that the girls, both juniors on the squad, are off somewhere on a lark. Tiffany knows Sarah's parents are out of town and that Jen spent last night at Sarah's house. For a moment, Tiffany imagines her two friends doing something slightly wicked, like joy-riding around Syracuse. "But then I'm like, 'Wait a minute. . . .'"
"Being a cheerleader at Dryden is the closest thing to being a movie star as you can get," says Tiffany's sister Amber. "It's like being a worldclass gymnast, movie star, and model all in one. It is fabulous! _Fab-u-lous_! It's so much fun! Because _we rule_."
The Dryden High girls have won their region's cheerleading championships twelve years in a row. The girls' pyramids are such a thrill, the crowd doesn't like it when the cheer ends and the game begins.
"I'm like, 'Hold on, Jen and Sarah would _never_ miss a game,'" Tiffany continues. "So the only thing we can do is just wait for them to arrive. And we wait and we wait. And finally, we walk out to the football game and sit down in the bleachers. We don't cheer that day. Well, we may do some sidelines, but we don't do any big cheers because you can't do the big cheers when you're missing girls."
Jen Bolduc is a "base" in the pyramids (meaning she stands on the ground and supports tiers of girls above her), and Sarah Hajney is a "flyer" (meaning she's hurled into the air). At sixteen, Jen is tall and shapely, a strong, pretty, lovable girl with a crazy grin and a powerful mind. She is a varsity track star, a champion baton-twirler, and a volunteer at Cortland Memorial Hospital.
"Jen is a great athlete and a wonderful cheerleader," says Tiffany. "Really strong. And she's so happy! All the time. She's constantly giggling. And she's very creative. When we make Spirit Bags for the football players and fill them up with candy, Jen's Spirit Bags are always the best. And she's silly. Joyful. Goofy. But she's a very determined person."
"Jen is always doing funny things," says Amanda Burdick, a fellow cheerleader, "and she's smart. She helps me do my homework. I never once heard her talk crap about people."
Sarah Hajney is an adorable little version of a Botticelli Venus. She's on varsity track and does volunteer work for children with special needs. "She's a knockout," says former Dryden football player Johnny Lopinto. "I remember being at a pool party, and all the girls, like Tiffany and Sarah, had changed into their bathing suits. And I was walking around, and I just like bumped into Sarah and saw her in a bathing suit, and I was just like, 'Oh my God, Sarah! You're so beautiful!'"
As the football game winds down to a loss, and Sarah does not suddenly, in the fourth quarter, come racing across the field with a hilarious story about how Jen got lost in the Banana Republic in Syracuse, the anxious cheerleaders decide to spend the night at their coach's house. "And we go there, and we begin to wait." says Tiffany. "And we wait and we wait and we wait and we wait."
Before the game is over, a New York state trooper is in Sarah Hajney's house. "I get a phone call on Friday night, October 4, at about—I should say, my _wife_ gets a phone call, because I'm taking the kids to a football game and dropping them off," says Major William Foley of the New York State Police.
Major Foley (at the time of the girls' disappearance he is Captain Foley, zone commander of Troop C Barracks, which heads up the hunt) is a trim man in enormous aviators, a purple tie modeled after the sash of the Roman Praetorian guard, and a crisply ironed, slate-gray uniform. The creases in his trousers are so fierce they look like crowbars are sewn into them.
Sitting with Foley in the state trooper headquarters in Sidney, New York, is the young, nattily dressed Lieutenant Eric Janie, a lead investigator on the girls' disappearance. "I know Mr. and Mrs. Bolduc because I lived in Dryden," says Foley. "Ron Bolduc calls me because he's concerned he's not going to get the appropriate response from the state police. A missing sixteen-year-old girl— _this happens all the time_. So I call Mr. Bolduc back and say I will look into it. And what I do is, I ask that a fellow by the name of Investigator Bill Bean be sent. This is unusual for us to send an investigator for a missing girl. We'd normally send a uniformed trooper who'd assess the situation, but in this case [as a favor to Mr. Bolduc], Investigator Bean is the first to arrive at the Hajney residence. And he quickly determines there's cause for concern."
The Hajney house, a snug, one-story dwelling with a big backyard, is outside Dryden, in McLean, a hilly old village settled in 1796. The village houses are done up in pale gray and mauve and preside over lawns so neat and green they look like carpeting. Wishing wells and statues of geese decorate the yards, flags flutter on porches, and there's a farm in the middle of town.
"There are a lot of people, concerned family members, inside the house," says Janie. "And the first obvious fact is: There's a problem in the bathroom."
"There are signs of a struggle," says Foley. "The shower curtain has been pulled down: the soap dish is broken off." On the towel rack is Jen's freshly washed purple-and-white cheerleading skirt. Sarah's skirt is discovered twirled over a drying rack in the basement.
"We start treating it as a crime scene," says Janie. "Sarah's parents have gotten the call [they are in Bar Harbor, Maine, for a four-day vacation] and are on their way back."
The first break in the case occurs almost immediately: The Hajneys' Chevy Lumina, which was missing, is found about seven miles from the house in a parking lot of the Cortland Line Company, a well-known maker of fly-fishing equipment. "The trunk is forced open by one of the uniformed sergeants," says Foley, "because we don't know, of course: _Are the girls in the trunk?_ "
The trunk reveals that the girls have, in fact, been inside. Investigators tear the car apart and find, among other things, mud, pine needles, charred wood, blood, and diamond-patterned fingerprints suggesting the kidnapper wore gloves, meaning this wasn't some freak accident or a hotheaded crime of passion. This was planned.
Outside the Hajney home, waiting behind the yellow police tape in the cold night, is the other flyer on the cheerleading squad, Katie Savino. Small, with sparkling dark eyes and the merriest laugh, more like a sylph than a human girl, Katie is Sarah's best friend. She watches the troopers go in and out of the house, and waits—full of hope—to speak to an official. What no one knows yet is that Katie could have been the third girl in the trunk. She had made plans to spend the night with Sarah and Jen but, at the last moment, decided to stay home.
Saturday dawns with diaphanous skies. The day is so sunny, so clear, that the natives, accustomed to clouds, find the silver-blue blaze almost disorienting. "It's a _beautiful_ day," says Kevin Pristash, a student affairs administrator at State University of New York at Cortland, which is near McLean and Dryden. "And suddenly these posters go up all over town. GIRLS MISSING! It's very eerie. Rumors are rampant. State troopers are everywhere. Helicopters are flying overhead. I go to get gas, and an unmarked car pulls up, and two guys from different police units get out. They're _everywhere_."
Gary Gelinger, an investigator with the state police, is in McLean interviewing the neighbors of the Hajney family. The first kitchen table at which he is invited to sit on Saturday morning belongs to John and Patricia Andrews. Their six-year-old son, Nicholas, attends Dryden Elementary. From an upstairs bedroom, one can look down into the Hajneys' bathroom.
"John Andrews is _not_ behaving appropriately," says Janie. "Isn't answering questions appropriately, doesn't seem to be aware of what's going on in the neighborhood. Investigator Gelinger reports back and just says: 'Nah, this isn't good. The next-door neighbor isn't good at all.'"
Back when he attends Dryden High, John Andrews is a bashful boy. The love of his life is cars. His old man has won a Purple Heart during one of his three tours in Vietnam: he's a "USA all the way" kind of religious alcoholic who believes in the belt and is strict about his rules. He beats John and his sisters, Ann and Deborah.
At Dryden, John finds a sweetheart, classmate Patricia McGory. They marry, and John joins the Air Force. At his German base, John allegedly, on two separate occasions, dons a ski mask and gloves and viciously attacks women who are young, attractive, and petite. They have long, fair hair and are his neighbors. He's found guilty of the second assault, dishonorably discharged, and sent to Leavenworth.
When John is released, he and Patricia (who, along with his family, insists on his innocence) buy a house in McLean, and he begins working the third shift as a lathe operator at the same company where his mother is employed, the Pall Trinity Micro Corporation, in Cortland. A year later, in August 1996, the Hajneys purchase the house next door to the Andrews, and John quickly becomes obsessed with their beautiful and dashing daughter.
While the troopers are trying to get ahold of military justice records and follow up leads on other suspects, the massive search has alarmed Tiffany Starr and the cheerleading squad. "We keep hearing different rumors all day Saturday after we go home from the coach's," says Tiffany. "The house where I live is five minutes from the place where Sarah and Jen have been kidnapped. Of course I go wild, thinking they're coming to get me next. We've been imagining that they're after cheerleaders. And Saturday night and Sunday it's just me and my mom at home [her twin sisters, Amber and Amy, are away at college], and everybody knows that. By Sunday, I'm freaking out. And I say, 'Mom, we have to leave now! We have to get out of here!' And my mom says, 'Okay, let's go.' And we throw our stuff in a bag. I can't be in that house another minute. I'm terrified. I'm sure somebody is gonna break in, and we just get in the car and go."
To fully understand Tiffany's dread, we must turn the clock back two years, to 1994, when Tiffany is a sophomore, her sisters are seniors, and their father is the Dryden High football coach. . . .
The Starrs live in a lovely two-story house at the end of a wooded cul-de-sac in the country village of Cortlandville, which, like McLean, feeds into Dryden High. In the backyard is a swimming pool where neighborhood kids scramble and laugh, and on the garage is a basketball hoop, where Stephen Starr shoots baskets with his girls. Coach Starr is admired; his wife, Judy, is clever and good-looking, and his three daughters are the goddesses of Dryden High.
"My family is perfect," says Tiffany. "Besides being the Dryden High School football coach, my dad is the assistant Dryden High girls' track coach, and he is a sixth-grade teacher at Dryden Elementary. With all his jobs, it's years and years before he finishes his master's degree, and I remember the day he comes home; he brings champagne, and he pops it, and my mother and he are so excited! They dream about growing old together and sitting out on our back porch. Mom wants to get one of these swings so they can sit out there while Amy, Amber, and I are at college."
"Dad's so funny," says Amy Starr.
"Dad sitting at dinner—" says Tiffany, laughing.
"The hat backward," says Amy.
"One of those mesh hats," says Tiffany, "backward, kind of sideways backward—"
"He calls me Pinny because I was so skinny," says Amy.
"Amber he calls Amber Bambi," says Tiffany, "and I'm Shrimp or Shrimper."
"And mom's Turtle, and he's Turkey," says Amber.
"Dad loves cookies," says Amy. "You come down to the kitchen, and there he is in the middle of the night, standing with the refrigerator door open. He can eat a whole bag of Oreos or Nutter Butters. He loves peanut butter."
"He dips the peanut butter out of the jar," says Tiffany, "and then dips the spoon into the vanilla ice cream. He's a very happy man."
"So I'm on my way up to bed," says Amber, "and he's on his way downstairs, he has a glass of milk and a plate of cookies, and for some reason this really overwhelming feeling comes over me. And I say, 'Dad! Wait!' And I say, 'Stop! I love you!' And I give him this _really_ big hug, and he's like, 'I love you too, kiddo.' And he goes on downstairs. And that's the last time I see him alive."
In the fall of '94, a moody young boy from Truxton, New York, appears on the scene. A sulky rogue with dead-poet good looks, his name is J. P. Merchant and, needless to say, he's irresistible to young women. But romance has a trick of turning ugly when it comes J. P.'s way, and his last high school love affair ended in catastrophe.
Then he meets Amber Starr. She is not like the clingy, docile girls he'd known before. Amber is a Dryden cheerleader and a queen. They start dating. He falls in love; she doesn't. She breaks it off; a hole is burned into his life.
Merchant starts calling. He shows up. He knows Amber's schedule, her whereabouts, her friends. He tells her if they do not get back together he will kill himself. Amber is kind; she speaks with him for hours on the phone, "letting him down gently." In late December, he threatens to kill Amber's new boyfriend. Coach Starr is out of town, playing in a basketball tournament at his old high school, so Tiffany and Judy go to the Cortland County Sheriff on December 27 and file a complaint.
"Merchant is stalking my daughter!" says Judy. She asks for an order of protection. The sheriff arrests Merchant. Merchant's family posts bail: $500. Upon his release, he calls Amber and threatens her. Again, Tiffany and Judy go to the sheriff's department, this time with Amber. It is December 28. Judy begs the sheriff's department for help and protection.
On December 29, a sheriff's officer watches the Starrs' house. The officer goes home when his shift ends. No officer replaces him.
"Our dad raised us to be aggressive, says Tiffany. She lowers her voice in an impression of her father: "'Where's the aggression? _Dive_ for the ball! Get in there!'"
"I don't know bow many times I heard _that_!" says Amy.
"'I don't want to hear the word _can't_ ,'" says Amber, imitating her dad. "'That's not part of our vocabulary in _this_ house.'"
Late on December 29, Stephen Starr returns home, eats a plate of cookies, drinks a beer, and goes to bed. Early the next morning, as the family sleeps, J. P. Merchant shoots the locks off the Starrs' back door, climbs the stairs, and is startled to see Tiffany standing in her bedroom doorway.
He aims the Ithaca 20-gauge shotgun at her. "I am ready to die," Tiffany recalls. "I think for sure this is it. But something as simple as shutting my door keeps me alive. He is not after me. He wants Amber. He just isn't going to let anyone get in his way. And I don't try. I shut my door and let him go."
Forever after, Tiffany dreams of stepping into her closet, retrieving her baton, surging up behind him and striking him over the head. But J. P. Merchant moves on quickly—a matter of mere seconds—to Amber's bedroom. As he tells Amber to wake up, her father comes running to protect her.
J. P. shoots Stephen Starr dead with two blasts of the gun.
Somehow the girls and their mother manage to flee the house in their nightclothes. Merchant reloads his shotgun and follows. He fires into the woods at the edge of their house, believing they are hiding there. But the family goes in the opposite direction instead, racing across the yard to a neighbor's. J. P. starts to follow. . . .
Amy Starr suddenly grabs the tape recorder out of my hand and yells into it. "This is reality, people!" she says. "This _really_ happened! OK? We were straight-A students! We had friends. We were cheerleaders. We played sports. We had great lives!"
The Starr sisters are visiting my room at the Best Western Hotel outside Dryden. We have been out for an Italian dinner at the A-1 restaurant, and now the girls are sitting on the huge double-king bed in my room, looking through their high school scrapbooks, doing their best to sort through the painful memories. They've since moved on, entered college (Tiffany is graduating this month from the University of Maryland), and they work every day. "We've not done one thing to mess up," says Amy, who is engaged to marry a "terrific" young man next spring.
But the girls carry scars. They do not talk to strangers now. They do not give out their telephone numbers. They fasten their seat belts to drive one hundred yards across a parking lot. They bolt their bedroom doors. If Russell Crowe appears with a sword, they walk out of the theater. It's six years later, and they still wake in the middle of the night, their hearts beating wildly. But the Starrs are prevailing. Not the growing-up sort of prevailing that most twenty-one-year-olds experience, but the kind of prevailing that comes from being trampled and standing back up.
As for J. P. Merchant, he leaves the cul-de-sac by the Starr home and drives to the grave of his high school sweetheart, Shari Fitts. Shari had committed suicide three years earlier, while she was dating Merchant. There, he puts the gun to his head, pulls the trigger, and kills himself.
"The biggest mistake I made was not cutting off contact with J. P.," says Amber, who is dating now and seems quite happy. She takes the tape recorder out of Amy's hand and starts looking for the volume control. "Now I know, and I can tell other people." She finds the control, turns it up as high as possible, and yells: " _Cut off contact and get professional help!_ "
There is silence for a moment. The girls are huddled together over the recorder, surrounded by pictures of themselves in their purple and white track uniforms, basketball uniforms, and cheerleading outfits, their long _Alice in Wonderland_ hair tied up in white ribbons. But one picture, from early 1997, is different. It is of Tiffany's cheerleading squad. On each of their uniforms, the ribbons are black.
So is it any wonder Tiffany and Judy pack their bags and drive all the way to Tiffany's grandparents' house in Pennsylvania when Jen and Sarah disappear? As they're driving, the police are narrowing the suspects down to four—the Hajneys' neighbor, John Andrews, and three others. The hour is now approaching 10 p.m. on Sunday. A call comes in . . . like hundreds of other calls. It's a woman in her early thirties named Ann Erxleben, and she holds the key that will solve the case.
Ann is a pleasant brunette, a former class officer, yearbook editor, and member of the softball team at Dryden High. "I'm working at the hospital with Cheryl Bolduc, who is a nurse," says Ann. "And when I hear about the girls missing, I can't even _begin_ to imagine the pain Mrs. Bolduc's going through. Then something strange happens.
"My fiancé, Bruno Couture, and I own a hunting camp out in Otselic. [In this part of the country, the word _camp_ is used to describe a cabin or lodge on rustic acreage.] A friend of ours, Marcus Hutcheon, has gone up to stay there Friday night. And when he walks in, the place is dark, but he notices a puddle on the floor. A friend of his comes in and shines a flashlight on it and says it looks like blood.
"So I say, 'I think we need to go up there and check it out.' So we get a hold of Marcus, and we drive up to the camp. It's a small place—a basic hunting camp, one room, a loft, a wood stove. Marcus shows us the spot on the floor. It looks like somebody—" Ann's voice falter. "There's been a puddle, a dried puddle, and I'm scared.
"So we drive to the troopers' barracks in Norwich. There isn't anybody there, so we have to call somebody to come. I'm the one who calls. I say, 'Look, we've found blood in our camp.' I feel suddenly guilty. Call it instinct.
"So a trooper arrives, and we drive back up to the camp. The trooper goes inside. He's very nonchalant. He comes out and asks, 'Do you know any people from McLean?' Well, obviously, Bruno has been raised there, and I grew up around there. And he asks us if anybody from McLean has been up there. And I answer 'friends and family.' And the trooper says, 'Well, I've called the barracks in Cortland, and we need to wait for them to come.'
"The Cortland troopers come. It's very dark now. They take a look in the camp and start interviewing Marcus. Then they interview Bruno. Then they turn to me and ask me who I am. I say I'm Bruno's fiancée. And one of the troopers asks if any of my family and friends live near the girls.
"Both Bruno and Marcus look at me. They're waiting for me to make the call as to what to say. I've decided beforehand—it's the only way I can live with my conscience—that I will volunteer no information unless they ask me _directly_. And I look at the trooper and I say, 'Yes, my brother.' And the trooper says, 'Has anybody you know that lives near the girls been up to this camp?' And I say, 'Yes, my brother.' And he says, 'Who is your brother?' And I say, 'John Andrews.'
"And the trooper flies by me so quickly he almost knocks me down. He runs into the camp and starts screaming for the senior investigator. And at that point I just want to vomit. Because my gut instinct is right. I love him, but the kidnapper is my brother, John."
"Ann's done the right thing," says Major Foley from behind his oak desk in the state trooper headquarters. "When the sun comes up at the camp, of course, it's obvious. Because we start to find. . . ."
He stops.
"Parts of the girls," says Lieutenant Janie. "Body parts."
Foley adjusts himself in his chair and tilts his head away with a rush of emotion. "Well, I will tell you what," he says, quietly. "Here is something we will never go into. The details of the torture of those two lovely girls."
Silence.
"We arrested John Andrews," concludes Janie, "Monday at work."
Three days earlier, the day the girls never show up to the football game, John Benjamin Andrews, wearing a dark T-shirt and jeans, ducks under the Hajneys' garage door. He cuts the phone wires. Over his thinning dark hair and fleshy cheeks, he pulls a brown ski mask. He knows there is going to be a mess, so he puts on yellow rubber gloves, the kind people wear to wash dishes. The door to the kitchen is unlocked. He enters, turns, and creeps down the steps to Sarah's room.
What does this grotesque, greasy-eyed nightmare carrying a bag holding duct tape, extra yellow gloves, and six knife blades look like to her? He weighs close to 250 pounds. His bulk must overpower the small, vibrant girl. He binds the little flyer with black plastic ties and seals her mouth with duct tape.
Is he surprised to hear the shower running? Does he realize two girls are in the house? Does he know that Jen Bolduc—whose might and muscle have tossed entire squads of cheerleaders in the air—does he know that courageous Jen will stand and fight? He must be amazed when he lurches into the bathroom and Jen claws him, kicks him, and, who knows, slams him in the face with the shower caddy. John Andrews is out of shape, but he has many knives; she is naked and outweighed by well over a hundred pounds. Sarah and Jennifer are soon trapped in the trunk of the Lumina.
Going the speed limit, the trip to the Otselic camp takes an hour. It is a curvy, up-and-down road. One of Sarah's greatest pleasures in life is to lie down full-length in the back of her brother's pickup, gaze up at the stars, and, as he drives round and round, guess where she is. Now they are passing June's Country Store in Otselic. Now they are turning up Reit Road. It is bumpy. They are passing a farm. The farmer's dog must be barking. The girls are disciplined athletes, trained to think under pressure. Are they planning an escape? Are they making a pact? They are folded together like fawns, and no matter what, as Tiffany and the cheerleading squad say, "These two girls are _there_ for each other."
The cabin and its pond are about a thousand yards off Reit Road in Otselic, on the edge of Muller Hill State Forest. At some point John Andrews builds a bonfire. At some point he tortures the girls. He cuts Jen and Sarah into small pieces. He drives back down Reit Road, throwing bloody body parts out the window. He heads toward a state game land and disposes of more. He sloshes motor oil over himself, the front seat, and the dash to conceal clues and leaves the car at Cortland Line Company. He tosses the yellow gloves in a trash can.
"Well, what can I tell you?" says Major Foley. "There's a driving force. A lust. A desire. Mr. Andrews was _going_ to attack those girls. Whether he knew Jennifer was there, we'll never know. But he was _going_ to commit this crime. What drove him to do it? The easiest answer is a three-letter word: Sin. People do things that are wrong because they _want_ to. That's all."
"What makes us do things?" says Ann Erxleben. "What makes us _not_ do things? What pushed my brother over the edge? The police tell us it was some kind of woman-hate crime. Because of the way the bodies were mutilated. But John _idolized_ my mother."
In 1985, John Andrews's father, Jack, was accused of sexually abusing young girls. He killed himself three years later with a 12-gauge shotgun. Did the son blame the girls? Was he so ashamed and angry that he took revenge against young women for his father's suicide?
Looking for answers about her brother and father has not been easy for Ann. But she is not grim, not somber. She smiles and says what doesn't kill her makes her stronger. She has baked delicious blueberry muffins for me to eat during this interview. She is relatively happy now, the mother of four comely young daughters—a toddler, twins who are athletes, and her oldest daughter, now in college, who was a cheerleader. "It's a little scary for me to think that, in a lot of ways, we both were caring, giving people," Ann says of her brother. "We both were raised the same way; we both were taught the same values, we both were told to do unto others as you would have them do unto you. I said it's scary because I don't know what would make him do what he did."
When word comes on Monday, October 7, Dryden High decides to send notes to the classrooms. "Each teacher has to read to the students that Sarah and Jen have been found and that they are definitely dead," says Tiffany. "When the teachers read the notes to the classes, people jump out of their seats and run down the hallways, screaming. Everybody gathers in the gym and just screams and just cries and cries. And then people speed out to the parking lots, and they just, like . . . _leave_."
Superintendent of Schools Donald Trombley is quoted in the _Ithaca Journal_ : "It is unbelievable hysteria."
"I'll never get over it," says Tiffany. "As a female, it's the most terrifying thing to imagine happening to you. Sixteen! They are sixteen! Young women are so protective of their bodies, about being touched . . . and then the way they're killed is so bad. And the question we keep asking is: _Why does it keep happening to us, our town, our group of people_?"
Before the school makes the announcement to the students, Katie Savino, Sarah's best friend, the raven-haired, high-bouncing flyer, the _third girl_ , is taken out of class and told privately. On hearing the news, she runs toward Sarah's locker and collapses.
On Saturday, November 2, one day after being indicted on twenty- six counts of murder, kidnapping, aggravated sexual abuse, auto theft, burglary, and criminal possession of a weapon, John Andrews hangs himself in his jail cell with his shoelaces.
Scott and Tiffany's class graduates in 1997. Sarah and Jen's class graduates in 1998. In June 1999, Gary Cassell, the young Dryden High athletic director and the man who became a surrogate father to the Starr sisters, dies of a sudden heart attack. Three days later, Judy comes home from work and softly knocks on Tiffany's bedroom door. She asks Tiffany to get Amy and to come out to the living room. One glance at her mom's pale, twisted face, and Tiffany is terrified.
"And we come out in the living room and we sit down. And mom just says . . . 'Katie Savino.'"
Only two prisoners are receiving visitors today at the Tioga County Jail in Oswego, New York. One prisoner is a young curly-haired woman who is accused of killing her three-year-old child. The other is Cheryl Thayer, who has pleaded guilty to killing Katie Elizabeth Savino.
Katie graduated from Dryden and went on to the State University of New York at Oswego. When news of her death roared across the Finger Lakes region on the morning of June 11, 1999, the home of the Purple Lions was forced to shut down completely. Students simply could not believe Katie was dead.
"We felt like we're living in the Village of the Damned," says a student who described Katie as "the most popular girl who ever lived." "We were mad," says Tiffany. After standing strong through her father, Billy, and Scotty, this one was "just way too much"—she became physically ill upon hearing the news. " _We're like, 'When is this going to stop?_ '"
Twenty-three hundred people attended the memorial service for the cheerleader who pulled a whole school back to something like normality after Jen's and Sarah's deaths. "She really believed Sarah and Jen were with her," says her mother, Liz Savino. "She was always smiling. I mean, she always _glowed_. Katie didn't make friends; she took hostages. She never left a room without a hug and a 'Bye, I love you!' I miss her terribly. I miss her horribly."
"Before Jen and Sarah died, Katie was so innocent," says Tiffany. "I don't think she'd kissed a boy until she was a senior in high school. _If_ then. She was very smart, did really well in school, and she was friends with _everybody_. Then when Sarah died, Katie took a lot of her clothes and wore them. She wore Sarah's belt every day. I think it really terrified her that she was supposed to have been [at Sarah's house the night of the kidnapping]. And then on top of it, she lost her best friend in the most painful way that you could possibly imagine."
Katie's killer is tall and slender with lovely, dark, deep-set eyes, black eyebrows, and dark hair pulled high in a ponytail. Long wisps fall across her forehead as she sits very straight on her stool, her narrow shoulder blades drawn back elegantly. She is nineteen and pretty enough that, even in her orange prison pants and top, she looks like she stepped out of a Tommy Hilfiger ad.
"Katie was my best friend," Cheryl says, and immediately a large tear fills the comer of her eye. "I was leaving for California the next day, so Katie stayed and partied with me at a place in Cortland that serves kids drinks."
The tear falls against the side of her nose and begins to roll down— not down the type of burly, pockmarked face one sees in prison movies, but the face of a young girl with her hair pulled up in a scrunchy. It is disconcerting. "Katie and I were refused service because of our age." Cheryl says. "So we both just drank out of our friends' drinks. We left around two o'clock in the morning. When we got to the car. I could feel alcohol in my system, so I called shotgun. And Katie would never drive if she's even had one sip of a drink.
The tear falls against the side of her nose and begins to roll down— not down the type of burly, pockmarked face one sees in prison movies, but the face of a young girl with her hair pulled up in a scrunchy. It is disconcerting. "Katie and I were refused service because of our age." Cheryl says. "So we both just drank out of our friends' drinks. We left around two o'clock in the morning. When we got to the car. I could feel alcohol in my system, so I called shotgun. And Katie would never drive if she's even had one sip of a drink.
"I told the three guys we were taking home that one of them should drive," she continues. "But the guys all said they were too wasted. So that's how I ended up behind the wheel, even though I'm from Ithaca and I didn't know the roads. Also the seating arrangement was weird. Katie was sitting in the seat behind me. The guy in the middle was huge. Normally, Katie would have been in the middle.
"I was driving her home first. She told me to take the back roads because they were quicker. I had no idea where we were going." It was so dark, Cheryl had the creepy feeling that if she stuck her arm out the window she would never see it again. Curves appeared suddenly, but even worse were the hills. She missed a turn. Katie laughed and made her stop the car and turn around. Cheryl lost all sense of direction but dutifully took the road Katie told her to take. A minute later. . . .
"I didn't see the stop sign," Cheryl says, "and we got hit by the truck. It was so _dark_!" It's half a cry, and it strikes terror in my heart to hear it. "I didn't know the roads! I didn't see the sign! It's two-thirty in the morning. The roads are deserted. And here comes this _truck_ out of nowhere! We were dragged a couple hundred yards under the truck and the car caught on fire. As soon as the truck got stopped, the three guys climbed out. There were flames. My door was wedged closed. The truck driver pulled me out. The moment I was taken out of the car, it exploded."
"Cheryl," I say, "people in Dryden are saying Katie's screams could be heard as the flames shot through the car."
"No," Cheryl says. She waves her hand in vigorous denial. A yellow plastic ID band circles her thin, girlish wrist. Burns are still visible on her slender arms.
"I know Katie didn't die afraid," says Liz Savino. "But I have many, many nightmares about whether she was awake at the end. If she was, that would have been horrific. Absolutely horrific."
"Was Katie conscious at the end, Cheryl?"
Her upper lip trembles, but she speaks with certainty. "I think she was killed the moment the truck hit us. Katie was my best friend. I loved Katie. Everybody loved Katie. Katie was always laughing or shouting. We would have heard her if she were alive."
Liz, a small, personable woman, says she does not want to punish Cheryl Thayer. She remembers that when Katie was applying to colleges, one of her essays talked about sitting at Scott Pace's funeral and holding Jen and Sarah's hands. Liz Savino would like to think "that Katie's life was not in vain," and she believes that if Cheryl is given a chance, she will "teach others a lesson": Don't get in a car with someone who's been drinking. So Liz and her ex-husband, Jim Savino, working with the Cortland district attorney, have asked that Cheryl be released from prison in six months and begin five years' probation. (She was released last summer and is taking classes at Tompkins-Cortland Community College in Dryden.)
"I tried to do what Katie would have wanted," says Liz. "Katie was a true, loyal friend. My way of handling my daughter's death is to live the legacy she would have wanted . . . to try to open myself up to others and be less judgmental. I'm not certain I'm as successful as she was, but I'm certainly trying. I truly believe she is guiding me."
Visiting hour is over. Cheryl must return to her cell. She stands with reluctance. She squares her slender shoulders and turns to go. There is a half moment to ask one last question: Katie escaped fate the first time by not spending the night at Sarah's. . . .
"But fate made _sure_ it met Katie," says Cheryl.
Three months after attending Katie's memorial service, her good friend Mike Vogt, the class clown and Dryden High's IAC Division All Star middle linebacker, walks out to a cabin in the woods. Mike is red-haired, big-muscled, fast, born to play football. He's funny, a musician, and absolutely notorious in Dryden for his pranks. Mike drinks real beer onstage in a school play. Mike takes Chris Fox's car, parks it at the school's archery center, and covers it with condoms he steals out of the nurse's office. Mike loves "mudding" and buries all kinds of vehicles up to their axles in the big open fields around Dryden.
"Mikey's my best friend since first grade," says Johnny Lopinto, who played football with him. "I never remember doing _anything_ without him. We could be in the shittiest place in the world, and we would hate to be there, but as long as we were together, it was like everything was a big show and we were the only ones watching it. But Mikey was complicated," Johnny adds.
Mike was depressed by Katie's death and probably never got over the loss of his Dryden teammate Scott Pace three years earlier. "Maybe he wanted to protect us from his pain." says Johnny. "The morning after my twenty-first birthday, he walked out to the woods to the cabin that we built when we were younger, and he put a 12-gauge to his head."
"Jill [Yaeger, Tiffany's best friend and fellow Dryden cheerleader] called me at school," says Tiffany. "She was hysterically crying. She was like, 'I'm gonna tell you straight out: Mike killed himself.' It was the last thing I thought I was ever going to hear. I _never_ prepared myself to have one of my friends kill themselves." She sighs. "When I think about Mike," she says with a sad chuckle, "I can't think about anything but his red hair." That is the end of the story.
That is the end of the story.
The last Dryden High class that really knew Billy, Scott, Sarah, Jen, Katie, and Mike is graduating this year. And the town? "It's weird, but young death almost seems to be the norm here," says the mother of a Dryden Elementary School student. The town's dead boys and girls live on in legend now. How mythic, how beloved they've become is seen at the graves of the three cheerleaders. They are buried together high on a hill outside McLean.
The graves are simple, but they're laden with a blanket of every kind of memento the townspeople can carry up to the cemetery—stuffed bears, angels, flowers, lighted candles, crosses, butterflies, letters wrapped in see-through sandwich bags, photographs, lip balms [Katie was known for wearing three or four different flavors at a time], poems, ribbons, purple lions, megaphones, sparkle nail polish, and on and on.
On a cold, gray day, Tiffany and Jill agree to take me on a drive. As we go, Tiffany and Jill stare dejectedly out the windows.
"It's gloomy here," says Tiffany.
"It has to do with the elevation or something," says Jill.
"It's usually overcast," says Tiffany.
"Too many corn fields," says Jill.
"Tiffany," I say, "when you get married, do you want to live here?"
"No!" says Tiffany. She smacks the steering wheel lightly.
"Jill, do you want to live here when you—"
" _Absolutely not!_ " says Jill.
Indeed, when Tiffany pictures the future—she's a 4.0 student with several job offers—the town of Dryden doesn't even enter into it. She can't afford to buy a car at present, but her "biggest freedom thought," she says, is this:
"I see myself flying down some highway in my new Mustang convertible with the Verve's 'Bitter Sweet Symphony' blasting. I can just see myself flying down the highway, far away, with my hair blowing and just being happy and free! That will be the day that I take this cleansing breath. And life will have been good for a while. And it will be forever."
_Joan Didion_
Joan Didion is an award-winning writer whose work has appeared in _Vogue_ , _Life_ , _Esquire_ , _The Saturday Evening Post_ , _The New York Times_ , and _The New York Review of Books_ , among others.
Didion is the author of a dozen nonfiction books, including _Slouching Towards Bethlehem_ , _Political Fictions_ , and _Where I Was From_. Her 2005 memoir, _The Year of Magical Thinking_ , which chronicles the year following the death of her husband John Gregory Dunne, received a number of awards, including the 2005 National Book Award for Nonfiction. It was also a finalist for the Pulitzer Prize for biography/ autobiography. She adapted the book into a play, which ran for twenty-four weeks in New York and was performed around the world.
In addition to her memoirs, Didion is the author of six screenplays and five works of fiction. She holds honorary Doctor of Letters degrees from both Harvard University and Yale University.
Didion was born and raised in Sacramento, California, where she lived for a good deal of her life. She attended University of California, Berkeley, and graduated with a bachelor of arts degree in English. She currently resides in New York.
Some Dreamers of the Golden Dream
In this New Journalism classic, a suburban California woman is convicted of burning her husband to death in their family Volkswagen.
This is a story about love and death in the golden land, and begins with the country. The San Bernardino Valley lies only an hour east of Los Angeles by way of the San Bernardino Freeway but is in certain ways an alien place: not the coastal the California of subtropical twilights and the soft westerlies off the Pacific but a harsher California, haunted by the Mohave just beyond the mountains, devastated by the hot dry Santa Ana wind that comes down through the passes at one hundred miles an hour and whines through the eucalyptus windbreaks and works on the nerves. October is the bad month for the wind, the month when breathing is difficult and the hills blaze up spontaneously. There has been no rain since April. Every voice seems a scream. It is the season of suicide and divorce and prickly dread, wherever the wind blows.
The Mormons settled this ominous country, and then they abandoned it, but by the time they left the first orange tree had been planted and for the next hundred years the San Bernardino Valley would draw a kind of people who imagined they might live among the talismanic fruit and prosper in the dry air, people who brought with them Midwestern ways of building and cooking and praying and who tried to graft those ways upon the land. The graft took in curious ways. This is the California where it is possible to live and die without ever eating an artichoke, without ever meeting a Catholic or a Jew. This is the California where it is easy to Dial A-Devotion, but hard to buy a book. This is the country in which a belief in the literal interpretation of Genesis has slipped imperceptibly into a belief in the literal interpretation of _Double Indemnity_ , the country of the teased hair and the Capris and the girls for whom all life's promise comes down to a waltz-length white wedding dress and the birth of a Kimberly or a Sherry or a Debbi and a Tijuana divorce and return to hairdressers' school. "We were just crazy kids," they say without regret, and look to the future. The future always looks good in the golden land, because no one remembers the past. Here is where the hot wind blows and the old ways do not seem relevant, where the divorce rate is double the national average and where one person in every thirty-eight lives in a trailer. Here is the last stop for all those who come from somewhere else, for all those who drifted away from the cold and the past and the old ways. Here is where they are trying to find a new life style, trying to find it in the only places they know to look: the movies and the newspapers. The case of Lucille Marie Maxwell Miller is a tabloid monument to that new life style.
Imagine Banyan Street first, because Banyan is where it happened. The way to Banyan is to drive west from San Bernardino out Foothill Boulevard, Route 66: past the Santa Fe switching yards, the Forty Winks Motel. Past the motel that is nineteen stucco tepees: "SLEEP IN A WIGWAM—GET MORE FOR YOUR WAMPUM." Past Fontana Drag City and the Fontana Church of the Nazarene and the Pit Stop A Go-Go; past Kaiser Steel, through Cucamonga, out to the Kapu Kai Restaurant-Bar and Coffee Shop, at the corner of Route 66 and Carnelian Avenue. Up Carnelian Avenue from the Kapu Kai, which means "Forbidden Seas," the subdivision flags whip in the harsh wind. "HALF-ACRE RANCHES! SNACK BARS! TRAVERTINE ENTRIES! $95 DOWN." It is the trail of an intention gone haywire, the flotsam of the New California. But after a while the signs thin out on Carnelian Avenue, and the houses are no longer the bright pastels of the Springtime Home owners but the faded bungalows of the people who grow a few grapes and keep a few chickens out here, and then the hill gets steeper and the road climbs and even the bungalows are few, and here—desolate, roughly surfaced, lined with eucalyptus and lemon groves—is Banyan Street.
Like so much of this country, Banyan suggests something curious and unnatural. The lemon groves are sunken, down a three- or four-foot retaining wall, so that one looks directly into their dense foliage, too lush, unsettlingly glossy, the greenery of nightmare; the fallen eucalyptus bark is too dusty, a place for snakes to breed. The stones look not like natural stones but like the rubble of some unmentioned upheaval. There are smudge pots, and a closed cistern. To one side of Banyan there is the flat valley, and to the other the San Bernardino Mountains, a dark mass looming too high, too fast, nine, ten, eleven thousand feet, right there above the lemon groves. At midnight on Banyan Street there is no light at all, and no sound except the wind in the eucalyptus and a muffled barking of dogs. There may be a kennel somewhere, or the dogs may be coyotes.
Banyan Street was the route Lucille Miller took home from the twenty-four-hour Mayfair Market on the night of October 7, 1969, a night when the moon was dark and the wind was blowing and she was out of milk, and Banyan Street was where, at about 12:30 a.m., her 1964 Volkswagen came to a sudden stop, caught fire, and began to burn. For an hour and fifteen minutes Lucille Miller ran up and down Banyan calling for help, but no cars passed and no help came. At three o'clock that morning, when the fire had been put out and the California Highway Patrol officers were completing their report, Lucille Miller was still sobbing and incoherent, for her husband had been asleep in the Volkswagen. "What will I tell the children, when there's nothing left, nothing left in the casket," she cried to the friend called to comfort her. "How can I tell them there's nothing left?"
In fact there was something left, and a week later it lay in the Draper Mortuary Chapel in a closed bronze coffin blanketed with pink carnations. Some two hundred mourners heard Elder Robert E. Denton of the Seventh-Day Adventist Church of Ontario speak of "the temper of fury that has broken out among us." For Gordon Miller, he said, there would be "no more death, no more heartache, and no more misunderstandings." Elder Ansel Bristol mentioned the "peculiar" grief of the hour. Elder Fred Jensen asked "what shall it profit a man, if he shall gain the whole world, and lose his own soul?" A light rain fell, a blessing in a dry season, and a female vocalist sang "Safe in the Arms of Jesus." A tape recording of the service was made for the widow, who was being held without bail in the San Bernardino County jail on a charge of first-degree murder.
Of course she came from somewhere else, came off the prairie, in search of something she had seen in a movie or heard on the radio, for this is a Southern California story. She was born on January 17, 1930, in Winnipeg, Manitoba, the only child of Gordon and Lily Maxwell, both schoolteachers and both dedicated to the Seventh-Day Adventist Church whose members observe the Sabbath on Saturday, believe in an apocalyptic Second Coming, have a strong missionary tendency, and, if they are strict, do not smoke, drink, eat meat, use makeup, or wear jewelry, including wedding rings. By the time Lucille Maxwell enrolled at Walla Walla College in College Place, Washington, the Adventist school where her parents then taught, she was an eighteen-year-old possessed of unremarkable good looks and remarkable high spirits. "Lucille wanted to see the world," her father would say in retrospect, "and I guess she found out."
The high spirits did not seem to lend themselves to an extended course of study at Walla Walla College, and in the spring of 1949 Lucille Maxwell met and married Gordon ("Cork") Miller, a twenty-four-yearold graduate of Walla Walla and of the University of Oregon dental school, then stationed at Fort Lewis as a medical officer. "Maybe you could say it was love at first sight," Mr. Maxwell recalls. "Before they were ever formally introduced, he sent Lucille a dozen and a half roses with a card that said even if she didn't come out on a date with him, he hoped she'd find the roses pretty anyway." The Maxwells remember their daughter as a "radiant" bride.
Unhappy marriages so resemble one another that we do not need to know too much about the course of this one. There may or may not have been trouble on Guam, where Cork and Lucille Miller lived while he finished his Army duty. There may or may not have been problems in the small Oregon town where he first set up private practice. There appears to have been some disappointment about their move to California: Cork Miller had told friends that he wanted to become a doctor, that he was unhappy as a dentist and planned to enter the Seventh-Day Adventist College of Medical Evangelists at Loma Linda, a few miles south of San Bernardino. Instead he bought a dental practice in the west end of San Bernardino County, and the family settled there, in a modest house on the kind of street where there are always tricycles and revolving credit and dreams about bigger houses, better streets. That was 1957. By the summer of 1964 they had achieved the bigger house on the better street and the familiar accouterments of a family on its way up: the $30,000 a year, the three children for the Christmas card, the picture window, the family room, the newspaper photographs that showed "Mrs. Gordon Miller, Ontario Heart Fund Chairman. . . ." They were paying the familiar price for it. And they had reached the familiar season of divorce.
It might have been anyone's bad summer, anyone's siege of heat and nerves and migraine and money worries, but this one began particularly early and particularly badly. On April 24 an old friend, Elaine Hayton, died suddenly; Lucille Miller had seen her only the night before. During the month of May, Cork Miller was hospitalized briefly with a bleeding ulcer, and his usual reserve deepened into depression. He told his accountant that he was "sick of looking at open mouths" and threatened suicide. By July 8, the conventional tensions of love and money had reached the conventional impasse in the new house on the acre lot at 8488 Bella Vista, and Lucille Miller filed for divorce. Within a month, however, the Millers seemed reconciled. They saw a marriage counselor. They talked about a fourth child. It seemed that the marriage had reached the traditional truce, the point at which so many resign themselves to cutting both their losses and their hopes.
But the Millers' season of trouble was not to end that easily. October 7 began as a commonplace enough day, one of those days that sets the teeth on edge with its tedium, its small frustrations. The temperature reached 102 degrees in San Bernardino that afternoon, and the Miller children were home from school because of Teachers' Institute. There was ironing to be dropped off. There was a trip to pick up a prescription for Nembutal, a trip to a self-service dry cleaner. In the early evening, an unpleasant accident with the Volkswagen: Cork Miller hit and killed a German shepherd, and afterward said that his head felt "like it had a Mack truck on it." It was something he often said. As of that evening Cork Miller was $63,479 in debt, including the $29,637 mortgage on the new house, a debt load which seemed oppressive to him. He was a man who wore his responsibilities uneasily, and complained of migraine headaches almost constantly.
He ate alone that night, from a TV tray in the living room. Later the Millers watched John Forsythe and Senta Berger in _See How They Run_ , and when the movie ended, about eleven, Cork Miller suggested that they go out for milk. He wanted some hot chocolate. He took a blanket and pillow from the couch and climbed into the passenger seat of the Volkswagen. Lucille Miller remembers reaching over to lock his door as she backed down the driveway. By the time she left the Mayfair Market, and long before they reached Banyan Street, Cork Miller appeared to be asleep.
There is some confusion in Lucille Miller's mind about what happened between 12:30 a.m.., when the fire broke out, and 1:50 a.m., when it was reported. She says that she was driving east on Banyan Street at about 35 m.p.h. when she felt the Volkswagen pull sharply to the right. The next thing she knew the car was on the embankment, quite near the edge of the retaining wall, and flames were shooting up behind her. She does not remember jumping out. She does remember prying up a stone with which she broke the window next to her husband, and then scrambling down the retaining wall to try to find a stick. "I don't know how I was going to push him out," she says. "I just thought if I had a stick, I'd push him out." She could not, and after a while she ran to the intersection of Banyan and Carnelian Avenue. There are no houses at that corner, and almost no traffic. After one car had passed without stopping, Lucille Miller ran back down Banyan toward the burning Volkswagen. She did not stop, but she slowed down, and in the flames she could see her husband. He was, she said, "just black."
At the first house up Sapphire Avenue, half a mile from the Volkswagen, Lucille Miller finally found help. There Mrs. Robert Swenson called the sheriff, and then, at Lucille Miller's request, she called Harold Lance, the Millers' lawyer and their close friend. When Harold Lance arrived he took Lucille Miller home to his wife, Joan. Twice Harold Lance and Lucille Miller returned to Banyan Street and talked to the Highway Patrol officers. A third time Harold Lance returned alone, and when he came back he said to Lucille Miller, "O.K. . . . you don't talk any more."
When Lucille Miller was arrested the next afternoon, Sandy Slagle was with her. Sandy Slagle was the intense, relentlessly loyal medical student who used to babysit for the Millers, and had been living as a member of the family since she graduated from high school in 1959. The Millers took her away from a difficult home situation, and she thinks of Lucille Miller not only as "more or less a mother or a sister" but as "the most wonderful character" she has ever known. On the night of the accident, Sandy Slagle was in her dormitory at Loma Linda University, but Lucille Miller called her early in the morning and asked her to come home. The doctor was there when Sandy Slagle arrived, giving Lucille Miller an injection of Nembutal. "She was crying as she was going under," Sandy Slagle recalls. "Over and over she'd say, 'Sandy, all the hours I spent trying to save him and now what are they trying to _do_ to me?'"
At 1:30 that afternoon, Sergeant William Paterson and Detectives Charles Callahan and Joseph Karr of the Central Homicide Division arrived at 8488 Bella Vista. "One of them appeared at the bedroom door," Sandy Slagle remembers, "and said to Lucille, 'You've got ten minutes to get dressed or we'll take you as you are.' She was in her nightgown, you know, so I tried to get her dressed."
Sandy Slagle tells the story now as if by rote and her eyes do not waver. "So I had her panties and bra on her and they opened the door again, so I got some Capris on her, you know, and a scarf." Her voice drops. "And then they just took her."
The arrest took place just twelve hours after the first report that there had been an accident on Banyan Street, a rapidity which would later prompt Lucille Miller's attorney to say that the entire case was an instance of trying to justify a reckless arrest. Actually what first caused the detectives who arrived on Banyan Street toward dawn that morning to give the accident more than routine attention were certain apparent physical inconsistencies. While Lucille Miller had said that she was driving about 35 m.p.h. when the car swerved to a stop, an examination of the cooling Volkswagen showed that it was in low gear, and that the parking rather than the driving lights were on. The front wheels, moreover, did not seem to be in exactly the position that Lucille Miller's description of the accident would suggest, and the right rear wheel was dug in deep, as if it had been spun in place. It seemed curious to the detectives, too, that a sudden stop from 35 m.p.h.—the same jolt which was presumed to have knocked over a gasoline can in the back seat and somehow started the fire—should have left two milk cartons upright on the back floorboard, and the remains of a Polaroid camera box lying apparently undisturbed on the back seat.
No one, however, could be expected to give a precise account of what did and did not happen in a moment of terror, and none of these inconsistencies seemed in themselves incontrovertible evidence of criminal intent. But they did interest the Sheriff's Office, as did Gordon Miller's apparent unconsciousness at the time of the accident, and the length of time it had taken Lucille Miller to get help. Something, moreover, struck the investigators as wrong about Harold Lance's attitude when he came back to Banyan Street the third time and found the investigation by no means over. "The way Lance was acting," the prosecuting attorney said later, "they thought maybe they'd hit a nerve."
And so it was that on the morning of October 8, even before the doctor had come to give Lucille Miller an injection to calm her, the San Bernardino County Sheriff's Office was trying to construct another version of what might have happened between 12:30 and 1:50 a.m. The hypothesis they would eventually present was based on the somewhat tortuous premise that Lucille Miller had undertaken a plan which failed: a plan to stop the car on the lonely road, spread gasoline over her presumably drugged husband, and, with a stick on the accelerator, gently "walk" the Volkswagen over the embankment, where it would tumble four feet down the retaining wall into the lemon grove and almost certainly explode. If this happened, Lucille Miller might then have somehow negotiated the two miles up Carnelian to Bella Vista in time to be home when the accident was discovered. This plan went awry, according to the Sheriff's Office hypothesis, when the car would not go over the rise of the embankment. Lucille Miller might have panicked then—after she had killed the engine the third or fourth time, say, out there on the dark road with the gasoline already spread and the dogs baying and the wind blowing and the unspeakable apprehension that a pair of headlights would suddenly light up Banyan Street and expose her there—and set the fire herself.
Although this version accounted for some of the physical evidence— the car in low because it had been started from a dead stop, the parking lights on because she could not do what needed doing without some light, a rear wheel spun in repeated attempts to get the car over the embankment, the milk cartons upright because there had been no sudden stop—it did not seem on its own any more or less credible than Lucille Miller's own story. Moreover, some of the physical evidence did seem to support her story: a nail in a front tire, a nine-pound rock found in the car, presumably the one with which she had broken the window in an attempt to save her husband. Within a few days an autopsy had established that Gordon Miller was alive when he burned, which did not particularly help the State's case, and that he had enough Nembutal and Sandoptal in his blood to put the average person to sleep, which did: on the other hand Gordon Miller habitually took both Nembutal and Fiorinal (a common headache prescription which contains Sandoptal), and had been ill besides.
It was a spotty case, and to make it work at all the State was going to have to find a motive. These was talk of unhappiness, talk of another man. That kind of motive, during the next few weeks, was what they set out to establish. They set out to find it in accountants' ledgers and double-indemnity clauses and motel registers, set out to determine what might move a woman who believed in all the promises of the middle class—a woman who had been chairman of the Heart Fund and who always knew a reasonable little dressmaker and who had come out of the bleak wild of prairie fundamentalism to find what she imagined to be the good life—what would drive such a woman to sit on street called Bella Vista and look out her new picture window into the empty California sun and calculate how to burn her husband alive in a Volkswagen. They found the wedge they wanted closer at hand than they might have at first expected, for, as testimony would reveal later at the trial, it seemed that in December of 1963 Lucille Miller had begun an affair with the husband of one of her friends, a man whose daughter called her "Auntie Lucille," a man who might have seemed to have the gift for people and money and the good life that Cork Miller so noticeably lacked. The man was Arthwell Hayton, a well-known San Bernardino attorney and at one time a member of the district attorney's staff.
In some ways it was the conventional clandestine affair in a place like San Bernardino, a place where little is bright or graceful, where it is routine to misplace the future and easy to start looking for it in bed. Over the seven weeks that it would take to try Lucille Miller for murder, Assistant District Attorney Don A. Turner and defense attorney Edward P. Foley would between them unfold a curiously predictable story. There were the falsified motel registrations. There were the lunch dates, the afternoon drives in Arthwell Hayton's red Cadillac convertible. There were the interminable discussions of the wronged partners. There were the confidantes ("I knew everything," Sandy Slagle would insist fiercely later. "I knew every time, places, everything") and there were the words remembered from bad magazine stories ("Don't kiss me, it will trigger things," Lucille Miller remembered telling Arthwell Hayton in the parking lot of Harold's Club in Fontana after lunch one day) and there were the notes, the sweet exchanges: "Hi Sweetie Pie! You are my cup of tea!! Happy Birthday—you don't look a day over twenty-nine!! Your baby, Arthwell."
And, toward the end, there was the acrimony. It was April 24, 1964, when Arthwell Hayton's wife, Elaine, died suddenly, and nothing good happened after that. Arthwell Hayton had taken his cruiser, Captain's Lady, over to Catalina that weekend; he called home at nine o'clock Friday night, but did not talk to his wife because Lucille Miller answered the telephone and said that Elaine was showering. The next morning the Haytons' daughter found her mother in bed, dead. The newspapers reported the death as accidental, perhaps the result of an allergy to hair spray. When Arthwell Hayton flew home from Catalina that weekend, Lucille Miller met him at the airport, but the finish had already been written.
It was in the breakup that the affair ceased to be in the conventional mode and began to resemble instead the novels of James M. Cain, the movies of the late 1930s, all the dreams in which violence and threats and blackmail are made to seem commonplaces of middle-class life. What was most startling about the case the State of California was preparing against Lucille Miller was something that had nothing to do with law at all, something that never appeared in the eight-column afternoon headlines but was always there between them: the revelation that the dream was teaching the dreamers how to live. Here is Lucille Miller talking to her lover sometime in the early summer of 1964, after he had indicated that, on the advice of his minister, he did not intend to see her any more: "First, I'm going to go to that dear pastor of yours and tell him a few things. . . . When I do tell him that, you won't be in the Redlands Church any more. . . . Look, Sonny Boy, if you think your reputation is going to be ruined, your life won't be worth two cents." Here is Arthwell Hayton, to Lucille Miller: "I'll go to Sheriff Frank Bland and tell him some things that I know about you until you'll wish you'd never heard of Arthwell Hayton." For an affair between a Seventh- Day Adventist dentist's wife and a Seventh-Day Adventist personal-injury lawyer, it seems a curious kind of dialogue.
"Boy, I could get that little boy coming and going," Lucille Miller later confided to Erwin Sprengle, a Riverside contractor who was a business partner of Arthwell Hayton's and a friend to both the lovers. (Friend or no, on this occasion he happened to have an induction coil attached to his telephone in order to tape Lucille Miller's call.) "And he hasn't got one thing on me that he can prove. I mean, I've got concrete—he has nothing concrete." In the same taped conversation with Erwin Sprengle, Lucille Miller mentioned a tape that she herself had surreptitiously made, months before, in Arthwell Hayton's car.
"I said to him, I said 'Arthwell, I just feel like I'm being used!'. . . He started sucking his thumb and he said 'I love you. . . . This isn't something that happened yesterday. I'd marry you tomorrow if I could. I don't love Elaine.' He'd love to hear that played back, wouldn't he?"
"Yeah," drawled Sprengle's voice on the tape. "That would be just a little incriminating, wouldn't it?"
"Just a _little_ incriminating," Lucille Miller agreed. "It really _is_."
Later on the tape, Sprengle asked where Cork Miller was.
"He took the children down to the church."
"You didn't go?"
"No."
"You're naughty."
It was all, moreover, in the name of "love"; everyone involved placed a magical faith in the efficacy of the very word. There was the significance that Lucille Miller saw in Arthwell's saying that he "loved" her, that he did not "love" Elaine. There was Arthwell insisting, later, at the trial, that he had never said it, that he may have "whispered sweet nothings in her ear" (as her defense hinted that he had whispered in many ears), but he did not remember bestowing upon her the special seal, saying the word, declaring "love." There was the summer evening when Lucille Miller and Sandy Slagle followed Arthwell Hayton down to his new boat in its mooring at Newport Beach and untied the lines with Arthwell aboard, Arthwell and a girl with whom he later testified he was drinking hot chocolate and watching television. "I did that on purpose," Lucille Miller told Erwin Sprengle later, "to save myself from letting my heart do something crazy."
January 11, 1965, was a bright warm day in Southern California, the kind of day when Catalina floats on the Pacific horizon and the air smells of orange blossoms and it is a long way from the bleak and difficult East, a long way from the cold, a long way from the past. A woman in Hollywood staged an all-night sit-in on the hood of her car to prevent repossession by a finance company. A seventy-year-old pensioner drove his station wagon at five miles an hour past three Gardena poker parlors and emptied three pistols and a twelve-gauge shotgun through their windows, wounding twenty-nine people. "Many young women become prostitutes just to have enough money to play cards," he explained in a note. Mrs. Nick Adams said that she was "not surprised" to hear her husband announce his divorce plans on the Les Crane Show, and, farther north, a sixteen-year-old jumped off the Golden Gate Bridge and lived.
And, in the San Bernardino County Courthouse, the Miller trial opened. The crowds were so bad that the glass courtroom doors were shattered in the crush, and from then on identification disks were issued to the first forty-three spectators in line. The line began forming at 6 a.m., and college girls camped at the courthouse all night, with stores of graham crackers and No-Cal.
All they were doing was picking a jury, those first few days, but the sensational nature of the case had already suggested itself. Early in December there had been an abortive first trial, a trial at which no evidence was ever presented because on the day the jury was seated the San Bernardino _Sun-Telegram_ ran an "inside" story quoting Assistant District Attorney Don Turner, the prosecutor, as saying, "We are looking into the circumstances of Mrs. Hayton's death. In view of the current trial concerning the death of Dr. Miller, I do not feel I should comment on Mrs. Hayton's death." It seemed that there had been barbiturates in Elaine Hayton's blood, and there had seemed some irregularity about the way she was dressed on that morning when she was found under the covers, dead. Any doubts about the death at the time, however, had never gotten as far as the Sheriff's Office. "I guess somebody didn't want to rock the boat," Turner said later. 'These were prominent people."
Although all of that had not been in the _Sun-Telegram's_ story, an immediate mistrial had been declared. Almost as immediately, there had been another development: Arthwell Hayton had asked newspapermen to an 11 a.m. Sunday morning press conference in his office. There had been television cameras, and flash bulbs popping. "As you gentlemen may know," Hayton had said, striking a note of stiff bonhomie, "there are very often women who become amorous toward their doctor or lawyer. This does not mean on the physician's or lawyer's part that there is any romance toward the patient or client."
"Would you deny that you were having an affair with Mrs. Miller?" a reporter had asked.
"I would deny that there was any romance on my part whatsoever."
It was a distinction he would maintain through all the wearing weeks to come.
So they had come to see Arthwell, these crowds who now milled beneath the dusty palms outside the courthouse, and they had also come to see Lucille, who appeared as a slight, intermittently pretty woman, already pale from lack of sun, a woman who would turn thirty-five before the trial was over and whose tendency toward haggardness was beginning to show, a meticulous woman who insisted against her lawyer's advice, on coming to court with her hair piled high and lacquered. "I would've been happy if she'd come in with it hanging loose, but Lucille wouldn't do that," her lawyer said. He was Edward P. Foley, a small, emotional Irish Catholic who several times wept in the courtroom. "She has a great honesty, this woman," he added, "but this honesty about her appearance always worked against her."
By the time the trial opened, Lucille Miller's appearance included maternity clothes, for an official examination on December 18 had revealed that she was then three and a half months pregnant, a fact which made picking a jury even more difficult than usual, for Turner was asking the death penalty. "It's unfortunate but there it is," he would say of the pregnancy to each juror in turn, and finally twelve were seated, seven of them women, the youngest forty-one, an assembly of the very peers— housewives, a machinist, a truck driver, a grocery-store manager, a filing clerk—above whom Lucille Miller had wanted so badly to rise.
That was the sin, more than the adultery, which tended to reinforce the one for which she was being tried. It was implicit in both the defense and the prosecution that Lucille Miller was an erring woman, a woman who perhaps wanted too much. But to the prosecution she was not merely a woman who would want a new house and want to go to parties and run up high telephone bills ($1,152 in two months), but a woman who would go so far as to murder her husband for his $80,000 in insurance, making it appear an accident in order to collect another $40,000 in double indemnity and straight accident policies. To Turner she was a woman who did not want simply her freedom and a reasonable alimony (she could have had that, the defense contended, by going through with her divorce suit), but wanted everything, a woman motivated by "love and greed." She was a "manipulator." She was a "user of people."
To Edward Foley, on the other hand, she was an impulsive woman who "couldn't control her foolish little heart." Where Turner skirted the pregnancy, Foley dwelt upon it, even calling the dead man's mother down from Washington to testify that her son had told her they were going to have another baby because Lucille felt that it would "do much to weld our home again in the pleasant relations that we used to have." Where the prosecution saw a "calculator," the defense saw a "blabbermouth," and in fact Lucille Miller did emerge as an ingenuous conversationalist. Just as, before her husband's death, she had confided in her friends about her love affair, so she chatted about it after his death, with the arresting sergeant. "Of course Cork lived with it for years, you know," her voice was heard to tell Sergeant Paterson on a tape made the morning after her arrest." After Elaine died, he pushed the panic button one night and just asked me right out, and that, I think, was when he really—the first time he really faced it." When the sergeant asked why she had agreed to talk to him, against the specific instructions of her lawyers, Lucille Miller said airily, "Oh, I've always been basically quite an honest person. . . . I mean I can put a hat in the cupboard and say it cost ten dollars less, but basically I've always kind of just lived my life the way I wanted to, and if you don't like it you can take off."
The prosecution hinted at men other than Arthwell, and even, over Foley's objections, managed to name one. The defense called Miller suicidal. The prosecution produced experts who said that the Volkswagen fire could not have been accidental. Foley produced witnesses who said that it could have been. Lucille's father, now a junior-high-school teacher in Oregon, quoted Isaiah to reporters: " _Every tongue that shall rise against thee in judgment thou shalt condemn._ " "Lucille did wrong, her affair," her mother said judiciously. "With her it was love. But with some I guess it's just passion." There was Debbie, the Millers' fourteen-year-old, testifying in a steady voice about how she and her mother had gone to a supermarket to buy the gasoline can the week before the accident. There was Sandy Slagle, in the courtroom every day, declaring that on at least one occasion Lucille Miller had prevented her husband not only from committing suicide but from committing suicide in such a way that it would appear an accident and ensure the double-indemnity payment. There was Wenche Berg, the pretty twenty-seven-year-old Norwegian governess to Arthwell Hayton's children, testifying that Arthwell had instructed her not to allow Lucille Miller to see or talk to the children.
Two months dragged by, and the headlines never stopped. Southern California's crime reporters were headquartered in San Bernardino for the duration: Howard Hertel from the Times, Jim Bennett and Eddy Jo Bernal from the _Herald-Examiner_. Two months in which the Miller trial was pushed off the _Examiner_ front page only by the Academy Award nominations and Stan Laurel's death. And finally, on March 2, after Turner had reiterated that it was a case of "love and greed," and Foley had protested that his client was being tried for adultery, the case went to the jury.
They brought in the verdict, guilty of murder in the first degree, at 4:50 p.m. on March 5. "She didn't do it," Debbie Miller cried, jumping up from the spectators' section. "She didn't _do_ it." Sandy Slagle collapsed in her seat and began to scream. "Sandy, for God's sake please _don't_ ," Lucille Miller said in a voice that carried across the courtroom, and Sandy Slagle was momentarily subdued. But as the jurors left the courtroom she screamed again: "You're murderers. . . . Every last one of you is a _murderer_." Sheriff's deputies moved in then, each wearing a string tie that read "1965 SHERIFF'S RODEO," and Lucille Miller's father, that sad-faced juniorhigh- school teacher who believed in the word of Christ and the dangers of wanting to see the world, blew her a kiss off his fingertips.
The California Institution for Women at Frontera, where Lucille Miller is now, lies down where Euclid Avenue turns into country road, not too many miles from where she once lived and shopped and organized the Heart Fund Ball. Cattle graze across the road, and Rainbirds sprinkle the alfalfa. Frontera has a softball field and tennis courts, and looks as if it might be a California junior college, except that the trees are not yet high enough to conceal the concertina wire around the top of the Cyclone fence. On visitors' day there are big cars in the parking area, big Buicks and Pontiacs that belong to grandparents and sisters and fathers (not many of them belong to husbands), and some of them have bumper stickers that say "SUPPORT YOUR LOCAL POLICE."
A lot of California murderesses live here, a lot of girls who somehow misunderstood the promise. Don Turner put Sandra Garner here (and her husband in the gas chamber at San Quentin) after the 1959 desert killings known to crime reporters as "the soda-pop murders." Carole Tregoff is here, and has been ever since she was convicted of conspiring to murder Dr. Finch's wife in West Covina, which is not too far from San Bernardino. Carole Tregoff is in fact a nurse's aide in the prison hospital, and might have attended Lucille Miller had her baby been born at Frontera; Lucille Miller chose instead to have it outside, and paid for the guard who stood outside the delivery room in St. Bernadine's Hospital. Debbie Miller came to take the baby home from the hospital, in a white dress with pink ribbons, and Debbie was allowed to choose a name. She named the baby Kimi Kai. The children live with Harold and Joan Lance now, because Lucille Miller will probably spend ten years at Frontera. Don Turner waived his original request for the death penalty (it was generally agreed that he had demanded it only, in Edward Foley's words, "to get anybody with the slightest trace of human kindness in their veins off the jury"), and settled for life imprisonment with the possibility of parole. Lucille Miller does not like it at Frontera, and has had trouble adjusting. "She's going to have to learn humility," Turner says. "She's going to have to use her ability to charm, to manipulate."
The new house is empty now, the house on the street with the sign that says
PRIVATE ROAD
BELLA VISTA
DEAD END
The Millers never did get it landscaped, and weeds grow up around the fieldstone siding. The television aerial has toppled on the roof, and a trash can is stuffed with the debris of family life: a cheap suitcase, a child's game called "Lie Detector." There is a sign on what would have been the lawn, and the sign reads "ESTATE SALE." Edward Foley is trying to get Lucille Miller's case appealed, but there have been delays. "A trial always comes down to a matter of sympathy," Foley says wearily now. "I couldn't create sympathy for her." Everyone is a little weary now, weary and resigned, everyone except Sandy Slagle, whose bitterness is still raw. She lives in an apartment near the medical school in Loma Linda, and studies reports of the case in _True Police Cases_ and _Official Detective Stories_. "I'd much rather we not talk about the Hayton business too much,"' she tells visitors, and she keeps a tape recorder running. "I'd rather talk about Lucille and what a wonderful person she is and how her rights were violated." Harold Lance does not talk to visitors at all. "We don't want to give away what we can sell," he explains pleasantly; an attempt was made to sell Lucille Miller's personal story to _Life_ , but _Life_ did not want to buy it. In the district attorney's offices they are prosecuting other murders now, and do not see why the Miller trial attracted so much attention. "It wasn't a very interesting murder as murders go," Don Turner says laconically. Elaine Hayton's death is no longer under investigation. "We know everything we want to know," Turner says.
Arthwell Hayton's office is directly below Edward Foley's. Some people around San Bernardino say that Arthwell Hayton suffered; others say that he did not suffer at all. Perhaps he did not, for time past is not believed to have any bearing upon time present or future, out in the golden land where every day the world is born anew. In any case, on October 17, 1965, Arthwell Hayton married again, married his children's pretty governess, Wenche Berg, at a service in the Chapel of the Roses at a retirement village near Riverside. Later the newlyweds were feted at a reception for seventy- five in the dining room of Rose Garden Village. The bridegroom was in black tie, with a white carnation in his buttonhole. The bride wore a long white _peau de soie_ dress and carried a shower bouquet of sweetheart roses with stephanotis streamers. A coronet of seed pearls held her illusion veil.
_Melissa Fay Greene_
Melissa Fay Greene, a Macon, Georgia, native and 1975 graduate of Oberlin College, is the author of six books of nonfiction: _Praying for Sheetrock_ ; _The Temple Bombing_ ; _Last Man Out_ ; _There Is No Me Without You: One Woman's Odyssey to Rescue her Country's Children_ ; _No Biking in the House Without a Helmet_ ; and _The Underdogs: Children, Dogs, & the Power of Unconditional Love_.
Greene's books have been translated into fifteen languages and have been honored with two National Book Award nominations, a National Book Critics Circle Award nomination, the Robert F. Kennedy Book Award, the Southern Book Critics Circle Award, the Salon Book Prize, the ACLU Civil Liberties Award, and the Hadassah Myrtle Wreath Award. She holds an honorary doctorate from Emory University, was a 2011 Georgia Writers Hall of Fame inductee, and was a 2015–2016 Guggenheim Fellow.
In 1999, _Praying for Sheetrock_ was named one of the one hundred best works of English-language journalism of the twentieth century by the faculty of New York University. Entertainment Tonight named it one of the "New Classics: The Best One Hundred Books of the Last Twenty-Five Years."
Greene has also written for _The New Yorker_ , _The New York Times Magazine_ , _The Atlantic, Newsweek_ , _The Washington Post_ , _The Boston Globe_ , _LIFE_ , _Elle_ , _Reader's Digest_ , _Good Housekeeping_ , and other periodicals, and she has been a frequent guest on CNN, National Public Radio, and NPR-affiliate stations.
Greene and her husband, defense attorney Don Samuel, live in Atlanta. They are the parents of nine children.
Wonder Dog
How a family struggling with a child's serious cognitive and behavioral disabilities finds peace and reassurance with an assistance dog.
In May 1999, Donnie Kanter Winokur, forty-three, a writer and multimedia producer, and her husband, Rabbi Harvey Winokur, forty-nine, beheld the son of their dreams, the child infertility denied them. Andrey, a pale dark-eyed one-year-old in a cotton onesie, held in a standing position by a caregiver, appeared in a short videotape recorded in a Russian orphanage. If the couple liked the little boy, they could begin the legal process of adopting him. They liked the little boy very much.
Four months later, the Winokurs flew to Russia from their home in Atlanta to adopt Andrey, whom they renamed Iyal, and to adopt an unrelated little girl two days younger, whom they named Morasha. All four appear in another orphanage video: the beaming new parents on the happiest day of their lives, the toddlers passive in the arms of the strangers cradling and kissing them. In August 1999, the family arrived home to congratulations, gifts, and helium balloons.
"Sometime after their third birthdays, our wonderful fairy tale of adopting two Russian babies began to show cracks," said Donnie Winokur, who is now fifty-five. She is pert and trim, with cropped brown hair and a pursed-lips, lemony expression softened by wearying experience. Unlike bright and cheery Morasha, Iyal grew oppositional and explosive. He was a sturdy, big-hearted boy with a wide and open face, shiny black hair in a bowl cut, and a winning giggle. But, triggered by the sight of a cartoon image on a plastic cup, or an encounter with Morasha's Barbie dolls, he threw tantrums that shook the house. He stuffed himself at mealtimes with an inexplicable urgency. In a fast-moving car, he unfastened his seat belt and tried to jump out. He awoke every night in a rage. "I had panic attacks in the night when I heard him coming," she said. "I assumed everything was my fault, that I was not a good-enough mother." In preschool, Iyal plowed his tricycle into other children without remorse, or maybe without awareness. He tried to kiss strangers, or feel their toes. Friends and congregants (Harvey Winokur is the founding rabbi of Temple Kehillat Chaim in Roswell, Georgia) who had assured the Winokurs, "He's all boy!" or "Mine was the same way!" began to fall silent, out of shared concern.
The rabbi wears a carefully trimmed brown beard, wire-rim glasses and a commiserative expression. "Iyal's disabilities began to define our family's existence," he told me.
We sat in their high-ceilinged kitchen in a suburb of Atlanta on a summer Sunday morning; sliding glass doors opened onto a redwood deck filled with flowers and bird feeders. Morasha, thirteen, cute and sporty, packed her pool bag for an outing with friends; Iyal, thirteen, played a video game alone in the den but checked in frequently, anxiously, to know when his mother's crumb cake would come out of the oven. Earlier that day, Morasha played the video game with her brother. In a realm in which they searched side by side for hidden treasure, there was peace between them. "Iyal, push the green button!" she ordered. "The green button! Iyal, steer the canoe!" He obeyed. But outside this virtual kingdom, he doesn't listen to her, especially when she begs him to leave her alone or to get out of her room. It's stressful for a young teenage girl to navigate middle school with a large inept brother lumbering through the same hallways, the target of gibes and ridicule.
For more than a year after Iyal's third birthday, child psychiatrists, pediatricians, and specialists examined him without reaching consensus. Finally he was seen by Alan G. Weintraub, a developmental pediatrician, who noted his small head, the small and widely spaced eye openings, the extra skin folds close to the nose, and the way the middle area of his face appeared flattened. When the little boy became anxious during the exam, he began making animal noises and tried to escape. He detected scary themes in benign pictures. The doctor's conclusion was a blow the Winokurs had not seen coming: Iyal's brain and central nervous system had been severely, irreversibly damaged in utero by the teratogen of alcohol, resulting in an incurable birth defect. Though alcohol consumption by Iyal's birth mother could not be documented, the available evidence pointed to fetal alcohol syndrome, F.A.S., the most extensive form of the range of effects known as fetal alcohol spectrum disorders, or F.A.S.D.
It is well known that maternal drinking can lead to neurobehavioral and growth impairments in a fetus, as well as malformations in the face, palate, joints, kidneys, genitals, heart, brain, and nervous system. There is no known safe window during pregnancy for alcohol consumption of any kind or quantity, according to Dr. Jacquelyn Bertrand of the National Center on Birth Defects and Developmental Disabilities of the Centers for Disease Control and Prevention. The U. S. surgeon general calls for total abstinence. C.D.C. estimates of the prevalence of F.A.S. in America range from 0.2 to 1.5 children per thousand live births, but this data may represent chiefly those children whose facial dysmorphia render them recognizable; the rest may appear physically typical while contending with hidden neurological damage.
It's possible that as many as one in one hundred children are born with some exposure to fetal alcohol. A.D.H.D., learning disability, or mental illness are just a few of the accompanying disorders that may be diagnosed instead. Adults who presented symptoms before the syndrome's description in the U.S. medical literature in the 1970s may never have been given an accurate diagnosis.
Iyal Winokur was intellectually impaired and at high risk for a range of secondary disabilities, including poor judgment, impulsive behavior, social isolation, limited academic achievement, unemployment, drug and alcohol abuse, imprisonment, mental-health problems including suicidal ideation, inability to live independently, and inappropriate sexual behavior. Few medications or therapies could be recommended as truly effective.
At seven, eight, and nine years of age, Iyal often babbled a nonstop stream of senseless chatter and baby talk. He required a full-time aide at school and his mother's undivided attention in the house. Donnie put aside her production career. Harvey juggled the needs of hundreds of congregants while facing escalating mayhem at home. But if their friends wondered what their lives would have been like if they hadn't adopted Iyal, the Winokurs would have reacted with horror. "It's unbearable to imagine our child growing up without us," Donnie says. "We never considered dissolving the adoption! We fell in love with our son." Still, she admits: "Staying in love with him has been trickier. People with brain injuries aren't able to reciprocate love in the ways you expect. You're struggling with this cluster of emotions toward your child—love, but also anger, bewilderment, resentment, frustration, and yearning."
As Donnie found her footing in the parallel universe of special-needs families, she discovered that a nonprofit service-dog agency in rural Ohio placed autism assistance dogs with children. Could a service dog help Iyal? "Are you kidding me?" cried her husband. "We don't need a dog!" He felt that one more howl raised under their roof, one more living creature whining for attention, one more source of strife between the children would push him beyond endurance. "No, Donnie. It's too much. I couldn't take it."
"This could be the help we need," she persisted.
"A dog?" Harvey said. "Forget about it, please. It's me or a dog."
Karen Shirk operates a dog-training school in Xenia, Ohio, a charming antebellum village flattened twice by tornadoes. Dressed in baggy jeans and a man's white T-shirt, swaying deeply as she walked, breathing through the metal button of a tracheotomy tube, she led me into her office at the far end of a brick building that once served as the local V.F.W. Hall. We waded into a crowd of bouncing ecstatic Papillons—toy dogs whose wide, silky ears inspired the breed's name, the French word for "butterfly." Though she stepped away only a moment earlier, the dozen little dogs rejoiced as if they'd feared never seeing her again: some spun in excitement, others leapt onto her desk, and one tap-danced along the computer keyboard. They raised their pointed little faces and emitted high-pitched yips of hallelujah. When Shirk, who is forty-nine, reached her desk chair, they settled on the floor at her feet, folded up their ears like kites and watched her. When she laughed, they took out their ears and waved them around.
As a young woman, Shirk pursued a master's degree in social work and held a full-time job with cognitively impaired adults. She felt, she says, "carefree," until the day she collapsed in respiratory distress and was rushed to an emergency room. Hospitalized for months, she received the grim diagnosis, at age twenty-four, of myasthenia gravis, a rare neuromuscular disease. She left the hospital only to become a respirator-dependent patient in need of constant care.
"Why don't you get a service dog?" a new nurse asked Shirk, six years into her illness. Supine in front of the TV, Shirk seemed unaware of the hour, day, or month. A dog could offer mobility assistance, the nurse said, like opening a drawer and bringing clean socks. She seemed also to suspect that a dog might jump-start the life of this sad and lonely patient.
"How could I take care of a dog?" Shirk rasped. "I can't even take care of myself."
"You could take care of a dog," the nurse said.
Beginning in 1992, Shirk applied by mail to service-dog agencies around the country. Every program rejected her. "They're not going to give a dog to a respirator-dependent individual who will never lead a productive life," she told the nurse. Finally she won a spot at the bottom of a waiting list. In 1994, a trainer visited to prepare for the placement of a golden retriever. Shirk began to feel strangely hopeful. But then the agency sent a letter instead of a dog: "Our guidelines prohibit the placement of service animals with people on ventilators."
"I didn't care if I lived or died after that," she told me, placing a fingertip over the metal knob in her throat to enable speech, in a voice that is winningly husky. "All I could see ahead of me was a long, slow death. I started stockpiling morphine."
"Karen," the nurse said, "get out of this bed, and let's go get you a puppy."
Feebly, she dressed herself in bed and crumpled forward into a wheelchair. The nurse drove her to see a litter of black German shepherd pups, the breed of Shirk's childhood dog, and there she found Ben. "I didn't leap back into life with Ben so much as inch back into it," she said. The puppy had to be taken outside, and to obedience classes. Wherever he frolicked, strangers greeted the tubed-up woman in the motorized wheelchair, as they did not when she was alone. It was a lesson Shirk wouldn't forget.
When year-old Ben graduated from puppy classes, he was a gorgeous animal with a shiny coal-black pelt, orange-flecked brown eyes, and a feathery tail. He wasn't a complex thinker or problem-solver. But he was smart, and she loved him.
She acquired a wheelchair-adapted van and commuted to a dog-training school in Columbus, where Jeremy Dulebohn, a crew-cut man from rural Ohio, taught Ben the basics of mobility work: to open and close doors and drawers; to hand Shirk's wallet to retailers and return it with change to her lap; to brace her for balance as she moved from bed to wheelchair and back; and to remove her shoes, socks, and jeans at bedtime. "When I asked for water, Ben opened the refrigerator and brought me a bottle," she told me. "When I asked for laundry, he pulled my clean clothes out of the dryer, put them in a basket and dragged them over."
Dogs evolved over at least fifteen thousand years to know and like humankind as well as, or better than, we know and like ourselves. Like many German shepherds, Ben was a one-person dog. He seemed to watch Shirk closely when she returned to her apartment following open-heart surgery. "I had a daytime nurse but was alone at night," she says. "I was on a morphine pump and—though I didn't realize it—a deadly combination of drugs. I slipped into unconsciousness." When the phone rang, Ben waited— as he'd been trained to do—for Shirk's command to answer it rather than to let it ring into the answering machine. But that night, with his owner failing, Ben picked up the receiver without her command, dropped it on the bed and barked and barked. It was Shirk's father. Realizing something was wrong, her father hung up and called 911. The rescue team told Shirk she wouldn't have lived through the night.
With Ben at her side, Shirk became manager of a day care facility for cognitively impaired adults. Gaining in strength and confidence, with new medications allowing her to come off the ventilator during the day, she wondered how many other people were being told they were "too disabled" to get a dog. "I could start my own agency," she thought. "I could place four or five dogs a year with people rejected by the big agencies." She mentioned the idea to coworkers and almost instantly heard from a couple seeking a mobility dog for their twelve-year-old daughter, who'd been paralyzed by a spinal stroke. Their impression was that no service-dog agency worked with children.
Shirk studied the Americans With Disabilities Act governing service animals and found no legal impediment to placing a dog with a child if a parent served as cohandler. In October 1998, she assembled a board and founded 4 Paws for Ability, a nonprofit corporation. She rescued Butler, a German shepherd mix, from a shelter; hired a trainer to prepare him for mobility work with the twelve-year-old; and became a pioneer among service- dog agencies. "People started calling from all over to ask, Am I too young? Am I too old? Am I too disabled? Am I disabled enough?" she says. "I said, 'If your life can be improved by a dog, and if you and your family can take good care of a dog, we're going to give you a dog.'"
A couple with a ten-year-old son on the autism spectrum called 4 Paws. This was new ground. Placing dogs with adults with "invisible disabilities" like post-traumatic stress disorder or seizure disorder was the cutting edge of service-dog work; it hadn't been widely tried with children. Patches, a rescue-hound mix, became one of the world's first dogs trained for autism assistance for a child.
In 2001, Dulebohn joined 4 Paws full time as training director. Today he oversees an expanding staff of trainers, vets, groomers, and dog-walkers. The dogs are a mix of shelter dogs, donated dogs, and puppies bred in-house, and every one gets five hundred hours of training, well beyond the 120-hour industry standard. "Any breed can become a service dog," says Dulebohn, who is thirty-seven, "but, over time, we found that roughly 70 percent of Labradors, golden retrievers and German shepherds graduated from our program, while only about 2 percent of other breeds made the cut." It costs $22,000 to train a 4 Paws dog; clients are asked to contribute $13,000 to the organization, with the difference made up through charitable donations and grants. To date, 4 Paws has placed more than six hundred dogs.
For socialization, Dulebohn places foster puppies with local families, and for basic obedience training he places them with specially chosen inmates in regional prisons. "Convicted murderers cry when it's time to give back their dogs," Shirk says. "But we give them another one." Since most 4 Paws dogs go to children—and children want playmates more than they want therapists and trackers—Dulebohn asks the prisoners to teach their pups tricks, including "Roll over," "Speak," "Gimme five," and "Play dead."
"I learned with Ben that a dog helps you make friends," Shirk says. "We place dogs with kids in wheelchairs, kids on ventilators, kids with autism, kids with dwarfism, kids with seizure disorder and cognitive impairments; but if your dog does tricks, other kids want to meet you. Kids will ignore your disability if you've got a cool dog."
One prisoner with a sense of humor returned a dog who—upon hearing the command "Play dead"—lurched, as if shot, staggered across the floor, knelt, got up, buckled, whined piteously, and then dramatically collapsed.
Cool dog. Lucky kid.
In 2007, a phone call came into 4 Paws from an Atlanta mother of a boy with special needs. Iyal Winokur's doctors had tried twenty different medications without lasting success. Iyal was nine; his I.Q. was eighty and falling; his language was primitive. He got hooked on bizarre thoughts and repeated them endlessly. He still suffered from night terrors and bed-wetting. Sometimes Iyal touched his mother's shirt, sniffed his fingers, and tried to wipe off the smell. Aware that a majority of individuals living with fetal alcohol syndrome also fight mental illness, his parents feared impending schizophrenia or psychosis.
"Do you place dogs with children who have fetal alcohol syndrome?" the mother asked Shirk.
"Never heard of it."
Donnie Winokur, who had by then founded the Georgia affiliate of the National Organization on Fetal Alcohol Syndrome, explained with rapid and precise diction.
"Is your son likely to verbally abuse a dog?" Shirk asked.
"Well, yes," Donnie had to admit, at a reduced speed.
"Is he likely to try to physically abuse a dog?"
"It's not impossible," she said, now certain of rejection.
"O.K.," Shirk said. "We'll need a doctor's prescription and we'll need video. We want to see your son every day, everywhere—getting up in the morning, eating breakfast, getting in a car, at school, at bedtime. We need to hear his noises and see his tantrums."
"You'll give us a dog?" Donnie gasped.
That night at home, Harvey gasped, too. "Thousands of dollars for a dog?" he cried. "Instead of for a nanny, or respite care, or a private school? Does that make sense? A dog's not going to mean anything to Iyal."
"It might."
"You're talking about a dog with a vest like a seeing-eye dog? It will be embarrassing to go into public like that!"
"It's already embarrassing to go into public with Iyal, especially for Morasha."
"But a dog in a vest will make him seem so disabled."
"A dog in a vest will tell people that he acts like this because he's living with a disability."
She wore him down. He loved her, trusted her judgment, and knew she wasn't going to give up.
In January 2008, Donnie, her father, her first cousin, and her children drove to Xenia for a ten-day class with other special-needs families and their new dogs. In the parking lot at 4 Paws, parents carried, coaxed, dragged, pushed, chased, and wheeled their children toward the front door. A circle of threadbare sofas, sunken love seats, and canvas sports chairs surrounded a training area in the former V.F.W. social hall. In an inner room, pet crates and pens held dogs of all ages and sizes—two hundred dogs are in training at all times—while dog-walkers, vets, and groomers came and went through the side door. The building smelled fragrantly of dog, with undertones of ammonia.
For children with autism or behavior disorders, dogs were trained in "behavior disruption." For children with seizure disorder or diabetes or respiratory issues, dogs were trained to alert the parents at the onset of an episode, and there have been a few able to predict the medical incidents six to twenty-four hours in advance. (How they do this is something of a mystery.)
"The dogs are nonjudgmental," Dulebohn tells each class. "You've got a kid who's picking his nose? The dog isn't thinking, That is gross. He's thinking, Save one for me! Or your child has disappeared and you say: 'Find Jeffrey.' The dog isn't thinking, Jeffrey's in danger! The dog thinks: _Game on!_ "
About 10 percent of 4 Paws placements fail. "Some fail because parents weren't prepared for how much extra work a dog would be," Shirk says. "They can barely get themselves and their special-needs child out the door; adding a dog feels overwhelming." Others fail because it's not a good match. A family's video may not have reflected the severity of behavior. "A child looks gentle on his video, so we place a soft dog," Shirk says. "Then the child's violent meltdowns scare the dog, and he starts avoiding the child." Dulebohn and Shirk try to discourage clients from engaging in "the Lassie syndrome": the belief that a devoted, sensitive, and brilliant dog will gallop into their lives and make everyone feel better.
And yet, sometimes, that's what families get.
Dulebohn matched Iyal with Chancer, a big, good-humored golden retriever with "high self-esteem" who wouldn't be hurt or insulted by the boy. Chancer was originally purchased as a puppy from Mervar Kennel in Youngstown by a family that lost interest in him; he was returned, after a year, overweight, matted, undisciplined, and lonely. Knowing that 4 Paws successfully placed Mervar dogs in the past, Judy Mervar donated Chancer. Like all 4 Paws dogs, he was shown kindness and affection in the course of his training, but he was not offered a long-term close human friendship. "Once dogs have been matched with families, we pet them and love them, but we don't give them that intense, 'I love you so much, you're my baby' kind of one-on-one attention," Shirk says. "We don't take them home with us at night. Every one of our dogs wants that closeness, is primed for that closeness; but we want them to find it with their families, not with our staff." Chancer didn't know what he was missing. But his trainers knew. "Chancer," Dulebohn says, "really needed a boy."
The dog's deeply encoded desire to attach to humans came alive when he was introduced to the Winokurs. A shaggy, tawny giant, he panted and wagged with pleasure. Something similar, on the human side, was sparked, too. Morasha dropped to her knees and embraced Chancer's big neck. Donnie felt like doing the same. "Hi, hi, hi good boy," she cooed, stroking his broad handsome head. Iyal was briefly interested but then wandered off.
They were in a hard stretch with Iyal. He was throwing tremendous rages daily, and instantly did it here. "I'm so sorry," Donnie said, mortified, unable to budge the explosive boy from the dog-training circle on the very first morning. But she was among friends; special-needs parents all, they patiently waited for Iyal's tantrum to die down. Unfortunately, on a lunch break in town, Iyal lost control again and threw a fit in the drivethrough lane at Wendy's. He crossed his arms, sat down hard and bawled. The backed-up drivers looked at Donnie with less empathy than had the 4 Paws parents. Shirk says, "Iyal really needed a dog."
At the conclusion of the second day's class, the families were invited to keep their dogs overnight for the first time. At the hotel, Donnie's cousin took Chancer outside for a walk while Donnie supervised Iyal and Morasha in a hot tub in the indoor pool area. "When they came back from their walk," she says, "Chancer looked around, and then broke away! I thought: Oh, my God, he's escaping. We're going to lose him. He streaked past everybody in the solarium and took a flying leap into the hot tub. He was saving Iyal!"
Chancer had not been trained to do water rescue. Why he leapt unnecessarily into a hot tub is hard to know. Shirk thinks that after thirty-six hours, Chancer had bonded to Iyal. The reverse, though, may not have been true yet. Part of the havoc wreaked by alcohol on a child's brain is to scramble the emotional pathways. The routes to friendship, fun, intimacy, and love are underdeveloped or buried under cognitive roadblocks. But Iyal's burst of laughter when the big yellow dog came sailing through the air and clumsily exploded into the hot tub was the greatest sound his mother had heard out of him in a long time.
The morning after Chancer's first night in the house outside Atlanta, the Winokurs woke up after a full night's sleep for almost the first time since 1999. They looked at each other in semihorror: was Iyal still alive? They found him snoozing beside the big yellow dog, the latter hogging the mattress. Since Chancer's arrival in the house, they've rarely been disturbed in the night. Iyal may still wake up, but he's evidently reassured by the dog's presence and returns to sleep.
"The moment he walked in the house with Chancer, I knew something had changed," Harvey says. "I could feel it instantly, the magnetism between Iyal and the dog. . . . Chancer was an emotional and physical anchor for a kid who was pretty lost in the world."
When Iyal is distressed, Chancer is distressed. Unlike Iyal, Chancer knows what to do about it. Iyal rages by crossing his arms, sitting down hard on the floor, and screaming and kicking. Chancer unknots the crossed arms by inserting his wide muzzle through the locked arms from below, opening them up and nuzzling toward Iyal's face, licking and slobbering, until the boy's screams turn to tears of remorse or to laughter.
Chancer sometimes heads off tantrums before they start. If a tutor or a therapist has worked with Iyal in the dining room a bit too long, Chancer moves between the visitor and the boy, clearly relaying: We're done for today. From two floors away, he will alert, flicking his ears, tuning in. Sensing that Iyal is nearing a breaking point, he gallops up or down the stairs to find him, playfully head-butts and pushes him down to the floor, gets on top of him, stretches out and relaxes with a satisfied groan. Helplessly pinned under Chancer, Iyal resists, squawks, and then relaxes, too. The big dog lies on top of the boy he loves, and seals him off from the dizzying and incomprehensible world for a while.
When I ask Dulebohn about Chancer's preternatural sensitivity, he says: "We trained Chancer to disrupt tantrums. Being able to prevent tantrums is coming from subtle training within the family. He may be reading Donnie's body language or facial expressions, or he may be smelling some chemical changes in Iyal or hearing some noises from him that predict a tantrum. He feels rewarded when Iyal stabilizes."
Donnie says: "Lately, and this is the best yet: if Iyal gets distressed, he goes to find Chancer, and he curls up next to him. He picks up Chancer's big paw and gets under it." It's the closest the boy has come to mood self-regulation.
Two weeks after Chancer's arrival, Iyal startled his parents by using multisyllabic words. He was suddenly possessed of opinions, judgments, and important questions, and he expressed them.
"B.C., Before Chancer," Donnie says, "which is how we refer to our life then, Iyal echoed Morasha word for word. It drove her nuts. Every morning I asked, 'Do you want to take your lunch today or eat lunch at school?' and every morning Iyal parroted whatever Morasha said. If she said, 'School,' he said, 'School.' With his frontal-lobe damage, decision-making like that was difficult for him. One morning, A.C., when I asked about lunch, Morasha said, 'School,' and Iyal said, 'I'd rather have lunch from home than a school lunch.' It was a more sophisticated expression of his thoughts than we'd ever heard.
"B.C., driving in the car with Iyal, if I turned down an unfamiliar route, he might say, 'What happened?' A.C., sensing I'd taken a wrong turn, Iyal asked: 'Were you distracted by Chancer and that's why you made a bad turn?' That showed an understanding of cause and effect, and a high-level word choice.
"B.C., Iyal never mentioned his disability, although we have educated him about it. A.C., he suddenly started asking things like, 'Did Chancer's birth mother drink alcohol?' and 'Does Chancer have a boo-boo on his brain?' and 'Why did my birth mother drink alcohol?'"
Before Chancer, Iyal didn't seem to possess "theory of mind," the insight, usually achieved by age four, that other people have points of view different from your own. But Chancer has inspired him to think about what Chancer likes and what Chancer wants and what Chancer thinks. Only since the dog's arrival has Iyal shown sheepishness or regret following a tantrum, signaling a new awareness that his outbursts may affect others. "Is Chancer mad at me?" he asks his parents. "Mommy, tell Chancer I love him, O.K.?"
"The sad flip side of 'theory of mind,'" Donnie says, "is that Iyal is deathly afraid that if he misbehaves too much, Chancer will want to be someone else's dog. We'll be in a park, and he'll tell me that Chancer is smiling at another kid and wants to be his dog."
The science behind Iyal's cognitive leaps is still in its infancy. Alan M. Beck, the director of the Center for the Human-Animal Bond at Purdue University's College of Veterinary Medicine, is among those intrigued by it. "There is a real bond between children and animals," he told me. "The younger the child, the greater the suspension of disbelief about what an animal understands or doesn't understand." According to Beck, more than 70 percent of children confide in their dogs, and 48 percent of adults do. "The absolutely nonjudgmental responses from animals are especially important to children," he says. "If your child with F.A.S.D. starts to misbehave, your face may show disapproval, but the dog doesn't show disapproval. The performance anxiety this child may feel all the time is absent when he's with his dog. Suddenly he's relaxed, he's with a peer who doesn't criticize him."
The hypothesis is that the sudden drop in Iyal's anxiety level—the sudden decrease in his hypervigilance, the lowering of his cortisol level, and the disarming of the fight-flight physiology—frees up cognitive energy that he can use for thought and speech. "A child with a disability feels freer not to suppress his ideas and behaviors when he's with his dog," Beck says. "There's a level of trust and confidentiality he has with no one else. And it's a good choice: the dog is his true confidant and friend."
Chancer has not cured Iyal.
"From the moment Iyal wakes up in the morning, there's tension in the house," Donnie says. "He has neurological and psychological damage Chancer's paws can't reach. But Chancer mitigates the disability. It's like we have a nanny."
Last fall the Winokur family wrestled with the likelihood that Iyal was being bullied at middle school. "Some boys told me to hump a chair," he reported to his mother and psychiatrist a few days into eighth grade.
"Hump a chair?" Donnie said. "I'm not sure what that means,"
Iyal stood to re-enact it.
"Look at Chancer," murmured the psychiatrist to Donnie. As Iyal engaged provocatively with a chair, Chancer rose, distressed. Whining, he tried to block Iyal's lunges.
On another day, a distraught Iyal told his parents that the boys said: "Go kiss that boy or we'll hang your dog."
"This is the classic setup for people with impaired judgment," Donnie says. "They're at risk of being exploited criminally and sexually. They can become both victim and perpetrator." It may not be classic bullying either. "Iyal may have pursued those boys," Donnie says. "He desperately wants friends. He doesn't understand personal space or social distance. He might have been annoying them, and they reacted."
The school principal was instantly responsive; his staff spoke to the other boys. "But Iyal keeps talking about it," Donnie says. "It's hard for Harvey and me to know if the bullying is still happening or if Iyal has just fixated on the trauma of it. Past, present, and future get confused in his mind."
Chancer doesn't accompany Iyal to school because the boy can't take the reins as Chancer's handler. "He can't even take Chancer for a walk around the block," Donnie says. "He might drop the leash, and Chancer might interpret that release as permission to track a hamburger. Chancer's an amazing service dog, but he is a dog, and he loves meat."
If Iyal ordered Chancer to do a wrong or dangerous thing, or to join him in reckless behavior, would Chancer recognize that they were transgressing? Would Chancer disobey Iyal? "When a dog puts a vest on, it changes his persona," Donnie says. "He knows he's working. In the service- dog world, they call it the halo effect. Guide dogs for the blind are trained in 'intelligent disobedience' for dangerous situations, like traffic. But I don't know if a dog can reason between right and wrong."
With every passing year, the challenges to Iyal's safety, and to the well-being of those around him, multiply. Iyal's attempts to touch his mother inappropriately are escalating; the Winokurs fear that soon the principal's office will be calling them, rather than vice versa. "Harvey and I feel like we're sitting on a volcano," Donnie says. "Iyal is a thirteen-year-old who functions cognitively, emotionally and socially like an eight-year-old. That gap will widen. He will never catch up to his chronological age. And few outsiders perceive the difference between 'neurological noncompetence' and 'behavioral noncompliance,' in other words, that Iyal's doing the best he can."
Iyal will never drive. He will never hold a regular job. He doesn't understand money or time. Experts say that the transition from adolescence to adulthood is particularly difficult for individuals with F.A.S.D. And Chancer won't be around forever. For as long as they live, the Winokurs hope to make sure there is a 4 Paws dog at Iyal's side; for now, they cannot conceive of a life without Chancer.
Chancer doesn't know that Iyal is cognitively impaired. What he knows is that Iyal is his boy. Chancer loves Iyal in a perfect way, with an unconditional love beyond what even the family can offer him. Chancer never feels disappointed in Iyal or embarrassed by Iyal. Beyond cognitive ability or disability, beyond predictions of a bright future or a dismal one, on a field of grass and hard-packed dirt, between the playground and the baseball diamond, you can see them sometimes, the two of them, running, laughing their heads off, sharing a moment of enormous happiness, just a boy and his dog.
_Lis Harris_
Lis Harris is a writer and critic who worked for more than two decades as a staff writer at The New Yorker. She is the author of _Holy Days: The World of a Hasidic Family_ ; _Rules of Engagement: Four American Marriages_ ; and _Tilting at Mills: Green Dreams_ , _Dirty Dealings, and the Corporate Squeeze_.
Her writing has also appeared in publications such as _The New York Times_ , _World Policy Journal_ , _Guernica, Du_ , and _The Wilson Quarterly_.
Harris is a two-time Woodrow Wilson Lila Acheson Wallace Fellowship recipient, and was awarded grants from the J. M. Kaplan Fund, the Fund for the City of New York, the Rockefeller Fund and the German Marshall Fund. Harris is a graduate of Bennington College and has lectured at Yale University, Cornell University, Dartmouth College, New York University, and Wesleyan University, among others.
_Holy Days_ is one of her best-known works. In his _New York Times_ review of the book, Christopher Lehmann-Haupt wrote: "What is special about Miss Harris is her combination of openness and skepticism toward her subject. These qualities in various combinations enable her to ask the hard questions without putting off the people who had taken her into their lives."
Harris is a full-time writing professor at Columbia University's School of the Arts. Her forthcoming book is titled _Dream-Land: Three Generations of a Palestinian Family and an Israeli Family in Jerusalem_. She lives with her husband, the painter Martin Washburn, in New York City, where she was born and raised.
Holy Days
A rare glimpse inside the Hasidic Jewish community of Brooklyn, New York—how an ancient and traditional culture is adapting to life the modern age.
_EDITOR'S NOTE: "Holy Days" was first published by The New Yorker in three installments. The following is an excerpt from the second installment._
Some time after my visit to the new _mikvah_ with Sheina, and after the birth of my second child, I return to pay a visit by myself. It is about a forty-five-minute subway ride from lower Manhattan, where I live, to the Kingston Avenue I.R.T. stop. After Park Slope, mine is the only white face in the subway car. It is sundown, the prescribed time for taking a _mikvah_ , when I surface, and, as usual, there is a large group of bearded, black-hatted, sombrely dressed men in front of 770, which is just across the street from the subway exit. The neighborhood still seems as dreamlike as it did on my first visit, but the faces above the beards have begun to look more individualized to me; the perceptual trick seems to be to "read" them from the nose up.
As I walk down Union Street toward the _mikvah_ , I find myself trailing a black teen-age boy who is bopping along the sidewalk with an enormous radio blasting out reggae music. Formalizing the occasion, I imagine that the music is a kind of fanfare for the ritual I am about to experience for the first time, and that the boy and I form a procession. The farther we get from Kingston Avenue, the more deserted and run-down the street becomes. Across from the tan brickface _mikvah_ , which glows faintly in the fading light, is a large, deserted apartment house with broken windows. There are no windows at all in the new _mikvah_ building. A fancy brown canvas canopy arches above the entranceway, giving the place the appearance of an elegant salon. I am buzzed in, ascend a flight of stairs, and give eight dollars to a Slavic-looking attendant, who tells me that she can't remember ever seeing me before. Many women both shower and bathe at the _mikvah_ in preparation for the ritual immersion. I explain to the attendant that I have already bathed at home, so she leads me to a dressing room with only a shower in it. The shower has a sliding frosted-glass door. There is burgundy-white-and-gold Art Deco paper on the walls, and a bevelled diamond- shaped mirror over a Formica dressing table. There is also a wig stand, a bright-red Clairol Son of a Gun fourteen-hundred-watt hair dryer, a long cream-colored terry-cloth bathrobe, and a pair of multi-colored rubber sandals. The attendant shows me a tan intercom and tells me to pick it up when my preparations are completed, and let the _mikvah_ lady know that I am ready.
As I close the door, I notice several women flitting down the corridor. The _mikvah_ is considered an extremely private ritual, and there is no socializing here—none of the affectionate banter so prevalent in feminine Hasidic society. Women generally go to the _mikvah_ alone, and even Sheina, who has always been eager to act as my cicerone on visits to landmarks of Jewish Orthodoxy, did not offer to accompany me to this one. The lower half of the dressing room is covered with gleaming off-white tile. I had always imagined _mikvahs_ to be rather Oriental—vaporous, dim places, chaste seraglios. Nothing could be farther from my preconception than this sleek Swiss spa.
The toilette one makes in preparation for the _mikvah_ is elaborate. Having read about (and followed) the complicated instructions before I came, I am not surprised to see a small tray holding Q-tips, cotton balls, tissues, shampoo, baby oil, toothpicks, bleach, alcohol, soap, and Adwe New Fluoride Formula Kosher toothpaste. It is considered extremely important that the waters of the _mikvah_ touch every part of a woman's body. Any particle of matter, however minute, that prevents this from happening makes the immersion invalid. The kinds of matter that might—literally— gum things up have been spelled out in excruciating detail by rabbinical authorities; for example, "A splinter which protrudes from the skin, or even if it does not protrude from the skin but nevertheless is on the same level with the outer skin, _must_ be removed." Even bits of food that get caught in one's teeth are considered impurities, so a box of dental floss is provided. But, then again, the floss might get stuck, so taped to the little plastic box is the typed message "Don't floss if your teeth are tight together." Most of the people who come to the _mikvah_ already know everything there is to know about preparing for it. As a precaution, however, a lengthy checklist has been taped to the wall. Among other things, it suggests that "rabbinic advice should be sought for temporary fillings, root-canal work or capping in progress, nits in the hair, stitches, casts . . . unremovable scabs, unusual skin eruptions." It is further suggested that one ask oneself, "Have I cut finger and toe nails and removed dirt in crevices (bleach helps)? Removed all foreign bodies: false teeth, contact lenses, paint and makeup, nail polish, artificial nails, Band-Aids, bobby pins . . . ?" The list goes on, and suffice it to say that no crevice or orifice of the body is neglected. Scrubbing myself in the shower, I remember a woman who, when I told her of my visits to Crown Heights, commented on the "dirtiness" of the Hasidim, and I laugh out loud. The last words on the checklist are "Now you are ready for the great mitzvah of _Tevilah_ [immersion]." A French version of the list has been taped below the English one (many French-speaking Jews have been visiting the community recently), and, with characteristic Gallic grace, the French version adds a little coda to this: Now, it says, you are ready to perform the great _mitzvah_ of _Tevilah "avec joie el assurance._ " I pick up the intercom and call the _mikvah_ matron. I do not feel " _joie et assurance._ " I feel nervous. I find myself wishing my mother were here. Then one of the kindest, most benign faces I have ever seen appears smiling at the door, and my worries evaporate. The woman, who is middle-aged and wears a dark wig, tells me as she leads me through the climate-controlled corridor that her name is Brachah. _Brachah_ , of course, means "blessing." These Lubavitchers really know how to do things. I tell Brachah, as we enter a small, ivory-tiled, antiseptic-looking room, that I've never taken a _mikvah_ before. She folds her hands over her stomach and beams. "Well, then, we'll treat you like a _kallah_ [a bride]," she says, and proceeds to explain some of the basics of the ritual to me. Then she asks, enumerating the various items on the checklist, if I've remembered to do all of them. I haven't. I've forgotten to comb my wet hair, and I've forgotten my nose, which I now blow, rather showily. Then, after blotting my eyes with a linen cloth to make sure no mascara lingers on my lashes, Brachah leads me over to the _mikvah_. I take off the robe and stand expectantly in the chest-deep warm green water. Brachah tells me to keep my eyes and lips closed but not too tightly and to keep my feet and arms apart, so that the water will touch my whole body. When I go underwater, I curl naturally into the fetal position. When I come up, Brachah places a linen cloth over my head, and I repeat the _mikvah_ blessing after her. Then, the cloth having been removed, I go down two more times. The second time down, I see a little speeded-up movie of all the religious people I know performing this ritual. I think of all the generations of people I haven't known who have considered the impurities of the world dissolvable. My grandmother floats by, curled up, like me, like a little pink shrimp. I see her as she was in her very old age, senile and mute, curled up in the same position on her bed. The third time down, I think of my boys suspended inside me, waiting to join the world. I look up and see Brachah's smiling face through the water. I feel good. As I am climbing out, Brachah tells me that some people prefer to immerse themselves with their bodies in a horizontal position, and asks me if I'd like to try it that way. I try it, but find it less satisfactory. It's too much like going for a swim.
When I finish dressing, I find Brachah and the other attendant huddled together at the reception desk. I've told Brachah that I live in Manhattan and came over by subway. "We don't think you should go home on the subway," Brachah says. "It's really not safe to walk alone out there now." It's only a little after eight, but I take their word for it and call Black Pearl, a local car service. (No city radio cabs will come to Crown Heights. No yellow cabs cruise the neighborhood streets.) Five minutes later, a blue Chevrolet station wagon pulls up. I thank the women for their help, say goodbye, and climb into the car. The driver is a garrulous, handsome Haitian. Loud, monotonous music blasts out of the radio. Hanging from his rearview mirror is an air deodorizer, which fills the car with an overpowering sickly-sweet smell. Competing with the air deodorizer is the sharp scent of his after-shave, which he keeps in a kit on the front seat. At a long red light, he splashes a little extra on. My ablutions have made me feel tender, almost porous, and the harsh smells are overwhelming. Is this how Moshe and Sheina feel when they traffic with the outside world? The driver tells me that his company is often called upon to pick up women at that spot, which he seems to believe is a kind of shrine.
"You're Jewish, right?" he says, shouting to be heard over the music. "You have a lot of rules you have to follow?"
I hesitate. I am not up to any discussion.
"Yes, I'm Jewish," I say, as I roll the window down and point my nose toward the fresh air.
That evening, the Lubavitchers were having one of their _farbrengens_ , or gatherings, at the synagogue, and Sheina was eager for me to attend. The Rebbe is the only speaker at the _farbrengens_ , and he talks extemporaneously and for many hours. Some of these gatherings have been known to last all night. Sometimes they are announced days in advance; sometimes only a few hours' notice is given. When the word gets out that a _farbrengen_ is going to be held, everybody telephones somebody else, and in an astonishingly short time the whole community knows about it. I looked forward to the _farbrengen_ , which was to begin at nine o'clock. It would be my first chance to see and hear the Rebbe, and also to see the Lubavitcher community assembled in one place. The Rebbe spoke in Yiddish, I had been told, but in deference to the many visitors who came to hear him a simultaneous English translation was provided, and little transistor radios with earphones would be available at the door.
Walking along Eastern Parkway toward the synagogue, Sheina and I encountered Moshe's two sons, Mendel and Shmuel, who were both bachelors in their early twenties. Before we arrived on the scene, they had been having an animated conversation, which our presence somehow stifled. We all walked along together for a while, but in the absence of their father, who had gone on earlier, they seemed uneasy in Sheina's company and downright uncomfortable in mine, and when we arrived at the synagogue they bounded toward the men's entrance like puppies let off the leash.
All Orthodox synagogues have separate women's and men's sections— a practice that was initiated in ancient times by the priests of the Temple, some say, because they thought that otherwise the congregants would be distracted from the religious ceremonies. Whether the priests' plan proved to be effective then I cannot say, but the women's gallery of the Lubavitcher synagogue, which is upstairs, would surely have given them some second thoughts. About four thousand people were crammed into the synagogue—a space that, by a generous estimate, could have comfortably sheltered two thousand. About a third of those present were women, but the area consigned to them seemed hardly large enough to contain half their number. To enter the women's gallery—two adjoining windowless, airless balconies screened off from the men's section, downstairs, by black Plexiglas panels—one simply allowed oneself to be swept forward by a tidal wave of female Lubavitchers of all sizes, shapes, and ages flowing through the women's entrance. Once inside, the lucky found seats, but most either stood in the aisles or scrambled up onto the narrow back ledges of six stepped rows of wobbly wooden pews and teetered this way and that to find a spot that would afford a glimpse of the Rebbe, who had not yet arrived. There was a lot of jockeying for position, some fairly alarming shoving, and sporadic skirmishes inspired—it seemed— by territorial disputes. It was a scene that suggested equally the paintings of Hieronymus Bosch and the Times Square I.R.T. station at rush hour. Sheina quickly found her accustomed place—a minute stretch of bench in the left front corner of the gallery—and somehow tucked us both into it. From this vantage point, if you craned your neck down toward the fourinch span that separated the Plexiglas and the rim of the balcony you had a clear, bird's-eye view of the dais and, on it, a long table covered with a white cloth, where the Rebbe always spoke. Below, a black sea of male Hasidim swayed back and forth. Above, a tremendous din. Downstairs, little boys raced between the men's legs and around the dais, and little girls drowsed on their fathers' shoulders. Upstairs, innocent-looking teen-age girls and delicately powdered old ladies mingled with plump young mothers and innumerable babies. A large contingent of grandmothers, their arms moving like steam shovels, popped small morsels of food into the mouths of restless toddlers. The smell of sour milk and wet diapers was faint but pervasive. Knowing that the Rebbe's talks usually went on for hours, mothers came forearmed, and plastic bags filled with apple slices, pretzels, granola bars, and peanut-butter sandwiches were much in evidence. Ignoring the confusion, a few gallant souls, eyes tightly shut, attempted to pray. Sheina exchanged hugs and hellos with people nearby, and settled down with a bright-colored needlepoint wall decoration (a Hebrew alphabet) that she was working on.
After about half an hour, a regal white-bearded figure with a brisk, almost military gait and kindly, penetrating blue eyes entered the hall, and, except for the lip-smacking sound of sucking babies, it fell completely silent. The Rebbe. Like most of the other men, he wore a wide-brimmed black fedora, a long black coat, black trousers, and a white shirt. Unlike most of them, he wore a tie, also black. Taking a seat at the table, in front of several rows of whitebeards, he gazed placidly at his followers and began speaking, in the manner of a teacher picking up the threads of an earlier discussion with his students. No audience-grabbing anecdotes, no uplifting chorale. He spoke for more than four hours, in a voice that never wavered. The children dropped off one by one, but the rest of his followers listened raptly until the end. I listened carefully, too—to the translated version of his talk—but most of what was said went by me in a billowing gray cloud of words. The Yiddish accent of the translator was so thick, the translation itself so poor, and the nature of the discourse so elusive that the few intelligible statements I caught—"A truly free man is one who studies Torah," "We should dissociate ourselves from idolatry, from things that are foreign to Jews"—were unlinkable. The Rebbe, who is in his early eighties, spoke without notes, pausing for a few minutes every half hour to rest and to hold up a small wine-filled Dixie cup and toast his flock. The men also toasted him, holding paper cups aloft and waiting until the Rebbe looked and nodded in their direction before downing the wine. During these brief breaks, the men sang beautiful Hasidic songs that were thematically related to the Rebbe's discourse. Late in the evening, while scanning the expanse of pale faces downstairs during one such interlude, I happened to catch a glimpse of Moshe—hat pushed back on his head, eyes closed, mouth curved in a slight smile—singing. He looked as if he were crooning a lullaby to a baby.
I have attended many _farbrengens_ since then. Often, the Rebbe's talks have been well translated (the translations broadcast on cable television, for example, are models of clarity), the themes broad-ranging and graspable. But one has to learn how to listen to them, and for anyone with a secular background this means acquiring a religious one. Like so many Jewish texts, the Rebbe's talks are not aimed at the uninitiated, nor are they readily accessible to those who have a mere passing familiarity with the subjects touched upon—though his followers insist that the Rebbe can be understood on one level or another by everybody. For a non-Yiddish-speaking layman, the constant references to Talmudic sources, persons, and principles, the Yiddish phrases that no translator—however deft—could even attempt to Anglicize, and the nonsequential, didactic style of discourse can be daunting. During the first two or three _farbrengens_ I attended, I found myself constantly straining, but failing, to make out through the thicket of Hebrew and Yiddish references something more than an occasional "thus," "so," or "obviously." One year and many hours' immersion in Jewish texts later, I came away from the _farbrengens_ feeling that I'd understood about half of what the Rebbe was talking about.
_Robin Marantz Henig_
Robin Marantz Henig is a contributing writer for _The New York Times Magazine_ and has written nine books, most recently _Twentysomething: Why Do Young Adults Seem Stuck?_ , which she coauthored with her daughter, Samantha Henig. Her previous books include _Pandora's Baby: How the First Test Tube Babies Sparked the Reproductive Revolution_ and _The Monk in the Garden: The Lost and Found Genius of Gregor Mendel_. She also co-edited _A Field Guide for Science Writers_. Her articles have appeared in _National Geographic, Civilization_ , _Discover_ , _OnEarth_ , _Scientific American, Smithsonian_ , and the _Best American Science Writing_ anthologies. She writes book reviews and opinion pieces for _The New York Times_ and _The Washington Post_ and is science editor of _The Virginia Quarterly Review_.
Henig's book _Pandora's Baby_ was named Book of the Year in 2005 by the American Society of Journalists and Authors (ASJA) and won NASW's Science in Society Award that same year. The Monk in the Garden was a finalist for a National Book Critics Circle Award in 2001. In 2010, she received ASJA's highest honor, its Career Achievement Award, as well as a Guggenheim Fellowship. Her other honors include a fellowship from the Alicia Patterson Foundation, where she now serves on the board of directors; a Sloan Foundation grant; and a fellowship from the Marine Biological Laboratory in Woods Hole, Massachusetts, where she is currently co-director of its Science Journalism Program.
Henig lives in New York with her husband, Jeff, a professor at Teachers College, Columbia University.
The Last Day
A compelling profile of prominent psychologist and professor Sandy Bem, who voluntarily ended her life before succumbing to Alzheimer's disease.
Sandy Bem, a Cornell psychology professor one month shy of her sixty-fifth birthday, was alone in her bedroom one night in May 2009, watching an HBO documentary called "The Alzheimer's Project." For two years, she had been experiencing what she called "cognitive oddities": forgetting the names of things or confusing words that sounded similar. She once complained about a "blizzard" on her foot, when she meant a blister; she brought home a bag of plums and, standing in her kitchen, pulled one out and said to a friend: "Is this a plum? I can't quite seem to fully know."
Sandy was a small woman, just four-foot-nine and ninety-four pounds, with an androgynous-pixie look: cropped hair, glasses, and a wardrobe that skewed toward jeans and comfortable sweaters she knit herself in the 1990s. As she watched the documentary, her pulse thrumming in her ears, a woman on screen took a memory test. Sandy decided to take it along with her. Listen to three words, the examiner said, write a sentence of your choice and then try to remember the three words. Sandy heard the three words: "apple," "table," "penny." She wrote a brief sentence: "I was born in Pittsburgh." She said aloud the words she could remember: "apple," "penny." . . . The simplest of memory tests, and she had failed.
The next month, Sandy's husband, Daryl, from whom she had been amicably separated for fifteen years, drove her from Ithaca to the University of Rochester Medical Center for cognitive testing by a neuropsychologist named Mark Mapstone. Mapstone showed Sandy a line drawing and asked her to copy it, and then to draw it from memory ten minutes later. He read her a list of words and had her recall as many as she could. He gave her two numbers and two letters and asked her to rearrange them in a particular order: low letter, high letter, low number, high number. Thank goodness that last one wasn't timed, she thought to herself, as she focused all her mental energy on the task. She felt as gleeful as a kid who had earned a gold star when Mapstone said, "Yes, that's right."
After three hours, Mapstone gave a preliminary diagnosis: amnestic mild cognitive impairment. At first Sandy was relieved—he had said mild, hadn't he?—but then she caught the look on his face. This is not a good thing, Mapstone told her gently; most cases of amnestic M.C.I. progress to full-blown Alzheimer's disease within ten years.
When Sandy went back to the waiting room to meet Daryl, she was weeping uncontrollably. Between sobs, she explained the diagnosis and the inevitable decline on the horizon. She felt terror at the prospect of becoming a hollowed-out person with no memory, mind, or sense of identity, as well as fury that she was powerless to do anything but endure it. With Alzheimer's disease, she would write, it is "extraordinarily difficult for one's body to die in tandem with the death of one's self." That day at Mapstone's office, she vowed that she would figure out a way to take her own life before the disease took it from her.
Later that month, Sandy sat down in her upstairs study—painted a rich burgundy, as the rest of the house was, to make the sprawling old place feel cozy—and looked at her Mac desktop computer screen. She had some trepidation about her plan to keep a journal of her own deterioration. But she opened a new document, gave it a file name—"Memoir"—and began to type. She tried to describe the maddening capriciousness of "a mind that could be so alive one moment with thought and feeling building toward a next step and then someone erases the blackboard. It's all gone and I can't even reconstruct what the topic was. It's just gone. And I sit with the dark, the blank."
The prospect of mental decay was particularly painful for Sandy, whose idea of herself was intimately entwined with her ability to think deeply and originally. She was a pioneer in the field of gender studies: She created the Bem Sex Role Inventory in 1974, which assesses a person's traits along a traditional gender continuum; led Cornell's fledgling women's studies program from 1978 to 1985; wrote a groundbreaking book, _The Lenses of Gender_ , in 1993; published a memoir, An Unconventional Family, in 1998; became a licensed psychotherapist in 2000; and returned for a second term as the director of Cornell's renamed feminist, gender, and sexuality studies program in 2001. Friends and colleagues knew Sandy to be intensely observant, a person who spoke her mind with a bluntness that could be off-putting. Her best friend, Karen Gilovich, a psychotherapist who lived around the corner, said that one of Sandy's favorite conversational openers was: "I find myself thinking. . . ." You never knew what would follow. She once wondered aloud, for instance, where the line was between acceptable and unacceptable behavior between parents and their children. Would it automatically be wrong for a waitress who comes home exhausted to ask her young son to rub her feet? Massage her back? Cuddle her? "She was the most clear thinker I have ever seen," Karen said, "with the ability to cut to the core of any messy issue."
On June 22, her sixty-fifth birthday, Sandy returned to the University of Rochester for another three-hour consultation. This time it was with a senior neurologist, Charles Duffy, to evaluate not only her cognitive abilities but also her mood and functional status. At one point, Sandy told Duffy she didn't intend to live out her life with dementia. "I want to live only for as long as I continue to be myself," she said.
To her surprise, Duffy began to reminisce about his own life. His mother had had Alzheimer's, he said, and his time nurturing her through her decline profoundly shaped him as a physician, as a researcher, and as a man. He said that Sandy—who had spent her career examining and describing her own life with frank insight and clarity—might have a lot to contribute to the world just by experiencing her disease and giving others a glimpse of how it felt to have it.
Sandy was touched by Duffy's empathy and by his willingness to reveal the private details of his life. Others had stories like his, about people who watch their loved ones slip away, or people who go through that slipping away themselves, and are surprised to find a kind of grace in it: the Zen-like existence in an eternal now, the softening of hard edges, the glorification of simple pleasures. But Sandy knew that wasn't right for her. Not for a moment was she swayed by Duffy's arguments.
Over the next several weeks, Sandy told those closest to her about her diagnosis and her plan to end her life before she became incapable of doing so. She told her two adult children, Emily and Jeremy, both in their thirties, and a handful of others: Karen; Daryl's sister, Robyn Bem; and Sandy's sister, Bev Lipsitz, who lived in Oregon. No one in that inner circle tried to talk her out of suicide; they knew how fierce she could be once her mind was made up. All they asked was that she promise not to choose a method that would be particularly disturbing—using a gun or jumping off a bridge into one of Ithaca's famously beautiful gorges. Sandy had contemplated both of those options, but she didn't want that sort of death either. "What I want," she typed in her journal in an emphatic boldface font, "is to die on my own timetable and in my own nonviolent way."
But when would that be? Sandy knew that the Alzheimer's decline itself was predictable—it usually moves from mild (misplacing things, repeating questions) to moderate (being unable to learn something new, getting lost, failing to recognize loved ones) to severe (losing the ability to speak, swallow or remain continent; needing help with every function of day-to-day life). In the immediate aftermath of a diagnosis of amnestic M.C.I., however, she couldn't know how long each stage might last. She wanted to squeeze in as much intellectual and emotional joy as she could before she died, but she wanted to make sure she didn't wait too long. She needed to be engaged enough in her life to be able to end it.
In early July, Daryl drove Sandy back to Rochester to see Duffy. The doctor suggested that she start on Namenda, one of the few drugs approved by the F.D.A. to treat Alzheimer's. Namenda works by increasing the level of glutamate in the brain and is thought to interfere with cell death. Researchers say that by the time someone exhibits mild cognitive impairment, the brain has been degenerating for years, and the drugs, even if they can slow the decline, are too little too late. But doctors and patients hope that starting a drug soon after diagnosis might make a slight but tangible difference— slowing memory loss enough to forestall total dependence. Sandy quickly agreed, which surprised Daryl, because she usually avoided medication, other than a low dose of Prozac that she had been taking for years for depression.
That summer, Daryl began taking her to most of her doctors' appointments. Sandy could have done the driving herself, but it was good to have Daryl along, so he could recount each visit's details on the ride home as Sandy took painstaking notes. Besides, she was enjoying Daryl's company. Something about her diagnosis had opened up an emotional vein in him.
"Who is this wondrous Daryl?" she wrote in her journal that month, after a drive in which Daryl talked with unexpected empathy about Sandy's early years with a harsh and mercurial mother. She loved the tender, attentive husband who seemed to have emerged from the ruins of her diagnosis. "If some devil had asked whether I would be willing to buy Daryl's deeper self at the cost of my developing dementia," she wrote, "I would say NO without hesitation. But if it comes free with my unstoppable decline into hell, I'm thankful for the gift."
It was a bit like the earliest days of their relationship, back in 1965, when they met at Carnegie Tech in Pittsburgh. She was a senior psychology major; he was a new assistant professor of psychology. Just four months after being introduced by Sandy's roommate, they married.
They vowed at the time to share everything fifty-fifty: the housework, the child-rearing, the inevitable career compromises. For a while, this worked well. So well, in fact, that in 1972 they were featured in the inaugural issue of Ms. magazine, in an article titled "A Marriage of Equals."
The Bems were both psychology professors, at Stanford and then Cornell, and they traveled around the country giving tandem talks about society's creation of sex-role stereotypes. They were a slightly odd couple. Sandy was petite and not the least interested in fashion. Daryl was bigger, dapper, six years older and already a bit stooped, with a scholar's pallor, a kind face, and a courtly manner cultivated over his years of performing as a magician. (He would also come to be known, later in his career, for some controversial experiments involving ESP.)
They turned their politics into a way of life, raising their two children, Emily, born in 1974, and Jeremy, born two years later, in what they described as a gender-neutral way. "Many other feminist couples have experimented with egalitarian relationships and feminist child-rearing," Sandy wrote in _An Unconventional Family_. But few "have shared the details of their daily lives as exuberantly as Daryl and I." She talked about everything, in print and on the lecture circuit: letting Jeremy wear pink barrettes to kindergarten; driving Emily past the same construction site every day because a woman was on the crew; hanging a chart on a kitchen cabinet to let the children know which parent was "on duty" that week.
Despite their good intentions, though, the marriage grew strained. As their children went through adolescence, Sandy complained that she felt like a single parent, with Daryl not fully engaging with the family's needs. They both saw the paradox in their supposedly egalitarian marriage floundering in such a gender-stereotypical way. In 1994, when the children were nineteen and seventeen, the Bems separated.
After the split, Daryl acted on his attraction to men, a part of his sexuality that he never hid from Sandy. He liked to joke that on their first date, he told her there were three things she should know about him—"I'm a stage magician, I'm from Colorado and I'm primarily homoerotic"—and that she calmly replied that she had never met anyone from Colorado.
About a year after the separation, Daryl began a long-term relationship with a communications professor at Ithaca College. Yet he and Sandy never divorced, and he remained a frequent visitor to the big house in Cornell Heights where they raised their children. He ate dinner there a few times a week and stayed involved in the lives of Emily and Jeremy—even more involved, in a way, than when he lived with them. He also remained one of Sandy's best friends and one of her few close confidants. (She had a short-lived relationship with a woman soon after Daryl moved out and remained single after that.) Daryl wrote in the epilogue to _An Unconventional Family_ , which was published four years after they separated, "Sandy and I are still kin."
On a quiet Friday morning in November 2010, Sandy sat down with a mug of honey-ginger tea to read two books that Daryl had brought her. By this point, a year and a half after her amnestic M.C.I. diagnosis, she had progressed to what Duffy said was Alzheimer's disease. She had retired from Cornell, but she was doing well. She could still travel alone to familiar destinations, including Austin, Texas, where Emily was living. Jeremy had temporarily moved back home to be with her. She could read novels, even difficult ones like Cormac McCarthy's _The Road_. She played tennis, gardened, and went for walks around Ithaca with a handful of friends, most of them former colleagues from Cornell. She saw a few psychotherapy patients. One would later say that even though Sandy was having some trouble remembering words, "it didn't really matter. In a therapy relationship you're talking more about emotions—and in that regard, she didn't miss a beat."
The first book on her table that Friday morning was _Final Exit_. Sandy read it in the early 1990s when it was published; even then she was intrigued by the argument of the author, Derek Humphry, in favor of self-directed "death with dignity" for people who were terminally ill. The second was a newer book by the Australian right-to-die advocate Philip Nitschke called _The Peaceful Pill Handbook_. The pill in the title (though not literally a pill; it comes in liquid form) was Nembutal, a brand name for pentobarbital, a barbiturate that is used by veterinarians to euthanize animals and that is also used in state-sanctioned physician-assisted suicides. After reading about it, Sandy thought pentobarbital was what she was looking for. It was reliable, fast-acting, and—most important to her—a gentle way to die. It causes swift but not sudden unconsciousness and then a gradual slowing of the heart.
There could be complications, of course, like vomiting; Nitschke and his coauthor, Fiona Stewart, recommended taking an antinausea drug a few hours before taking the fatal dose to minimize that risk. They warned that pentobarbital is detectable in a person's body after death—but that didn't matter to Sandy. In fact, she preferred having people know that she died by her own hand.
One morning during one of Sandy's frequent phone calls to her sister in Oregon, she told her about the decision to use pentobarbital. Sandy had a special relationship with Bev, who was six years younger. When Sandy married Daryl, Bev was fourteen, and Sandy invited her sister to live with them rather than with their parents, whose unhappy marriage made it feel, as Sandy put it in her memoir, as if "chaos could erupt at any moment."
A year before Sandy received her diagnosis, Bev was found to have Stage 4 ovarian cancer. The sisters had discussed the fact that Oregon law allows people with terminal illnesses to take their own lives. Sandy now envied Bev's situation. "I don't think I have ever been as jealous about anything in my life as I am about this," she wrote in her journal shortly after she saw Mapstone. It was weeks before she could get past that jealousy and take Bev into her confidence.
But even if Sandy had lived in Oregon, her Alzheimer's disease would have precluded her from getting help in taking her own life. States that allow for assisted dying require two doctors to certify that the person has a prognosis of less than six months to live, and most people with Alzheimer's have no such prognosis. They also require that the person be declared "of sound mind," a difficult hurdle for someone whose brain is deteriorating.
On the phone that day, Sandy told Bev that pentobarbital was a controlled substance in the United States. She would have to write to one of the foreign suppliers listed in the book and hope for the best. Bev suggested an alternative: When the time came, she could request the drug from her own doctors in Oregon and then give it to Sandy. She didn't think she would need it herself—her cancer seemed to be in remission, and all she really cared about was eventually dying without pain. Like almost everyone else in Sandy's inner circle, Bev was devastated by the prospect of her sister's decline and death, but she tried to tune out her own anticipatory grief so she could focus on helping Sandy die the way she wanted to.
On Dec. 9, 2012, Sandy's daughter, Emily, and her partner, Julius Viksne, had a baby boy, whom they named Felix. Sandy went down to Austin, still able to make the familiar trip on her own. During the previous two years, her life had become more limited, but she continued to enjoy it. She spent time during Ithaca's abbreviated growing season gardening in her backyard, either alone or in the company of Karen or Daryl's sister, Robyn. Although she had abandoned writing in her journal, she could still read novels on her iPad, but nothing quite as complex as The Road anymore. She managed her day-to-day needs in part because she was such a creature of habit. She ate almost the same thing every day: a bagel for breakfast, a sandwich for lunch, a piece of salmon for dinner, mugs and mugs of tea throughout the day. Her freezer was always stuffed with ten-pound bags of almonds, so she could roast them by the handful and sprinkle a few onto chocolate frozen yogurt for her nightly treat. By then, she was also taking a second F.D.A.-approved drug, Aricept, which inhibits the chemicals that break down acetylcholine, a neurotransmitter linked to short-term memory; people with Alzheimer's often have lower concentrations of acetylcholine in their brains. Daryl wasn't sure that either drug was making a difference. It was impossible to tell without knowing what Sandy would have been like without them.
Becoming a grandmother was never something Sandy had cared much about. But when Felix was born, she was thrilled. He was in the neonatal intensive-care unit when she arrived in Austin; doctors had detected a bacterial infection in his urine and were administering antibiotics. Sandy sat in a rocking chair alongside the bassinet, and Emily handed the infant to her, naked except for his diaper, the IV port in his tiny hand capped off until the next infusion. She gazed down at her grandson, placid and perfect. She cooed and babbled. For weeks afterward, she talked about those first moments holding Felix. "I don't know what I was saying or what I was doing," she would say. "But he just looked into my eyes."
Emily was surprised to see her mother so at ease in the traditional role of Felix's _bubbe_ (Yiddish for "grandmother"). As a parent in the 1970s, Sandy turned every interaction with her children into a political act. During story time, she would go through their picture books with a bottle of Wite-Out and a Magic Marker, changing a hero's name from male to female, revising plot lines, adding long hair or breasts to some of the drawings. Story time was a different experience with Felix. Sandy would cuddle with the baby and turn pages. If she couldn't remember the word for "zebra" or "lion," she wouldn't fuss about it. "Oh, it's some animal," she would say.
She told Emily that her "new brain" might actually make her better suited to being a grandmother than her focused, hyperanalytical "old brain." She seemed to have found a way of being that she liked, content to sing silly songs and make nonsense sounds for hours on end.
Emily liked her mother this way, too. It had sometimes been difficult to be Sandy's daughter. As a child, Emily wanted to wear her hair long and take ballet lessons; Sandy, ever vigilant about gender stereotypes, nudged her to cut her hair and play soccer instead. But now Sandy didn't seem to care about such things. Emily thought that her mother was taking pleasure in life in a way that the old Sandy could not have anticipated—and she found herself hoping that the joy her mother took in Felix might make her reconsider her intention to end her life quite so soon.
The others in Sandy's inner circle saw her relationship with Felix and wondered what it would mean for her original plan. The old Sandy, who valued her rationality and her agency, had been clear that she would be unwilling to keep living when she could no longer articulate coherent thoughts. But this newer Sandy didn't seem unhappy living her life in this compromised way. Ultimately, who should make the decision to die, the old Sandy or the new one?
Ronald Dworkin, an influential legal scholar and the author of _Life's Dominion: An Argument About Abortion_ , _Euthanasia and Individual Freedom_ , wrote about a kind of hierarchy of needs for people in Sandy's situation, who want their autonomy to be respected even as disease changes the essence of who they are and what autonomy means. He differentiated between "critical interests" (personal goals and desires that make life worth living) and "experiential interests" (enjoying listening to music, for instance, or eating chocolate ice cream). Sandy was appreciating her experiential interests—playing with Felix and working in her garden—but her critical interests were far more sophisticated and were moving out of her reach. Critical interests should take priority when making end-of-life choices on behalf of someone whose changed state renders her less capable of deciding on her own, Dworkin wrote, because critical interests reflect your true identity. The new Sandy seemed to love being a grandmother, but it was important to take into account what the old Sandy would have wanted.
Granting priority to critical interests is difficult even in a society that tries to do so. In the Netherlands, the Termination of Life on Request and Assisted Suicide Act makes it possible for a doctor to end a person's life when she is not cognitively able to do it herself, as long as she laid out her intention while she was still competent. According to the 2002 law, if someone with Alzheimer's disease has an advance directive declaring her wish to die when her dementia reaches a point she considers intolerable—when she has to be spoon-fed, for example, or put in diapers—that document is sufficient to allow a doctor to perform euthanasia. Nevertheless, it is rare for a doctor in the Netherlands to actually euthanize a patient who has dementia. In fact, one recent survey of 110 Dutch physicians treating dementia patients with advance directives asking to be euthanized found not a single one who had carried out the request. And of the 4,829 people who died in 2013 under the Dutch euthanasia act, just 97 of them, or 2 percent, had dementia.
"You know I plan to kill myself," Sandy said all through 2013, whenever the thought occurred to her. She seemed to say it partly for the sake of others, so they could get used to the idea and steel themselves against pain and grief when the time came. But it seemed that it was also for her own sake, to keep her plan at the forefront of her disintegrating mind. Emily and Felix were living with Sandy at the time, so that Sandy could help with child care while Julius attended nursing school in Colorado. (Jeremy had recently moved out. He was going through a rocky time and was not communicating with the family, though he supported his mother's plans.) It drove Emily crazy to hear her mother continually bring up suicide. "Stop saying that!" she would tell her.
One night in August 2013, when Sandy was home alone, she pulled out a yellow legal pad and sat down at the tile-topped table in her big, oaktrimmed kitchen, where she had eaten thousands of dinners. She had just heard of two experimental treatments for Alzheimer's that she hoped might, in addition to the Namenda and Aricept, keep her functioning so that she could help care for Felix until August 2014, when Julius was scheduled to finish his nursing program and move back home. But the drugs were prohibitively expensive, and she would have to pay for them out-of-pocket, because her insurance wouldn't cover the cost. On the pad, she started to make calculations. The treatment—a combination of IVIG (intravenous immunoglobulin), a drug approved for other neurodegenerative diseases, and repetitive TMS (transcranial magnetic stimulation), which was usually prescribed for depression—cost $6,000 every two weeks at the New York Memory and Healthy Aging Services. What if she could persuade the center to charge less, because at her size she would need less medication? And what if she received the treatment less frequently, maybe every three weeks? It was still a lot of money, but she had never touched her I.R.A., and she was already sixty-nine and was clearly not going to live much longer. She calculated that her savings could cover about $4,000 every three weeks until Julius graduated in a year. At the bottom of the page filled with numbers, she wrote to remind herself not to fret too much over the staggering dollar amount. "Expensive: but now money is _not_ an issue (because of imily)." The previously meticulous scholar had misspelled her daughter's name.
Over the next months, Sandy and Daryl boarded a bus early in the morning every few weeks and rode down to Manhattan for the treatments. "I still feel as though I'm me," she told him on one ride. "Do you agree?" He did, sort of. In fact, he was surprised by how much herself Sandy could still be, even as she became less and less the formidable thinker he had always known. He was surprised too to discover that it didn't matter to him. "I realized how little of the fact that she was an intellectual played into my feelings for her," he said. "They were feelings for _her_ , not her intelligence. And they were still all there."
Daryl proved himself steadfast, and as her more casual friends fell away, either because Sandy shut them out or because they were unwilling to witness her decline, he became more central to her life. He and Karen were the ones who saw her frequently, and they were the ones she kept checking in with to be sure her suicide window was not about to close.
In October, Sandy wrote to an address in Mexico listed on the website of _The Peaceful Pill Handbook_. Weeks passed, and she fretted that her order had been confiscated at the border. But at last it arrived: a cardboard box, no bigger than a softball, wrapped in brown paper. Sandy eagerly took scissors to the packaging and retrieved two one hundred–milliliter bottles of pentobarbital—she had bought an extra one just in case, even though she believed that one bottle would be enough for a person her size. The drug needed to be kept in a cool place, so she took the bottles down to the basement. For the time being, she could leave the pentobarbital on a shelf, comforted by the knowledge that it was there.
Now that the matter of "How?" was taken care of, the Bems turned back to the elusive question of "When?" They still generally agreed that Sandy would probably be alive until the end of 2014. But even with the treatments in Manhattan, her cognitive deficits were becoming more pronounced. When Bev came from Oregon to visit, Sandy couldn't understand how Bev and she could possibly have had the same parents. She didn't recognize Robyn's name in conversation, and when Emily tried to explain that Robyn was "Dad's sister," Sandy asked who, exactly, was Dad?
Daryl noticed something else disheartening. He had been following Sandy's progress as she took up the piano again. The Bems had always had one in their home, though it was mostly for Daryl, who played and accompanied Emily, whose clear, resonant singing voice carried her through student productions and into a career in musical theater. Sandy had worked her way slowly and steadily through the lesson book, moving from the simplest "Twinkle, Twinkle" tunes to études that were slightly more complex.
Late in 2013, Daryl began to see that Sandy could teach herself to play up to a certain point in a piece, but the next time she sat down at the piano she had to flip back several pages and work her way up all over again.
A few months later, whenever she sat down and flipped back several pages, she could make it only part of the way to where she had been the last time she played. And a few months after that, she would sit down at the piano, flip back to the beginning and get stuck at "Twinkle, Twinkle."
Even that was O.K., Daryl thought. She seemed to be enjoying whatever it was she _could_ play. And then one day, she didn't even have that. Even the easiest pieces had become too difficult. Almost simultaneously, she was no longer able to read novels on her iPad or follow movie plots with complex flashbacks. Eventually there were only two movies she enjoyed: _Mary Poppins_ and _Funny Girl_.
Right around Christmas, Julius dropped out of nursing school and rejoined his family. Sandy no longer felt it made sense to spend thousands of dollars in the hope of staying a little more functional a bit longer for Felix's sake. She told Daryl she wanted to stop the treatments. Shortly afterward, over dinner in her kitchen, she told him something else: She wanted to talk about when might be a good date to die.
O.K., Daryl said mildly, trying to stay focused on the task at hand, pre-emptively shutting down thoughts about what it would really mean to lose her. How about June? He knew how much Sandy would want to get back to her garden when the dreary Ithaca winter finally ended, and he hoped this would be a way to make her last few months happy ones. She agreed that June sounded right.
At the time, Emily, who was splitting time between her mother's house and her own home, was back in Austin. She and Julius were planning to buy a house a few miles beyond the Ithaca city limits, and she was preparing for the move. One day in April, Emily returned to Ithaca and was driving home from a bank with her father, who was helping her secure a mortgage. She was at the wheel of Sandy's car.
Should we change the car registration and put it in your name? Daryl asked. Emily started talking about wanting to wait until late summer, when she would be in her new house and would need to change the address only once. Maybe in July she could ask Sandy to sign over the car to her.
"Oh, she'll be dead by June," Daryl said.
Emily struggled to keep the car on the road. "What did you just say?"
Daryl looked at her, surprised by her surprise. He assumed Sandy had talked to Emily. But she hadn't, and Emily thought June was much too soon. She was looking forward to the summer with her mother and her son—Felix running around his _bubbe_ 's backyard in the late afternoon light while Sandy puttered in the garden, digging a hole here, moving a rose bush there, pulling up weeds around the lilacs.
Emily was angry at her father for speaking so pragmatically about her mother's death. She was angry too at her mother for choosing a date that was so soon, and at her mother's inner circle for allowing all of it to happen. That night, she sat with her parents and Robyn while they discussed the situation. Emily felt as though she was defending her mother's life against everyone who wanted her to end it.
"O.K., so maybe not June," Daryl said, backing off. He had spent his life avoiding conflict. "We just thought that with your mother turning seventy- seven on June 22, that might be a good time."
"Well, that's nuts," Emily said. "How can you just pick a month like that?"
"What month did we say, again?" Sandy asked.
"June," Daryl said.
"Why don't you just say August?" Emily said. "It could just as easily be August as June."
"What month did we say, again?" Sandy asked.
"June," Daryl said.
"August, June—you can't just draw an equation," Emily said.
"What month did we say, again?" Sandy asked.
Emily turned to her mother. "You slogged through the winter," Emily told her. "By late May, it will be gorgeous around here." She wanted Sandy to hold out for one more summer. She wanted Sandy to _want_ to hold out.
"I'm sure it would be nice," Sandy agreed. But her voice was flat; the prospect of one more summer, even with Emily, even with Felix, even with her garden, no longer seemed to be enough.
The nagging sense that there might be more time, that there should be more time, is inevitable for those close to a person with dementia who wants to end her life. At an annual conference in Chicago last summer of the World Federation of Right to Die Societies, a collection of fifty-one member organizations that push for right-to-die laws in twenty-three countries, the problem of timing was a running theme. When someone has dementia, said Rodney Syme, an Australian right-to-die advocate, "if you want to have personal control over what's happening, it means that you need to show considerable courage and considerable maturity." And it means, he said, that a person in a position like Sandy's might have to give up some period of time in which she might still be able to take some small pleasures in her life, just to be certain of ending it while she still could.
One night in April when Daryl was over for dinner, Sandy said to him out of the blue, "You're so smart."
"Is that it, or is it that as you get dumber, I feel smarter to you?" Daryl asked with a smile, taking a risk that Sandy still retained her puckish sense of humor.
She laughed. "I guess that's it," she said.
But she told him that she could feel herself slipping and that the day of her death would need to be "sooner rather than later." Daryl pulled the 2014 calendar down from the kitchen wall. He chose a date. How about Tuesday, May 20?
Sandy agreed, and Daryl wrote it down. She told Karen, Bev, and Robyn what she had decided. She planned to tell Emily but wanted to do it in person when her daughter returned from Austin. Emily and Felix arrived in Ithaca on May 13, and her parents told her that Sandy would die the following Tuesday. Emily was appalled. Just one week from today? She was sure it was much too soon.
That evening, she sat with her parents on the big L-shaped white couch in Sandy's oversize living room, the site of many serious family conversations over the years. Bev, Karen, and Robyn were there as well. Emily had no problem with the general idea of Sandy's ending her life soon, she said angrily, but now? _Now_? Karen and Robyn tried to explain how deeply changed Sandy was: She rarely laughed and seemed to find little joy in people or experiences. Emily thought everyone was thinking too narrowly. They assumed that her mother's joylessness was a result of the encroaching disease. Emily saw it as depression. The neurologist had recently taken Sandy off Prozac and started her on Zyprexa, an antipsychotic. Maybe her dose should be adjusted. Or the doctor should prescribe a different drug. Emily felt that they all needed more time to sort out what was causing Sandy's change of mood.
As the weekend approached, there was another discussion on the big white couch. Karen and Robyn wanted to make it clear to Emily how much Sandy had declined in the month she was away. "You didn't see this moment, Em," Robyn said, "when your mom was standing in the kitchen. She looked a bit lost and turned to me and said: 'I'm hungry. What do I do when I'm hungry?'"
And just the other day, Robyn went on, Emily had been chatting in the kitchen with her mother and Bev. After Emily walked out, Sandy turned to Bev and asked, "Who is that person's mother?"
"You are," Bev said, trying not to cry.
"I thought so," Sandy said. "I thought it might be me."
Emily now understood that Sandy was deteriorating rapidly. Karen organized a small gathering on Sunday, May 18, to celebrate Sandy's life. It was very intimate, just Sandy, Daryl, Emily, Karen, Bev, and Robyn. (Jeremy had moved out West and still wasn't in contact with his family, though Sandy and Daryl left a voice mail message to let him know when Sandy would die.)
At the gathering, Daryl talked about a lawsuit that the Equal Employment Opportunity Commission filed in 1972 against AT&T for sex discrimination in its recruitment practices, in which he and Sandy took the stand together to testify as a team.
"Did I really do that?" Sandy asked, pleased.
He talked about her expert testimony in another lawsuit, filed by the National Organization for Women against The Pittsburgh Press, that went all the way to the Supreme Court in 1973 and made it illegal to categorize classified job listings by gender.
"Did I really do that?" Sandy asked again.
Karen talked about how, relatively late in life, Sandy decided to go back to school at Rutgers and get a doctor of psychology degree while still teaching part time at Cornell. In 2000, at fifty-six, she opened a psychotherapy practice, turning the den in her house into an office that she made comfortable with big pillows. Emily said her mother had always been fearless and had raised her and her brother to always think for themselves.
The next day was warm and sunny, so a small family entourage went to Stewart Park, on the shores of Cayuga Lake, for what they all knew would be a final outing. Julius pushed Felix in a stroller, with Bev and her partner alongside them; Emily walked behind, holding her mother's hand—something she hadn't done since she was a child. When they all stopped at the playground to let Felix run around, Emily pulled Sandy over to sit with her on a bench nearby.
Emily said she understood why her mother felt she needed to take her life now. She spoke admiringly of Sandy's ability to find exactly the right moment—not too early, not too late. "I think you nailed it," she said. Sandy was quiet while Emily spoke, looking into her eyes. Her relief was deep and obvious, but all she said before they hugged was, "I'm so glad."
On her last day, five years after she first went to see Mapstone, Sandy set about creating a tidy paper trail to make sure no one else would be held responsible for her death. She found the printout of an email, with the subject line "ENDING," that she sent to Daryl nine months earlier, stating why she wanted to die and saying that no one—not her physician, not her attorney, not anyone—had offered help or advice. In the email, she had written that she would add the date of her death, "and perhaps other thoughts," when the time came, another way to make it plain that her death was her decision alone.
When the time did come, there were really no other thoughts. All the ideas Sandy might have wanted to express in writing—the sophisticated musings, the incisive arguments, the upending of conventional thinking— were already beyond her. Sandy sat down with the printout of the "ENDING" email and in longhand wrote out the date, May 20, 2014, the one she had been repeating to herself ever since Daryl wrote it on the calendar. She followed it with a simple declaration: "The time has come to end my life. I love you, Daryl." She signed it, formally, Sandra L. Bem. Then she and Daryl went for a walk in the Fall Creek Gorge, a ruggedly beautiful spot, 110 stone steps down from the noise of Stewart Avenue.
When they got back, Sandy and Daryl watched Mary Poppins. Emily, Bev, and their partners had assembled at Karen's house. Karen wanted Sandy to feel as if "a loving net was around her" as she prepared to die— though it had to be love at a distance, because Sandy wanted no one but Daryl in the room.
Around 5:30 p.m., Sandy took antinausea medicine and poured herself a glass of wine; she had read that drinking alcohol after taking the pentobarbital would mask its bitter taste and speed its action. With Daryl accompanying her, she carried the glass of wine upstairs to her bedroom and set it on the nightstand. Next she had to trim the foil off the collar of the 100milliliter vial of pentobarbital using manicure scissors and remove the small rubber stopper. Daryl held his breath, unsure whether Sandy could manage all these maneuvers on her own. She did. She poured the pentobarbital into a glass and set it next to the wineglass.
"Now what?" she asked. Daryl didn't know what to say. They had expected the preparations to take an hour or so, yet just fifteen minutes had passed.
"Have you decided what to wear?" he asked. She said she was happy with what she already had on. She got into bed and looked at the two glasses on the nightstand. She asked which was the drug and which was the wine.
"The drug is clear, and the wine is red," Daryl said.
Sandy nodded and looked around the room and then at the two glasses again. She asked which was the drug and which was the wine. Daryl told her again.
"Can I sip some of the drug and then drink some of the wine?" Sandy asked. "That's not a good idea," Daryl said. "You don't want to fall asleep before you've drunk it all."
"O.K., I'll drink the whole thing," she said, and she did. He asked if it tasted terrible. "No," she said. "It's intense, but it's not bitter. I'm not having any problem with it."
She put the glass down. "How much wine am I supposed to drink?" she asked. Daryl told her she could have as much as she wanted. She took a sip.
"I have to go pee," she said.
"You can't go pee," Daryl said. "I'm afraid you'll fall asleep."
"Can you come with me?" she asked.
So Daryl and Sandy walked to the bathroom together, and Daryl sat outside the door while his dying wife sat on the toilet.
He helped Sandy back into bed, and within five minutes she was unconscious. Daryl watched her for a while, not quite feeling anything. Still to come were the calls to 911 and the coroner and the undertaker, and the writing up of the death notice, highlighting the reasons for Sandy's decision. Still to come, too, was the brutal reality of what it would feel like for Sandy to be completely gone from his life. "How powerful a presence is
her absence," Daryl would say at a memorial service that summer, quoting from a poem by Fred Chappell. "The rooms were quiet when she was resident./ Now they lie silent. That is different."
For now, though, Daryl simply gazed at his unconscious wife. Around 8:30, he telephoned Bev at Karen's house around the corner. Bev came over to sit with him at Sandy's bedside. They were quiet, watching the sheet go up and down with each breath. Over the next hour, the sheet's rise and fall began to slow. Then it stopped.
_Gerri Hirshey_
Gerri Hirshey has worked as a features writer, columnist, reporter, and essayist for more than thirty years at publications including _The Washington Post_ , _The New York Times Magazine_ , _Vanity Fair_ , _GQ_ , _Esquire_ , and _New York_. In the 1980s, she was the first female contributing editor at _Rolling Stone_ , writing celebrity profiles of entertainers, including B.B. King, Michael Jackson, Jodie Foster, and James Brown.
Her work has also appeared in _O: The Oprah Magazine, More_ , The Nation, Food & _Wine_ , _Ladies' Home Journal_ , and _Parade_.
Hirshey is the author of several books, including _Nowhere to Run: The Story of Soul Music_ , now in its seventh reprinting, and _We Gotta Get Outta This Place: The True, Tough Story of Women in Rock_. Her most recent book is _Not Pretty Enough: The Unlikely Triumph of Helen Gurley Brown_ , a biography of the longtime editor of Cosmopolitan.
Hirshey lives in New York City with her husband, Mark Zwonitzer, a writer and documentary filmmaker. They have two young adult children.
On the Bus with B.B. King
On the road with the great blues man and his beloved guitar, Lucille. Classic music journalism by Rolling Stone's first female contributing editor.
The trucker is squinting hard through the downpour of a sudden tempest, steering his load of appliances along the slick Connecticut four-lane. He's nattering on the CB radio as the mammoth motor coach pulls alongside. The lights are on in the bus's private lounge, and despite the rain, the windows have been pushed open. Looking out at the trucker is a seventyish black man popping a Diet Coke and a smile. _Bwwaaaaaaaaa_. The trucker leans on his horn in a delighted blast of recognition. There's an American vision sweeping past, and the driver is hollering the news into his mouthpiece: _It's B.B. King!_
B.B. grins at the honking salute. Even when he was in his early twenties, plying the rural juke joints and eye-blink towns outside Memphis, he says, he loved to hear his imminent arrival announced on local radio, to have excitable women trill the news as he sauntered past with his guitar: _B.B. King's in town!_
At seventy-three, he is King of the Blues Worldwide, according to the custom tour jacket tossed on the seat; a national treasure by virtue of his Presidential Medal of the Arts—and a real gone daddy if you count his fifteen children by as many women. Shanghaied by a fatal attraction to one Gibson guitar, now speeding headlong into his fiftieth year on the road, B.B. King has to be the hardiest, the most mythic, the almighty rollingest stone.
Tonight, ours is a procession befitting the elder statesman of blues highways. Our "chase" vehicle is B.B.'s band bus, with his name and the likeness of his guitar, Lucille, painted on the side. B.B. leads the way in his private Fortress of Solitude, a custom-configured, Belgian-built VanHool motor coach rigged with a 120-channel satellite dish, a kitchen, a shower, six TVs and a 500-horse engine smart enough to diagnose its own ailments via computer printout. "This is home," says B.B., waving at the stacks of his beloved electronic gizmos, the milk crate full of videocassettes, the sound system playing Dexter Gordon.
He leans back into the buttery leather of the horseshoe-shape banquette, looking relaxed in his traveling clothes. As usual, B.B.'s cool will frost your eyeballs: burgundy silk shirt, knife-pleated black slacks and soft, boaty loafers with woven black-and-burgundy leather insets. He's conjuring road stories, shaking his head as he recalls some of the wheezy highway schooners that brought him so far: "We were in Louisiana one morning. I had a big bus, my name on it. We broke down in this small town . . . . "
B.B. is off, reaching into his bottomless sack of anecdotes and B'isms. Painted from life with the broad strokes of a Mississippi Delta impressionist, these are the twelve-bar picture stories that have made his blues believable for so long:
_He's trudging down a dusty two-lane, a lone, worried black man in too-fine clothes. Dawn is breaking as he comes to a whites-only cafe and walks around, instinctively, to the back door. The owner is just opening up when B.B. identifies himself and explains his problem: a breakdown, a busload of hungry musicians. Sure would help if they could come in and sit down. The man says OK and sets up a table. As the band wolfs biscuits and gravy, the jumpy owner stands at the front door, greeting his white regulars. To each he blurts apologetically: "That's B.B. King and his band. The bus broke down and . . . ._ "
B.B. laughs at the vision of the nervous but kindly man who dared serve him a square meal beneath the menacing wings of Jim Crow. "I'm glad that's changed," he says. "Thank God for the change."
We're bumping over a dark, grassy area on a rain-swept peninsula in southern Connecticut, toward an outdoor gig made possible by a last-minute letup in the rain. Summer is almost gone, but the B.B. King Blues Festival, barnstorming tents and amphitheaters from Toronto to Tulsa, still has another month to go. For the headliner, that's the briefest moment in time.
"Fifty years—really?" a fan is squeaking.
B.B. is doing his ritual meet-and-greet after the show. Tonight's dressing room is a cramped RV behind the tented stage. Outside, the fitful strobe of heat lightning reveals a line of autograph seekers about a hundred feet long.
"Feels like I just got here sometimes," B.B. says. "But, yeah, I got serious about things in '48."
He was twenty-three then and had been doing farm work since he was six. B.B. has great recall for the significant numbers in his life. He likes to roll them off:
$22.50: his salary driving a tractor on a Mississippi plantation six days a week, in 1946;
Twelve percent: the alcohol content of the Pep-Ti-Kon "family" tonic that sponsored his first ten-minute spots on Memphis' legendary black radio station, WDIA, in 1948;
A penny a pound: The '48 rate for picking cotton across the river in Arkansas every afternoon after B.B.'s Pep-Ti-Kon gig (this, he notes, was triple the rate in his native Indianola, Mississippi);
$12: the cash a certain Miss Annie gave him for his first paying gig, a couple of hours at her West Memphis juke joint. She'd happily turn off the box, she told B.B., "'cause the ladies like to dance to a live man."
Which Miss Annie saw he was.
"So you go ahead and do the math," B.B. says by way of explaining his career choice. But even he falters trying to calculate his lifetime one-nighters. The miles traveled between them would bankrupt any latter-day frequent-flier program. Riley B. King, billed as the Beale Street Blues Boy in post–World War II Memphis, then Blues Boy King, then Bee Bee, then B.B. (and now, to his nearest and dearest, just B), has carried his blues to eighty-eight countries. Most years he averaged 340 shows until recently, when he "cut back" to 250. He has released seventy-six albums (the latest, _Blues on the Bayou_ , came out in October).
Volumes have been written limning B.B.'s legend. But it takes a serious road trip—submersion in B.B.'s diesel-scented, day-for-night existence—to appreciate his astonishing lust for the Life. B.B.'s story is the road, as American as Woody Guthrie's in its tuneful observations, hipper than Kerouac's in its outsider wisdom. And in a nation known for producing wanderers of distinction, no one—save perhaps Lewis and Clark—can match B.B. for sheer endurance.
Having held the floor for maybe fifteen thousand-plus smoky, hip-twitching nights, he has allowed himself a total of three months' scheduled vacation time. In half a century, he says, he's missed only eighteen gigs. Most of them, he points out, were promoter mix-ups or "acts of God—you know, the weather and such."
Though he is in most ways a modest man, B.B. is very proud of his golden anniversary. Traveling with him through New England, Manhattan, Maryland, Texas, and Oklahoma, I've seen him celebrate it on-stage every night, fists raised, eyes closed, with a mighty bellow: "You've kept me out here for FIFTY YEARS! THANKYEW!"
In quieter moments, B.B. admits he's been married more to the road than to the two fine women who tried to be his wife—and to the many others who bore his children. He'll tell you that Lucille, his darkly gorgeous Gibson guitar, has always been the home-wrecker, that only one woman has ever come close to bringing him out of himself the way Lucille can every night. Named for the vixen who ignited a brawl and a roaring house fire at one of B.B.'s early gigs, the current Lucille (the sixteenth) is waiting for him now in the bus. She drew her own applause as she was escorted back there, gently, via golf cart. Watching Lucille's stately departure, some boys just stared, slack-jawed and silent, as the lightning flashed Excalibur-like off her gleaming gold frets and pickups.
The kids always come around: lanky Hanson look-a-likes, dreadlocked tenth-graders in surfer jams—the "guitar kids," B.B. calls them. They understand that whatever fills arenas nowadays can be traced directly to the big flat fingers now curled around an autograph pen. Theirs is a global tribe; they show up in Moscow, Kyoto, Rio. B.B.'s kids are virtually all male, downylipped dreamers who lie in their childhood beds and practice the most urgent fingerings in the dark. Tonight, once they reach his plastic-slipcovered inner sanctum, they stare unabashedly at his hands. Across the blasé "whatever" faces of the Info Age's first spawn, I detect the refreshing bloom of awe.
"I have this one really important question—OK?"
A female voice cuts through the basso murmur. She says her name is Ann. She looks eighteen or so; there is a feisty DiFranco cut to her jib: frizzy hair flying from a leather clip, baggy shorts, and sandals. She says she's a singer-songwriter.
"Do you ever get tired of singing and playing?"
"No. Never."
Ann isn't buying it. "Look, even I get tired, because it's so emotionally draining," she says. "So painful sometimes. I put so much of me in my songs, things that really hurt me. And to do it over and over . . . . "
For the first time tonight, B.B.'s smile has vanished. He is almost stern as he leans across the table and tells her: "Don't think about the hurt. Don't think people are responding to the hurt when you play to them. _Don't . . . do . . . that_. It's about the music. You don't know what they find in it—some connection. Who knows? Leave it at that. Don't think of being _hurt_ . . . . "
Morning is breaking civilly enough on the Connecticut shoreline as we settle into the private lounge of B.B.'s bus to talk en route to Baltimore. He slips off his shoes, which he always buys at Rochester Big and Tall in New York for his "big old weird feet." Reflexively, he shuts down the throat-killing AC vents and slides the windows open to a powerful highway racket.
Morning is breaking civilly enough on the Connecticut shoreline as we settle into the private lounge of B.B.'s bus to talk en route to Baltimore. He slips off his shoes, which he always buys at Rochester Big and Tall in New York for his "big old weird feet." Reflexively, he shuts down the throat-killing AC vents and slides the windows open to a powerful highway racket.
Traveling with B.B. is not for the fainthearted. The company is genial, but the pace is brutal by any standards. After every show, B.B. spends up to two hours talking to those who clamor to see him up close. At 2 a.m., with six hundred miles to be covered for the next gig, even when B.B.'s diabetes is working its sneaky hoodoo on his blood sugar, no one is turned away. The band is often abed or winding down in some hotel lounge before the boss finishes work. Lately, B.B. has been suffering from insomnia—the damned diabetes, he thinks—so he sits in his room, clicking at his laptop computer, answering letters, playing solitaire or his beloved cybergame, Freespace, past dawn.
Creep down to the hotel coffee shop after a couple of hours sleep, and that sound—the low, impatient diesel rumble—is always right outside. B.B. owns the band bus, but his private leased coach is a recent indulgence. "There's two other things I thought I owed myself the last five or six years," he says. "A suite—not a huge one, but a parlor and a bedroom. And a firstclass ticket on the airplane."
The object of all these splurges: privacy. He says he's such a creature of group travel that he catches himself closing his bedroom door behind him during the rare times he's at home—alone—in his Las Vegas town house. Having the coach to himself gives him a great deal more solitude. On a full tank it has a cruising range of more than twelve hundred miles. "You need that to keep up with the man," says today's driver, Hap Arnold.
The remarkable engine that is B.B. has been sputtering a bit lately, owing to a flare-up of the diabetes that was diagnosed several years ago. He has been wondering if it's the same thing that might have blinded and then killed his mother when he was just nine and she was in her late twenties. With great effort, he pared more than thirty pounds off his ample self. His pockets rattle with Ziploc bags of prescribed medications. He does not drink alcohol or eat meat. B.B. carries no masseurs or personal trainers on tour, just a small space heater to ward off air conditioning in his dressing rooms. This morning he's salting the first of a few stabilizing light meals—a couple of hard-boiled eggs. He pops open a wake-up Diet Coke.
"I've been thinking about that girl last night," he says, meaning Ann. Hers wasn't a new question, but it nettled. Why do folks always fixate on the pain? B.B. just hates it when people come up to him and confess how they held fast to his music when their lives and loves fell apart. He is always polite, but the truth is, he'd like to pound a wall.
"People say, 'Oh, me and my girl broke up last week and, boy, I've been playing you ever since.' It's not really flattering. Any way I can get them to listen to me, I guess I should be happy. I shouldn't be surprised that they reach out to the hurt in it. But don't play it simply because you lost someone or you _hurt_. I'd still like very much to know that I'm a pretty good musician. And you can hear it even if you haven't been hurt."
Pain is hardly a taboo subject for B.B. It keeps cropping up in conversation as it does in life, as natural and persistent as crabgrass. He'll finish an anecdote: It hurt me. You don't know how that hurt. B.B. even speaks the word with the same upward-sliding glissando that makes his guitar licks instantly recognizable. In hours of conversation, he demonstrates great intimacy with all kinds of hurt: romantic, racial, professional. He admits that his own hurts have compelled and propelled him. He's played them, shouted, stomped, and roared them. He's autopsied and orchestrated them.
_I've been downhearted, baby . . . ._
_Have you ever been mistreated?_
_How blue can one man get?_
Growing up poor and motherless in the segregated South, B.B. says, he has lived some very deep blues. And he has no problem with people thinking that pain may have pushed and shaped his art. But he would like this understood: B.B. King also owns the copyrights. All this time and mileage he has put in are not about the triumph of misery as much as its subjugation. His favorite blues are about survival, not submission. He says he's tickled at the way audiences are reacting now when he powers through a song from his new CD called "I'll Survive." Night after night, they rise to their feet when he brings it home with those two words. Respected, well paid, draped with honorary degrees and awards, he relishes his role as the poet laureate of Enlightened Complaint.
"Can you imagine," he asks me, "how a seventy-three-year-old guy has all these young people coming out nightly? Just to see me? How lucky can you be?" He's just as ready to admit that, yes, every day he also has the blues, if only for a moment. That's life. We haven't been talking five minutes when B.B. mentions what he calls his education hang-up.
"By coming out of school in the tenth grade, I've never caught up," he says. "I'm always trying to learn what everybody else knows. For example, if I'm with somebody and they say, 'Yeah, when I was going to college, I learned so and so . . . . ' Well, I feel a little inferior at the time. I have that little complex. I always feel that I'm the one that has to learn."
Press him about his honorary degrees from the likes of mighty Yale and tiny Tougaloo College, the Kennedy Center Honors, his 1987 induction into the Rock & Roll Hall of Fame; remind him of the seminars he gives at universities like Northwestern to demonstrate his philosophy of funk and technique, and he'll finally acknowledge: "I feel I have a reasonable degree in art, in what I'm doing now. I started feeling this way not awful long ago. People praise you—I love it. But they don't know my limits as I do. I know my limitations _very_ well."
Not that he is one to rest easy with them. He classifies himself as "very aggressive" in trying to learn things. To illustrate his point, B.B. hands me a zippered case full of CD-ROM disks with the invitation, "Go on, check it out." I find Success Builder disks for Algebra 2. Geometry. Elementary Spanish and French. Reptiles of the World. Architecture. What else is in B.B. King's Compaq laptop? Letters, compositions in progress.
All this talk of sophisticated communications leads him to remember a very basic message sent to him one night in Chicago. It came when he felt like a big deal—recording star, lady's man, King of Saturday night—until this blues blindsided him and he came undone:
_B.B.'s got a band, paying it pretty good, too. They're playing a joint called Roberts, and B.B. is strong, loose, and juiced with the promise of a new release. He tells the crowd he's got a new ah-blum out. "Ah-blum." After the show there's the rasp of a note sliding under his dressing-room door. It says, "The word is 'al-bum,' not 'ah-blum.' "Walking by the band room, he hears the guys saying "ah-blum." They're laughing . . . ._
"That hurt me deeply," B.B. says. "One of the guys that was doing it I thought was a friend. To be a friend, he should have told me, straightened me out."
Suddenly he brightens with a coda to the story. About a decade later, in St. Louis, someone sent a note backstage asking whether he remembered the ah-blum message, which was meant kindly. Then the writer stood revealed: "It was a beautiful lady." B.B.'s smile bespeaks a happy ending to that blues.
_"As a tractor driver, sex was always on my mind. It didn't take much to get me going. If I drove past a girl picking cotton, I'd notice the way she bent down. The way her buttocks outlined the back of her dress could fire me up for hours. On a scorching summer day, the sight of beads of perspiration on a soft feminine neck would arouse my imagination."_
In his 1996 autobiography, _Blues All Around Me_ , B.B. often admits to his runaway lusts, complete with a description of his senior-citizen circumcision ("and my penis, thank you very much, has been in good working condition ever since"). There have been so very many women to bounce him headlong between desperation and divine inspiration. "It took me a long time to realize that you can't have _all_ of the women," he says now. "I've _always_ loved the girls."
B.B. pronounces the word more like _gulls_. And by grammar school, he had figured out how to get to them. He was shy. He stuttered. He was no Romeo. So: "I played my harmonica. And the girl that I was crazy about she was crazy about one of my cousins. But when I would play the harmonica, she would listen, seem to soften up a little bit. Others would say, 'Oh, Riley can really play, can't he?' Well, that's like you pat a little dog on the head if he bring you the paper. He's ready to go back and get you another one."
Petting—the female kind—was what B.B. figures he was after. Losing at an early age the only people who loved him truly and unconditionally (his mother, then his grandmother) might have marked young Riley as a needy young man. His parents divorced when he was five, and his father would not find him for some years. An aunt and uncle would have taken him in, but he was steadfast in his intent to live alone at ten.
"Everybody cared about me," he says. "But how much? The house where we lived—my room, my grandmother and I—I wanted to stay there. I felt that it was nobody but me. I just felt it was me against the world. So there was something about the place—the way they left it. And I can have that now. This is mine. If nothing else."
He says he still craves the comforts of some imagined domestic hearth. Despite his two divorces, he insists that "the times I was married were the happiest times of my life." He likes the idea of a woman waiting for him at the end of the day, looking down the road for him. And when he was a teenager, the beauty of that vision threw him full force at Miss Martha Lee. She was the most enticing of the women he noticed as he jounced along high above the cotton rows, one of nine big-shot tractor drivers on Mr. Johnson Barnett's mammoth plantation. Theirs was a spare, quiet Delta wedding of seventeen-year-olds, in 1942.
"I guess I was looking for love, because I'd never had anybody I believed truly loved me," he says. "More than to tolerate and put up with me—to truly love me. So when I did get married, young, I was crazy about the girl. It seemed to me that I finally had love."
Things were fine for a few years. Lucille wasn't in the picture; B.B. had a series of raggedy, cheap guitars and a nettlesome little itch: "For some reason, my mind was not settled at being a husband like a husband should be. Now I wanted to play an instrument. Now I'm thinking that there's something out there for me. I don't know what to do, but I'm seeking. I'm looking for it. She'd fight me on it."
He remembers the night he saw the end coming.
_The house supper sounds common enough by Delta standards—hamburgers in the kitchen, bed taken out for dancing, some live music courtesy of Riley B. King. B.B. doesn't know the people or the place. It could be rough, but Martha is hardheaded about coming along: "It don't make no difference. You go and I go." But he sneaks off alone. He's at the house supper an hour when the woman, a stranger, starts to flirt, sits in his lap, just talking. Now, a man can't be ungentlemanly and toss the girl off . . . boom! The door opens. Martha._
"I swear to you, I didn't know the lady," B.B. says now. "And that's when my wife looked at me differently. We lived together four or five more years, and she never believed I was telling the truth."
After their divorce, he was on the road nearly a decade and saw that look plenty of times. "I kept searching, searching. Seemed to be me alone. I'd run into ladies, thought we had a thing going real good. Then we'd start to talk about my music." His voice rises to the descant timbre of Female Complaint: " _Get another job._ " Then, one night in a club back home, Sue Hall hit him like a wrecking ball. She was lovely, he says, bright enough to go to business college and to help manage his affairs, light-skinned enough to get arrested in Ocala, Florida, for being with a man as dark as B.B.
In 1958, Sue married him on the condition that she could come with him, at least part of the time. They tried it, but after nearly a decade they came apart, too. And it's clear when he talks about his second wife that she is his great regret. He says she was the one who came closest to giving him what Lucille is so generous with: release. He figures he's still looking for the feeling: "Since my early childhood, I have had a problem trying to open up. _Please_ open me up. Look inside! 'Cause I can't. I don't know how to."
It's no surprise, then, that B.B. is such an accomplished poet of Yearning. Longing and regret pepper his lyrics like birdshot. "The Thrill Is Gone," his signature piece, was written in the wake of his divorce from Sue. The melody, like the woman, confounded him for eight years. "I think I understand women better than a lot of men do," he says. "But no man, I believe—even King David—really understands ladies. And maybe that's the way it should be. Because ladies are _verrrrry_ mysterious as far as I'm concerned."
These days, he says, he's getting to appreciate his conversations with women. In fact, he'd rather talk or go to a movie. For so many years, talk—if it came at all—was but a dinner mint after the real feast. "I haven't been no angel," he says. "Haven't had a halo around my head. I've always liked girls. I don't throw 'em out in the street, I don't treat them badly. I do see a person, and I'd like to be married again. But I'm not ready. l can't even open up to her, not like I would like to. I think I'm starting to see the problem."
_"When I was five and my daddy was coming to Gainesville, my mother dressed me in this beautiful crinoline dress. She would allow me to stand in front of the big picture window. And I would wait and wait. And then I would hear the bus coming. And I would get so excited, my little heart would just pound. He'd come maybe four or five times a year, whenever he was performing in the area."_
Patty Elizabeth King, born forty-one years ago to B.B. and Essie Williams, who owned the Blue Note in Gainesville, Florida, gave this interview to People magazine in 1993 from a prison in that city. She was serving time for cocaine trafficking, and B.B.'s bus was arriving again, to play his sixty-ninth prison show. Patty cried when he left.
"My father was always in my life," Patty King says now. "He supported us. We've always been able to get in touch with him. If there was ever a problem, he was trying to solve it. I've called him in the middle of the night with problems, or just needing to talk to someone, and he's always there."
Released from prison and reunited with her four children, she now works with an elder-care organization in Gainesville to support the two young sons still at home. She is writing a memoir: _Blues Baby_. And, if anything, it will be a love song to her father. She says she's made it a point to reassure B.B. that papa's rolling-stone lifestyle was not an issue: "I tell him that he's done well by us. I don't want him to think that things that have happened to me in the past had any kind of bearing on him. He's just a great dad. He's my father, and he's my friend."
B.B. says that most of his children are doing well. They are spread out all along the road, from California to New York. Among the four sons and eleven daughters are a preacher, a blues singer, homemakers. He has helped support all of them, sent those who wanted to college, and is sending their children now—seven at present. They come out to see him as he passes nearby, generally on his dime, and fuss over him in hotel suites, cooking fish and greens, bringing along grandchildren to spoil. "I haven't been there all the time," B.B. says. "I've been the loner; I've run here, ran there. But I've always kept a place where they could get in touch with me, and they do."
His youngest, Riley B. King Jr., wrote from a prison in Huntsville, Texas. B.B. says he drew eighteen years for drug-related thefts. Once he had served five years, B.B. set his attorneys to pressing for an early release. "He's my son, and I love him," he says of the boy who used to come out on the road with him. But Riley's early letters from prison were angry and accusatory. "He seemed to think that his mother and me didn't do everything we could have done," B.B. says. He tried tough love— _don't bother writing back if that's all you have to say_. They settled things; the lawyers went to work. Once or twice a year, B.B. drove through the prison gates to visit his namesake. (On November 3rd, Riley B. King Jr. was paroled; B.B. got the news on the road.)
"I wasn't much of a father," B.B. says. We spend a good deal of time discussing B.B.'s will, which is a source of pride and comfort to him. His blues will provide a substantial legacy. The education fund he started for his descendants is close to $2 million. "This is my way of trying to show them that Dad did love them," he says. "For those days that I wasn't there to hold their hands when they needed me."
He feels he's changed a lot—for the better—in the last ten years: "I think I'm a better father. My kids say, 'You're all right.' Some of them are very outspoken. They wouldn't say it if they didn't think so."
It took him years to make peace with his own father. Albert Lee King was also a traveling man, since his work as a tractor driver kept him away from home all week. "My dad never told me that he loved me," B.B. says. "Never. But when he was pleased with me, he called me Jack. I could always tell when my dad loved me. That's when my little heart would bust."
The ladies still like to dance to a live man. I watch them nightly from a gloriously loud spot behind the horn section. Dancing women always ring the front, jostling moony guitar kids. As one female security guard at a harborside Baltimore venue explains: "Lucille _understands_. Lucille is a girlfriend." And all the churning hips are sassing back: teen queens in twin-scoop halter tops, bifocaled soccer moms, a hard-rockin' grandma who's rumbling up the center aisle, fists pumping: _Do it, pretty baby. Rock me hard!_
It's hard to sit still with B.B.'s fine, tight eight-piece band at full throttle, with the huge beat of two drum sets, a bass rumble that walks right up your pant legs, a rhythm guitar courtesy of "The Fabulous" Leon Warren that is anything but utility. The cables and curtain pulls vibrate; flecks of abused drumstick fly up toward the lights. Bandleader and trumpet player James "Boogaloo" Bolden must weigh three hundred pounds, but he never stops dancing, a syncopated Frigidaire with his COOL setting on permafrost.
Live! is the key to B.B. King's longevity. He noticed long ago that record sales always went up in towns he had played. Crowds got bigger every time he went back, which is probably why, wherever we go, people holler at B.B., "See you next year!" It's a marker as reliable as an equinox: _B.B. King's in town!_
Throughout this string of one-night stands, I make sure to claim my lookout in time to see B.B. walk into the noise and light. There is always a chair in the wings for him to wait for the moment. As he sits there, profiled in the half light, his festival-casual silk shirt falling over his tux pants, there is something calm and Mandela-like about him. The usually expressive face is inscrutable as he surveys the pandemonium of his own making, then steps out to face its consequence as saxman Melvin Jackson hollers:
"Mr. Beeee Beeeeee _King_!"
They're on their feet before he's played a note. It happens nearly every night. And this is where the long road leads, to pangenerational crowds, solid ticket sales, and a robust stock portfolio—B.B. favors blue-chips. At this level, the road becomes its own destination. It affords a nightly renewal that keeps Bob Dylan plying the gyms of backwater academe, propels the rich-as-Croesus Stones, grandfathers now, gigging uproariously toward the millennium.
"You say you want _more_?"
In the glow of an hour's unconditional love, B.B. forgets how much his damned feet hurt, and Lucille sounds forever young. She sails, seduces, stings when the left fingers slip into the trademark perpendicular slide that stamps a B.B. King note. Sitting down casually with his bass and rhythm guitarists, he belts "Rock Me" with enough body language and hormonal glee to make the jaded drink vendors scream here on the Baltimore harbor.
"Domestic disturbance in Aisle Three," crackles a security walkie- talkie. "Man gettin' b'lligerent with his lady, all units . . . . "
Unaware of the wee squall of mace that results, B.B. leans back and roars into the lights.
"Rock me, baby . . . like my back ain't got no bone."
It's tough to get B.B. to own up to his own eloquence. Bulldog him with it, as I'm trying to this muggy Chesapeake morning, and he puts all his achievements at the foot of an entrenched Inferiority Thing. "Stupid fingers"—not agile or fast enough—is the explanation he has long given for the development of his signature style: "I always liked the steel guitar. I also love the guys that play the bottleneck. Just love the sound of it. But I could never do it; I never made it do what I want. So every time I would pick up the guitar, I'd shake my hand and trill it a bit. For some strange reason my ears would say to me that that sounds similar to what those guys were doing. I can't pick up the guitar now without doing it. So that's how I got into making my sound. It was nothing pretty. Just trying to please myself. I heard that sound . . . . "
B.B.'s signature vibrato is the stringed equivalent of the best soul singer's melisma—the art of drawing out a note, pulling it like taffy, releasing it a scant heartbeat before the ear can say, "Yes, _now_!" It satisfies like the perfect, unpredictable logic of Miss Ella's scat. In its fiercer moments, the kind that upend the hairs on your neck, such art does nothing less than make a fragmented world whole.
He has said that money was his first muse, the one that freed him from plantation work. I ask him when the music itself began to shape the life, and he says that it was pretty early on: "I was starting to hear the sound. It was more like a little itch—yeah. Any time I played, I enjoyed playing. The music was soothing to me. Just the notes from a guitar sounded like somebody bringing a kid candy. It was good for me."
After five decades and millions of permutations on six strings, is he satisfied with his own music?
"It wasn't what I wanted it to be. I still every so often would search for it. When I'm practicing now, I'm really chasing that sound." B.B. is dead certain that a perfect sound exists in his universe, though he can't possibly describe it: "I've got close to it, but I haven't heard it yet." Ask him whether he's OK with the idea that he might never hear it, and his ferocity is startling: "No! No, I'm not. As long as you're alive, there's a chance. As long as you study and try, there's a better chance. So I still study, I still practice."
B.B. is cracking another breakfast egg; we've left the dark of the harbor tunnel, and the shipyards of Baltimore shimmer in a morning haze. A big tanker horn blasts farewell. B.B. says he knows this East Coast corridor well. Baltimore had the Royal Theater, part of the Grand Slam in the restless and sophisticated black music market. Playing there, as well as the Apollo in Harlem, the Howard in Washington, DC, and Chicago's Regal, certified you as a "made" act. Provided you survived. It was right here in Baltimore that B.B. endured the greatest public hurt that he has ever suffered.
_The pretty men are all on the bill, Sixties soul singers so handsome, so clean in soft alpaca and vented sharkskin: Sam Cooke, Jackie Wilson, the Drifters. B.B.'s capable band backs them all; from backstage, he hears the screeching—the kind of uncontrolled, love-me-thrill-me-take-me female wailing that a man only dreams of. Finally, the blues singer is introduced, and those pretty little things are booing. B.B. dresses sharp, but he does not look pretty when he plays, and Lucille makes him grimace, wince, and go all guppy-mouthed. Louder comes their cruel descant: Booooooooo. Tears are rolling down his cheeks as he sings "Sweet Sixteen." He cries hardest when he reaches this line: "Treat me mean, but I'll keep on loving you just the same . . . . One of these days you'll give a lot of money to hear someone call my name . . . . "_
"They got quiet then," B.B. says. "I guess they saw the tears. And they applauded me. But that one time, I was hurt like never before. I've felt it many times, been cut down to size for being a blues singer."
He has taken it from the other side, too, for not being blue enough. B.B.'s 1970 "crossover" hit, "The Thrill Is Gone," raised hackles with some blues purists for its gusting studio strings. The very eclecticism that has made his blues popular music—the covers of Ellington classics like "Don't Get Around Much Anymore," the duets with the likes of George Jones and Joe Cocker, the bits of comic onstage shtick that have roots in Ma Rainey's black vaudeville—can set some root-blues aficionados to fanning themselves with a Blind Lemon Jefferson album sleeve.
"Blues purists have never cared for me anyway," B.B. says a bit crossly. "I don't worry about it. I think of it this way: When I made 'Three O'Clock Blues' [his first hit on the R&B charts, in 1951], they were not there. They were not. The people out there made the tune. And blues purists just wrote about it. The people is who I'm trying to satisfy."
Likewise, he has no patience for the self-styled ethnologists who seem to think that a guy who started singing on a Delta street corner shouldn't cover a swank Sinatra tune. B.B. can still sing his uncle's stentorian field hollers, loosed at the end of the workday:
_If I feel like this tomorrow . . . ._
_Feel like I'm gonna make my getaway . . . ._
B.B. honors the memory, but he's not going to drag that out on _The Tonight Show_.
"For God's sake, why Vegas?"
This is the question that B.B.'s manager, Sid Seidenberg, fairly shrieked when B.B. decided to make that neon Gomorrah his putative home, in 1975. On the surface it made some sense—Elvis was wowing capacity crowds at the Hilton's showroom, and B.B. rocked those high rollers at the hotel's big lounge hard enough to secure a five-year contract. But at the time, B.B. also had a big IRS debt and an alarming weakness: "I used to gamble like mad. Keno is the thing that got me hooked." There's still a glint in his eye as B.B. describes his biggest score: "They paid me $50,000 in crisp $100 bills!"
Seidenberg, a CPA turned manager who has been with B.B. for more than thirty years now, has guided his sole client into ambitious "five-year plans," hammered out the corporate endorsements that long eluded bluesmen and heckled record companies into respectful box sets, "superstar" duet projects and the like. Seidenberg also suggested a gambling cure to B.B.: Don't draw your salary in Vegas—have them bank it. And don't take casino credit—write checks. Write them out with all those scary zeros and you'll realize how much you're spending. "I did that," says B.B., "and when I noticed how big some of those checks were, that cured my gambling. I go home now; I don't want to go to the casinos."
He prefers to stay at home, in the town house that two secretaries watch over during his long absences. B.B. admits it's more like a warehouse, chockablock with souvenirs, electronic gadgets, and the largest collection of hats this side of Elton John's closet. One of his secretaries, Laverne Toney, has instructions to leave all of B.B.'s toys where they lie when she goes in to empty the six VCRs of his favorite soap-opera tapes ( _The Bold and the Beautiful_ and _The Young and the Restless_ ); she ships them out wherever he is. The boss's other secret vice is really kinky: The man can't stop buying office supplies—pens, computer disks. Toney giggles when she conjures the image: big bad old bluesman cruising Staples for a deal on double-A batteries. . . .
"Home? Can you believe that? I'm going home. Gonna stay in my pajamas till I have to get on the next plane. Just sit there . . ."
B.B. is fantasizing about his imminent three-day stopover in Las Vegas. Having rejoined the tour in Dallas after a couple of weeks' absence, I find B.B. looking tired, a bit gray and worrying aloud. "I've always had confidence," he says as he waits to mount the outdoor stage amid a cloud of Texas-size crickets. "I've never thought of going onstage and passing out or anything, just going out trying to make people happy and having fun myself. But it started two months ago, when my diabetes got really out of whack—I started not having the confidence I've had. Each night is like, 'God, can I make it? Can I make it?' So if I can make it tonight and tomorrow . . . . "
We have two nights left out here. And B.B. says he knows he could not make it—anywhere, any time—if not for his cadre of men. Their loyalty and honor have sustained him on the road. Despite our sue-me, sue-you times, B.B. says, their working arrangements are simple. "Just this," he says, extending a big hand and shaking it in the air. Bandleader James Bolden has made that handshake last more than twenty years; few have been with B.B. less than a decade.
B.B. calls it a family: bass player, rhythm guitarist, keyboard player, two trumpets, one saxman, a pair of drummers, roadies, assistants, drivers, security men, and Sherman Darby, the calm, durable tour manager given to watching Truffaut movies—in French—while he does paperwork on the bus. And Norman.
Every busy statesman must have his attendant, and for close to forty- five years, on and off, B.B. has been aided, annoyed, and abetted by one Norman Matthews, a fellow Mississippian and one of his very oldest and truest friends. They met on Norman's birthday in April 1947. They have dodged whining bullets in boardinghouses together, drowned happily in the company of willing women and—before the days when B could command a nightly plate of smoked salmon backstage—shared the last of the tinned sardines and pork and beans.
"You don't think that's good, young lady, you try it when you're hungry," Norman says. He still has a photo taken on a day they were "just plain starvin'" somewhere in the Carolinas. A desperate B.B. went fishing—and ended up in the creek, drenched and still hungry. Norman says he could always finesse a little something. "I dealt a little three-card monte, picked up some change . . . . "
Now that B.B. is mainly vegetarian, and in need of many small meals, Norman is ever at his elbow with a Diet Coke, a sandwich, a Tupperware container rolling with hard-boiled eggs. Norman accompanies B.B. on his yearly tune-up at the Pritikin Longevity Center, in Santa Monica, and absorbs the directives of its dietitians and doctors. Norman cooks a mean plate of butter beans; he knows the names and locations of all B.B.'s lady friends, children, and grandchildren who show up along the road. The two don't have to talk much. Hand signals, shrugs, the semaphore of a raised eyebrow do the job. "I have one brother that's as close as he is," B.B. says. "Norman knows everything. And he tells me _everything_ —whether I want to hear it or not."
The loyal bodyman is an honored tradition in soulful circuits: In the Sixties the self-styled King of Rock & Soul, Solomon Burke, employed a midget—Little Sammy—to boogaloo beneath the fifteen-foot ermine-trimmed train of his cape and keep it snapping neatly behind him. Ship out with James "Butane" Brown and you'd better know how to handle a hysterical woman and a hot comb.
The key to a bodyman's strength and longevity is in his very transparency. Much of the time, Norman is onstage but unseen. "Nobody knows who Norman Matthews is, and ain't nobody about to find out," Norman tells me. Even when B.B. and the band are in summer casuals (tuxedo pants and silk print shirts), Norman tucks himself into a fly tuxedo studded with B.B. King lapel pins and American-flag pins. You can always tell where Norman has been sitting by the small heaps of souvenir guitar picks and Lucilleshaped pins that trickle from his pockets.
Toward the end of every show, Norman walks onstage during the final ovation and gives the boss a cup of cold water and handfuls of the giveaways to toss out. Willie King, B.B.'s son, wraps a snowy towel around his father's wet neck and helps him into a warm tour jacket for the final bow. Norman wields a flashlight to get the boss safely backstage. They make a slow, stately procession, two elderly men stepping cautiously out of the bright clamor and into the darkened wings.
One more show to go—Tulsa, Oklahoma—and B.B. is down. He hasn't slept at all. Didn't yesterday, either. As we load up to leave Dallas, a fretful Norman says the boss isn't even eating. Almost before the city limits, B.B. is finally out, slumbering past the newly carved condo buttes, the grazing cattle, the highway that narrows as we hit Oklahoma and becomes a twolane bordered by trailer parks and ammo shops. He dreams past Okmulgee, where leathery dudes in pickups hoot, honk, and raise arms ringed with tattoos of barbed wire as they recognize the bus with Lucille's likeness. Tulsa is announced by the hiss of brakes. B.B. moves slowly through the front lounge; pillow marks still crease his cheek as he follows Lucille into the hotel lobby. "Gonna sleep some more," he mumbles. "Feel like hell."
Yet four hours later, B.B. is transformed: jaunty Kangol cap, big grin, a wicked cackle as he light-fingers a forbidden bag of chips from the bus kitchenette. He's well enough to talk, strong enough in the hour we sit outside the riverside venue to accept the wild embrace of bluesman Buddy Guy, who has bounded onto the bus after his set in crisp white overalls, bandanna, and glistening Jheri curls. "Where is my King?" Guy is yelling. "I have come to honor my King!"
Along with tour stalwart Dr. John and muscular newcomer Susan Tedeschi, Guy has put the crowd gathered here on the banks of the Arkansas River in a party mood. This being big-oil/cheap-gas country, rackety five-dollar helicopter rides departing from the festival grounds compete with B.B.'s set, which is full, strong, and celebratory. "FIFTY YEARS!" I hear again over the chopper's thwuck thwuck. "THANKYEWWWWW!"
As the eternal flame—Norman's flashlight—guides him back down to the grass, I hear B.B. say to his old friend, "Guess we made it, man."
He is ravenous after the show, chomping on a plate of broccoli, beets, baby corn, carrots. Between bites, he's talking about tonight's walk into the light: "Even when I'm sick, it makes me feel better. I swear to God, it's like therapy for me when they say, 'Here's B.B. King.' And if I'm able to get to that microphone, I feel better. I don't want to stop. Never. I'll retire five or ten years after I'm dead."
He says he will slow down enough to do the things he enjoys. He will spend more time at home in Las Vegas. He plans to fix up the fifty acres he has bought near Indianola, so he can kick back there with the three generations he has begat. And he will keep practicing, straining to hear that sound he hasn't heard yet, coming from himself. If he and Lucille can agree to see a bit less of each other, maybe he'll even get married, provided that his current flame waits. B.B. has a hope chest—nearly $1 million set aside so that any surviving spouse won't have to tussle with that huge crowd of kin when he's gone.
Whenever he gets the chance, when anything crowds him—the business, the women, the kids—he says he will grab a set of keys and do his very favorite, soulsoothing thing. B.B.'s final picture story is not a blues but an American reverie—a rippling mirage, perhaps, since it must take place in the Nevada desert some miles from his town house.
_It is hot—hotter than ever it got in Mississippi cotton fields—but blessedly free of the weepy humidity that darkened work clothes half an hour into the day. The dead quiet is broken by the growing roar of an engine powering up the desert road. The motor was built in 1984, the last year that Chevy made its low-riding half-car/half-pickup, the El Camino. This one has been overhauled and painted a baby blue that vibrates against the parched ochers and reds. An elderly black man in soft, expensive clothes is driving; his only air conditioning is the masterful cool of Mitt Jackson's vibraphone floating from the dashboard. Painted on the truck's side are two small musical notes that announce to the rattlesnakes, the roadrunners, the sleepy buzzards:_
_B.B. King's in town!_
_Elizabeth Kaye_
Elizabeth Kaye is the award-winning author of _The New York Times_ number one bestselling e-book _Lifeboat No. 8: An Untold Tale of Love, Loss_ , and _Surviving the Titanic_ , which was number one on Amazon Singles for a record two months.
A recipient of the Alicia Patterson Fellowship, she has been a contributing editor to _Esquire_ , _Rolling Stone_ , _Elle_ , _Harper's Bazaar_ , and John F. Kennedy Jr.'s George magazine, and a frequent contributor to the Arts pages of _The New York Times_.
She is the author of six books whose subjects range from American Ballet Theatre to the Los Angeles Lakers. The recent collection of her extensive magazine writing about men, _Men: What They Do, What They Think, and Why_ , features profiles of iconic figures ranging from Kobe Bryant and Phil Jackson to Sen. Edward Kennedy and Anthony Hopkins. Other books by Kaye include the memoirs _Mid-Life: Notes From the Halfway Mark and Seven Men_.
She lives at the beach in Los Angeles with her man and their cat.
Nureyev Dancing in His Own Shadow
Celebrated and famously volatile ballet dancer Rudolf Nureyev, captured intimately at the end of a glorious career.
_What would he seem to be if you did not know what he once was? He never was like the others, and he is not like others now. Long ago, Balanchine told him, "Go and dance the princes out of your head." It may be that he tried. Yet in all the ways that matter most, he was those princes and noblemen: He was Albrecht, compelled to dance himself to death; he was Siegfried, stunned and beckoned by a private, exquisite vision._
_There have been times when he felt like the Chosen One, and times when he wondered whether being chosen was a blessing or a curse. Either way, he was indentured by his gift. Ordinary men have careers; Rudolf Nureyev had a destiny. He is aware that dancers often say you cannot be born to dance. "I say you can," he says. "Witness with my life."_
_He was six years old when he discovered dance. Even now, he can summon every detail of that evening: the tasseled velvet curtain, the stage lights of orange and blue and green. And he remembers the deliberate way the curtain rose, and that the stage was huge and deep, and the way a fresh breeze wafted from it and glazed his cheek. "And then," he recalls, "the gods came dancing."_
_At that moment he knew that he would be a dancer. He aspired to be bold and unfettered, to inhabit the air. Whether he aspired to be a god was of scant importance. In every city of the world, audiences gathered to receive him as one. And they were not simply responding to a great dancer; they were responding to something infinitely more mystical and uncommon: an artist who was, as Pavlova and Nijinsky had been before him, the living embodiment of the Dance._
_He took his bows in a hailstorm of roses. He was courtly, then ardent, as he kissed his partners hand. He acknowledged the shouts of "bravo!" with one uplifted hand, a gesture steeped in majesty and insolence. Then he disappeared behind enfolding curtains, and the houselights went back on. And in taking his leave, he abandoned his audience to the pallid company of other mere mortals._
_He was twenty years old when these rites began. He is fifty-two now. And in those intervening years he came to know as much as any man may ever know about what it means to be consumed by a passion._
Rudolf Nureyev opens his eyes. It is eight in the morning. He is tired but cannot fall back to sleep. Last night, after a performance at the Orpheum Theater in San Francisco, he ate dinner, drank a beer and several glasses of wine, returned to his hotel room alone, played piano for an hour, then watched movies on television until 4:30 in the morning. There is nothing unusual in this chain of activity. It was developed over years of traversing the planet on a stateless passport, making his home in the presidential suites of every great and half-forgotten city of the world. Now it is his basic pattern, a proud man's insistence on fashioning a bulwark against loneliness and need.
Nureyev has never liked to be alone. Nighttime visitors to his home have always noted the book-strewn canopied bed, assorted volumes set out like provisions to nourish him through the long night ahead. And though he has come to cherish solitude, he often desires whatever he does not have: Alone, he is apt to crave company; in company he may crave seclusion. And there are no patches of gray; only the irreconcilable tension of black and white.
But even in the years when he could choose from an unending array of women and men, he gave precedence to the adoring strangers who composed his most satisfactory and enduring relationship. "You share so much of yourself onstage," he says, "afterwards you don't want to share. You want to keep yourself to yourself. At fifty-two I don't have a lot of time left to be with me. But if you say that it sounds heavy, _tragique_. It isn't. What it is . . . finally, you marry yourself."
Nureyev enters the kitchen of his hotel suite, padding across the linoleum as silent as a cat. The larder is stocked with his usual fare: packets of instant chicken soup, a five-pound bag of sugar, whole-wheat English muffins, and tea. Years ago his appetites were uniformly voracious, and in a single day he could consume numerous meals, attend half a dozen movies, make love to several people. "We want caviar," he would chant along with Cecil Beaton, "and we want it fresh and we want it now."
At that time, in his twenties, he was bemused by the abstemious habits of the great choreographer Sir Frederick Ashton. In his early fifties, Ashton lunched on boiled eggs. Now, at the age Ashton was then, Nureyev eats boiled eggs, and usually prepares them himself.
"With age you have less desires," Nureyev believes. "You want less and less and less. Eat less. Less love. Less everything."
A few hours later, Nureyev leaves his hotel. He is in a terrible mood. All around him, the air is fragrant and clear. Usually he appreciates fine weather, but now he doesn't notice. There is no special reason for this mood; it seems no more than a foray into the low-grade suffering he tends to enjoy; it has appeared and will depart, as have so many others, with all the meaning of passing clouds. But at this moment his lips are folded into a regal pout, his nostrils are flared, there is steel in his eyes. It is the face of a horse ready to buck, but the body retains its pantherine grace. His walk is deliberate, elegant, contained, his steps long and slow, his hands folded before him, one resting over the other. It is a walk replete with the slightly weary grandeur of a man who would never dream of changing places with anyone, yet has long experienced his own station as a great—and terrible—burden.
Nureyev's Chilean driver opens the door to the San Francisco opera house. Nureyev enters, trailed by his Russian masseur and a sixty-nineyear- old Armenian woman, a friend of many years who rose at six a.m. to make him soup. They proceed to the rehearsal hall, where the American Ballet Theatre company is taking class.
With younger dancers Nureyev will be sly and lascivious and playful, accessible at one moment, at the next, remote. To them, he is a master, a legend, a danseur noble, though most are too young to have seen him when he was, as his mentor Nigel Gosling wrote, "one of those strange haunted creatures that ballet throws up from time to time, through whom some intense, urgent message seems to be passing."
That message was conveyed with such scorching abandon that many critics assumed Nureyev would streak through the ballet world with all the brilliance and longevity of a shooting star. Instead, he stayed on to rouse people who had no serious interest in ballet, forging a vast new ballet audience, while also becoming the first pop icon of the sixties, harbinger of the most blatant and confused sexual era in recent memory. Untamable and defiant, he was part James Dean, part Mick Jagger, part Lord Byron, a blend of fierce sexual authority and the vulnerability of a waif. Later, Baryshnikov's virtuoso performances would athleticize ballet, but it was Nureyev who romanticized it, and without him many of today's young dancers might not be dancing now.
No one, these danseurs say, used the floor as Nureyev did, or had his authority or range of emotions. But lately they also say other things. Why, they ask, does he go on? Why does he dance now that he is no longer the fiery Tatar boy Margot Fonteyn recalls as "a young lion leaping"?
"It's so good to have you in class," a male dancer tells him.
"Because I look so good," asks Nureyev, "or the idea?"
"Well . . . the idea," says the dancer.
Nureyev smiles. "You bitch," he says.
But he pays no attention to what they think. If he paid attention, as he perceives it, he would lose and they would win. The prospect is untenable. He is still fighting his way as he did from the beginning, still embroiled in the furious battle that pitted him first against the father who opposed his ballet training, then against the ballet establishment, critics, and competitors, and finally and most fervently against his own limitations and gravity itself. He was a relentless fighter because he wished so desperately to win. And though he can no longer win in the glorious way that he once did, the battle itself has become a way of life. He can't relinquish it.
Nureyev takes his place at the barre. As he begins his pliés, his expression turns rapt, as if he were lost in some private music. From the outset, he was enchanted by the process of dancing, and class, where that process began, was a respite from the hunger and cold of Ufa, a frigid region dwarfed by snowless mountains, where his family subsisted on baked potatoes. "Class was extraordinary ritual," he recalls. "All unpleasant things vanished."
But today he is too tired to do a good class, and he knows it. And when the young dancers begin their pirouettes, he simply stops and stares. His gaze honed by an uncanny ability to perceive what is wrong in a step, he tells himself, "You could help them create something."
Even in his heyday he was unusually generous in the studio, ready to work endlessly with any dancer who desired his help at any hour of the day or night, his manner typically demanding and uncharacteristically patient. These days he often feels unappreciated. "Nobody's dying to delve into my baggage of knowledge," he will say. In a better mood, he feels differently. "I'm dying to teach somebody," he says with a grin, "but somebody grateful."
And in class, the danseurs practice a la seconde turns, their left legs extended at a 90 degree angle while the pumping motion of their right foot propels them around and around at warp speed. There was a time when Nureyev's execution of this bravura step roused an audience to its well-tended feet, as it did when he and Fonteyn performed _Swan Lake_ in Vienna and took eighty-nine curtain calls, still the most ever taken, according to the _Guinness Book of Records_. But now, as he studies the fast-spinning young men, his expression veers from quizzical to interested to contemptuous. Then he tilts his head, shuts his eyes for a millisecond, and arches his neck, a characteristic gesture. It is the gesture of a man who does not give up, who cannot be intimidated by loneliness or frailty or laughter, the gesture of a man who wills himself on, knowing that the nights of eighty-nine curtain calls are far behind him. Watching the young men dance before him, he knows they cannot approach his artistry just as he cannot approach their youth.
"But then suddenly you can do one movement, one gesture," he tells himself, "and you're younger than all of them."
He is a beguiling and difficult figure. His sense of self is as epic as his distrust of others; together, they render him as impenetrable and imperial as he desires to be.
"I used to be so intimidated by you," a younger male dancer tells him.
"But no more?" Nureyev asks.
"Well," says the dancer, "a little."
Nureyev smiles, satisfied. "Just enough," he says, "to be attractive."
Among friends, or after some wine, he will be unguarded and expansive, and as an evening wears on, he will seem youthful and impish. But more often he will be coiled and watchful, ruled by instinct, a cat prepared to pounce. His attitude toward others is ever changing: Warmth cools to indifference, or is supplanted by ice, and these alterations occur with a suddenness that is inexplicable even, at times, to Nureyev himself. Speaking with friends he can be happy, open, engaged. "Yes, sure, of course," he will say, and then he will change. "I don't know," he says abruptly, dismissively, retreating behind a curtain drawn through will or need.
He can be a saint, he can be a devil; it is difficult to ascertain which role he relishes more. "Stage is cathedral," he likes to say, and he means it. Yet onstage, his back to the audience, he works at making other dancers laugh, teasing them with funny faces and dirty names. "The stage is a cathedral for Rudolf," says a ballerina, not unkindly, "when he's facing front."
Because of his early obsession with dancing, he was never really a child; now that same obsession keeps him from seeming middle-aged. Even at fifty-two, and despite a strong attraction to caution and logic, he is rash, elemental, unrepentant. "You swine," he shouts at a recalcitrant dancer as he slaps him across the face. But there are equally sudden bursts of warmth: the enveloping bear hugs, a delighted "aha!" upon seeing an old friend. Outrageously theatrical and unabashedly genuine, his behavior is both an uncensored reflection of his emotions and his longest-running performance.
He loves to shock, to test others' limits. Speaking of the ballet _Blown by a Gentle Wind_ , he calls it _Blowjob in the Wind_. At a dinner party, he mentions a Russian public official. "He has large dick," he announces to twelve well-dressed, sedate diners, "which he is eager to share."
A woman leans across the bouillabaisse and crystal. "Mr. Nureyev," she says earnestly, "I've always had such respect for you."
Nureyev laughs. "But no more," he says.
At other times he will be humorless, unreasonable. And though he has ceased to underscore his points by hurling chairs across a stage, his reputation for being temperamental makes hirelings dart gingerly, placating him, anticipating his needs, their anxiety to avoid mistakes making mistakes inevitable. Two hours before he is to dance at the Kennedy Center, an assistant hands Nureyev's tea tray to a new driver. Mr. Nureyev will need this at the theater, he explains, with the subdued urgency that characterizes such instructions. Moments later, Nureyev's white stretch limousine pulls up at the stage door, the tea tray perched on the backseat in solitary splendor. A few minutes later, a dilapidated taxi pulls up behind the limousine and an annoyed Nureyev emerges from it.
Yet it is impossible to predict what will ruffle him: He takes bad news in stride just as often as he is incensed by a trifle. Easily hurt and quick to see slights, he can feel his anger coming, dark and forceful as a locomotive. "Don't speak of this," he will warn, "or I will get angry and then I get ugly." To ignore these warnings is to learn that he means them. But the sway he holds over others is not due to temper, but to a sweetness that emanates from him unexpectedly, like perfume from a thorny rose. When Margot Fonteyn was hospitalized he hastened to her bedside, carrying a bagful of _I Love Lucy_ videos and an armful of teddy bears. One night, after an engagement, wealthy guests vied for his attention at a dinner party, but he reserved the seat beside him for a shy and elderly friend. When she demurred, convinced he should sit with someone more prominent, he told her, "I don't need another performance."
Yet he can also be as harsh as Siberian weather. "That Yiddish bitch," he says of a woman whose largely favorable review of his performance mirrored his own analysis.
He exerts a force beyond the obvious pull of his gift and fame, and even those who encounter him in his more virulent moods and vow never to speak with him again keep his photographs displayed on their living- room tables.
His years in the Soviet Union convinced him that no one can be trusted; thirty years as an international celebrity has done little to erase that sense. And this abiding suspicion intensifies his fierce loyalty to the knot of friends he calls "family." Typically his relationships are unusually difficult and unusually sweet. Many of his strongest ties are with women who have the time and inclination to accompany him on the road or to movies and galleries or on shopping expeditions. Often he wants to see two movies and a play in a single day or to organize an elegant dinner for Jacqueline Onassis and Margot Fonteyn on one day's notice. They will exhaust themselves to bring these wishes into being, repaying his affection in ways that border on the pathological. Yet to fully gain his confidence is nearly impossible, and no one gains it in any measure without first passing one of his tests.
"I do not want crusts on my toast," he announced at the first breakfast served to him in the home of Phyllis Wyeth, who was to become a close friend.
"In this house, Rudolf," she told him, "you cut the crusts off yourself." But the most rigorous tests are reserved for himself, and he plunges into them with the fervor of one who has never doubted the therapeutic value of hardship. He was always arrogant, but years of independence and adversity have made him brave, and he insists on bravery in others. "Is good for you to have to struggle," he commented when his manager, who speaks no Russian, was attempting to navigate Leningrad. Years ago, arriving at La Scala to mount _The Nutcracker_ , he informed Italy's prima ballerina, Carla Fracci, that he intended to rechoreograph all of their pas de deux. There were only five days until the first performance. "It is quite impossible," said Fracci. Nureyev turned to the rehearsal pianist.
_"Maestro_ ," he said.
They worked hard for four days, and at the performance, during the final pas de deux, the audience began to applaud, and rose to its feet, shouting "bravo!" During the curtain call Nureyev beamed, knowing as Fracci also knew, that the evening was one of her most successful. Taking her hand, as he brought her forward for yet another bow, he turned to her and said, "You see what it is to have courage."
Even in the classics he was an intensely personal dancer. All that he was and never could be was manifested in his art. It may be that no successful man was ever more primitive or more extraordinarily refined. He was brutal and tender, spiritual and carnal; in the span of a few minutes, he could evoke Christ on the cross and Mephistopheles.
As a poor child trapped in a remote province, he dreamed of Leningrad, home of the Kirov Ballet, spawning ground for Pavlova, Balanchine, and Nijinsky. He was seventeen when he finally auditioned for the academy where Kirov dancers train, seeking to begin serious training at an age when it is customarily completed. His compact muscles were wrong for ballet; the late start presented serious problems. "Young man," a Kirov teacher told him, "you'll either become a brilliant dancer or a total failure. And more likely you'll be a failure."
Three years later, a young French writer attended one of his early performances. "I think I just saw the world's greatest dancer," she cabled her editors. "Master your emotions," read the reply. But soon came countless echoes of her assessment. "There is something of the genius in him," the great teacher Vera Volkova told Fonteyn, "He has the nostrils."
Yet dancing would never come easily to him. Even in his prime, his technique could be ragged, and though it lent his performances an intriguing edge absent from more consistent performers, it frustrated Nureyev beyond endurance. Rehearsing _La Bayadére_ , he practiced until his legs wore out, trying to perfect double assemblés. Performing them, he lost control; mortified, he fled the stage. The next day, he was at work again, applying himself with all the focused intent of Sisyphus pushing the stone uphill. "Everything that Rudy had," says Carla Fracci, "he really obtained with his will."
"Surely he ought to save somewhere," Margot Fonteyn thought when she first saw him dance, certain that the effort he expended on each motion would preclude finishing his solo. She was not yet acquainted with Nureyev's strength or his desperate drive to match the ideal that possessed him. Termed a perfectionist, he disliked the word. "Perfection is sterile," he says. "My ideal is not everybody's perfection."
His strivings to reach that ideal were more intense because he knew he would usually fail to achieve it. "You sense expectation of the audience," he recalls, "and you expect from yourself something extraordinary, you have to astonish yourself, you expect a miracle and you're not sure it's going to be there."
His own severest critic, he plunged into black moods, unsettling members of the Royal Ballet, his adopted company. "He's angry with himself," Margot Fonteyn would tell them, "and with nobody else."
But then, often when he was tired, his muscles would simply go his way. And later, attempting to describe those moments, he would speak of exultation and transcendence. But at the time there was only the leaping and the soaring and the inner voice that told him, _I am_.
And for all the work invested in technique, no work was required to convey his love of dance. Nor did he labor on the passionate urgency that animated every step he took onstage. "That passion is already embedded in me, you know," he says. "I don't have to work on it."
Rudolf Nureyev returned to Leningrad in November 1989. He was fifty-one years old. He had come to dance one final time with the Kirov Company before it was too late. In the considered opinion of many, too late had come and gone. "What variation of the variations will he do?" his former Kirov ballerina's wondered.
Nureyev still worshiped the Kirov, still viewed the company as the most lyrical and sublime. And he still revered the Kirov's former prima ballerina, Natalia Dudinskaya, who had been forty-eight years old when she honored the twenty-year-old Nureyev with an invitation to make his debut with her.
Dudinskaya understood the way that magic is created onstage, and from her Nureyev absorbed what cannot be taught. Speaking of her, he would always extol her power "to sparkle, to make performance," qualities he would revere and revel in for the rest of his life. And in all that she taught him about dancing, she was steeping him in an ideal. This was the nineteenth-century ideal of classicism, one Nureyev could mold himself to readily, for it was the absolute expression of what he had always been. "I am a romantic dancer," he used to say, and it was a precise description.
But then, in the silence after a performance, he would remove his makeup and costume and slip back into a Soviet world he instinctively opposed, where his refusal to join the communist party or go to indoctrinations or sit on the company bus singing Soviet songs made clear that he had no stomach for living in the expected manner. "Like amoeba," as he describes it now.
He defected in June 1961, a twenty-three-year-old with thirty-six francs in his pocket, ignorant of Western ways, commencing a life that would be forever haunted by nightmares in which he made desperate runs for the border. And now, returned to the Kirov, he was pleased at the prospect of dancing once more on its stage.
"It will be good for you," he told himself, "cleansing experience."
But for twenty years he had driven his body through as many as 250 performances a year, a feat just this side of masochism. And though his artistry was still matchless enough to cause the young Kirov ballerina he partnered to marvel, even Nureyev had long admitted his feet were shot and that his jump and elasticity were gone. But then artistry outlasts facility; that is the way things are. And thirty-two years had passed since the night of Nureyev's dazzling Kirov debut, and now he was forced to stop rehearsal in order to rest, and Dudinskaya walked with a cane.
Nureyev's Kirov engagement underscored that he had delayed the reckoning for as long as he could. "I have outlived my time," he would say a few weeks later, and it was true, of course, as it always becomes true for dancers. The process is swift and punishing, though considerably more so for some than for others. It had not been easy for Erik Bruhn, the brilliant Danish dancer who was the love of Nureyev's life. "You panic," Bruhn said, looking back at when he began to lose his powers. "Whether you talk about it or not it is with you day and night."
For Nureyev, it was no better, and likely far worse. Unlike Bruhn, or Baryshnikov, for whom performing was often painful, Nureyev was one of those theatrical creatures who found onstage a universe far stronger and more appealing than life.
"What you have onstage may be untrue," he often said, "but maybe it's more true than the other."
Over the years, knowing he needed to dance less, he had attempted other ventures, only to be disappointed. Lukewarm reviews greeted his brief foray into movies; faint praise or none was bestowed on most of his original choreography. Most recently, after an agonizing seven-year tenure, he resigned his prestigious post as artistic director of the Paris Opéra Ballet, a position widely viewed as ideal for him. And though his directorship breathed life back into that moribund company, he had been so determined to keep dancing that he dedicated six months a year to performing in any far-flung cranny with a stage. And his protracted absences from Paris— in violation of his contract—angered the company and the management. "Paris Opéra Ballet does not give sabbaticals," the board's president said.
And now he was touring the United States in a stage revival of _The King and I_. His drawing power was still phenomenal, and in nine months on the road, the production would make $11 million, but his reviews were poor and sometimes downright insulting, and in any case, he would rather have been dancing.
Years before, he told Bruhn that unless he danced incessantly he would lose belief in himself. Now, dancing much less, he had become more easily hurt and needful of reassurance, maladies apt to worsen as time passed.
For inexorably, his life as a dancer was ebbing away. It was, he would explain, a "gradual descent," the demand for him diminishing, fewer engagements. In the spring of 1990, his dance troupe, Nureyev and Friends, would play a dozen cities in Mexico, Canada, and the United States, and the tour, he assumed, would be his last. "There's a natural kind of death," he tells his manager. There are times when he is fatalistic about this and times when he isn't.
He has always dreaded this moment, steadfastly contriving to postpone or ignore it. Turning thirty-nine, he was scared and depressed. Never again, he told Bruhn, would he dance Siegfried in Swan Lake. But he continued for ten more years, until critics who had lavishly praised him began beseeching him in print to find another outlet for a talent that was still formidable, but utterly changed.
"You want me to stop? No, I'd die," he said to Clive Barnes eight years ago.
"I don't know. Maybe I will die," he says now, "God knows."
For Rudolf Nureyev, reaching middle age was bound to be uniquely traumatic, not only for him, but for his audience, who did not wish to age any more than he did. He had been beautiful, of course, but there was more to it than mere aesthetics. As Nijinsky had before him, Nureyev had come to represent human possibility. That was at the heart of his legend. His own youth had burned so intensely that it could ignite the forty-two-year-old Margot Fonteyn, who was about to retire when she met him, having danced her first _Swan Lake_ when Nureyev was three months old. And because he had embodied youth and seemed to bestow it, it was no wonder that dancers and audiences presumed that Nureyev would never grow old.
One night, preparing for the tour he assumes will be his last, he does a barre backstage. "This is the crux of my training," he says, executing a slow turn that requires utmost control. Then he stops, his face pensive. "Crux . . . crux . . . is that like crucifixion?" he asks.
Fourteen years ago, seeing the end of his career encroaching, the great Russian danseur Yuri Soloviev committed suicide. "You have to be a dancer to understand," Nureyev had said upon hearing of it. Every dancer could recite a grim litany of that single theme. And one night a Russianspeaking man comes backstage and tells Nureyev of a retired Kirov dancer who currently does wardrobe for college productions. Listening, Nureyev tightens his lips and shakes his head, as if trying to ward off what he is hearing. Later, repeating the story to a friend, a shudder goes through his body. "It is grotesque," he says. And now, at fifty-three, he is no longer being invited to dance leading parts. "So when I went to Kirov," he says one evening, suddenly sad, as if the truth were dawning on him at that very moment, "when I went to Kirov . . . that was last time."
"But I can still pull it off, no?" he asks. "Have to do longer classes and practice in nice warm weather in good climate. But I can pull it off. It would be no problem."
Curtain time for _The King and I_ is 8:30. At that time precisely, twenty minutes before his first entrance, Nureyev strides through the stage door. In the old days, to feel an adrenaline rush before a performance, he worked himself into a rage and became verbally abusive. "I always thought it was funny," recalls Margot Fonteyn. "But you could take it personally."
These days, the reason for his last-minute arrival is a source of speculation among crew and cast. "He does it to get that rush," says a stage manager.
"He does it," says an actor, "because he can."
Backstage, Nureyev is unfailingly sweet with the dozen youngsters who play his children. A few months into the tour, they wrote a newspaper and proudly showed it to him. "Is it good?" they asked.
"Is very good," he told them, "but you need gossip."
When he is not onstage, he retreats to his dressing room and plays the piano. Convinced he learns best from what is most difficult, he has been teaching himself Bach's _Das Wohltemperierte Clavier_. He plays haltingly, but prettily. "Is probably absurd," he tells a friend, "but it gives me thrill."
Since arriving in the West, he has talked of becoming a conductor, a career with the advantage of enduring past middle age. Recently, his manager investigated the possibility of his enrolling at Juilliard, and he has been encouraged by two European conductors. "I need third blessing," he told Leonard Bernstein a year ago.
And now, as he plays, a worn Neiman Marcus terry-cloth robe is draped over his shoulders, obscuring his bright silk costume. His feet, brutalized from the years of dancing, are covered by heavy gold clogs. On the dressing table, Lancôme mascaras and a sienna lip pencil are tossed beside bottles of back- and chest-pain remedies, and a pile of half-read fan letters filled with long-familiar phrases: _Thank you for all the joy . . . your genius will remain with us . . . I will never forget how brilliant you were . . . forever etched in my memory. . . ._
Hearing his cue, Nureyev hurries toward the stage, the robe still over his shoulders, like a boxer about to enter the ring. His dresser follows, carrying a Thermos filled with heavily sugared tea. Waiting in the wings, Nureyev sips the tea, winks at chorus boys, or pinches them, and calls the cutest ones Poopsie. Stepping out of the clogs, he steps onto the stage, and his dresser turns the shoes around so he can slip back into them easily.
Dancing, he loses himself in the thrill of movement. Acting, he watches and criticizes himself. "Is like a sports commentator," he tells a friend. "I am always thinking. Now he runs, now he turns."
"Your king enjoys being king," a fan tells him backstage.
"Why be king," he asks, "if you can't enjoy it?"
Nureyev is preparing to leave the theater. There is just the barest chill in the California night air. But he is always cold, and now he dresses in a wool sweater, buttons a heavy Missoni sweater over it, dons his large tweed coat, a multicolored Kenzo cape, and his beret.
Years ago, leaving a stage door, he required police protection. Rudimania was in full force then, that pre-Beatles orgy of mass hysteria, complete with wild-eyed fans straining against barricades. Now wherever he goes, a small group of fans still waits, and sometimes this pleases him, and sometimes it reminds him of how much things have changed. "You're ten years too late," he once told an autograph seeker. On this night, a dozen men and women stand at the stage door.
"Mr. Nureyev," a man calls to him, "I'd just like to tell you how wonderful you are."
Nureyev smiles. "Don't restrain yourself," he says.
He signs the programs, then bids the fans good night, moving swiftly to his waiting car. He gathers his heavy coat around him and glances up disapprovingly at the dark, starry sky. He pulls the coat still tighter. "I'm afraid of the cold," he says.
He does not look back. He never did. "Look back," he used to say, "and you fall downstairs." Still, at rare moments, it pleases him to recall his past.
"Nothing was extraordinary in a way," he says of the years following his defection. "I went to Paris Opera—I danced there already in my mind. I went to La Scala, to Covent Garden—I already danced in my mind everywhere."
Once, attending a play, he had somehow known the entire plot. He couldn't understand why. "Then it dawned on me," he says, "that a few years before, they gave me the script to read. In a way, coming to the West was like that. I already read that script, I wrote it myself, and now it is happening according to that."
As a young Kirov dancer, he dreamed of dancing with Fonteyn, though prior to meeting her in 1961, Nureyev knew neither what she looked like nor her age. "But name," he says, "was magic."
Their partnership was to be an inspired union of the world's most elegant ballerina and its most explosive danseur. And their eighteen-year age difference and the disparity in their backgrounds and training were subsumed by a shared aesthetic that permitted their onstage melding to be seamless and sublime. For fourteen years they were in such demand that when the Royal Ballet performed abroad, they danced every performance, while a generation of the Royal's ballerinas languished in the wings, and most of the company's danseurs, knowing their day would never come, disintegrated into alcoholics.
But for Nureyev and Fonteyn, those years were an enchantment intensified by knowing it could not last. "If I'm sensible," Fonteyn thought after their first few performances, "I would retire now on this high note." But she went on, as enthralled with him as he was with her, neither of them wanting it to end, and Nureyev dreading it particularly.
"When my time comes," she once asked him, "will you push me off the stage?"
And he had answered, "Never."
But in 1980 Fonteyn was sixty years old. Nureyev was forty-two and still dancing Romeo, while she had forsaken Juliet for Lady Capulet. She often spoke of retiring.
"Why don't you just keep going," he would say.
"I've done that," she finally told him. "I've kept going for a long time."
Then she was gone, and he found himself wishing that Gelsey Kirkland could provide for him the igniting force he had provided for Fonteyn. But that did not happen, of course. The glory of their years together was never to be replicated. And though many fine things came to Nureyev after Fonteyn stepped aside, nothing would ever be quite the same.
Like anyone, he has good nights and bad nights, and the bad nights can be bad indeed. Those are nights when it seems that no one is more alone than a man who owns six extraordinary dwellings but has forsaken his homeland.
Information was always his one real possession, his means to satiate curiosity and fill the void. "I want to be pupil," he always insisted to choreographers.
His passion for learning was obvious, and noted by anyone who saw him fixate on Balanchine or Ashton during rehearsals, or watched him navigate a museum, magnifying glass in hand, or observed his excitement upon being given the complete works of Shakespeare, or knew of the intent way he read all of Byron before choreographing _Manfred_. But for Nureyev, knowledge is like sex: It soothes and tantalizes but does not save, and on bad nights he is apt to dismiss his assiduously gathered information as "haphazard," and sink into his resident sadness.
In the beginning, there were countless distractions from that sorrow. "I was in love with three people," he recalls. "I was in love with Margot, in love with Erik Bruhn, in love with Ashton." And wherever he went, men and women desired him. "To know what it is to make love as a man and as a woman," he says now, "is special knowledge."
But within this libidinous mélange, Erik Bruhn was an exception: The only person perceived by Nureyev as a peer, he was also the only person who could wield power over him.
The West's leading danseur, Bruhn was an exquisite Dane mired in all the psychic baggage his nationality implies, a perfectionist who could give an impeccable performance and come offstage feeling suicidal. Nureyev was a nineteen-year-old student when he saw a film of Bruhn dancing the _Black Swan_ pas de deux, and was transported by a reserve and simplicity so counter to his own flamboyance.
"Bruhn is cold," a student commented.
"He's so cold he's like ice," Nureyev answered. "Touch it and it burns you."
And from the moment he saw Bruhn dance, his defection was assured. "Whether as friend, lover, or enemy," he told himself, "I have to go to that camp and learn it all." Within a year, he was in Copenhagen studying with Bruhn's teacher, and he had become Bruhn's lover.
Nureyev was twenty-three then, young and very raw, still developing his technique, while the thirty-three-year-old Bruhn was polishing what critics uniformly hailed as technical and artistic perfection. But Nureyev was hungry, while Bruhn, having always lacked the stimulus of competition, had long sensed himself at a dead end. And now he found a tonic in Nureyev's dancing, and inspiration in the unabashed romanticism he had never had. At the same time, he found a threat, and he was not alone in that.
"Some people," Bruhn said later, "said that Rudik came out of Russia for the express purpose of killing me." He never believed that himself, but his mood often turned morbid and dark. And he accused Nureyev of trying to unseat him.
"How can you be so evil?" Nureyev answered, and burst into tears.
Still, ambivalent as he must have been, Bruhn helped Nureyev in every possible way and even coached him for his first Giselle with Fonteyn, an event swamped in such endless hype that it was destined to either break Nureyev or make his career in the West. But afterward, when the audience response turned to pandemonium, and Nureyev, overcome by the triumph of the moment, knelt and kissed Fonteyn's hand, Bruhn fled the theater. "And I was running after him and fans were running after me," Nureyev recalls. "It was a mess."
And when Nureyev came to be perceived as the West's leading danseur, they were pitted against each other. "I mean, the press saw us together," Bruhn said, "and they watched us like hawks, and it was as if they had placed bets on which of us was going to survive."
Their romance was too tempestuous to last, though their working relationship continued, always strangely diabolical, as if each existed to be a challenge to the other, and to fill the other with despair. "After seeing your Albrecht," Nureyev once told Bruhn despondently, "l don't think I can ever dance it again." Yet, their professional alliance remained productive for a quarter of a century.
Erik Bruhn was fifty-eight when he was diagnosed with lung cancer. After that, there was not much time. Nureyev hastened to the hospital. Bruhn had always hoped he would not die in one of his depressions because, "Then, everything would seem meaningless," he once explained. But now, in a Canadian hospital, the two men spent the day together, and afterward it was said that it had been one of their very good days. The next morning, Nureyev returned to the hospital. Bruhn was in a coma.
Later, Nureyev's bedside visit was related to Carla Fracci, Bruhn's primary partner and one who knew of their hold on each other. "Erik waited for Rudy," said Fracci, "before he would let himself die."
It is almost two in the morning. Nureyev sips wine in the hotel bar. Gradually the night turns into one of those occasions when each conversation takes a dark turn and all triumphs attenuate into sorrow.
There is the Royal Ballet's misuse of him after Fonteyn retired, the Boston Ballet's failure to credit him for his production of _Don Quixote_ , which assured the company's survival. "Very ungrateful of Boston Ballet," he says. "However. What other cheerful matters?"
Lately he has been preoccupied by his replacement at the Paris Opéra Ballet by the thirty-one-year-old Patrick Dupond. "He's very nice boy," Nureyev says. "He's charming at the dinner. But he doesn't know classical dance."
Not all that Nureyev did as the Paris Opéra Ballet's artistic director was successful but it was all the product of an original and intriguing mind: a Cinderella set in the Hollywood of the 1930s; a production of Henry James's _Washington Square_ set to music by Charles Ives; the commissioning of works from all the great modern choreographers, from Martha Graham to Paul Taylor to Merce Cunningham to Glen Tetley.
"It isn't nothing," he says. "It isn't shit. They _know_ it. But they won't say it."
A ragged woman enters the lobby. She approaches Nureyev, asking him for money in thickly accented English.
"You are from where?" he asks.
"From France," she says.
Nureyev pulls nine crumpled dollar bills from his pocket, slowly smoothing each one against his leg. He hands the bills to the woman. As she walks away, he shakes his head. "Why I help the French?" he wonders.
He finishes his glass of wine. He is silent for a moment. "Nobody remembers me," he says finally. All evening, people have asked for his autograph, but the attention did not register. In the distance he hears Elvis singing "Suspicious Minds." "What happened to Elvis?" he asks.
And suddenly he seems bereaved, and the glamour of his past seems very long ago. Once, there were trips with Bobby Kennedy and Margot Fonteyn on Onassis's yacht. And there was the night Montgomery Clift cooked him a steak, and the evening Princess Margaret refused to stand up because her sable coat was shorter than his, and there were the dinners he gave for Maria Callas at his villa in Monte Carlo. Now, so many years later, his New York City living room is dominated by two mustard-colored velvet couches purchased from the Callas estate.
Nureyev raises himself up from the banquette. As he slowly crosses the empty lobby, his loneliness seems grim and basic and unassuageable. He rings for the elevator, and once inside, moves into the corner, holding on to the brass rail as if it were a ballet barre. Then the elevator door closes and he ascends to his vast suite, a solitary traveler in a world without memory.
In February, Nureyev is in San Francisco and no longer certain that his ballet troupe will go on tour.
"It's off, it's on," he tells a friend. "Every day there is big question if it's happening or not happening." He doesn't bother going to class. Without incentive, he sees no point in subjecting himself to the strain. "After three days off, I'd faint," he said nearly twenty years ago. Now he stays away for a week.
"How can you?" a dancer asks.
"With pleasure," he says.
Then the tour is set, and each day he calls the presenters to complain about the scheduled program. "Is too thin," he keeps saying. He wants to add Flemming Flindt's _The Lesson_ but is told it is too expensive. "Then I just will not go to any official receptions," he replies. "You don't respect my choice, I don't respect your effort." A week later, he gets what he wants.
At month's end, American Ballet Theatre comes to San Francisco for a fiftieth-anniversary gala and two weeks of engagements. The company's new executive director, Jane Hermann, has long been a friend of Nureyev's, briefly fueling speculation that she would ask him to take over the company as artistic director when Baryshnikov quit. Then it is rumored that she would invite him to take part in the gala, but she doesn't. In private, he speaks bitterly of ballet politics and of feeling passed over; in public he maintains his diva-like exterior. The evening of the gala, his chauffeur drives him to the restaurant where he is to join the stars who took part in the event at a celebratory dinner given by Hermann. The chauffeur circles the block of the restaurant, unable to find a place to park. "There's no parking till midnight," he explains to Nureyev.
"It's midnight in Paris," Nureyev says.
Having arrived before the other guests, he drinks a beer in the bar. He hates to wait but does so now with equanimity. Soon people in evening clothes trickle in, among them the columnist Herb Caen, who often saw Nureyev dance in San Francisco. Now he tells Nureyev about the gala. "Best performance I've ever seen," says Caen.
Nureyev's mouth tightens. "Good," he says.
And as Caen describes the evening's program, which included a brief film clip of Nureyev dancing _Raymonda_ , Nureyev listens politely, his proud face upturned as if posing for Scavullo. In a few days his frustration will boil over in a nasty public scene with Jane Hermann, but for now, a man who has known Nureyev since the early days watches him from a corner, "How gracious Rudolf is," he thinks, "about his star having fallen."
The Rudolf Nureyev Farewell Tour opened in Querétaro, Mexico, in March 1990. Nureyev arrived in Mexico feeling ill, shuffling as he walked, and calling his condition pneumonia.
"How will he ever recover in time?" the dancers asked one another.
This is the second year that Nureyev will dance only character parts, having scaled down a repertoire that previously included Balanchine's _Apollo_. Recently the Balanchine estate refused to give him permission to dance it, one more sign that it is time to accept what has long been obvious. On opening night, while Nureyev dances, two of the company's ballerinas watch him from the wings. Among themselves, they describe his movements as "poetic . . . profound . . . distilled," and as the tour continues they watch him every night, and sometimes they weep, moved by how much he loves the process of dance.
Long ago Nureyev said, "The only voice you listen to is that of your talent." Now he says, "The only critic is a full house," and as he travels to a succession of what one dancer calls "suburbs and outskirts," the houses are full consistently.
As the tour reaches its end, presenters call Nureyev's manager, requesting Nureyev and Friends for 1991. The offers are from Grand Rapids, Philadelphia, Athens, Ohio—hardly the world's great venues. But to Nureyev, all that matters is that there is still a demand for him.
All that he had ever done implied that the end would come with a bang, but it is not working out that way. And maybe that is just as well, for a whimper might be kinder. And maybe the reckoning can be staved off a little more, and in the time newly bought maybe the inevitable would seem more tolerable.
And if he is not the most cautious custodian of his legend, perhaps he has come to believe what Erik Bruhn said several years before: "lf we go on beyond the age when things become difficult, it doesn't really matter, because even if we lose face our achievements have already been recorded in history."
One night, standing backstage, he watches his protégé Charles Jude, a French-Vietnamese dancer with a distinct resemblance to the young Nureyev. "After you," Nureyev once told Fonteyn, "he's my favorite partner."
Jude is dancing _The Sleeping Beauty_ grand pas de deux, which Nureyev used to dance in a manner as thrilling as it was exquisite. And as Jude completes a series of grand jetés, the audience begins to cheer. Nureyev smiles. "Fireworks are for youth," he says.
Moments later, it is time for him to perform Othello in _The Moor's Pavane_. When he is finished, the ovation begins, as it has each night. The audience is fervent. And it is not merely a performance they applaud. They are applauding a life given over to the Dance. Nureyev stands before them, regal as he was as a youth. Still they applaud him. He bows once more. The slight smile he allows himself is incandescent.
Rudolf Hametovitch Nureyev, contract player who never cancels, enters yet another airport. It is eight in the morning, and he is headed from Washington, DC, to Thunder Bay, Ontario. He has just learned there will be a layover in upstate New York. He is not pleased. "Fucking Buffalo," he says.
He sits and reads the paper. Visible behind him, through a window, there are huge jets like those he emulated in the days when he could leap and soar. "Is very impressive when jumbo jet moves on the runway," he says, "and goes more to the ground before it goes up, and then—this extraordinary thing—it takes off and flies."
But now Nureyev walks deliberately onto the airplane. He regrets nothing. He wants only to go on, to move forward as he has since he was born, fifty-three years ago, on a train speeding through the Ural Mountains.
_No figure in dance is more admired by Nureyev than the American choreographer Paul Taylor. A year ago, outside Cairo, Taylor found a tiny beetle pushing on Cheop's pyramid, as if trying to scale a structure that has become synonymous with the human spirit. Taylor brought the beetle home and placed it in a cricket cage. Like all beetles, this one makes a ball of grass and twigs and then rolls it. "But they don't just roll it the easy way," says Taylor, "they roll it uphill. And they're programmed to do that. It's not that they decide. Every day the beetle rolls that ball. And every day it tries to reach the top of the cage. It can't. It's impossible. But that doesn't stop him from trying."_
_With love, and without irony, he named the beetle Rudolf._
_Jeanne Marie Laskas_
Jeanne Marie Laskas is the bestselling author of seven books, including her most recent, _Concussion_ , which was made into the 2015 Golden Globe-nominated movie, _Concussion_ , produced by Ridley Scott and starring Will Smith, and was longlisted for a PEN Literary award. Her other books include _Hidden America_ , based on her GQ story about coal miners, for which she was nominated for a National Magazine Award, and a trilogy of memoirs: _Fifty Acres and a Poodle_ , _The Exact Same Moon_ , and _Growing Girls_ , which grew out of her syndicated personal essay column in _The Washington Post Magazine_.
Laskas writes regularly for _The New York Times Magazine_ , _The New Yorker_ , and _GQ_ about subjects ranging from migrant workers and fake hit men, to cowboys, airships, mules, and the White House. Her work has appeared in _Esquire_ , _Smithsonian_ , and numerous other magazines and anthologies including many in the _Best American_ series. She has won more than a dozen Gold Quill awards for Excellence in Journalism.
A professor at the University of Pittsburgh, where she teaches and directs The Writing Program, Laskas is the founding director of the university's Center for Creativity. She lives on a horse farm with her husband and two daughters in Scenery Hill, Pennsylvania.
The New Face of Richard Norris
Richard Norris undergoes complex transplant surgery, giving him a new face and a new life, but leaving him with many of the same problems.
_Before we bring him in, maybe we can open the floor to some questions. This will be your first time meeting him. He's very comfortable with people evaluating him. Because right now he's being looked at almost as an experiment. Which he is. He's a human-subjects experiment._
Richard Norris was twenty-two when he shot himself in the face. This was back in 1997. He doesn't remember how or why it happened, but his mom, who was three feet away, said it was an accident. She remembers pieces of Richard's face showering her body. This was in the living room. The gunshot had blown off his nose, cheekbones, lips, tongue, teeth, jaw, and chin, leaving just his wide brown eyes and a swirl of nameless twisted flesh.
The miracle that would come to define Richard's life begins with these tragic details. Like most miracles, with each retelling, the edges of the story sharpen, the colors become more vibrant, and the shadows disappear. Ashamed of his appearance, Richard became a hermit, living for nearly a decade on a foggy mountaintop in rural Virginia with his parents. They covered the mirrors in the house so Richard wouldn't have to look at his hideous face. He stayed in his room even to eat, wore a black mask on the rare occasions he came out. According to legend, one time the cops stopped him at gunpoint, mistaking him for a robber.
Then one day, searching on the Internet, his mom found Eduardo Rodriguez, a Baltimore reconstructive facial surgeon. He promised Richard he would make him normal. Over the next few years, Rodriguez performed dozens of surgeries using Richard's own flesh, fashioning a nose-shaped appendage out of tissue from his forearm and a small chin out of flesh from his legs, but these crude approximations failed to make Richard normal. Meantime, Rodriguez had a grander idea in mind. He was driven to achieve perfection. He had been practicing face transplants on cadavers. What he envisioned for Richard was the most extensive transplant any surgeon had ever attempted: He would give Richard a whole new face.
"It's showtime," Rodriguez said one day.
"You're my godsend," Richard's mom said.
"Let's do this thing," Richard said.
The surgery started at dawn on March 19, 2012. The face of a recently deceased twenty-one-year-old man came off as one solid flap, skin, muscle, bone, nerves, blood vessels, tongue—everything as one piece. Rodriguez removed what was left of Richard's disfigured face, dissected down to the skull. He attached the new face midway back on Richard's scalp. He stabilized it with screws, tapped the jaw together, and finally draped the skin and sewed it down like a patch on a coat or a pair of jeans.
_You can see the junction; the incision actually goes here in the coronal, extends in front of the ear, and goes posteriorly all the way down, uh, to the neck._
Rodriguez and his team worked nonstop for thirty-six hours, and when they were finished, Richard's mom looked at her son and felt like he was somehow resurrected. "We have Richard back!" she said on the phone to Richard's dad, who had not had much of anything to say for many years. With his new face, Richard, now thirty-nine, became a media sensation for a time, the story of the miracle told many times over until it hardened even in Richard's mind into a kind of precious jewel.
_Maybe we can scroll through some of the clinical photographs while we're talking. I feel very happy about the bony union here. That's the donor palate. And that's the donor floor mouth. The donor hair, it's a little darker than his. His is a little bit more salt-and-pepper._
Most of the thirty or so people gathered in the conference room are wearing white coats or lanyards or both, and they sit visibly captivated by the photographs Rodriguez is describing. The mood is electric, scrambled, like a show on opening night. The pictures show Richard, who's waiting in an adjacent room for his cue to enter. The expectation lends an extra edge of drama to the presentation. He's flown up here to New York from the foggy mountaintop where he still lives, so that the assembled doctors and other clinicians at NYU Langone Medical Center can meet him.
Rodriguez is an imposing figure, tall and broad, with a big dimple on his giant chin, wide pinstripes, cuff links, and unbuckled galoshes affecting a disheveled nonchalance. In the wake of his world-famous work on Richard, he was just named NYU's chair of plastic surgery, a substantial professional promotion ("like I've just been handed the keys to the starship _Enterprise_ ," he told me). In part, he's been hired to get NYU into the face-transplant business, and today, as the hospital begins the process for its first one, he's brought his star patient before his new colleagues.
_One of the things you'll notice is he has a couple of these scratches. He tends to pick and scratch a bit._
Since the first face transplant, in 2005, only three American hospitals have performed the procedure. Many of the twenty-eight transplants were partial, sections of the face transplanted from deceased donors. Richard's transplant was a full face and is said to be the most ambitious ever. Rodriguez likens the medically complex procedure to the Apollo moon landing.
Surgical difficulty aside, the fact that Richard didn't need a new face to survive raised an ethically grim question: Is a "life-enhancing" surgery worth the risk? There was a good chance he'd die—either on the operating table or later, if his body rejected the face. Of course, for people disfigured like Richard, the breakthrough represents something far beyond a mere enhancement. Here was new hope for millions of people disfigured by trauma, burns, disease, or birth defects. Wounded warriors suffering ballistic facial injuries would now have a surgical option that would go light-years beyond the currently available treatments. No more Band-Aid cosmetic surgeries. No more skin grafts that might only complicate your appearance. Now you could get rid of that face and replace it whole. "We've gone beyond the boundary of what we thought was even possible," Rodriguez tells me.
One by one, some of the specialists in the conference room who had a chance to evaluate Richard earlier today stand up to speak of their findings. Concerns emerge, principally about Richard's state of mind. Has he become too emotionally attached to Rodriguez, the medical attention, the fame? What will happen to him now that Rodriguez has moved on to a new hospital, new face transplants, new miracles?
_We were both struck by how good he looks and the really excellent aesthetic result._
_He reported to me no chewing or swallowing problems._
_I was having trouble understanding him._
_I asked: Do people understand you? Are you mostly intelligible?_
_He hasn't done any exercises._
_He said he just wants to move on, do his own thing._
_I think he's maybe overwhelmed, like you said._
_He is not in any kind of psychotherapy._
_He seems to have somewhat habituated to all the media attention._
_He's sort of had this Mick Jagger status._
_He feels there's a sense of abandonment._
_I did not get the sense that he was open to therapy at this time._
_In terms of any concerns about suicidality or low mood, severe depression, I would say that he denies it. I don't know fully if that's exactly accurate. I would want to speak with him again._
When Rodriguez gives the nod, the door flies open and Richard saunters in, dressed in a bright purple Baltimore Ravens hat and jacket. He's been living with his new face for two years now, and he's undeniably attractive—clean-shaven, youthful, the kind of guy you would hire to run the front office. He takes a seat facing the crowd, arms splayed out, cool as Justin Bieber on a late-night talk show. Everyone stares at him, and some cock their heads. He's used to this; sometimes people applaud. Is he smiling? His new face doesn't move a lot. Does it move at all? He might be smiling, or it might just be the will of the room. His eyes, the one part of his original face still intact, dart like anyone's eyes, and I find myself chasing them, the only reliable clue as to what might really be going on in there.
One of the things I wanted to know when I first reached out to Richard was how he felt about the miracle. What was it like to walk around with someone else's face? I thought it might be kind of unsettling, or confusing. You're chewing with another man's teeth? When I wrote to him to ask, he told me he had an agent. Cal Ripken's agent, he pointed out. He said everything about his new face was great. He has received thousands of letters from fans. One of the fans is now his girlfriend. She lives in New Orleans. He said he was planning to go meet her in person. He said he was in college now, wanted to focus on school, on being normal. Then he invited me to the foggy mountaintop. The fog was famous, he said, had recently made national news when it caused seventeen pileups involving ninety-five vehicles in one night.
When I get there, the sun has already burned the fog off the morning, which is oddly disappointing. Richard's house isn't actually at the top of the mountain. There's a street carved about midway up with a dozen or so homes, and his is a small yellow double-wide with red trim, a carport, and a for sale sign out front. The storm door has a bear etched into the glass. Richard opens it and welcomes me inside. He's wearing a black Under Armour shirt and cargo pants and he's thin, old-man thin. His posture has curved into a slump from years of hanging his head low, from years spent feeling he was hideous to look at, so now he has to make a conscious effort to stand up straight.
He seems nervous. His hands tremble, bringing constant sips of water to his mouth. His lips can't quite grip the bottle, so each sip is more a little pour. He fights a constant drool with the help of a towel. His new face is a marvel nonetheless. _It's a new face_. Wide and open, the cheekbones of an Irishman and the wrinkle-free complexion of a college kid. It's difficult to reconcile the youthful face with the body of a man nearly forty. I am trying not to stare. I am trying to stop looking for the seams, where the new connects to the old, the eyelids, the neck, the scar in front of his ears. I am trying to stop thinking about his beard, which isn't really _his_ beard, except now it is, and it grows. I'm distracted by a thousand little thoughts like these. Coupled with his lack of facial expression—a solid, largely unmoving veneer—in all these ways the barrier to getting to know Richard feels to me immediately and appreciably steep. Microexpressions, split-second movements of the face, are said to communicate wide arrays of meaning. Even infants who are blind are said to use facial cues to tell their parents how they feel. You don't recognize how true these theories likely are until you are with someone with a face frozen in place.
"Here you go," Richard says, picking up a DVD. It's a copy of the hourlong TV special Ann Curry did about him. He takes a Sharpie from his pocket, signs his name on the DVD, hands it to me.
It's a little bit awkward. I don't know where to begin. A face is a surprisingly intimate and complex subject. Part personal, part public. Partly a thing, partly an idea. Part physiology, part psychology.
I spin the Ann Curry DVD on my finger.
Richard leads me through the living room past his mother, who is on a recliner, staring into a laptop. She does not look up. He shows me his room. It's neat as a hotel. No clutter. Just pictures on the walls, every newspaper and magazine article ever written about him, each of them framed. "They even did me on _Ripley's Believe It or Not!_ ," he says, pointing to one of his clippings. "In Japan I got rated in the top fifty miracles."
I ask him about school, how it's going, how it feels to interact with students. Do they know he has a new face?
"I don't have any classes right now," he says. His voice is muffled, like it's coming from the same place his eyes are, somewhere deep inside. New lips, new mouth, new tongue—it's remarkable he can form words at all. He says he's between classes. At the moment. Well, he isn't actually in school. He's taken some online courses. A lot of what Richard presents to the world is vague. The girlfriend. He says they're soul mates, but so far she's still just a Facebook profile. He says they text all the time. He can't wait to meet her in person.
He leads me back into the living room, where his mom is poking her keyboard angrily. He leaves me here. Two dachshunds sniff at my feet.
"So you like my two little wieners, huh?" his mom says to me, closing her laptop. She has a round face surrounded by gray curls, a soft neck, wide arms. "That one is Raven and that one is Mark—after the race-car driver. She's spoiled rotten. Mark is, too. He's got cancer."
Raven climbs a set of doggie steps up to the couch, digs intently at a blanket, around and around, until she has made herself a cocoon fully covering her body, with just her little nose popping out.
I compliment Richard's mom on the house, the homey feeling, the beautiful views off the back deck.
"You want to buy it?" she asks. It was a mistake moving up here, she says. She never liked it, and neither did Richard. The old house, they gave it to Richard's sister, because she was having trouble making rent. "We're below poverty line," she says. She invites me to sit down on an adjacent recliner. "I have fibromyalgia," she says. "That's why I have these heated blankets."
Richard comes back, carrying pill bottles. "This is what I take every day," he says. "These are my pills." It's a five-pill maintenance regimen he'll need to keep up for the rest of his life; his body will always regard his new face as a foreign object, prompting his immune system to constantly attack it. The drugs trick the immune system by kicking it into its lowest possible gear. This leaves Richard vulnerable to every conceivable health problem down the line. Cancer, diabetes—all the majors.
"He's not supposed to smoke," his mom says. He can't get sunburn. He can't get a cold. He can't drink. He can't fall and risk injury. He can't afford to tax his immune system at all. Even a cut could trigger rejection. It starts as a blotchy rash; it means his body is winning the fight to reject the transplant, and Richard has to be flown to the hospital to receive rounds of emergency drugs intravenously. Uncontrollable rejection would mean an almost certain death; the only things left of Richard's old face are his eyes and the back of his throat. Everything else is now gone for good. "I have to keep watch that his face doesn't go yellow," his mom says. "He's had two rejections so far."
"I'll leave you two talking," Richard says, and he heads outside for a smoke.
His mom motions toward the mantel on the fireplace, where two framed photographs stand side by side. One is Richard's high school portrait. The other is Josh, the twenty-one-year-old donor, who used to have the face Richard now has.
"Isn't that amazing?" she says of the resemblance.
It really is. I don't know which is which, who is which, or what. Pronoun problems emerge. I didn't know Richard as a young man, and now the young face of one is attached to the aged body of the other.
"The likeness?" she says. "It's _Richard_." She tells me she met Josh's mom, visited her at her home. "Real down-to-earth person. I said, 'I really like your kitchen.' That was the kitchen I wanted. Island in the middle of it. Her cabinets had glass in them. I said, 'I'm gonna have to get Eddie to make me a kitchen like that.'"
Eddie is Richard's dad. He used to be a long-haul trucker, but he had to quit when he started needing insulin. He was not in favor of the idea of Richard getting a face transplant. "I like your regular face," he said at the time. Richard's mom told him to back off. "It's Richard's choice," she said. Rodriguez told Richard and his parents that the surgery would be extremely risky—it would take a day and a half—and that Richard would have only a 50 percent chance of surviving it. "And he told us that if the face transplant didn't take, Richard would die because there would be nothing of his old face left," his mother tells me. "But it worked out great."
I ask her if she thinks Richard has changed since his surgery. Does she see a big difference in his personality?
"Yeah, he gets out a little more than what he did," she says. Which still isn't a whole lot. He can't drive, because he could have seizures. She can't drive on account of her fibromyalgia. So the two of them are mostly stuck here, dependent on Eddie.
I lean up in the recliner, stretch my legs out straight. Right now this story is not screaming: miracle.
"What Richard is, he's a lab rat," she says. "He gets to be a brat sometimes. Gets on my nerves so bad. I've always told my kids I don't care if I'm ninety-five years old, if you do something I don't think you should, I'll climb up on a chair and I'll slap you good."
I ask her what she means by "lab rat," and she says exactly that: an animal people do experiments on. "Lab rat," she says. "I don't think he'll ever be able to work like in a normal life. He spends his time in hospitals, everybody poking and prodding, studying him. A boss don't want somebody that's gonna be absent 99 percent of the time."
She says Richard doesn't complain about being a lab rat. He'll do anything for Rodriguez, and so will she.
"Did you meet Dr. Rodriguez?" she asks me. "Me and the nurses, we said, 'Yeah, he sure is good candy-looking stuff.'"
Rodriguez, forty-seven, didn't start out wanting to be great. Or not this great. He was in dental school. His parents had emigrated from Cuba. A Miami drill, fill, and bill dentist—that was his destination.
He speaks in the present tense when he talks about his past. "It's this pursuit of understanding," he tells me, sucking on a peppermint. "Pursuit of knowledge." He tells me about medical school after dental
school. General surgery. Plastic surgery. Microsurgery. "Pursuit of being better. Aim for excellence. I think humility is an important factor." He discovered in himself all the components of a star surgeon, and he could not quiet his urge to learn: "Like being in a library and you keep looking, and it leads you to another thing and you keep going." Soon he's in surgery heaven, in Taipei, Taiwan, Chang Gung Memorial Hospital, a mecca for craniofacial and microsurgery, ninety-nine operating rooms, reattaching fingers, attaching toes to hands, round-the-clock microsurgery, free flaps—taking tissue from one part of a person's body and attaching it to another— _a hundred free flaps every single month_. "Crank it up," Rodriguez tells me, bouncing his head with the memory. "Push and push and push."
Eventually he comes home, accepts a post in Baltimore at the University of Maryland Shock Trauma Center. "Now I'm treating soldiers. Gunshot wounds. Explosions," he tells me. He's fixing faces. A guy needs a nose; Rodriguez takes flesh from his leg and makes him a nose, a chin, a cheek. "Everything is, like, happening now. These patients start coming out of the woodwork. 'There's this crazy guy in Baltimore; I think he might help you.' It's like Doctor Dolittle. People just come from everywhere."
But there are limitations, sculpture-wise, when it comes to making a nose. It would be so much better to put a real nose on someone. People had of course been doing organ transplants for years. But a face is a whole new ball game.
A face isn't an organ, like a liver or a heart. A face is muscles, nerves, bones, and skin. A face is more like a hand or a foot. These kinds of transplants, composite-tissue transplants, sparked a fiery ethical debate from the beginning, back in 1998, when the first successful hand transplant occurred. This sort of surgery was, after all, elective surgery. Would it be worth the risk that a lifetime of immunosuppressant drugs would present? Composite-tissue transplantation became a reality when Clint Hallam, a forty-eight-year-old man from New Zealand, got a new hand that was bigger and pinker than his other hand. _But it was a hand_. If they could do a hand, what about a face? Should they?
Hallam's own experience was not particularly encouraging. The new hand freaked him out. One hand his, one hand somebody else's. He couldn't handle it. "Take it off," he said to his doctors. They refused. He persisted. They refused. So he stopped taking his meds, hid the hand from view so no one could tell what was happening to it. Doctors ended up having to amputate what was left. A mess. _How stupid can you get?_ Hallam, they said, was a psychopath. Physical rejection may be a conundrum, but psychological rejection was the stuff of madmen.
In 2003, the Royal College of Surgeons of England, and a year later, a national ethics committee in France, said that face transplantation would be going too far. The risk of complication would far outweigh the benefits. A new hand might be a reversible decision. But a new face? Once you take your old face off, you're committed. You live your whole life like that, taking medications to keep your new face on so you don't die. All this, for elective surgery?
Nevertheless, the first face transplant was performed in France in 2005. It caught the surgical world by surprise. It was a partial, a triangle. The nose and lips from a deceased donor were removed and sewn onto a woman whose face had been bitten off by a dog. Everyone had thought the first face transplant would be done in the United States. Just a year earlier, the Cleveland Clinic had gotten ethical approval and had started testing prospective patients. But then France did it first, and the worldwide race to do something bigger, better, was on.
Meanwhile, Rodriguez—with all the cowboys in his field now talking face transplants—is imagining something bolder than what's been tried. He's practicing on cadavers, taking the face off one and putting it on another, not just the triangle but the whole face. He isn't sure about the procedure. Could he do it? Like a fighter pilot on a flight simulator, he practices the surgery on his computer. He's got all his training behind him, and all the technology before him, and he's looking at Richard. He's looking at two or three other people who might be good candidates. "But I'm not totally convinced yet," he tells me, still engrossed in the unfolding dramatization of his life. "I'm apprehensive about it. I think it's a big deal. Taking someone's face off." There is no turning back once you do something like that. The old face is mangled and useless; you can't put it back on. "I'm thinking about it very systematically."
He tells me about going to Paris, having lunch with Pascal Coler, a thirty-year-old man who got a new face in 2008. Coler had lived with big tumors all over his face. A terrible life. "And I have a very good steak tartare," Rodriguez says. "And next to me is Pascal, chowing down the biggest porterhouse. As normal as can be. No one looks at him. He has a job, a potential girlfriend. I'm like, holy mackerel, this is huge. We've gone beyond clinical; now we can change someone's life."
I want to ask Rodriguez how one patient who seemed reasonably content during one meal is enough to quiet an entire internal moral crisis, but he's on fast-forward, the adventure playing out like an action movie.
"Let's go!" he tells me. "It's on!"
So in Baltimore, a team of five surgeons is mobilized—two to work on the donor, three on Richard—and the operating rooms are readied. "I go, 'Richard, if you want to pull the plug right now, do it,'" says Rodriguez. "'I don't want you to feel any pressure; there's a lot going on here, and I just need you to tell me right now if you're ready, and if you aren't, we just walk away and it's all good.'"
The surgery, which started with incisions from Richard's scalp to his neck, began at dawn on March 19, 2012, and continued nonstop for the next thirty- six hours. Richard's old face was removed to expose his skull, eye sockets, and throat and neck muscles. The donor face was applied in one solid piece that included the jaw, tongue, muscles, skin, nose, teeth, and eyelids. Hammers and saws were used to shape Richard's new jaw. Metal plates stabilized the donor face and allowed doctors to attach it to Richard's skull with screws.
"He said, 'Let's go. We're doing it.' That's it. You get yourself in your mental game. It's fourth and goal, and we are going into the end zone. We're going to win the Super Bowl. This is the moment of no return. Complete silence in the room; like, holy mackerel. Tough moment. No failure. This is it. We're going to make this thing. The face is one unit. I'm picking it up. I'm laying it on him. I've got to get everything centered, it can't be off-kilter, the nose has gotta be straight, everything's gotta land perfectly.
"Now, this organ is not receiving blood. It needs to receive blood. The longer it takes to receive blood increases the chance for acute rejection. We gotta get this thing drinking.
"Connecting the artery, then the vein, and then we release, and it really is like a miracle. Impressive. You see the blood just coming from the neck, crossing the lips. Seeing the nose—it's white and it turns pink, like normal flesh. It takes seconds. And now we're thinking, Okay, I can take a little breather, but we still have to push ahead, I have to get Richard out of the operating room. We still have to connect the nerves, a lot of nerves. We have to suture the inside of his mouth and tongue; everything has to be sutured. And fix the rest of the bones. Line up the soft tissue. Line up his hairline. Line up his eyebrows. Get the eyelids. So all that needs to happen.
"We're at about eighteen to twenty hours. We've still got a lot to go. I predicted twenty-four to thirty-six hours. I think the unique thing about this face transplant is that it was so extensive and comprehensive. I say this very humble, up to that point, nothing of that level had been performed."
Doubt, when it comes to miracles, is like steam on a mirror. You have to wipe it off if you want to see anything. And what choice do you have? You've already moved forward. You've given a guy a new face. You've gotten a new face. Your kid has a new face. There is no turning back.
One day on the foggy mountaintop, Richard and I get bored sitting around his house, so we decide to take a drive. He is a man of hard-earned platitudes. "Sometimes God will put you on your back to make you look up. Sometimes you need that nudge." He's grateful for his new face. He's grateful to Josh, the donor. Five other people are now living with organs from Josh. "It helps you understand . . . I'm not going to say the afterlife, but what you do here on earth and what you leave here on earth—it's totally two different things." He tells me about his efforts to raise awareness for organ donation; he's become something of a national spokesman for the cause.
In the car, we talk about all the fun things we'll do on our drive. His sister lives about an hour away, in the house Richard grew up in, the house where this whole mess started fifteen years ago, when he shot himself in the face. We haven't talked much about the accident. He doesn't remember anything. Morning or night, nothing. "It's just a cobweb." He suggests we stop at a beautiful lookout place, too. First, though, he asks me to stop at a store. "Something for my throat," he says. He comes out of the store carrying a brown bag. We continue our journey, go up into the voluminous mountains rolling every which way you look.
We talk about cracked iPhones. He's going to start a business fixing cracked iPhones. We talk about his girlfriend. "Melanie," he says, pulling up a picture of her as if to offer proof. "She's real." He can't wait to meet her in person. We talk about being a lab rat. On this matter he says he's honored. In a way, his whole life has been volunteering. When he was in high school, he was a volunteer firefighter. Now, with his new face, doctors are learning so much about how to treat soldiers suffering ballistic facial injuries. He likes helping people. He likes giving people hope. "A drop of hope can create an ocean," he says. "But a bucket of faith can create an entire world."
"That's true," I say.
My throat," he says. We're getting higher, and my ears have popped. He's stretched out, relaxed. He reaches into the brown bag, pulls out a bottle of Wild Turkey.
"For my throat," he says. He can't take over-the-counter medications, he says. Too risky with his meds. There's a backpack at his feet. He opens it, pulls out some tubing and a wide syringe, about a half-inch across, the kind you use to give medicine to horses.
"I don't like the taste," he says.
He hooks the syringe to the tubing, lifts up his shirt. I think I'm supposed to pretend this is not happening. I'm driving. There's a port under his shirt, connected to his stomach. I was not aware of that. He hooks the tubing to the port, so now the fat syringe is standing straight up.
He opens the Wild Turkey and starts pouring.
"Richard, I don't think Dr. Rodriguez would want you doing this—"
All this talk of risk, all the meds, no smoking, no drinking, no falling. A lab rat. Everything measured and quantified and documented.
The Wild Turkey isn't going down. It's clogged. He jiggles the syringe. Nothing is happening.
"Can you pull over?" he asks. "This isn't working right."
I pull over. There are no cars anywhere. We are deep into the mountain range.
He gets out, flips the syringe, emptying it. He gets back in and examines all the tubing, pinches this and that, allowing some stomach juices to squirt out.
"So that's how you eat?" I ask. "You use the tube?"
"No," he says. He used to. He doesn't need it anymore.
He reconnects everything, pours again. The whiskey goes quickly inside him, like water down a drain.
He looks at me. "This is how it's supposed to work."
"This is for your _throat_?"
"I can't take over-the-counter medication," he says again.
Everything about him is vague. Except this isn't vague. He refills the syringe, lets another round drain into him, and another. I don't know how much. It seems like a lot. I wonder what it will be like to be with a drunk Richard. I imagine his sister, the beautiful lookout place. It doesn't happen. Within five minutes he folds, like a shot animal folds, over himself, folded, eyes open, his body deflated, the tube hanging out, dignity depleted.
"Richard?" I say. "Richard?" I shove his shoulder and nothing happens. He is dead. He is on my watch and he is dead. I hear gurgling. Breathing. He's on my watch and he is not dead.
My watch?
The breathing is going in his mouth and out his nose. I am studying his face to make sure air is moving. In his mouth. Out his nose. This didn't used to be his mouth or his nose. He can smell with that nose. He can chew food with a dead man's jaw and teeth. There is no denying how fantastic that is.
I don't know where to go. I don't know one foggy mountaintop from another. I can't wake him up.
I have his address in my GPS. I hit "previous destination." When we get to his house, I drag him out of the car, exactly the way you drag a drunk out of a car, one arm over my shoulder, little steps. I kick the etched bear on the storm door to open it.
"Hello? Hello?" I deposit him in his mom's recliner, wait for someone to come home, or for him to wake up.
One day I ask Rodriguez how he picked Richard over the other candidates he had for his epic heroic story. "I had developed a relationship with Richard, so I knew the kind of person he is," he tells me. "This is an individual that I can trust as someone that can really care for this gift. Keep in mind, someone had to die for him to receive this face. So there's a certain sense of responsibility and burden that I need to make sure that this is not going to be wasted. This has to be a responsible person that will share the precious gift and take this gift and make something with it, and of it."
"A lot of responsibility," I say.
"A lot," he says.
I ask him if he thinks people can get overly invested in the happily- ever-after in his line of work.
"Of course," he says. "Of course."
A recent article by Rodriguez in the medical journal _The Lancet_ features before-and-after pictures of Richard. Side by side, the change is stunning, from a man with a mangled swirl of a face to regular guy. The more I look at the picture of Richard's disfigured face, the more I wish I knew it. His eyes are bigger, rounder, provide a wider window. His eyebrows are all mid up, one curved sharply, the other a gentle swoop, thick scars in between. His lower face is cartoonish, like a drawing of an old guy who took his dentures out. There is so much to find in this face, so many avenues of inquiry.
Patient selection, Rodriguez writes in The Lancet, is the key to success when it comes to face transplantation. "Patient selection by a thorough screening process . . . serves as the best safeguard against ethical challenges," he writes. "Rigorous preoperative psychiatric and psychological selection of patients deemed to be stable, motivated, and compliant by a multidisciplinary team is a crucial determinant of a safe and rapid recovery."
The article looks back at all twenty-eight face transplants that have occurred and represents Rodriguez's full circle from doubt to certainty. Three people have died from complications. Everyone else is said to be doing great.
NYU hospital issued a press release when the article was published, extolling Rodriguez's call for a "moral imperative" to offer face transplants, the "Mount Everest" of medical-surgical treatments.
And yet while the debate about the ethics of face transplantation has shifted dramatically since the early days of harsh warnings against the procedure, researchers in a recent academic survey of the "successful" transplants note a distinct paucity of data on the psychological outcomes for these patients, who, they point out, often suffer from PTSD, alcohol abuse, and opiate dependence as a result of the trauma leading to their initial disfigurement.
Another journal article, in _Anthropology Today_ , considered the topic under the heading "Ethical slippage and quiet death," with Richard's picture occupying an entire page.
I try to tell Richard's mom about the Wild Turkey, but this is a difficult conversation, and I'm not sure it's the right thing to do. She's in her recliner, and she has a migraine hangover. The medicine leaves her woozy. Eddie is over there feeding Slim Jims to Mark and Raven. Richard is in his room.
"Richard isn't supposed to smoke," I start.
"Oh, I know," she says. "And you know, sometimes he drinks until he passes out."
I'm relieved, but not.
She talks about God. She talks about Rodriguez. Everything comes back to Rodriguez: "My godsend." That's a lot of pressure on a savior. I wonder what exactly happened back in 1997 when Richard shot himself in the face. In an e-book version of the story, Richard says he was blindingly drunk, had come home and become verbally abusive with his mom, who sent him to his room to sleep it off. That part appears to be true, according to a police report filed in the Henry County Sheriff's Office. But the rest of the story—that a shotgun happened to be tilted in a gun case and, upon coming out of his room, Richard was asked by his mom to straighten it, causing it to fire accidentally—appears to be apocryphal. According to the report: "Mrs. Norris was standing in the doorway of Richard's bedroom; fussing at Richard about him wanting to go out again. Richard took a shotgun from his gun cabinet and told Mrs. Norris that he would just shoot himself. When Richard racked a shell into the shotgun's chamber, the gun fired.... There was what appeared to be human flesh, bone, and teeth on all four walls in Richard's bedroom." Mrs. Norris told police, "Richard's face exploded."
It would take fifteen years to fix what had gone wrong that night—or at least to try to.
"The really weird thing is, me and Richard's girlfriend found Dr. Rodriguez at the same time," his mom tells me. She pulls her blankets up. I'm sitting on the couch with Raven. Richard has joined us, is seated in the other recliner.
"Girlfriend?" I ask. Richard had a girlfriend before the transplant?
"She was looking online for doctors to help Richard same time I was," his mom says. "She found him same time as I did."
"Wait, who is this?"
"His girlfriend," she says. "Me and her could not get along."
"Girlfriend?" I say, looking over at Richard.
"An old one," he says. "Old, _old_. She was going to college to be a nurse."
I need a time-out. What year are we in? The accident happened in 1997. The new face didn't come until 2012.
"You had a girlfriend when you were disfigured?" I ask Richard.
"I lived with her," he says. "For two years."
"When you were disfigured?"
"Yeah."
"But I thought you were a hermit," I say. What about the foggy mountaintop? The covered mirrors and the black mask?
"This was during the whole stage of my disfigurement when I was working for race teams," Richard says. "When I was at the racetrack, it was like nobody didn't care. They didn't care what I looked like. Only thing they cared about was how good I could set that race car up."
"This was during the whole stage of my disfigurement when I was working for race teams," Richard says. "When I was at the racetrack, it was like nobody didn't care. They didn't care what I looked like. Only thing they cared about was how good I could set that race car up."
"You had a job? And a girlfriend? And an apartment? You were living a whole life?"
"Yeah."
So why did he need a new face? Why had he endured the complications involved in freeing him from disfigurement? The thirty-six-hour operation, a 50 percent chance of dying on the table, a life of antirejection medications. No sunburn. No falling. Watch out for yellow. Two rejections so far.
"I didn't like the girl," his mom says. "She tried to cut me out of everything."
There was a fight. At their apartment. Richard was convalescing after a surgery, and his mom came by. "She wouldn't let me in to see Richard," she says. "So I went through her. And she told me, get out of her house, the doctor said nobody could see Richard. I said, 'I ain't just nobody, I'm his mother. I'm gonna see him.' I whopped her. And I had her on the floor. I just told her, I said, 'You don't take my son away from me. Nobody takes my son.' And she said, 'I'm gonna call the law.' I said, ' _Well, just call the law.'"_
"I got rid of her," Richard says.
"Then we moved up here in 2005, away from everybody," his mom says.
Which is right about the time they found Rodriguez.
"We don't like it here," his mom says.
"It's the most boringest place on earth," Richard says.
"It's the most boringest place on earth," Richard says. Being famous is better. One of Japan's top fifty miracles. And he was willing to go through with getting a new face. And there was all that salvation his mom needed. And the wounded warriors needed him. And humanity needed him. And having a new face is better than some old disfigured one. You can't argue with that.
Richard wishes Rodriguez had stayed in Baltimore. He doesn't like New York. But wherever in the world Rodriguez needs him, he'll be there. And yes, as to smoking, he shouldn't. And yes, the incident in the car? "Uh-huh," he says. There's a certain amount of self-medication, he says. "But that's just self-medication." We're at NYU again, and he's not exactly sure what today's event is. Some kind of fund-raiser, he thinks. We're in the waiting room. He looks great in his suit. "I'm sure they'll be very nice people," he says. He likes people. He likes visiting patients. People come to talk to Rodriguez, or the team back in Baltimore, and Richard joins them. They want face transplants. They want to know what it's like. He tells them what it's like. He knows exactly how they feel. It's something positive he can do with his life. A face. "This is what I am," he tells me. "There is nothing more important than a face."
He brings up Hallam, the guy with the hand. "He couldn't take it, so he had to cut it off," he says. "So now they say, 'Is the face transplant going to have that identity crisis, too?' Well, if he does, _we're screwed_."
A woman with curly dark hair peeks into the waiting room. "We're ready!" she says. We head down the hall. Richard takes two steps into the conference room, and the people burst into spontaneous applause.
"Hi, Richard!"
"Hello, Richard!"
Rodriguez sits at the head of the table. Pretty little cookies are laid out. "This institute was made possible by these incredible people," he says to Richard.
"It's our pleasure, Richard."
"God bless you."
"You're very brave."
"You're a real ambassador."
Richard thanks them on behalf of all the people in the future who might be helped. "A drop of hope can create an entire ocean," he says. "A little speck of faith can create a world. You give that speck of faith to them."
"He never really thought about himself in all of this," Rodriguez says. "He's always thought about helping the wounded warriors and the other people, and providing hope. He's a remarkable man."
"And they can do research on me as well," Richard says.
And that about wraps it up?
A woman has a question. She has wide shoulders, blond hair.
"What about your family?" she says. "Do they have therapeutic resources available to them to support you?"
"Actually, the support I get mainly is from therapists," Richard says. "They send me home with homework, give me exercises to practice, to help with my speech, my swallowing, you know. Physical therapy helps a whole lot, played a huge role in my recovery."
The woman tries again.
"Emotionally, did you feel you got the support you needed?"
"Emotionally I can get lab work done no matter where I am, if I'm home, if I'm in Baltimore. If I'm here, I get it done here. That way the lab work is always current."
I want to jump in here. I want to tell the woman with the questions about Wild Turkey, the poverty line, lab rats and wiener dogs, illness and violence.
Rodriguez jumps in. "The answer is yes and no," he says. "This is a new field. Every transplant has to be well thought out. One thing that you can appreciate with Richard is there was not a step missed in this rehearsal and this practice. And now Richard is our ambassador, and he can help us."
"You are one of the greatest people I've ever met," one in the crowd says to Richard.
"Terrific. So handsome."
"Thank you," Richard says, reaching for Rodriguez. "It comes from his hands. His hands."
Richard calls me shortly before he heads to New Orleans to meet Melanie, the new girlfriend. I wish him well. I feel nervous for him, want her to be real. For all his guile, he's a trusting soul. And he needs a girlfriend or some companion to assist him through life. If there's a future for him, it is almost assuredly off the foggy mountaintop. But under whose watch? Who will drive him around? He's a science experiment. He's a lab rat. He's not a normal sort of boyfriend.
One day Melanie appears on my Skype screen. She's real. She's lovely. Richard is sitting next to her, waving to me. He wipes constant drool with the help of a towel. She just came home from work and put a pot roast in the oven. Richard folded the laundry. She wants to care for him. She finds a well of kindness in him. She's been burned so many times. She says nothing about his medical-rock-star status, nothing about the drool. "Why does anyone fall in love?" she says with a shrug, when I ask about the relationship. She's the first person I meet who talks about Richard as just a regular person. He's not: a miracle. He's not: a medical circus act. He's not: an ambassador of anything. He's a guy with barely a tick of a functioning immune system, thanks to the inexorable march forward of technology and a worldwide race for surgical glory. But for now, at least, she doesn't need to have that conversation. I get the sense she would have liked Richard with his old face just the same.
In short order she introduces Richard to her kids, her mom; everybody likes everybody. She meets Richard's mom by phone, and so far so good. One day she notices maybe a rash? "What is that?" she asks Richard. "I don't think it's anything," he says. He doesn't want it to be anything. He's finally with her, and he wants to stay. "I'm worried," she says. "It's getting worse." She takes a picture of the rash, insists that Richard send it to the doctors in Baltimore keeping track of his medical care.
"Get here," they say. "Next plane out."
If it's acute rejection and they can't reverse it, he dies.
He's hospitalized for two weeks, pumped full of stuff to get the rejection under control, then sent back to the foggy mountaintop to recover.
"Come back," Melanie tells him. "Please come back. We'll eat crawfish. You'll wear your sunscreen. I'll keep you safe."
_Adrian Nicole LeBlanc_
Adrian Nicole LeBlanc is a journalist who is best known for her 2003 nonfiction book _Random Family: Love, Drugs, Trouble, and Coming of Age in the Bronx_ , which chronicles the struggles of two young women as they deal with love, growing families, and prison time. The book took more than ten years to research and write and has received many awards, among them the Anisfield-Wolf Book Award and the Ron Ridenhour Book Prize. In 2010, it was named one of the Top Ten Works of Journalism of the Decade by the Arthur L. Carter Institute of Journalism at New York University. Before turning to writing full time, LeBlanc was the fiction editor at Seventeen. Her work was also featured on NPR's All Things Considered in an audio documentary about her last days with her father.
As a freelancer, LeBlanc has written for many publications including _The Village Voice, Harper's, The New York Times Magazine_ , and _The New Yorker_. Grants, fellowships, and residencies have been essential to continuing independent work, including: The Barbara Deming Women's Memorial Fund, the Open Society Institute, the Dorothy and Lewis B. Cullman Center for Scholars and Writers at the New York Public Library, Blue Mountain Center, Cottages at Hedgebrook, and the MacDowell Colony. In 2006, she was a recipient of a MacArthur Fellowship.
LeBlanc was raised in Leominster, Massachusetts, where plastics originated. Her hometown informs everything she writes about. She is currently completing another decade's long project, a nonfiction book about stand-up comedy.
The Troubled Life of Boys: The Outsiders
A look at high school boys as they struggle with the changing perceptions of masculinity in the feminist culture of the 1990s.
The fight takes place in the bright light of adult view—on a weekday afternoon, on a tree-lined residential street, within sight of the police station and a block from the middle school. The smaller boy, about twelve, waits until there is a safe distance between himself and the other boy, about thirteen. Then he sends a curse. It lands. He waits. No response. He follows with a homophobic slur. His opponent—a chubby boy nicknamed Sex Machine—finally turns around.
"C'mon, Sex Machine!" one shouts, then whispers to another, alarmed: "Look at him. He keeps backing up!"
Whatever started the fight is irrelevant. The friends clamber up a nearby wire fence to get a good view, hyper spiders clinging to the mesh.
Sex Machine is frightened. Despite his oversize T-shirt, you can see the rise and fall of his heaving chest. A man's voice chimes in and shouts encouragement to the smaller boy from the driveway.
"That's his father!" a boy says. "Can you believe it? He's telling him to fight!"
"That's not right," says a girl.
Borrowing from the man's confidence, the smaller boy rushes forward and swings. Sex Machine stumbles backward as he tries to duck. A woman leans out from the second-floor window of a ranch house and says, "Come in, come in," without sounding as though she means it, a weary Juliet.
Sex Machine looks desperate, flailing his arms frantically, trying to flag down a car. Luckily, one stops. Apparently, it's his mother. All the tension and fear that his body has been holding bursts into punctuated sobs. He storms around the car to the passenger side. His freckled friend, who had been cheering within inches of the action, cycles over and dismounts to say goodbye. With all the fury raging inside him, Sex Machine bellows, "You didn't help me!" then shoves him to the ground.
Antrim, New Hampshire, where the fight took place, is a long way away from Littleton, Colorado, as well as from Conyers, Georgia, where a fifteen-year-old boy shot six classmates at his high school in May. It is one of nine towns whose regional high school, ConVal, sits in Peterborough, New Hampshire, the setting that inspired Thornton Wilder's _Our Town_. But what it shares with those other places, and with countless others across the country, is a brutally enforced teenage social structure.
Boys at the bottom of the pyramid use different strategies to cope— turning inward and outward, sometimes in highly destructive ways. (There has been a fivefold jump in the homicide and suicide rates of boys in the last forty years, a rise some experts attribute to increasing male depression and anger as well as access to guns, among other factors.) Most boys live through it, suffer, survive. But the journey may be especially deadly now because, as the avalanche of new "male identity" literature demonstrates, the old prescriptions for behavior no longer hold, and the new ones are ambivalent. Today's young males may be feminism's children, but no one is comfortable with openhearted or vulnerable boys.
ConVal is in some ways progressive. There are about nine hundred students and an administration that consciously works to minimize the ultramacho sports culture that dominates many schools. Says Bob Marshall, the head of the social studies department, who founded the football program in 1992: "We had to create a football culture. People didn't know when to cheer. We didn't even have a school song."
Even so, the traditional hierarchies operate: the popular kids tend to be wealthier and the boys among them tend to be jocks. The Gap Girls- Tommy Girls-Polo Girls compose the pool of desirable girlfriends, many of whom are athletes as well. Below the popular kids, in a shifting order of relative unimportance, are the druggies (stoners, deadheads, burnouts, hippies, or neo-hippies), trendies or Valley Girls, preppies, skateboarders and skateboarder chicks, nerds and techies, wiggers, rednecks and Goths, better known as freaks. There are troublemakers, losers and floaters—kids who move from group to group. Real losers are invisible.
Bullying, here as elsewhere, is rampant. Even in small-town, supposedly safe environments like Peterborough, a 1994 study found, the vast majority of kids from middle school up are bullied by their peers. The shaming is sex-based, but the taunting is more intense for boys—an average high-school student, according to another study, hears twenty-five antigay slurs a day.
To be an outcast boy is to be a "nonboy," to be feminine, to be weak. Bullies function as a kind of peer police enforcing the social code, and ConVal's freaks are accustomed to the daily onslaught. The revenge-of- the-nerds refrain—which assures unpopular boys that if they only hold on through high school, the roster of winners will change—does not question the hierarchy that puts the outcasts at risk. So boys survive by their stamina, sometimes by their fists, but mainly, if they're lucky, with the help of the family they've created among their friends.
A good day for Andrew, fourteen, occurs when R., a boy who torments him, is absent from school, like when he was suspended for ripping the hearing aid out of another classmate's ear. R., fifteen, weighs more than two hundred pounds. Andrew, a small boy with straight, dirty-blond hair and glasses, takes care to note R.'s better days—say, when Andrew helps him with an assignment, when he's in a good mood or distracted by harassing someone else.
The trouble started long before the appearance of R. "First people harassed me because I was really smart," Andrew says, presenting the sequence as self-evident. "I read all the time. I read through math class."
Back then, in middle school, he had the company of Tom Clancy and a best friend he could talk to about anything. He says things are better now; during school, he hangs out with the freaks. Yet the routine days he describes sound far from improvement—being body-slammed and shoved into chalkboards and dropped into trash cans headfirst. At a school dance, in the presence of chaperones and policemen, R. lifted Andrew and ripped a pocket off his pants. "One day I'll be a 'faggot,' the next day I'll be a 'retard,'" Andrew says. One girl who used to be his friend now sees him approaching and shouts, "Oh, get out of here, nobody wants you!"
Andrew joined the cross-country team but the misery trailed him on the practice runs. He won't rejoin next year although he loves the sport. Recently he and some other boys were suspended for suspected use of drugs. According to Andrew, he used to earn straight As; now he receives mostly Cs and Ds. He does not draw connections between the abuse and the changes in his life.
He also does not expect help from the adults around him. He suspects they have their reasons—some don't care, while others worry only about physical attack. When I point out that he's being physically attacked, he imagines that the teachers think it's horsing around, although he does wonder why the teachers can see the same kid pushing other kids every day and don't just tell the kid to stop. "Maybe have a talk with him or do something," he says. "One little push isn't that much, but when it's every day, it's something." He only wishes that someone had helped in middle school, before the contagion grew. "When it first starts to happen, there's definitely something you can do," he says. "But you can't turn a whole school."
Neither does Andrew tell his parents. He believes they think he is popular. "If I try to explain it to my parents," he says, "they'll say: 'Oh, but you have plenty of friends.' Oh, I don't think so. They don't really get it." His outcast friends, however, do.
One of them is Randy Tuck, a five-foot-four-inch sophomore with a thick head of hair and cheeks bright red with acne. He rescued Andrew from a "swirly" (two boys had him ankle up, and headed for the toilet bowl).
Randy moved from Alaska to New Hampshire almost three years ago. To his frustration, his classmates called him Eskimo Boy. Art is his solace, along with the occasional cigarette. He loves to draw. He used to sketch Ninja Turtles and now, with the help of an art teacher, he's studying anatomy. He associates with the freaks during school mainly because they let him. He says, "They are friendly, but not welcoming."
Classmates debate with Randy about his atheism, but he refuses to believe a God could arrange a life as unlucky as his. Andrew blames himself. Randy says, "Andrew's vulnerable and small and weak and R. takes advantage of that." Randy utilizes '"verbal bashing" as a defense, although he admits that its powers don't prevent physical attack. R. surprised him one day in the hallway. He passed Randy, then turned around and punched him in the spine. But Randy also notes that R. can be funny. "When he's not in a bad mood, he can be very entertaining."
Andrew says that the ostracizing "does build up inside. Sometimes you might get really mad at something that doesn't matter a lot, kinda like the last straw." He could understand the Columbine killers, Dylan Klebold and Eric Harris, if their misery had shown no signs of ending, but Andrew remains an optimist. After all, there are some people who have no friends. "Things are not going up really fast, but they are getting better," he says. "I might have a week where they get worse, but overall they are getting better, definitely."
The quips ricochet around the bedroom like friendly-fire darts. Myles Forrest, sixteen, a sophomore with baby fat and sweet eyes, is one of George Farley's closer friends. George, also sixteen, is a floater. He has set up camp with ConVal's freaks for now. George sees weakness everywhere— in women who look for milk cartons with the latest expiration date at the store where he works; in the unemployed drunk who receives an allowance from his working wife; in white girls who think they are cool because they date guys who are black. Softness arouses his contempt. He is no more gentle with himself. The volleying with Myles, who wears his Y2K T-shirt — "01-01-00"— relieves George of the clearly burdensome obligation of having so much edge.
"The end of Myles's life," George starts.
"The end of life as we know it," Myles says. The phone rings. George lifts the receiver. "Myles Forrest, loser," he announces, and so the afternoon begins.
Myles and George provide sustenance between insults. Myles fiddles with his computer—one of two—as George peers out over the street. "What's up with the dress?" George asks, spotting an exchange student from ConVal.
"What? He's Hindu," says Myles.
"I said, What's up with the dress?"
"It's like a cult thing," Myles says, somewhat sharply.
"That's a dress," George says, losing steam.
"It's like a cult thing. It's like a kilt."
"You know I'm messing with you, don't go getting all politically correct with me." (Later on, Myles will explain the theory of equal-opportunity hatred: "You ever notice that you can't hate a particular group, but if you generally hate everybody nobody seems to mind?") The sarcasm slows when the Quake competition begins.
It strikes me as I watch them in front of the famously violent video game that it is one way for the boys to enjoy closeness without it being threatening. The violence of the game, the state-of-siege mentality, the technical expertise required, supplant the macho expectations and give the boys a rest from the relentless one-upmanship. Rather than insult each other, they can attack the game. Soon enough, they are allies in the search for snacks, rushing down the stairs. They amble past the locked gun case behind the door leading through the playroom, to the kitchen. George sticks his head in the fridge.
"How about some of these worms?" he asks, holding a baggie of bait. "Fishing is like alcoholism. It's an excuse to drink. Or maybe they're trying to level the playing field. How hard can it be to outsmart a fish?"
"Catching it is kinda fun—" Myles tries.
"Now ice fishing—alcoholism in the extreme," George continues. "Cold, boring, worry about falling through the ice. Hey, my girlfriend dumped me. She dumped my slacker [expletive]."
"I thought you were gonna give her the—" Myles tries.
"Yeah, but she surprised me."
"Irony of ironies. So, technically, you're the loser."
"Shut up," George says, sounding sad.
"To the winner goes all the spoils of war," Myles appeals.
"Shut up," George says, relocating to the sunroom. He lifts Fido, Myles's lizard.
"Watch out, George," Myles says protectively. George presses Fido into the aquarium to make the wood chips fly.
"That's cruel, stop it," Myles says, retrieving his lizard, as George moves on to his lectern, the Stairmaster.
"That cat is wishing for a tail," George says, observing Myles's tailless cat.
"To the victor goes the spoils of war," Myles sighs, mock ruefully.
"Stop defending your tailless cat," says George. "Anyways, so Colleen broke up with me."
"You already told me that," Myles says. He glances at his buddy. "I thought that's what you wanted."
"I did," George says, sounding far from sure. Now that he has been rendered single, what will come of the flirtation he lost his girlfriend for?
The new girl, a computer skateboarder chick, likes to spar. George says, "We're both the same person, but it's hard when you have two sarcastic people making fun of each other, and then they get worse and worse until—"
"Until there's no place you can go," Myles says knowingly.
"Shut up, you slack [expletive]," George says, knowing, too.
Teenagers find heroes among their friends. Tyler Snitko, seventeen, pulls other outcasts in, functioning as a kind of human insulation for the freaks. To each taunt he quips, "Thank you." He booms, "These are my people," opening his arms, his fingernails polished black, to embrace his fellow freaks at lunch time in what has been labeled Mutant Hall. In the presence of someone like Tyler, more vulnerable teenagers are less likely to be picked on, and they intuitively know this.
Tyler's hero is his grandfather. Not only did the old man give him advice that he often quotes ("Sometimes there are going to be rat bastards in life, and you have to deal with them"), but he also backed up the talk with action: he gave Tyler his first set of weights.
Tyler kept his strategy secret, taking long, midnight runs because he did not want to jeopardize his affiliation with the freaks, who were supposed to be "all skinny and pale." He soon discovered that his best friend, Toffer, seventeen, studied jujitsu to control the anger building in himself.
Toffer knew what it was like to be excluded. His isolation began in elementary school, and only in high school, through his friendship with Tyler and with his girlfriend, Anne Baker, did the fog begin to lift. Through the worst of the ostracization, the boys had each other. Says Tyler: "Other people turned me away, like I'd bring the whole house down. He stood by me."
Toffer, whose name is Christopher Eppig, is a senior who looks very much like Jesus. He survived the solitary years by not showing emotion. He shows very little emotion now. "I think it was the fact that I couldn't completely control myself that scared me," he says flatly. "I didn't like myself because I didn't have anything. No athletics, no grades. The only thing that kept me going was that I hated them more than I hated myself.
"Before, all I knew was what people were telling me about myself, and it wasn't a positive image, and I wasn't interested in who I was," he continues. "Jujitsu gave me something else that I was, that was better and more believable."
The friendship with Tyler created elbow room. They joined the wrestling team. They formed a band named Gawd. It helps that Tyler's parents encourage his use of their renovated colonial as a social center, and that his dad quit his job as an executive to stay home full time. His parents call the arrangement a luxury, a decision they made around the time when Tyler's mother was promoted to assistant principal of a middle school. Then Tyler had the great good fortune of several growing spurts, which, at last measure, topped six feet to match the hard-earned bulk.
His upbeat personality may defuse hostility, but his physical presence is a moat. A friend who has known Tyler since childhood, who will only give his online name, Bladerunner, says: "He is just really nice and he sticks up for people."
Bladerunner, seventeen, has had his own troubles. A boy he'd met in the hospital after a suicide attempt wanted to beat him up, and for months, the tranquil New Hampshire town became a minefield for him. Bladerunner stopped visiting the park and dreaded school. The restaurant where he washes dishes was the only place he anticipated with some pleasure because his boss treats him "like a person." Otherwise, he met Tyler at the Incubator, a room where students go when they have a free period. They would get passes to the weight-lifting room.
Bladerunner didn't stick with the weights, but it mattered that Tyler encouraged him. Recently, Tyler invited Bladerunner to be a vocalist for Gawd. "I realized I was walking around people on eggshells, because I'm always afraid of what's going to happen to me, or what people are going to think," Bladerunner says. "I am going to try to take what I am afraid of and look it in the face, as much as it might physically hurt."
Even as it helps in the day-to-day of high school, bodily renovation perpetuates the hierarchy. Bulking up—or being near someone who does— just means the pyramid starts lower down. Tyler sees similarities between R. and himself: "He gets respected because he throws his weight around. I get respected because I don't have to." He also recognizes how the pressure to prove his masculinity drove him to objectify girls. "I treated my girlfriends really bad," he says. "I admit it. I was like, Oh, there's a pair of boobs, I'll go stand next to it. I think I'll go talk to it."
Of course, trivializing girls is a most likely result of a pecking order in which girls represent "femininity," the perceived threat to conventional masculinity, the mix of which leaves boys so confused these days. The fear of feminizing boys is embedded in the hierarchy of the social cliques: winner-loser, popular-outcast, boy-girl. "This fear of sissifying boys," says Olga Silverstein, author of _The Courage to Raise Good Men_ , "I think it's going to be the last prejudice to go."
The danger signs are everywhere, but only if you want to see. Banning trench coats, installing metal detectors and security guards—the quick-fix solutions to the problem of seemingly rampant boy violence—"becomes a weird kind of McCarthyism," says Russell Novotny, a 1999 ConVal graduate. "The only way to get kids not to hurt each other is to get kids not to want to hurt each other," a process he compares to a road. "It's the whole little-step thing. You take a little step and suddenly you are in the woods. How did I get here? We are so far into the woods. For every mile you walk, you have to walk a mile back. You can't look too far ahead or you trip over what's in front of you." Or you look at what's in front of you, a boy like J., and you don't really see him.
J., who doesn't want his name to be used, ranks as a loser. He finds temporary refuge with the burnouts, but his precarious welcome depends upon their mood and whether or not he has weed. His greatest flaw seems to be his willingness to try anything to fit in.
"That kid does whatever you tell him to do," says Josh Guide, a classmate. Past instructions are rumored to include wading knee-high in a running brook, with his sneakers and socks on, fetching sticks. He doesn't fight back when people shake him down for money. He claims to get high when a classmate sells him oregano with chives. He falls off his bicycle when the other boys are done using it and ignores the bleeding, which, during a game of basketball, stains another boy's new Tommy Hilfiger shirt.
"Now I have AIDS," the boy says, disgusted. J.'s distress is so apparent that the boy says, "I'm kidding," but his hostility is clear. This particular afternoon, J., who has ragged black hair and a crumpling smile, opens his mouth as if to speak, but doesn't. He saves his mouth for his teachers.
The week after the shootings in Littleton, Colorado, ConVal High School held an assembly about school safety. J. recounts what happened in his class next period. "I said, 'I wish those kids would come over here and blow away the teachers,"' especially an assistant principal, with whom J. had a long history. J. says, "I am always in trouble, every day, for my attitude."
According to J., the classroom teacher said, "I'm kind of concerned about you."
"Nothing to be concerned about," J. replied. "Everyone hates him anyway."
"Do you want to go to the office?" she asked.
"Hell, no," he said. Then she sent him there.
Ordinarily, J. would have been sent home for cursing. He knew the drill. This time, however, he waited for the Peterborough police, who, he says, searched his knapsack and escorted him to the station, where he was charged in juvenile court with disorderly conduct. (ConVal officials cannot comment on J.'s case because he is a juvenile and because it is pending.)
That night, the police appeared at the homes of members of J.'s family with a search warrant and collected handguns and sporting rifles. The next day, news cameras greeted ConVal students in the parking lot. The print media continued the story, and J. became known as a copycat in a wider world. "It's retarded," he says. "I shouldn't have got in trouble. If it was some good kid that did it, they wouldn't have gotten in trouble." Many students feel that the administration overreacted, less because of Columbine than for the fact that even if he had meant what he said, he was an unlikely candidate to carry out that particular kind of plan. Says one parent, sighing, summing up a typical adult response, "That's just J. being J. again."
Being J., according to J., is as inevitable as his difficulty in school, which he compares to his unhappiness in his family life. He says: "I try not to spend much time at home. It's like I'm a failure. My sister is a straight-A student and everything." He doesn't get along with his stepfather. Right now, his relationship with his mother isn't good. "Whenever I get in trouble, she yells at me for ten minutes, then she stops," J. says. "They yell nonstop, then they forget what they are yelling for. They don't even punish me. It's like a habit with them."
J. spends his days watching television. In the afternoons, he goes to the nearby basketball court. His mother tracks him down. (She declined to comment for this story.) J. says, "Then she yells at me all the way home, then I fall asleep and get up and do it all over again."
More upsetting to J. than his threat to an assistant principal—and more memorable to many ConVal students—was an event for which he was suspended earlier. He stepped into a bathroom to smoke a joint. It wasn't getting high, or even getting high during school, that was so problematic to the other students, but that he had selected a bathroom without ventilation that led directly into the hall where a teacher stood. J. heard the teacher but still kept smoking. "I just finished cuz I knew I was gonna get caught," he says.
"How stupid can you get?" George asks. "He just proved to everyone that he's the [expletive] everyone thinks he is."
Andrew ventures, "Not to be mean to J., but that's plain old stupid."
Even Tyler, who tried to defend what was left of J.'s eroding reputation, admits: "That was the stupidest thing I ever heard of. I don't even know why I tried to protect that kid."
Drugs—at least temporarily—blur the social lines. Boys and girls from different groups get high together; says George: "Polar opposites— they are bound together by drugs." James Key-Wallace, a 1999 graduate, attributes the social leveling to limited distribution: "The drugs come from the same half-dozen sources. You're going to come in contact on grounds that demand respect." Says Hayden Draper, who also just graduated: "Popular kids do drugs. Unpopular kids do drugs. Everyone has their own place to get high." J., however, was all alone.
Since Columbine, the Safe School Committee at ConVal has undergone a renaissance. The Peterborough police have stationed an officer at the entrance. But many of the students believe that a shooting spree like that of Klebold and Harris's could happen anywhere. Says Toffer: "It certainly didn't happen because of the lack of a safe-school committee. Their problem was, they weren't accepted, and they weren't going to be accepted, and that's the way that our society is. There are always people that are going to be cast out and people that are cast in."
Colleen, George's ex, a slim girl with short straight hair and an easy smile, grew up down the street from J. He's generally annoying, she says. He used to sing Christmas carols on the bus in June, but he is not cruel.
Everyone, she says, has their days. What J. hates is people talking down to him, so she takes care not to do that. She feels the same way when people talk down to her because she is a girl. "There are times I can talk to him about things, without it being weird and without him being a pervert," she says. It's all relative. When you are close to the bottom, there's not much room left to fit. She recalls J. at his happiest during a class he described to her, in the high school's on-site preschool, how content he felt playing among the little kids.
_Jill Lepore_
Jill Lepore is the David Woods Kemper '41 Professor of American History at Harvard University and a staff writer at _The New Yorker_. She also teaches at Harvard Law School.
A prize-winning professor, she teaches classes in evidence, historical methods, the humanities, and American political history. Much of Lepore's scholarship explores absences and asymmetries of evidence in the historical record. As an essayist, she writes about American history, law, literature, and politics. Her essays and reviews have been translated into German, Spanish, Italian, Portuguese, Latvian, Swedish, French, Chinese, and Japanese, and have been widely anthologized. Her books include a political history of early America in the form of a trilogy, _The Name of War_ , winner of the Bancroft Prize, the Ralph Waldo Emerson Award, and the Berkshire Prize; _New York Burning_ , a finalist for the Pulitzer Prize; and _Book of Ages_ , a finalist for the 2013 National Book Award for Nonfiction. Her other books include _The Mansion of Happiness_ , a finalist for the Carnegie Medal for Excellence in Nonfiction; _The Story of America_ , shortlisted for the PEN Literary Award for the Art of the Essay; Blindspot, a novel jointly written with Jane Kamensky; and _The Secret History of Wonder Woman_ , a _New York Times_ bestseller and winner of the 2015 American History Book Prize.
Lepore has been contributing to _The New Yorker_ since 2005. A recent series of essays examines the Election of 2016, considering, for instance, the role of polls, facts, parties, and the conventions. She is currently writing a history of the United States.
Prodigal Daughter
A writer researching a book about Ben Franklin's extraordinary sister Jane finds illuminating connections to her own spirited, artistic mother.
In the trunk of her car, my mother used to keep a collapsible easel, a clutch of brushes, a little wooden case stocked with tubes of paint, and, tucked into the spare-tire well, one of my father's old, tobacco-stained shirts, for a smock. She'd be out running errands, see something wonderful, pull over, and pop the trunk. I never knew anyone better prepared to meet with beauty.
"Fingers nimble, brush or thimble," my mother's college yearbook said about her. The cabinets in our kitchen used to be a murky green. One day, I came home from kindergarten to find that my mother had painted every cabinet sunflower yellow. "I was just so sick of that green," she said, washing up, briskly, at the kitchen sink. She stitched quilts; she painted murals. She had one dresser drawer filled with buttons and another with crayons. She once built me a dollhouse out of a stack of shoeboxes. She papered the rooms with scraps of wallpaper and lit them with strings of colored Christmas-tree lights as brightly as she lit my childhood with her trapped passion.
She'd grown up in a small town in Massachusetts, a devout Catholic. After college—a Catholic school in New Rochelle—she'd wanted to go places. It was 1949: the war was over; the world was wide. She got engaged to a man named Winstanley, who had a job with the State Department; she wanted to marry him because he was about to be posted to Berlin. That fell apart. For a while, she worked as a designer for the Milton Bradley Company, in Springfield, but she couldn't stand how, from her apartment, she could hear the keening of the polar bear in the city zoo. ("He had the smallest cage you ever saw," she told me. "All night, he cried.") Then she drove across the country in a jalopy and took a slow boat from San Diego to Honolulu. After that, she became a stewardess, so that she could see Europe. In 1955, she had to quit and come back home, to Massachusetts, to take care of her mother, who was dying. That's when she met my father, a junior-high-school principal: he hired her as an art teacher. Every day, he left a poem in her mailbox in the teachers' room. The filthiest ones are the best. ("Marjorie, Marjorie, let me park my car in your garagery.") He told her he wanted to live in Spain. He was courting; he was lying; no one hated to go anywhere more than my father; he almost never left town. Except for during the war, he had never lived anywhere but his mother's house.
My mother married my father in 1956. She was twenty-eight and he was thirty-one. She loved him with a fierce steadiness borne of loyalty, determination, and an unyielding dignity. On their honeymoon, in a cabin in Maine, for their first breakfast together, she made him blueberry pancakes. Pushing back his plate, he told her he didn't like blueberries. In fifty-five years of marriage, she never again cooked him breakfast.
Before I was born, my parents bought a house on Franklin Street. (My mother promptly planted a blueberry bush in the back yard.) The year I learned the alphabet, the letter "J," the fishing hook, was my favorite, except for "F." "I am four and my mother is forty-four and my father's name is Frank and my house is 44 Franklin Street," I would whisper, when no one was listening: I was the youngest.
The street I grew up on was named for Benjamin Franklin. For a long time, no name was more famous. "There have been as great Souls unknown to fame as any of the most famous," the man himself liked to say, shrugging it off.
Benjamin Franklin was born in Boston in 1706. He was the youngest of his father's ten sons. His sister Jane was born in 1712. She was the youngest of their father's seven daughters. Benny and Jenny, they were called, when they were little.
I never heard of Jane Franklin when I lived on Franklin Street. I only came across her name on a day, much later, when I sat down on the floor of a library to read the first thirty-odd volumes of Benjamin Franklin's published papers. I pulled one volume after another off the shelf, and turned the pages, astonished. She was everywhere, threaded through his life, like a slip stitch.
We "had sildom any contention among us," she wrote him, looking back at their childhood. "All was Harmony."
He remembered it differently. "I think our Family were always subject to being a little Miffy."
She took his hint. "You Introduce your Reproof of my Miffy temper so Politely," she wrote back, slyly, "won cant a Void wishing to have conquered it as you have."
He loved no one longer. She loved no one better. She thought of him as her "Second Self." No two people in their family were more alike. Their lives could hardly have been more different. Boys were taught to read and write, girls to read and stitch. Three in five women in New England couldn't even sign their names, and those who could sign usually couldn't actually write. Signing is mechanical; writing is an art.
Benjamin Franklin taught himself to write with wit and force and style. His sister never learned how to spell. What she did learn, he taught her. It was a little cruel, in its kindness, because when he left the lessons ended.
He ran away in 1723, when he was seventeen and she was eleven. The day he turned twenty-one, he wrote her a letter—she was fourteen— beginning a correspondence that would last until his death. (He wrote more letters to her than he wrote to anyone else.) He became a printer, a philosopher, and a statesman. She became a wife, a mother, and a widow. He signed the Declaration of Independence, the Treaty of Paris, and the Constitution. She strained to form the letters of her name.
He wrote the story of his life, about a boy who runs away from a life of poverty and obscurity in cramped, pious Boston to become an enlightened, independent man of the world: a free man. He meant it as an allegory about America.
"One Half the World does not know how the other Half lives," he once wrote. Jane Franklin was his other half. If his life is an allegory, so is hers.
"Write a book about her!" my mother said, when I told her about Jane Franklin. I thought she was joking. It would be like painting a phantom.
History's written from what can be found; what isn't saved is lost, sunken and rotted, eaten by earth. Jane kept the letters her brother sent her. But half the letters she sent him—three decades' worth—are missing. Most likely, he threw them away. Maybe someone burned them. It hardly matters. A one-sided correspondence is a house without windows, a left shoe, a pair of spectacles, smashed.
My mother liked to command me to do things I found scary. I always wanted to stay home and read. My mother only ever wanted me to get away. She brought me with her wherever she went. She once sent me to live with my aunt in Connecticut. ("Just to see someplace different.") One year, she saved up to send me to a week of Girl Scout camp, the most exotic adventure I had ever heard of. I got homesick, and begged her to let me come home. "Oh, stop," she said. "And don't you dare call me again, either." When I was eleven, she took me to New York City, a place no one else in my family had ever been. It was the weekend of the annual gay-pride parade, on Christopher Street. "Isn't that interesting?" she said. She took a picture of me next to five men dressed in black leather carrying a banner that read "Cocksuckers Unite"—this was 1978—so that I'd remember the existence of a world beyond Franklin Street. No one else in my family left home to go to college. My mother made sure I did. She might as well have written me a letter: "Run away, run away." By then, I didn't need a push.
Jane Franklin never ran away, and never wrote the story of her life. But she did once stitch four sheets of foolscap between two covers to make a little book of sixteen pages. In an archive in Boston, I held it in my hands. I pictured her making it. Her paper was made from rags, soaked and pulped and strained and dried. Her thread was made from flax, combed and spun and dyed and twisted. She dipped the nib of a pen slit from the feather of a bird into a pot of ink boiled of oil mixed with soot and, on the first page, wrote three words: "Book of Ages"—a lavish, calligraphic letter "B," a graceful, slender, artful "A." She would have learned this—an Italian round hand— out of a writing manual, like "The American Instructor: Or, Young Man's Best Companion," a book her brother printed in Philadelphia.
At the top of the next page, in a much smaller and plainer hand, she began her chronicle:
Jane Franklin Born on March 27-1712
Edward Mecom Marryed to Jane Franklin the 27th of July 1727.
The Book of Ages: _her_ age. Born, March 27, 1712; married, July 27, 1727. Fifteen years four months. She was a child. The legal age for marriage in Massachusetts was sixteen; the average age was twenty-four, which is the age at which her brother Benjamin married and, excepting Jane, the average age at which her sisters were married.
The man she married, Edward Mecom, was twenty-two. He was poor, he was a saddler, and he was a Scot. He wore a wig and a beaver hat. She never once wrote anything about him expressing the least affection.
She added another line:
Josiah Mecom their first Born on Wednesday June the 4: 1729
She named this child, her first, for her father.
and Died May the 18-1730.
The child of her childhood died three weeks shy of his first birthday.
"A Dead Child is a sight no more surprizing than a broken Pitcher," Cotton Mather preached, in a sermon called "Right Thoughts in Sad Hours." One in four children died before the age of ten. The dead were wrapped in linen dipped in melted wax while a box made of pine was built and painted black. Puritans banned prayers for the dead: at the grave, there would be no sermon. Nor, ministers warned, ought there to be tears. "A Token for Mourners or, The Advice of Christ to a Distressed Mother, bewailing the Death of her Dear and Only Son" cited Luke 7:13: "Weepe not."
What remains of a life? "Remains" means what remains of the body after death. But remains are also unpublished papers. And descendants are remains, too. The Boston Puritan poet Anne Bradstreet wrote about her children as "my little babes, my dear remains." But Bradstreet's poems were her children, as well: "Thou ill-form'd offspring of my feeble brain." Her words were all that her children would, one day, have left of her. "If chance to thine eyes shall bring this verse," she told them, "kiss this paper."
Jane didn't know how to write a poem. She couldn't have afforded a headstone. Instead, she went home, and wrote a book of remembrance. _Kiss this paper_.
College was something of a bust. It was the nineteen-eighties. On the one hand, Andrea Dworkin; on the other, Jacques Derrida. I took a job as a secretary, on the theory that it would give me more time to read what I wanted. "Is that working out?" my mother wanted to know. I wrote a graduate- school application essay about a short story of Isak Dinesen's called "The Blank Page." Very "A Room of One's Own." Very "The Madwoman in the Attic." ("I write now in my own litle chamber," Jane wrote, when she was sixty-four, "& nobod up in the house near me to Desturb me." She was very happy to have that room, but not having it sooner isn't why she didn't write more or better.) Then, suddenly, I realized that my life plan—bashful daughter of shackled artist reads "The Yellow Wallpaper"—was narrow, hackneyed, daffy.
I was sick of silence, sick of attics and wallpaper, sick of blank pages and miniature rooms, sick of blighted girlhood. I wanted to study war. I wanted to investigate atrocity. I wanted to write about politics. Really, I wanted to write about anything but Jane Franklin.
"What about beauty?" my mother pressed. I kept making excuses. I was pregnant. ("Edward Mecom Born on Munday the 29 March 1731.") I was too busy. ("My mind is keept in a contineual Agitation that I Dont know how to write," Jane once apologized.) I was pregnant again. ("Benjamin Mecom Born on Fryday the 29 of December 1732.") I was so tired. ("My litle wons are Interupting me Every miniut.") I was pregnant again. ("Ebenezer Mecom Born on May the 2 1735.") I felt rebuked, even by Jane herself. ("I was almost Tempted to think you had forgot me but I check those thoughts with the consideration of the dificulties you must labour under.") I hadn't forgotten her. I just couldn't bear to think about her, trapped in that house.
But Benjamin Franklin: I adored him. He was funny and brilliant and generous and fortunate. Every year of his life, his world got bigger. So did mine. When he had something to say, he said it. So did I. My mother and I had got tangled up, like skeins. I wasn't the one who identified with Jane.
The more I thought about Jane, the sadder it got. I tried to picture it. Her belly swelled, and emptied, and swelled again. Her breasts filled, and emptied, and filled again. Her days were days of flesh: the little legs and little arms, the little hands, clutched around her neck, the softness. A baby in her arms, she stared into kettles and tubs, swaying. The days passed to months, the months to years, and, in her Book of Ages, she pressed her children between the pages.
Her husband fell into debt. He may have gone mad. (Two of her sons became violently insane; they had to be locked up.) Jane and her children lived with her parents; she nursed them, in their old age. Josiah Franklin died in 1744. He was eighty-seven. In his will, he had divided his estate among his surviving children. Benjamin Franklin refused his portion: he gave it to Jane.
In 1751, Jane gave birth for the twelfth time. She was thirty-nine. She'd named her first baby for her father; she named this baby, her last, for her mother, Abiah Franklin:
Abiah mecom born augst 1st 1751.
The month Abiah Mecom was born, Benjamin Franklin took a seat in the Pennsylvania Assembly. His eighty-four-year-old mother wrote him a letter, with her daughter at her side. "I am very weeke and short bretht so that I Cant set to rite much," Abiah Franklin explained. She asked her daughter to write for her. Aside from Jane's Book of Ages, and notes she made in books she read, this is the only writing in her hand to survive, for the first four decades of her life:
P S Mother says she an't able and so I must tell you my self that I rejoyce with you and bles god for you in all your prosperity and doubt not but you will be grater blessings to the world as he bestows upon you grater honers.
_Mother says she an't able and so I must_. They'd got tangled, too.
Jane's baby, Abiah Mecom, died within the year. So did Abiah Franklin:
my dear Mother Died May 8 1752.
She loved her; she washed her. She buried her. But it was Benjamin Franklin who paid for a gravestone, and wrote an epitaph:
Josiah Franklin
And Abiah his Wife
Lie here interred.
They lived lovingly together in Wedlock
Fifty-five Years.
. . . .
From this Instance, Reader,
Be encouraged to Diligence in thy Calling,
And distrust not Providence.
He was a pious & prudent Man,
She a discreet and virtuous Woman.
Their youngest Son,
In filial Regard to their Memory,
Places this Stone.
This book of remembrance was a monument, not to his parents but to Franklin himself: prodigal son.
"Do the right thing with Spirit," Jane once wrote. It's just the kind of thing my mother liked to say. One of Jane's sons became a printer. He once printed a poem called "The Prodigal Daughter": "She from her Mother in a Passion went, / Filling her aged Heart with Discontent."
My mother's heart began to fail. She had one heart attack, and then another. Surgery, and more surgery. Eventually, she had a defibrillator implanted in her chest. I'd visit her in the hospital; she'd send me away. All I wanted was to be there, with her, but that only made her remember going home to watch her mother die. "See? I'm fine," she'd say. "Now. Please: go. You have things to do."
I decided I had better read whatever of Jane's letters had survived. The first is one she wrote in 1758, when she was forty-five years old, to Franklin's wife, Deborah. This is her voice, gabby, frank, and vexed:
Dear Sister
for so I must call you come what will & If I dont Express my self proper you must Excuse it seeing I have not been acostomed to Pay my Complements to Governer & Baronets Ladys I am in the midst of a grate wash & Sarah still sick, & would gladly been Excused writing this Post but my husband says I must write & Give you Joy which we searly Joyn in; I sopose it will not be news to you, but I will tell you how I came by it, Mr Fluker Tould Cousen Williams & he Docter Perkins who Brought it to my Poor Son nedey who has a nother Relaps into Raising Blood & has not Done won stroke of work this month but was Just a going to begin when he was again taken Ill pray Pardon my Bad writing & confused composure & acept it as coming from your Ladyships affectionat Sister & most obedent
Humble Servant
Jane Mecom
She was in the middle of a great wash. One of her lodgers, Sarah, was ailing. Her poor son Edward (Neddy), who was married and a father, was sick again—weak and listless and coughing blood. But she had heard from Neddy, who heard it from her doctor, John Perkins, who heard it from Jonathan Williams, Sr., the husband of Jane's friend Grace, who heard it from a Boston merchant, Thomas Flucker, that Benjamin Franklin had been given a baronetcy. Jane's husband told her she must send her congratulations, "searly"—surely. She was miffed. If this ridiculous rumor was true, why, for heaven's sake, was she the last to know about it?
"Your loving Sister," or "Your affectionate Sister," is how Jane usually signed off—not "your Ladyships affectionat Sister & most obedient Humble Servant." That was a jab. Must she curtsey?
By words on a page, she wanted to be carried away—out of her house, out of Boston, out into the world. The more details the better. "The Sow has Piged," a friend reported from Rhode Island, reminding her, "You told me to write you all." She loved gossip. "Cousen willames Looks soon to Lyin," she told Deborah, "she is so big I tell Her she will have two." She once scolded her niece for writing letters that she found insufficiently chatty. "I want to know a Thousand litle Perticulars about your self yr Husband & the children such as your mother used to write me," Jane commanded, adding, "it would be Next to Seeing the little things to hear some of there Prattle (Speaches If you Pleas) & have you Describe there persons & actions tell me who they Look like." Stories, likenesses, characters: speeches.
My mother wasn't much of a letter writer. If she telephoned, she would yell, "This is your mother calling!" My sisters and my brother and I got her an iPad for her birthday. "If you call here keep talking as it takes us time to get to the phone," she e-mailed me. She had a cell phone, for emergencies; she brought it with her when she had to go to the hospital. Once, she was kept waiting on a gurney, in a hallway, for hours.
"This is your mother calling!"
"I know, Mom. Why haven't they gotten you a room?"
"Oh, I don't mind," she said. "The people here in the hallway are just _fascinating_." She was giggling.
"Mom. Did they give you something for the pain?"
"Oh, yes, it's wonderful."
"The people are interesting?"
"Oh, yes. It's like a soap opera here."
Jane, writing to her brother, worried that she had spelled so badly, and expressed herself so poorly—"my Blundering way of Expresing my self," she called it—that he wouldn't be able to understand what she meant to say. "I know I have wrote and speld this worse than I do sometimes," she wrote him, "but I hope you will find it out."
To "find out" a letter was to decipher it, to turn writing back into speech. Jane knew that letters weren't supposed to be speeches written down; they were supposed to be more formal. Her brother warned her that she was too free with him. "You Long ago convinced me that there is many things Proper to convers with a Friend about that is not Proper to write," she confessed. But, then, he scolded her, too, for not being free enough. "I was allways too Difident," she said he had told her.
"Dont let it mortifie you that such a Scraw came from your Sister," she begged him.
"Is there not a little Affectation in your Apology for the Incorrectness of your Writing?" he teased her. "Perhaps it is rather fishing for Commendation. You write better, in my Opinion, than most American Women."
This was, miserably, true.
It was the diffidence that got to me. Female demurral isn't charming. It's maddening. Half the time, I wanted to throttle her. Could she ever, would she never, express a political opinion?
I read on. And then, in the seventeen-sixties, a decade of riots, protests, and boycotts, something changed. Jane's whole family was sick. Her daughter Sally died; Jane took in Sally's four young children; then two of them died, only to be followed by Jane's husband and, not long after, by
Jane's favorite daughter. "The Lord Giveth & the Lord taketh away," she wrote in her Book of Ages. "Oh may I never be so Rebelious as to Refuse Acquiesing & & saying from my hart Blessed be the Name of the Lord." And then: she put down her pen. She never wrote in her Book of Ages again.
"Realy my Spirits are so much Broken with this Last Hevey Stroak of Provedenc that I am not capeble of Expresing my self," she wrote to her brother. She did not think she could bear it. In the depth of her despair, she began to question Providence. Maybe her sons had failed not for lack of merit but because they were unable to overcome the disadvantage of an unsteadiness inherited from their father. Maybe her daughters and grandchildren had died because they were poor, and lived lives of squalor. Maybe not Providence but men in power—politics—determined the course of human events.
She wrote to her brother. She wanted to read "all the political Pieces" he had ever written. Could he please send them to her?
"I could as easily make a Collection for you of all the past Parings of my Nails," he wrote back. He sent what he could. She read, and I read.
She left home in 1775, after the battles of Lexington and Concord, when the British occupied Boston. For a while, she lived with her brother in Philadelphia. After he left for France, she spent the war as a fugitive. "I am Grown such a Vagrant," she wrote him. When peace came—after he helped negotiate it—he returned to Philadelphia, and she to Boston.
He gave her a house in the North End. She loved it. "I have this Spring been new planking the yard," she one day boasted, and "am Painting the Front of the House to make it look Decent that I may not be Ashamed when any Boddy Inquiers for Dr Franklins Sister."
She knew, for the first time, contentment. Except that she was starved for company. "I Injoy all the Agreable conversation I can come at Properly," she wrote to her brother, "but I find Litle, very Litle, Equal to that I have a Right to by Nature but am deprived of by Provedence."
It was a shocking thing to say: that she had a right to intelligent conversation— a natural right—but that Providence had deprived her of it. Before the war, she had favored independence from Britain. After it, she found her own kind of freedom. Once she started writing down her opinions, she could scarcely stop.
"I can not find in my Hart to be Pleasd at your Accepting the Government of the State and Therefore have not congratulated you on it," she wrote to her brother in 1786, when he accepted yet another political appointment. "I fear it will Fatigue you two much."
"We have all of us Wisdom enough to judge what others ought to do, or not to do in the Management of their Affairs," he wrote back. "Tis possible I might blame you as much if you were to accept the Offer of a young Husband."
She let that pass. "I have two favours to Ask of you," she begged him: "your New Alphabet of the English Language, and the Petition of the Letter Z."
"My new Alphabet is in a printed Book of my Pieces, which I will send you the first Opportunity I have," he answered. "The Petition of Z is enclos'd."
In "The Petition of the Letter Z," a satire about inequality, Z complains "That he is not only plac'd at the Tail of the Alphabet, when he had as much Right as any other to be at the Head but is, by the Injustice of his Enemies totally excluded from the Word WISE, and his Place injuriously filled by a little, hissing, crooked, serpentine, venomous Letter called S." In another essay, Franklin proposed a new alphabet. Jane found it cunning, especially since, as she explained, "I am but won of the Thousands & thousands, that write on to old Age and cant Learn."
"You need not be concern'd in writing to me about your bad Spelling," he wrote back, "for in my Opinion as our Alphabet now Stands, the bad Spelling, or what is call'd so, is generally the best." To illustrate, he told her a story: A gentleman receives a note that reads, "Not finding Brown at hom, I delivered your Meseg to his yf." When both the gentleman and his wife are unable to decipher the note, they consult their chambermaid, Betty. "Why, says she, y. f spells Wife, what else can it spell?"
Jane loved that. "I think Sir & madam were deficient in Sagasity that they could not find out y f as well as Bety," she wrote her brother, "but some times the Betys has the Brightst understanding."
"How's that book about Jane Franklin coming along?" my mother asked, every time I took her out. (We'd go to art museums, mostly. I'd race her around, in a wheelchair.) "Better," I said.
When Jane was seventy-four, she read a book called "Four Dissertations," written by Richard Price, a Welsh clergyman and political radical. The first dissertation was called "On Providence." One objection to the idea that everything in life is fated by Providence, Price wrote, is the failure to thrive: "Many perish in the womb," and even more "are nipped in their bloom." An elm produces three hundred and thirty thousand seeds a year, but very few of those seeds ever grow into trees. A spider lays as many as six hundred eggs, and yet very few grow into spiders. So, too, for humans: "Thousands of Boyles, Clarks and Newtons have probably been lost to the world, and lived and died in ignorance and meanness, merely for want of being placed in favourable situations, and enjoying proper advantages." No one dies for naught, Price believed, but that doesn't mean suffering is fair, or can't be protested.
At her desk, with Price's "Four Dissertations" pressed open, Jane wrote a letter to her brother. "Dr Price thinks Thousand of Boyles Clarks and Newtons have Probably been lost to the world," she wrote. To this, she added an opinion of her own: "very few we know is Able to beat thro all Impedements and Arive to any Grat Degre of superiority in Understanding."
Benjamin Franklin knew, and his sister knew, that very few ever beat through. Three hundred thousand seeds to make one elm. Six hundred eggs to make one spider. Of seventeen children of Josiah Franklin, how many? Very few. Nearly none. Only one. Or, possibly: two.
Benjamin Franklin died in 1790. He was eighty-four. Jane died four years later. She was eighty-three. If she ever had a gravestone, it's long since sunken underground. She believed in one truth, above all: "The most Insignificant creature on Earth may be made some use of in the scale of Beings."
It wasn't until last year, sitting by my father in a room in intensive care in a hospital in the town where he'd been born eighty-seven years before, that I realized I had waited too long to write the only book my mother ever wanted me to write. From this Instance, Reader, Be encouraged to Diligence in thy Calling.
We buried my father. My mother ordered a single gravestone, engraved with both their names. I wrote as fast as I could. Meanwhile, I read my mother letters and told my mother stories. In a museum, I found a mourning ring Jane had owned; I told my mother about how, when no one was looking, I'd tried it on. (I didn't tell her that it didn't fit, and that I'd found this an incredible relief.) In a library not a dozen miles from Franklin Street, I found a long-lost portrait of Jane's favorite granddaughter: another Jenny, age nine. I brought my mother a photograph. She looked, for a long time, into that little girl's eyes. "She's beautiful," she said. She smiled. "I'm so glad you found her."
"That mother of yours," my father used to say, shaking his head, besotted. He knew he could never live without her. I never knew—never saw, never in the least suspected—that she couldn't live without him, either. But, after his death, she didn't last out the year. She died at home, unexpectedly, and alone. She kept her paintbrushes in glass jars in my old bedroom. She was eighty-five.
I finished the last revisions. Too late for her to read it. I wrote the dedication.
Their youngest daughter. In filial regard. Places this stone.
_Suzannah Lessard_
Suzannah Lessard is a journalist, author, and professor whose work has appeared in _The New York Times Magazine_ , _Architectural Record_ , _Architectural Digest_ , and _The Wilson Quarterly_. She was one of the first editors of the _Washington Monthly_ , from 1971 to 1974, and has taught at Columbia School of the Arts, Wesleyan University, The New School, George Mason University, George Washington University, and Goucher College. She spent twenty years as a staff writer for _The New Yorker_.
Lessard is a recipient of the 1995 Whiting Award as well as the 2003 Mark Lynton History Prize for _Mapping the New World: An Inquiry into the Meaning of Sprawl_. She also received a 2001 fellowship at the Woodrow Wilson International Center for Scholars in Washington, DC, and a 2002 Jenny McKean Moore Fellowship for creative nonfiction at George Washington University.
One of Lessard's best-known works is her memoir, _The Architect of Desire: Beauty and Danger in the Stanford White Family_. Her next book, _The Crack of Dawn: A Meditation on the American Landscape_ , will be published in 2018.
The Split
In Manhattan, the opulent Upper East Side, one of the nation's richest neighborhoods, gives way to Harlem, one of the poorest.
In the middle of Park Avenue, just north of Ninety-Sixth Street in Manhattan, there is a little nowhere of a paved-over spot that is the last in a sequence of otherwise lushly cultivated malls running up the center of the avenue from Forty-Sixth Street. Looking south from that island, one sees the parapets of Park silhouetted against the sky—two phalanxes of Italianate apartment buildings, built in the nineteen-twenties, quoined, rusticated, and corniced, in many instances, and from twelve to fifteen stories high. Facing each other, they stand at attention, chests swelling, the cornices creating the effect of a martial hat with a brim and generally making a statement of high-bourgeois solidarity which brooks no contradiction. This is the Upper East Side.
Turning to the north, one sees a porous hodgepodge of nineteenth-century tenements and high-rise public housing divided by four sets of train tracks which shoot out from under the hill at Ninety-Seventh Street, continuing on a viaduct that cuts a wide swath through the vista. The viaduct bodychecks the two sides of Park Avenue, reducing it to a mingy, shadowed passage not even up to the standard of an ordinary street. This is Harlem. Here at Ninety-Sixth Street, the nation's richest neighborhood collides with one of the poorest so sharply that it is very nearly possible to stand with one foot in each.
The community immediately north of Ninety-Sixth Street is East Harlem, also known as the Barrio, because of the many Puerto Ricans who live there. In 1990, the median family income in the Barrio (somewhat skewed by residences attached to Mount Sinai Hospital and by a thin strip of wealth that continues up Fifth Avenue) was twenty thousand dollars a year; farther north, in the Bradhurst area of Harlem, it was six thousand dollars a year, the lowest in the United States, with the exception of a former leper colony in Hawaii. The neighborhood immediately to the south of Ninety-Sixth and Park is Carnegie Hill, an old-fashioned, traditionally Wasp neighborhood, where the median family income is a modest hundred and twenty-nine thousand a year. Carnegie Hill overlaps with the Upper East Side historic district, which runs from Fifth Avenue to Lexington Avenue and from Seventy-Ninth Street down to Fifty-Ninth. South of Seventy-Second Street there is an eight-block area where the median household income of homeowners is seven hundred and thirty thousand dollars a year. At Ninety-Seventh and Park, unemployed men of varying ages sit on milk crates outside a bodega drinking cerveza fria. A few blocks south on Madison, you can buy a little girl's dress for four hundred and seventy dollars.
For New Yorkers, this abrupt juncture of the Upper East Side and Harlem is like a formation of the Ice Age: dramatic, but so familiar that it is barely noticed. The divide reflects a split in American society that is usually blurred by geographic distance, yet even here, where it is abrupt and plainly evident, it is somehow unseen.
Sometimes, as a way of trying to take in more deeply what the split is, and what it is to cross it, I would take the subway to 125th Street and then walk back through East Harlem toward Ninety-Sixth. Or, conversely, I would get off at Fifty-Ninth Street and follow the procession of Park Avenue uptown. In either case, what I found was that, though I was alert and absorbed in the world around me right up to the boundary, at the moment of crossing I would go into a little blackout, a shutting down of awareness, as if at that moment I had stepped into a special chamber designed to equalize the effects of vastly differing pressures.
Katharine Hellman, a young woman who works as a therapist in an elementary school in Harlem, knew what I meant by a disjunction that was almost more than one could absorb. She grew up in a wealthy family on the Upper East Side, assuming, as a girl, that a normal home was a Park Avenue duplex with Magrittes on the wall, as she put it. When she first started working in Harlem, she always felt a tremendous pressure, on returning to her own world, to tell what she knew—to try to convey, for example, that that day she had visited the home of a student where there was no furniture. "I had never conceived of such a thing," she said. "I wanted to bring people from my world to the apartment with no furniture. It seemed like important knowledge. You have to see it—if you haven't seen it, there is something about it you might not get."
The school Hellman works at is on a nearly abandoned street where brownstones with punched-out or blocked-up windows have lost their stoops, so that the pedimented landings hang disconnected in the air. One June Friday after school in that early period of her career, she travelled up to her family's vacation home, a farm in Connecticut, for the weekend, and found herself standing in a lovely field. She thought it was a good thing that it took some time to get there, because "if you went from one place to the other instantly you would go crazy because you couldn't make the shift fast enough,"
The shift can be difficult to absorb in the reverse direction as well. A young woman I will call Manuela, a high-school student who had worked in a crack factory for a time, was taken by a neighborhood program on a day trip to Princeton University. She was enraged that she had been shown this beautiful college world, because she knew that she could never become a part of it. She attends a high school in East Harlem that was founded expressly to prepare students for college, but in actuality most of the kids are nowhere near that level. Manuela says that when she was very small her mother was almost certainly on drugs; she remembers dialing 911 once because she was alone in the apartment in the middle of the night. She was not yet in school when her mother died of AIDS, and she now lives with her grandmother.
Manuela was upset by her visit to Princeton. Yet it is also true that people who live in the inner city know very well how others live: if nothing else, television is a daily reminder. Yolanda Sanchez, who works in a clinic in the Bronx, felt no shift in pressure in the days when she still lived in East Harlem, where she had grown up, and was going to school on the Upper East Side, studying to become a social worker. "I might think, Isn't it amazing that a person would be able to afford to pay another to wash the street in front of their house," she said. Pedro Gomez is an affable young man who lives in the Barrio and works as a doorman at a Park Avenue apartment building that is right on the border. At work, in his uniform and white gloves, he seems to enjoy himself as he chats with the gentlemen leaving for their offices and crouches down to eye level with children on their way to school. Of the boundary that he crosses daily he said to me, "There's no difference—it's just people."
The grid was laid down over the countryside of Manhattan Island in 1811, and ran all the way up to what it designated as 155th Street, although for the most part the city had grown only as far north as Houston Street at that time. It was a neoclassical conception, a rational ideal that was typically nineteenth-century: the whole Midwest was gridded into states, counties, and homesteading lots. In New York, the primary function of the grid was to facilitate real-estate development. The grid-makers were so far ahead of the time, however, that there was nothing for decades after they laid it down but a dreary cross-hatching of muddy tracks in a wasteland of disturbed earth.
Though East Harlem is farther north than the Upper East Side, it was settled earlier, by Irish fugitives from overcrowded immigrant slums downtown. They built shanties along the Harlem River and were called goaters, after their favorite form of livestock. As conditions downtown worsened, the squatter community grew. By the eighteen-sixties, the first urban structures had gone up, and after 1880 most of the tenements were built in the "dumbbell" style—an improvement over downtown models in that the buildings had slight indentations on the sides to let in light and air. With their narrow facades obediently aligned with the gridded street front, the first tenements stood alone in the wasteland, or in twos or threes, accompanied only by occasional goaters' shanties.
The same surreal vistas of urban buildings exposed on the naked grid did not appear on the Upper East Side until 1894, when Caroline Schermerhorn Astor—the wife of William Astor and the cofounder of the society list known as the Four Hundred—built an imitation French chateau in the mud at Fifth Avenue and Sixty-Fifth Street, thereby signaling to Gilded Age society not only that this would be the next fashionable neighborhood but that architectural excess was as acceptable in the city as it had become out of town. East Harlem was at this point a teeming neighborhood, upon which wave after wave of immigrants—Irish, then German, then Jewish—crashed, and it soon rivaled the Lower East Side in horrific health conditions and overcrowding, as well as in a pulsing dynamism. On the Upper East Side, other palazzo builders followed Mrs. Astor, and a costume ball of town houses in a dazzling variety of architectural styles appeared on the side streets off Fifth Avenue.
Park Avenue, known as Fourth Avenue then, remained through all this a seedy neighborhood, because the New York Central Railroad ran up the middle in an open cut, spewing cinders and rattling with noise. Then William Wilgus, the vice-president and chief engineer of the railroad, saw that if the railroad switched from steam to electric power the cut could be covered up and the value of the land along the avenue—land owned by the New York Central and the Vanderbilts—would soar. The avenue was renamed Park, and development there was modeled on the grand nineteenth-century visions of Baron Haussmann, Napoleon III's master planner, who had rebuilt Paris, creating wide boulevards to reflect the imperial status of modern France.
It was not until 1921, however, that the prototypical Park Avenue apartment building went up—the Marguery, a colossus consuming the entire block between Forty-Seventh and Forty-Eighth on the west side of Park, offering seventeen- and eighteen-room apartments and containing several restaurants managed by the Ritz. One or two members of Society succumbed to the lure of luxury and convenience over the hidebound convention of the town house, and then there was a reverse domino effect—a sequential uprising of Marguery-type buildings, creating the consistent architectural panorama that is the Park Avenue we know today, continuing right on up to Ninety-Sixth Street, where it was brought to a halt by the emergence of the tracks from under the hill there, and by the crash of 1929.
By this time, East Harlem had become a predominantly Italian neighborhood, its residents still very poor, but working, and with middle- class people who offered them professional services living in their midst. Beginning in the twenties and thirties and gaining momentum in the fifties, the Italians of East Harlem moved on, yielding to Puerto Ricans. This turnover reflected an established pattern of successive upward mobility, except that Harlem generally was becoming an "inner city": it was more and more isolated from the mainstream economy and was losing its middle-class population as its lower-class population lost jobs, while crime and dissolution separated it further from the mainstream society.
Ordinary things about the Upper East Side became vivid, even astonishing, when I had walked down into the neighborhood from Harlem. I noticed how during the morning rush hour Park Avenue streams with taxis as if with yellow pigment, but the atmosphere is nonetheless serene, for trucks are banned and the lights are not staggered—the traffic is brought to a halt for the entire length of the avenue every few minutes. I was struck by the men in midlife issuing from their buildings with lobbies that looked small and cozy in comparison with the grandeur of the façades. I noted the playfulness in their clothing—a bright-colored cashmere scarf or a ski cap with a tassel. They looked very clean, and I saw in their faces an attractive worldly complexity—the well-being that comes of having one's powers fully engaged. I watched them take time for an easygoing exchange with the doorman, sometimes getting into a cab he had hailed, or into a chauffeured car, or walking on down spacious, uncrowded sidewalks with a light, elegantly shod step.
I became aware for the first time that the Upper East Side is rooted in midtown—one of the highest-priced shopping districts in the world. But in contrast to the shops in midtown those on the Upper East Side have an intimate quality. Some of them might almost be a part of someone's home. (Much on the Upper East Side beckons with a homelike allure while at the same time the way is barred: the lobby is guarded by a doorman; the shop has a buzzer on the door.) Here you can find a store devoted entirely to buttons, or to shaving equipment, or personalized stationery and engraved invitations. The window displays often include a rough, beat-up common object, like an old knocked-together wooden chest, little more than a crate, or an old scale, rusty and chipped—objects that might look shabby elsewhere but look luminously beautiful here because they are placed and lit so perfectly, their textures and irregularities becoming, in that pristine environment, aesthetically pleasurable.
The burnishing effects of care also have a way of disguising common objects, so that the display in a liquor-store window somehow makes it look like a bookstore, and the bookstore looks like an art gallery; and the dressy pumps in the shoe-store window somehow look like vintage brandies, while stylishly decorated eggs clustered in the window of a bakery look like fancy shoes. Inside, the bakery is humbly picturesque, as if it were situated in a remote European village—an effect so studiously managed that entering the store is almost like entering a painting.
Real estate in the Upper East Side historic district is perhaps the most valuable, block by block, in the United States. On the side streets, one is hard put to find even a square inch that is not polished or steamed or embellished by a planter bearing, say, meticulously groomed small evergreens of an exotic variety. There is no improvisation here. Not a sliver of space has been neglected; indeed, there are quite a few buildings that are themselves barely more than slivers—contemporary architects' ingenious designs for the use of lots that would ordinarily be considered too narrow to build on. There is in this a kind of massing that is like a popular social event—an effect of crisp, alert energy, of contained excitement, even of merriment. The town houses crowd each other like jostling children eager for sweets, barely holding a line: their different roofs (dormers, rounded slate, the slanting glass of studios), some peaked and others corniced, making up a playful visual text that streaks back and forth across the architectural vocabulary of Western civilization.
Stopping in at the offices of local political representatives—a member of the community board, a State Assembly person—I heard about recent concerns of the neighborhood: Double-parked cars. Whether to create a private security district. Cleanup around a public school. This is an activist community, highly sensitive to any inroads on quality of life, and it expects its political representatives—who tend to be of a lower socioeconomic class than their constituents—to act quickly and effectively on its complaints. Nevertheless, meetings of Community Board 11 are famously decorous. ("Wasps don't shout, they sue," one representative commented.) The most boisterous meeting in memory had to do with the children's zoo.
In the course of my walks across the split, I became acutely conscious of the purr in the atmosphere along these quiet streets where chores are diligently performed by a vast staff (garbage stacked neatly just before pickup, brass polished, clean clothing delivered), of a pervasive sense of contentment that is profoundly reassuring even for the passerby. A woman walks under trees toward the light-filled open space of Park Avenue, in a coat that is both generously cut and sharply shaped and sways from shoulders to midcalf; she has tidy ankles and trim, close-fitting shoes. It was the delineated wrist and ankle that, in the end, emerged for me as the hallmark of the Upper East Side: somewhat in defiance of fashion and in contrast to the muffled, bundled look of the parkas and sneakers that are ubiquitous in East Harlem and, when they are old and less than clean, can seem to be on a continuum with the pile of rags in a doorway beneath which one can barely discern the shape of a sleeping body.
At Lexington and 124th Street, I know that I am in the Barrio because I see oranges peeled in a decorative way hanging in a string bag from a vender's cart. Spanish crackles around me. Emmaus House, a shelter across the street, which also provides a variety of other services to the homeless, looks worn and trafficked, the walls of the entryway silvered as if by a million hands. Drug addicts nod out, a dealer deals, the vender greets passersby. Though nearly everyone I see, with the exception of some young people, looks battered or injured or in some way physically stressed, the atmosphere is gregarious, at moments rippling with fun.
To the east, activity falls off quickly. Many buildings are boarded up, and the traffic is so light that even on the avenues men can work on cars well away from the curb. Along 120th Street, a stand of housing projects opens up the sky. Shades are drawn. Inside, intercoms and elevators are usually broken, and graffiti-ridden halls smell of urine; they are the domain of drug users, and are unsafe for the families attempting to bring up children. The exterior is banal and bureaucratic, but when I step off the street and into the domain of the projects the hair rises on the back of my neck.
East Harlem has the highest density of public housing in New York, and New York has the highest density in the country. Most of the housing went up in the late fifties and the sixties, driven by powerful visions of urban renewal which ripped out organic neighborhoods and, with them, the generative, socializing life of the street. As architecture, the projects reflect Le Corbusier's dream of the Radiant City, in which people would live in high-rise buildings separated by green expanses, connected by high-speed rail running at about the third-story level—a democratic utopia where architecture would break away from the class-ridden traditions of the past.
In the sixties and seventies, the population of East Harlem, like that of inner cities across the country, declined drastically, and the neighborhood went from being one of the most populated areas in the city to being one of the least populated. For decades, the available housing stock decreased steadily as the city pursued a policy of slum clearance through demolition. Because of this policy, a porous, flattened landscape began to appear, in which little groups of town houses or tenements are left freestanding, surrounded by vacant lots—a landscape eerily similar to that of the goaters' time, when urban buildings, alone or in small packs, first started going up on the empty grid. One even comes across plywood shacks in vacant lots in East Harlem—they are called casitas—which echo the shanties of the previous century. There is in large stretches of East Harlem a marginal, maverick feeling, a faraway mood, a deadness in the air that seems incompatible with the charged, compressed atmosphere of Manhattan. One would not be altogether surprised to see a goat.
These are some of the community concerns in East Harlem: The third highest incidence of AIDS in the city (after Chelsea and the Greenwich Village area). A generation lost to crack and heroin. That generation's children, traumatized in early life, now filling the schools. Teenage pregnancy: Overcrowding. Apartments with no heat. Poor health. High unemployment. Visiting politicians on 116th Street, I came across the idea that the Barrio could become a mecca for tourists—a kind of Latin Quarter that would draw people from south of Ninety-Sixth Street with restaurants and night clubs—and the idea of turning the old Washburn Wire factory, closed in 1982, into a shopping mall, and a rumor that the "smart money" was buying above Ninety-Sixth Street now and soon there would be hard bargains to be driven with developers, who would be refused access to the neighborhood unless they hired residents. The Barrio is an Empowerment Zone, which means that there is federal money to support businesses there, and it is not to deny that good may come of this (though businesses don't generally thrive where people have little money), or that hope itself can be productive, to say that beyond these good things there is an abyss of need that seems to engulf them. The spirit of the age is that all problems can be solved by free enterprise, and this creates an atmosphere of boosterism in which it becomes a kind of betrayal to acknowledge that abyss.
To the west of Park Avenue and a few blocks up there is an area where weeds grow on unused land and a single ailanthus tree leans into two brownstones that cling to each other alone in a desert, their empty window frames on the top floor open to the sky. Following Madison downtown from there, I came upon blocks that are razed completely on either side—the naked grid as it was laid down in 1811. That morning, the wind blew the way it blows in bleak rural places. Shards of brick—remnants of demolished houses—gave the terrain the reddish look of the desert in Utah. The effect of the brick remnants was to blur the distinction between the architecture and its surroundings, as in places like the Moroccan desert, and northern Portugal, where the houses are made out of materials taken from the earth in that very site and so blend with the ground. Garbage had been blown into drifts, especially in the westernmost block, which was surrounded by a chain-link fence. Pieces clustered up against the fence assumed an iconographic intensity, a feeling of trapped souls or fugitive ghosts. Across the reddish shards of the unfenced block I saw, faraway-seeming, tiny, and straining into the wind, a woman pushing a baby in a stroller, to what destination it was hard to imagine.
South of this wasteland I came upon a casita, a rough plywood shelter sitting on cinder blocks in the middle of a vacant lot, also enclosed by chain link. The sides of this casita were open to waist level; it looked like a rustic structure in a remote tropical place near the sea. Hanging from the struts were various objects: a fancy sombrero, a small facsimile of a hot-air balloon with a mouse in the basket, Christmas wreaths. Steps up to the open doorway were guarded by two seated rag dolls, black-faced, in neon-pink hats, and clothes of a paisley material. On the ground around the shelter, other objects—plastic pumpkins, a Superman figure, a plaster version of a terra-cotta Indian head—were placed in a way that made them interesting, like the ritualistic placement of rocks in a Japanese garden. Most arresting of all, however, were three medium-sized reproductions of landscapes in ornate gold frames, covered in clear plastic for protection, hanging on a white-painted brick wall at the back of the lot. One was of a sylvan scene of cows in a French Academic composition; one was of horses with vast mountains in the background, in a style reminiscent of the Hudson River School; and the third, smaller, was of the interior of a redwood forest. A placard affixed to the chain link identified this little world as "Peaceful Valley."
Some casitas are simply shacks, and most of of them are like little clubs—they are places to socialize. I also saw several that looked as if they might be primary dwellings. Almost all are decorated in some way; but some, like Peaceful Valley, can be seen as artistic installations of a kind, in which banal, artificial objects become fascinating because of their context, though to say even this much belies the unself-conscious, spontaneous nature of the casitas. There is no irony in the use of these objects, as there is in the sophisticated art exhibited at the Museo del Barrio, in which one also sees, occasionally, trinkets, religious images, or banal commercial objects—a scavenger aesthetic that is both amusing and arresting.
The Museo, now at Fifth Avenue and 104th Street, sprang up in the heart of the Barrio—it was situated there for a long time and collected and exhibited only Puerto Rican artists. Consequently, it has roots in the community; it sponsors, for example, a Three Kings' Day parade, a popular yearly event in the neighborhood. Yet it also belongs on the Museum Mile that includes the Metropolitan and Guggenheim Museums and the Museum of the City of New York, which is right next to the Museo, at 103rd Street. Having expanded its mandate to include the work of artists from all over Latin America, the Museo fills a gap in the international spectrum of curatorship in the city. In this, as an institution, the Museo truly belongs to both worlds.
Civitas, a citizens' organization based on the Upper East Side, also spans the divide, but from the perspective of planning and zoning—taking on the physical nature of the split. It consistently examines the transition between the neighborhoods, conducting a study of Madison Avenue straight on up to 125th Street, for example, or considering the Ninety- Sixth Street corridor with the needs of both neighborhoods in mind.
Then, there is Central Park, which extends from Fifty-Ninth Street to 110th Street, well into Harlem—literally common ground. Engineered and designed by Frederick Law Olmsted and Calvert Vaux in the mid-nineteenth century, it is both an artifact and a work of art, the creation of a pastoral landscape in the English Romantic tradition—truly a painting into which one can step. For a long time, the Park above Ninety-Sixth Street, and especially in the Hundreds, was deserted, overgrown, and dangerous: the Harlem Meer, a lake at the northern edge, was silted up with garbage—the same shopping cart protruded from the muck for years. Today, because of the work of the Central Park Conservancy, a private charity, the northern part of the Park is of a piece with the rest. The Harlem Meer now looks like a picture out of a storybook, with a bright-painted cottagelike structure on its edge and a flock of ducks that banks perfectly over blue waters in which, on still days, a prison that towers over Central Park North is reflected. Signs prohibit fires: it was once so wild here that people roasted whole pigs—a favorite Puerto Rican dish—over bonfires with impunity.
The government, of course, is the only enterprise that is truly equal to the scale of the divide. But in the last decade the sense that government can and should strive to bring about social and economic justice has lapsed to the point where that aspect of our political imagination has fallen into neglect—into a dreary wilderness akin to that of the Harlem Meer before its restoration. Standing on the island at Park and Ninety-Sixth Street, one cannot contemplate the Harlem vista for long without taking in an atmospheric sadness, the residue of defeated visions—of the failure of good intentions, epitomized eloquently by the stands of public housing.
There is another kind of common ground that I discovered in my travels across the line, however, one that is more important than art, or architecture, or even political institutions, and is the ground on which political life dedicated to equality must be based. This is the commonality of experience. Elsie Aidinoff, née Vanderbilt, who has taught and volunteered in Harlem for three decades but lives on the Upper East Side and finds herself from time to time in the venues of the extremely wealthy, makes the observation that over the years she has come to see the similarities between the worlds rather than the differences—that she has seen extraordinary, inexplicable courage in the face of the hardships that life can bring on both sides of the line, and, equally, spinelessness. She has noticed, too, the deleterious effect that not needing to work has on both the very rich and the welfare-dependent poor, and also the way that she as a mother sometimes feels more closely connected with a mother in Harlem than with a person on the Upper East Side who is not a parent.
Stephen Kurtz, a psychotherapist who practiced for many years both in prosperous venues and in Harlem, told me that, in terms of the interior landscape of people's lives, the divide does not show up. Suffering, he said, is the same on both sides of it, as is his job, which is to be with people in their suffering. He didn't mean that as many people suffer the emotional anguish of dying prematurely from AIDS on the Upper East Side as in East Harlem. What he meant was that the agony of dying young in East Harlem is no different from that of dying young on the Upper East Side, or—more to the point, more subtly—that people respond in a wide variety of ways to the same conditions but that the variety of response seems to have nothing to do with class or economic circumstances.
Kurtz allowed that one difference did show up from his point of view: while middle-class kids have any number of second chances, inner-city kids often have none. One bad mistake and they are lost, as Elsie Aidinoff observed of adults as well: families in poverty live on a brink where one misfortune—the breakup of a marriage, an illness, the loss of a job—sends the family headlong into disaster. "The older couple of kids may have made it, but the others are lost," she says. ("Lost" is a common term among people who work with kids in this territory.) Kurtz's observation fillets the issue in an exact way: it is precisely _because_ it is the same for a rich mother to watch her child fail as it is for a poor one that it is intolerable for an Upper East Side child to be given any number of second chances while the Harlem child makes one mistake and ends up in jail—and thence, almost inevitably, on a destructive life course, or dead.
Approaching Ninety-Sixth Street and Park Avenue through Carnegie Hill, I became conscious of a secluded feeling there, a quietness that comes from being at a distance from midtown with little through traffic to and from East Harlem: in a way, the nearby border of East Harlem might be the shore of a sea. There is a sense of older, less aggressive wealth here, and of discretion. Perhaps because it is a neighborhood where there is some social continuity between the generations, in certain midday moments, in a shift in the light, or in the silence created by a pause in traffic, the apartment buildings can seem to sag a little with the emotional weight of all the family life that has taken place within them—more, really, than a building can reasonably be expected to absorb. In that passing moment of candor, of tiredness, the buildings seem childishly dated, their authority thin, the defense they offer against the elemental facts of life that befall us all pitifully penetrable.
Then, beginning around two-thirty in the afternoon, massing herds of children begin to appear outside the numerous private schools in Carnegie Hill. With their teachers standing around the doors, and their mothers and nannies on the outer edges, they seem to be enclosed in a ring of safety and belonging. The mothers and children, or nannies and children, or children alone drift out into the neighborhood, and then one sees that the ring of safety includes all of Carnegie Hill: they buy snacks and eat them as they walk along or, in clement weather, as they sit in chairs set outside stores. Two little boys press their noses against the window of William Greenberg's, a bakery of the highest order, looking at schnecken— Greenberg's famous cinnamon rolls and an after-school favorite.
In December, Christmas trees decorated with white lights are placed on the malls all the way up Park Avenue from Forty-Ninth Street to Ninety-Sixth Street. This is a New York tradition that is less famous, perhaps, than the tree at Rockefeller Center, yet more resonant in feeling. Certainly it belongs to all New Yorkers, not just residents of the Upper East Side. I learned only recently that the practice was initiated in 1945 by Mrs. Stephen Clark, a resident of Park Avenue, who was married to the heir to the Singer sewing-machine fortune. In that year, Mrs. Clark lost a son in the war; the trees were a memorial to all the men from the City of New York who had died overseas.
The trail of starry firs comes to an end at Ninety-Sixth Street, but no one would think for a moment that because of this they honor a casualty from East Harlem any the less, or that his mother's grief is in any way less than Mrs. Clark's was. Nor could anyone imagine the memorial being conceived in any other than an inclusive way. And yet when it comes to the less heroic kinds of suffering, imposed in larger doses by poverty but common to all, that sense of commonality often fails.
Of course, it is not just suffering that runs across the line like the grid: it is joy and humor; it is all of organic human life with its infinite richness, a richness that so far outdistances the richness that money confers that it, too, confounds the boundary. Although in one sense the Upper East Side is a beautiful neighborhood in contrast to the often raw ugliness of parts of East Harlem, it is even true that beauty, in the sense of the Joycean poetry of the world, travels across the line as obliviously as the grid does. A child emerging in wonder from a shop where plaster statuary is being made, the tiny lights that line the window of a video store blinking wildly in late-afternoon light, a woman in a kerchief sitting on a small, low bench outside a funeral home, almost lost in the pails of flowers around her: Life is life wherever it is, inherently equal in a way that mere social and economic differences could never alter.
The late afternoon is a time that is particularly susceptible to life's magic spells: farther along that day, I saw a brick wall at the back of a casita on which a blue sky with clouds had been painted. On this mural, tall, narrow birdhouses in variegated colors had been placed at slight angles, creating a vertiginous effect. The painted sky was honeyed with real sun, and real birds flew to and from the birdhouses. In a big scruffy lot next door, vacant so long that trees had grown to maturity; a bonfire was reaching the point where a base of red-hot coals had built up, and a small group of men stood around the entrance of the casita smoking, waiting till the moment was right to start roasting a pig.
_Janet Malcolm_
Janet Malcolm is a writer for _The New Yorker_ and the author of eight books. _Diana and Nikon_ , her first, is a collection of essays on photography. _Psychoanalysis: The Impossible Profession_ is an expanded version of her profile of the psychoanalyst Aaron Green, and _In the Freud Archives_ is based on her two-part article on the psychoanalyst Jeffrey Moussaieff Masson. The Journalist and the Murderer, about a lawsuit brought by a convicted murderer against the author of a book on his crime, examines the relationship between writer and subject; it was first published in 1989 as a two-part article in the magazine. _The Purloined Clinic_ is a collection of essays and criticism from _The New Yorker_ and _The New York Review of Books_. _The Silent Woman: Sylvia Plath & Ted Hughes explores the life and work of Sylvia Plath_ and is based on an article that originally appeared in The New Yorker. In The Crime of Sheila McGough, Malcolm focuses on the American legal system. _Reading Chekhov_ weaves together close readings of Chekhov's works with scenes from the Russian writer's life and her own travels in Russia. In the fall of 2007, Malcolm published a book titled _Two Lives: Gertrude and Alice_.
Malcolm was born in Prague and emigrated to the United States with her family in 1939. She currently lives in New York City.
Yuja Wang and the Art of Performance
The young pianist is known not only for the brilliance of her music, but also for her dramatic, often surprisingly sexy outfits.
What is one to think of the clothes the twenty-nine-year-old pianist Yuja Wang wears when she performs—extremely short and tight dresses that ride up as she plays, so that she has to tug at them when she has a free hand, or clinging backless gowns that give an impression of near-nakedness (accompanied in all cases by four-inch-high stiletto heels)? In 2011, Mark Swed, the music critic of the L.A. _Times_ , referring to the short and tight orange dress Yuja wore when she played Rachmaninoff's Third Piano Concerto at the Hollywood Bowl, wrote that "had there been any less of it, the Bowl might have been forced to restrict admission to any music lover under 18 not accompanied by an adult." Two years later, the _New Criterion_ critic Jay Nordlinger characterized the "shorter-than-short red dress, barely covering her rear," that Yuja wore for a Carnegie Hall recital as "stripper-wear." Never has the relationship between what we see at a concert and what we hear come under such perplexing scrutiny. Is the seeing part a distraction (Glenn Gould thought it was) or is it—can it be—a heightening of the musical experience?
During the intermission of a recital at Carnegie Hall in May, Yuja changed from the relatively conventional long gold sequinned gown she had worn for the first half, two Brahms Ballades and Schumann's "Kreisleriana," into something more characteristically outré. For the second half, Beethoven's extremely long and difficult Sonata No. 29 in B-Flat, known as the "Hammerklavier," she wore a dress that was neither short nor long but both: a dark-blue-green number, also sequinned, with a long train on one side—the side not facing the audience—and nothing on the other, so that her right thigh and leg were completely exposed.
As she performed, the thigh, splayed by the weight of the torso and the action of the toe working the pedal, looked startlingly large, almost fat, though Yuja is a very slender woman. Her back was bare, thin straps crossing it. She looked like a dominatrix or a lion tamer's assistant. She had come to tame the beast of a piece, this half-naked woman in sadistic high heels. Take that, and that, Beethoven!
A few months before the performance, I asked Yuja why, out of all Beethoven's sonatas, she had selected the "Hammerklavier," and she said that she had done it out of defiance. She wanted to prove that she could play the most difficult of Beethoven's sonatas. I said that I was probably not alone in finding the sonata hard, almost unpleasant, to listen to, and several days later she sent me a link to a video of a lecture about the "Hammerklavier" by the Hungarian-born pianist András Schiff. Schiff speaks in the slow, self-savoring way in which many Eastern European men speak, to let you know how interesting and amusing everything they say is—except in his case it is.
Schiff characterized the work as "the greatest" and "most monumental" of Beethoven's sonatas, "a work that everybody respects and reveres but very few people love." Schiff's object was to communicate his own "deep love for this piece," and he began by talking about Beethoven's metronome markings, which are "incredibly fast" and are ignored by most pianists, who play the piece slowly and ponderously. The piece "is not pretty," but it is not "heavy-handed. . . not made of lead." Schiff mocked the pianists who protract the long third movement to show that "we are very deep and profound. . . . You can have lunch and dinner and breakfast, and we are still sitting here." Schiff went on to say, "If you play this piece at Beethoven's tempi, then it's not ponderous anymore. . . . It is not a piece in marble. . . . It is incredibly human and alive."
At Carnegie, Yuja did not play the piece quite at Beethoven's tempi—these days, few pianists do apart from Schiff—but I found myself responding to it as I had not responded to recordings by the great Maurizio Pollini and Mitsuko Uchida. I had not been able to get past the music's unprettiness. But now I was electrified. The forty- or fifty-minute-long piece (depending on how ponderously or not ponderously you play it) seemed almost too short.
A communication from another audience member, the pianist Shai Wosner, helpfully explained the inexplicable: why a piece that is about struggle and difficulty should have given the pleasure it gave in Yuja's interpretation. "There is hardly any passage in it that is truly comfortable to the hand," he wrote, along with "a certain harmonic tension that runs pretty much throughout the piece between B-flat major and B minor, Beethoven's 'dark, forbidden' key." He went on:
With all the Beethovenian struggle, this piece is also a very "cleanly" conceived sonata, more faithful to the Classical sonata model than any of Beethoven's other late Sonatas. So what I loved about Yuja's performance was how this other aspect of the piece came across . . . her effortless approach brought out the brilliant, clear structure of Hammerklavier and highlighted it from another angle. Like a great monument that's not made of stone but of light-reflecting glass.
Anthony Tommasini, reviewing the performance in the Times, wrote, "Ms. Wang's virtuosity goes well beyond uncanny facility. . . . She wondrously brought out intricate details, inner voices and harmonic colorings. The first movement had élan and daring. The scherzo skipped along with mischievousness and rhythmic bite." Neither Tommasini nor Wosner mentioned Yuja's dress, but I wondered about its impact on their experience. I know that what I saw was intertwined with what I heard. Looking at her in that remarkable getup was part of the musical experience. But what part?
Yuja had played the "Hammerklavier" a week or so earlier in Santa Barbara, and Mark Swed had again not failed to notice what she wore. This time, perhaps not altogether seriously, he attributed her choice of costume to altruism. Six days earlier, Murray Perahia, who is sixty-nine, had played the "Hammerklavier" nearby, in Los Angeles. "Hers is a 40-year age advantage," Swed wrote, so "as if to level the field technically, she came out onto the stage . . . tightly squeezed into a red-orange gown and wearing platform heels so high that she could barely walk." Swed praised both performances. "Perahia's understanding, feeling and urgency produce a 'Hammerklavier' for the ages," while Wang, "with a flick of her dazzling fingers on the keys, sends an electric current through the 'Hammerklavier' that makes it modern music, Beethoven for the 21st century." And, while Perahia "emerged from his ordeal exhausted, hardly able to walk offstage" (in spite of his flat-heeled shoes), Wang "in the manner of the greatest virtuosos of yore . . . made this great effort seem almost effortless and was ready for three amazing encores."
In New York, as it happened, Perahia had once again played the "Hammerklavier" a few days before Yuja did and again had had the starch taken out of him. Tommasini returned to Perahia's performance in his review of Yuja's (he had enthusiastically reviewed the Perahia on May 9th) and held up the older pianist's exhaustion as a sort of necessary tribute to the piece's profundity and monumentality. "This was not a probing or profound Hammerklavier," he said of Yuja's interpretation, as if suddenly remembering himself and wishing that his praise of her had been more grudging. I could hear András Schiff laughing to himself. _We are very deep and profound. . . . You can have lunch and dinner and breakfast, and we are still sitting here_.
Tommasini ended his review by complaining about the five encores that Yuja played, each one making the "Hammerklavier" recede "further from memory." I have to say that I agreed with him. I had heard these encores before. Yuja habitually wheels them out at performances. They include Vladimir Horowitz's amusing high-speed "Carmen" Fantasy and an equally funny arrangement by various hands of the Alla Turca movement of Mozart's Sonata No. 11 in A Major. The audience, as Tommasini felt obliged to report, went mad with delight. When I first heard Yuja play these encores, I went mad with delight, too. But this time I wished she had left us with an unmediated memory of her "Hammerklavier." The roars that went up after the encores were greater than those after the "Hammerklavier." This seemed wrong. But in the split between the concert proper and the encores we may read the split in Yuja herself—her persona as a confident musical genius and as an uncertain young woman making her way through the maze of a treacherous marketplace.
She was born in Beijing to a mother who was a dancer and a father who was a percussionist. She is vague about her emergence as a prodigy. She likes to tell interviewers that her mother wanted her to be a dancer, but that she was lazy and chose the piano because she could sit down. She was performing publicly by the age of six, and entering competitions from which she always emerged with the first prize. When she was nine, her parents enrolled her in the Beijing conservatory, and when she was fourteen they sent her to a conservatory in Calgary, Canada, where she learned English. From there she went to the Curtis Institute, in Philadelphia, whose head, the pianist Gary Graffman, immediately recognized her quality, and took her on as his student, something he did only with the most outstanding talents, such as Lang Lang. Yuja hasn't lived in China since.
About a year ago, I began meeting with Yuja in the Sky Lounge, on the top floor of the building she lives in on Riverside Boulevard, in the West Sixties—a common space with a view of the Hudson River and the New Jersey shoreline, whose privileged-looking armchairs and little tables evoke first- and business-class waiting rooms at airports. When I say "the building she lives in," I am speaking loosely. Yuja tours the world, playing in premier halls, either in solo recitals or with leading orchestras, in London, Paris, St. Petersburg, Edinburgh, Bucharest, Caracas, Tokyo, Kyoto, Beijing, Tel Aviv, Jerusalem, Sydney, Amsterdam, Florence, Barcelona, and San Francisco, among other cities, and spends only a few weeks, between more than a hundred scheduled performances, in the apartment, a studio she bought in 2014.
When you walk into the apartment—which is small and dark—the first thing you see is a royal-blue nylon curtain suspended from the ceiling like a shower curtain and drawn around a lumpish object that turns out to be a Steinway grand piano. The curtain is there to muffle the piano's sound, to accommodate a neighbor for whom the practicing of a world-class pianist is not the thrill it would be for you and me. The rest of the apartment has the atmosphere of a college dormitory room, with its obligatory unpacked suitcase on the floor and haphazard strewings of books and papers and objects. There may be a few stuffed animals on the bed or maybe only a sense of them—I am not sure because I was at the apartment only once. Yuja prefers to see interviewers in the Sky Lounge. When I proposed visiting the apartment again—this time with a notebook—she politely demurred. It was too much of a mess, or the cleaning woman hadn't come.
Yuja speaks in fluent—more than fluent—English, punctuated by laughter that gives one to understand that what she is saying is not to be taken too seriously, and that she is not a pompous or pretentious person. Occasionally, there is the slightest trace of an accent (vaguely French) and a lapse into the present tense.
We talked about her life as a child prodigy. "Oh, yes, I'm a real prodigy," she said. "They still call me wunderkind. I remember when I went to the conservatory for the first time. All the other kids were looking at me like—by then I was already a child star—like I am another species in a zoo. Oh, my God, she's here."
"You seem so unspoiled," I said. "Were you more spoiled then? Or were you unspoiled even then?"
"I think unspoiled came later," Yuja said.
She recalled something I didn't and still don't completely understand about the effect that playing Mozart had on her as a child. She said that performing his Twelve Variations in C Major ("Twinkle, Twinkle, Little Star") permitted her to feel for the first time what it was like to have stagefright. She was eight or nine.
"I was always quiet before a concert, while the other kids were so nervous. They talk, some are very noisy. I don't understand. Why are you nervous? _Until_ the first time I played Mozart. I was not nervous until I was onstage. Then I felt I was in a completely different time and space. My fingers just played. And I thought there is a difference between practicing at home and playing onstage."
I asked if she could explain further what had happened to her when she performed the Variations.
"Maybe intuitively I was struck by the beauty, by the symmetry, by how like something inherent in nature it is. Before, I was, Oh, Mozart is so boring."
When I told her of my feeling of awe at the superhuman feat that is a concert performance, she said, "For me that's normal—like talking." She has the erroneous idea that writing a book is a similarly remarkable achievement. She became a serious reader in her teens. Among the books she recently read are Virginia Woolf's _The Waves_ and Immanuel Kant's _Critique of Pure Reason_. When I commented on the high-mindedness of her reading, she quickly said, "No, I'm always reading something trashy, too."
I asked about her home life in China. "Did your parents immediately realize that you were different from other children?"
"I don't know. They're very naïve people. Extremely conventional and traditional. Very Communist. If you read Dostoyevsky or Tolstoy, you will understand what kind of people they are. Just simple, extremely kind. My dad was really talented, and my mom also. They are extremely artistic— or autistic," she said, with a peal of laughter. "Their environment never allowed them to develop to their full potential."
"Is this what you mean by 'very Communist'?"
"Yeah. Because you have to go to Party meetings and talk about how to do well for society. Twenty-year plan. Five-year plan. You work for the common welfare rather than for the individual. Working for the individual is almost synonymous with being selfish. Which is not how I feel. I feel lucky that I came out when I was fourteen." Yuja's mother came for her graduation from Curtis and for her Carnegie Hall début; otherwise, Yuja sees her parents only when she performs in Beijing. She speaks of them in an affectionate but veiled way, always stressing their kindness.
When I asked Yuja to elaborate on her sense of the political differences between China and America, she paused before answering. After a while, listening to her, I realized that she was talking about an entirely different subject. I decided to persist. "I asked you about politics, and you have been talking about music," I said.
"You noticed?" she said, laughing.
My visit to Yuja's apartment had taken place after this conversation. It was around four on a hot August afternoon, and Yuja was dressed in denim shorts, very short ones, and a tank top. We had tickets to a fiveo'clock concert of advanced contemporary music at Alice Tully Hall, and Yuja was debating whether to change for it. She rummaged through the suitcase on the floor and extracted two garments—strapless black-andwhite minidresses made of a stretch fabric, called bandage dresses by their French designer, Hervé Léger, because that's how they fit, and characterized by Yuja as "modern and edgy" as well as practical, because they don't have to be ironed and lie nice and flat in a suitcase—and asked my opinion. Should she wear one of them or stay in the shorts? I asked what the issue was—was she interested in comfort or in how she looked? She stared at me as if I were crazy. What weird world was I living in where comfort could even be thought of? She wiggled into one of the bandage dresses, added her high heels, and we walked the three blocks to Lincoln Center at a brisk clip.
In February of this year, on four successive nights at Geffen Hall, Yuja played Mozart's Piano Concerto No. 9 in E-Flat Major, the "Jeunehomme"— written when Mozart was barely twenty-one and considered his first masterpiece—with the New York Philharmonic, under the Swiss conductor Charles Dutoit. This was a departure for Yuja. Her career has been built on her playing of the Russian Romantics, the "red-blooded" and "hot-blooded" composers, as she calls them, Tchaikovsky, Rachmaninoff, Prokofiev, for whose "passionate, emotional" pieces her short flame-red dresses seem to have been made. For a while, there was a picture of Yuja in front of Carnegie Hall in the flame-red dress she had worn at a recital in May, 2013, her arms raised high in the air in a gesture of culminating abandon. It stopped passersby. Now she was entering a new phase of engagement with Mozart and the nineteenth-century German classical composers. The picture of her in front of Geffen Hall was unremarkable.
A day before the first concert of the series at Geffen, I attended an open rehearsal at the hall. The Mozart concerto was on the first half of the program, to be followed by Respighi's orchestral pieces "Roman Festivals," "Fountains of Rome," and "Pines of Rome." The Respighi pieces were being rehearsed first, and when I arrived at the hall, around noon, much Respighi remained to be played. Yuja was waiting in the small room upstairs where soloists change clothes and receive visitors. She showed me a closet where the three dresses, designed by Roberto Cavalli, she would wear at the concerts were hanging. I took an immediate dislike to one of the garments—a short pink dress with black swirling lines on its gathered skirt and bodice. It was neither ultra-short and tight nor long and clinging. It was a kind of girlish summer dress. I did not like the idea of Yuja wearing it onstage. The two other dresses were a glamorous dark-blue long gown and a short, also concert-worthy dress.
Yuja curled up on a sofa—she was wearing tight-fitting black leather trousers—and laughingly recalled a newspaper headline she had seen during a tour: " 'Twenty-Eight-Year-Old Wunderkind.' Isn't that an oxymoron?" she said. I had arrived early at Lincoln Center, and stopped into a café for a sandwich, though not so early that there was time to eat the whole large overstuffed thing. When I offered Yuja the half sandwich the waiter had wrapped, she accepted. Predictably, she opened the sandwich and ate the chicken, then the tomato, then the lettuce, and then—unpredictably— the bread.
Dutoit, a tall man of seventy-nine, appeared with his fourth wife, Chantal Juillet. After husband and wife hugged Yuja, Dutoit stood back to look with elaborate mock lecherousness at her tight trousers. Dutoit and Yuja go back a long way. The infamous habit of Dutoit's second wife, Martha Argerich, of cancelling concerts at the last minute had given Yuja one of her early breaks. Argerich was one of the stars Yuja replaced while she was still a student at Curtis; Radu Lupu, Yefim Bronfman, Evgeny Kissin, and Murray Perahia were others. ("With Martha it was like, 'I'm tired . . . do you want to play with the Boston Symphony for me?' And I'm like 'Of course!— Wrong question!' " Yuja told an interviewer for the Australian magazine _Limelight_.) Yuja's ability both to learn fast and to turn the disgruntlement of audiences into amazed delight did not go unnoticed. "By the end of the final movement"—of Tchaikovsky's Piano Concerto No. 1—"the audience stood and roared," the _Philadelphia Inquirer_ critic David Patrick Stearns wrote in a review of the concert at which Yuja replaced Argerich.
After some cheerful banter, Dutoit left to rehearse the final Respighi, and Yuja excused herself to warm up in a large adjacent room that had a piano. She preferred that I not go into the large room with her but didn't object to my staying in the small room, where I could hear her play phrases over and over and feel that I was uselessly eavesdropping on coded artistic secrets.
At the concert proper, the following night, Yuja wore the glamorous dark-blue gown, and played with delicacy and beauty. She and Dutoit and the orchestra were in elating rapport. The first cadenza produced one of those you-could-hear-a-pin-drop hushes in the sold-out hall. She had gone very quiet, and the audience followed as if mesmerized. No one coughed.
"Who can play Mozart the way she did?" Graffman said afterward. "It was so natural, in such good taste. Not that she was doing anything. That's just the way it came out. Who can do that and also play the Horowitz 'Carmen' Fantasy?" In the _New Criterion_ , Nordlinger wrote, "Mozart ends with a rondo—and it should be fast, exuberant, and fun. It was. Wang ripped the notes out of the keyboard, as much as played them. At one point, I almost laughed out loud. That's how funny she was, and how funny Mozart is."
Yuja must have liked reading this. She had once talked about how funny Mozart is: "Mozart is like a party animal. I find I play him better when I am hung over or drunk." At the same time, she saw Mozart's music as "noble, tragic, like a great Greek play. The human emotion is there but with a lot of godliness in it." On the second night, my heart sank when Yuja walked onstage in the pink dress. Was it my imagination or was her playing less inspired than it had been the night before?
Meeting Yuja in the Sky Lounge a few weeks later on a rainy day, I told her of this impression, and she did not contradict it. "Because of that dress, the little pink one, because it's so different from everything I've ever worn, I didn't really feel myself, and maybe that came through. I liked the pink dress because it was different. Sometimes, the difference might become the style of my next season. It could be what's going to come. Or it could be something to discard. You don't know until you try it." She added, "They wanted to put in social media that I was dressed by the designer Roberto Cavalli."
"Were you feeling something related to the dress while playing?"
"No, not while playing. Just when I walked onstage. This was a cute little pink dress, and I thought, It's not me. It's about a young girl. Just the opposite of the nude dress."
In 2014, when an interviewer from the London _Telegraph_ asked Yuja about "her fondness for riskily short, clingy dresses," she gave a flippant reply: "I am 26 years old, so I dress for 26. I can dress in long skirts when I am forty." But in fact Yuja's penchant for the riskily short and clingy has less to do with allegiance to the dress code of her generation than with an awareness of her own "super-smallness," as she calls it. She knows that small tight clothes bring out her beauty and large loose garments don't. But she is not just a woman who knows how to dress. She is a woman who is constantly experimenting with how to dress when she is playing on a concert stage. She is keenly aware—as many soloists affect not to be—that she is being looked at as well as listened to. Reviewing the Carnegie Hall recital Yuja played in May, 2013, Zachary Woolfe wrote in the _Times_ , "I confess that while perhaps 90 percent of my attention was on her precise yet exuberant playing, a crucial 10 was on her skintight flame-colored dress." Woolfe went on to brilliantly anatomize the experience of simultaneously listening to and looking at Yuja: "Her alluring, surprising clothes don't just echo the allure and surprise of her musicianship, though they certainly do that. More crucial, the tiny dresses and spiky heels draw your focus to how petite Ms. Wang is, how stark the contrast between her body and the forcefulness she achieves at her instrument. That contrast creates drama. It turns a recital into a performance." When Yuja played the "Jeunehomme" in the girlish pink dress, that contrast was absent. The sense of a body set in urgent motion by musical imperatives requires that the body not be distractingly clothed. With her usually bared thighs, chest, and back demurely covered by the black-splotched pink fabric, this sense was lost.
Yuja's customary self-presentation as a kind of stripped-down car is, of course, only one way of appearing onstage to artistic advantage. When Maurizio Pollini plays in some nondescript suit, his body-aliveness is no less present for us. Martha Argerich's widow's-weeds black gowns heighten the beauty and mystery of her playing. Plainness is never a mistake on a concert stage. For the two remaining Mozart performances, Yuja, realizing her misstep, returned to the designer she regularly uses.
The "nude dress" was a long gown (in recent years, long gowns have been admitted into Yuja's concert-clothes closet, but they have to be slinky) made of body-stocking fabric with sparkling encrustations at bosom and stomach and a long swishing skirt. Yuja wore this fabulously gorgeous costume at the third concert—which had the electricity of the first one— and felt comfortable and happy in its defiant sexiness and her feeling of nakedness.
I looked out the window of the Sky Lounge and saw the New Jersey shoreline disappearing in a gray mist. Yuja herself was in a dark mood. She had recently returned from a European tour and was exhausted and dispirited. In Munich and Paris, she had played the Mozart piano concerto with Valery Gergiev and the Vienna Philharmonic, and the reception had been only O.K. A blog about the Paris concert saying, "Yuja Wang disappoints," had stayed with her. She paraphrased its words: " 'She didn't have emotion. She's not yet mature enough to play Mozart.' " She went on, "With Rachmaninoff, Prokofiev, Tchaikovsky I can blow them away. 'So amazing, so impressive!' But I went for the surprise, for the unexpected. I ask myself, Am I playing for the applauding, for the standing up, or am I playing because I really like something in the music and I just want to play?"
She talked in the same dark vein about her personal life. She spoke of the "too many people" she meets on tour: "Who are your real friends? I naturally give my love and friendship, but once the tour is over are they really your friends? What's always there, of course, is music. The other things come and go—except maybe your parents." She laughed. "And Gary."
Gary Graffman, who is eighty-seven and now retired as the head of Curtis, and his wife, Naomi, who is eighty-eight, are Yuja's best friends in New York and perhaps in the world. Graffman, you may recall, is the distinguished pianist whose career was disrupted in the late nineteen-seventies, when he lost the use of his right hand. When I visited the Graffmans in their apartment at the Osborne, on West Fifty-Seventh Street, they spoke of Yuja as of a beloved granddaughter of whom they are so proud they can hardly stand it. When I asked Graffman how she compared with the other prodigies at Curtis, he said, "She was remarkable among remarkable students. She didn't play like a prodigy. She played like a finished artist." Naomi recalled that when Yuja first arrived at Curtis, Gary asked her to take the new student to lunch, and she dutifully did so. "By the time lunch was over, I thought she has to be at least thirty-five or forty," Naomi said. "She was speaking so intelligently about so many things." Yuja was fifteen and a half.
As Yuja had been a musical wunderkind at six, at twenty-nine she is a kind of existential prodigy, already undergoing the crisis that ordinary people undergo in midlife. "I've been doing this for twenty-nine years. Do I want to go on doing it, or is there something else waiting for me?" She spoke of her sense of alienation from people who don't have to constantly and relentlessly study music and practice, of feeling like an outsider, sometimes even "I don't like to say but almost like a prisoner. I haven't ever enjoyed my free time. It's always like I am challenging myself. I must be a little masochistic." She would see people walking in the park on a beautiful day and long to join them. But by the time she had untied herself from the mast of her art it was midnight, and there was no one to join her in a walk in the park.
At the "millennials' parties" she had attended on the last two nights of the Mozart concerts (their purpose was to encourage young people to go to concerts), she had wearily answered questions from a stage. "I would have enjoyed these parties five years ago," she said. "I still enjoyed them. They were fun. Nice people. I had lots of drinks. But I get the same questions again and again. It's like water goes into the same spot. And then I become a little unpleasant. And then I feel guilty that I was unpleasant. They ask me things like"—she began speaking in a mocking singsong voice—"'Are you single?' 'How do you memorize your pieces?' 'How do you pedal with your heels?' 'Who do you buy your dresses from?' 'Why do you wear short dresses?' 'Why do you wear long dresses?' 'Why do you have short hair?' 'Do you like travelling?' 'Why don't you play more Prokofiev?' 'Why do you play Mozart? "
The room had darkened, and everything on the river was disappearing. When I drew Yuja's attention to the apparition of the sublime in the window, she was looking at her phone. "I'm just checking," she said. "I'm not being impolite." Yuja treats her phone the way almost every young (and not so young) person today treats it—as a transitional object. She and I have corresponded by e-mail (largely about chocolate), and the messages from her phone are filled with emoticons and LOL-like abbreviations. In deference to my age, she does not text me.
When I commented on her melancholy, she denied—and then acknowledged—it: "It's a very depressing thought. Just touring and playing— the same things or different things. But in society people don't allow you to be sad or depressed. It's like a bad thing. It's why I'm antisocial. I feel this negative energy. 'She just complains a lot.' Excuse me, that's part of what I do. You feel all these things. As a musician, you probably feel them more intensely. But society wants me to be happy. My parents. They are the most unintrusive parents. 'I don't care what you do—just be happy.' " She made an _urrrgghh_ sound and laughed.
Yuja has made changes in her professional life that she is not sure have solved the problems of doubt and restlessness by which they were impelled. Last year, she abruptly left her manager, Earl Blackburn, of the large Opus 3 Artists agency, with whom she had been since she was sixteen, and joined Mark Newbanks, whose London-based agency, Fidelio Arts, has only three other clients—but what clients!—the conductors Gustavo Dudamel, Lionel Bringuier, and Esa-Pekka Salonen. Although Yuja doesn't speak of it in such terms, the change of managers has the atmosphere of the dissolution of a marriage: a young wife leaves the dull, older husband for an exciting younger man. Naomi Graffman spoke of Blackburn's extraordinary devotion to Yuja: "He coddled her as no one had ever been coddled before. Every little thing she wanted or needed, he did it for her. He would brush her teeth for her if she wanted." The younger man is different. He does not take Yuja's clothes to the cleaners; recently, he did not offer to pick up a Russian visa for her, as Blackburn would have done. "She was furious," Gary said. "Never having had experience with anybody else, she thought that was what managers did." I happened to have heard about the Russian visa from Yuja. She had not mentioned Newbanks, just the fact of this and other annoying little errands she had to run, followed by the playful question "Shall I hire a boyfriend or an assistant?"
I proposed a boyfriend/assistant. Earlier, she had spoken of the obstacle her touring schedule put in the way of lasting romance. The boyfriend/ assistant—i.e., a muse—would always be in the next seat on the plane. "No," she said, "guys won't do that. It's O.K. for a woman to do that. It's harder for guys to get rid of their egos, to be even a little bit subservient." She added, "Of course, I want guys who are successful. Which means that they have their own work, that they're busy—and that I am the one who visits them."
I asked if her romances were with artists of her calibre.
"Not of my calibre," she said without hesitation (and the obligatory peal of laughter). "I never meet people of my calibre who are available."
She talked of the older and old people with whom she feels happy and comfortable (the conductor Michael Tilson Thomas is a kind of runner- up to Gary Graffman in the lovable-mentor sweepstakes): "people who have their whole life behind them"—as opposed to the young with their oppressive burden of futurity. Another older friend, Emanuel Ax, invited her for Thanksgiving last year, and she accepted, but in the end did not go, preferring to "be home and snuggle up and watch Netflix."
She spoke of leaving Earl Blackburn not regretfully, exactly, but with a kind of cold wisdom about the possible pointlessness of the gesture that people three times her age don't often achieve. "There was nothing wrong with the old manager. He really built my career. He was really caring. But I was, like, if I don't make a change, I'll never make a change. I'm bad at confrontation. So I just did it out of the blue. But nothing much has changed. It's a little better here and there. But it's still the same circus."
When I met for coffee with Newbanks—a suave, slender, elegantly dressed man of forty-eight, a former cellist—he told me that his aim as Yuja's manager was to cut back on her engagements and "put air" in her schedule. "She had three days free when I met her—that's impossible." Another goal was to steer her toward experimentation with repertoire, and one of these experiments has already taken place—in March, Yuja played for three nights with the New York Philharmonic in Messiaen's "Turangalîla-Symphonie," conducted by Salonen. "Turangalîla" is a thrilling, mad, loud piece that features two solo instruments, the piano and the ondes martenot, an early electronic instrument that makes unearthly wavering sounds not easy to hear over the orchestral pandemonium. Yuja's playing was brilliantly audible. She played from a score, and on the night I attended did her own page turning, which lent a certain suspense to the proceedings. The pages flew at a rate of about one every thirty seconds. Would they lie flat? Page turners usually give a little firm pat to the page they have just turned to make sure it will stay in place. Later, Yuja told me that she had put adhesive on the pages to insure that they would stay in place.
Newbanks told me that it is customary for management to take twenty percent of a soloist's fee and fifteen percent of a conductor's fee. I asked him, as I had asked Yuja, what her fee was, and, like her, he wouldn't tell me. "No one in the business talks about it," he said. The business evidently exacts a vow of omertà from its members. Newbanks laughingly (perhaps a little nervously) said that Yuja had alerted him to my unseemly interest in money. When I put the futile question to her, she had answered, "I don't usually like to talk about fees," and added, with uncharacteristic humorlessness, "I feel it is degrading to art to measure it with money."
As patches of blue and orange appeared in the sky of the Sky Lounge, Yuja's internal bad weather seemed to lift as well. She recalled her time in Europe with Gergiev: "He is amazing. This is the first time I am playing Mozart with him, and I was curious how he would do it. I did Russian stuff with him before—the energy for the Russian stuff was unbelievable. And he had the same energy for Mozart, which is scary, because it's overwhelming for Mozart. But it put us into a good place. He has that. Claudio"—the conductor Claudio Abbado—"had that. Claudio is like intense listening. It makes you feel so scrutinizingly uncomfortable. And that place of uncomfortableness is exactly where you want to be every time you are onstage. Because that makes you play better, and that is when you are growing. Feeling comfortable is always like O.K., I'll do the thing again. Been there done that."
Yuja reveres Abbado, who died in 2014. When, in the interview for _Limelight_ , she was asked what it was like to play under Abbado, she spoke of how "obscure and mysterious" the experience was. During rehearsals, "he didn't say a word—to me at least. And then in the concert, everything just came out. You don't really know what happens with the gestures or the energy field. . . . He made everyone play his or her best . . . without any words."
She spoke of her new repertoire: "It makes me happy playing 'Hammerklavier' rather than playing Rach 3 another twenty times. I used to only play pieces I was comfortable with and good at, Rachmaninoff, Prokofiev, Tchaikovsky. Now I propose music I won't be comfortable with. This is the only way to get out of my skin, out of myself, and to learn." She added, laughing, "But once in a while I crave those Russians. My heart is crying, _Where are they?_ "
A week or so after Yuja's "Hammerklavier" concert, the photograph that accompanies this piece was taken at the new Steinway piano showroom, on Sixth Avenue at Forty-third Street. When I arrived at the showroom, around noon, Yuja, wearing one of her bandage dresses, was sitting on a table, facing a mirror, as a hair-and-makeup man from Paris applied mascara to her eyelashes. She was patient and compliant and practiced. She had done this before. There are many beautiful portraits of Yuja floating around the print and Internet worlds. After greeting me, she began lighting into Tommasini for his comment about her encores. "If instead of feeling exhausted I feel exhilarated, and want to make people happy by giving them a gift, why not do it?" she said. "It feels like home to play those familiar pieces. People play encores after much more sublime pieces. Why can't you do it after climbing Mt. Everest? Stupid conservative doctrine."
We were on a below-street-level floor, filled with pianos. The photographer, Pari Dukovic, and his three assistants were placing lights and screens around one of them. They had been there since eight-thirty in the morning (catering and hair and makeup had followed at eleven-thirty). Several of Yuja's concert dresses were strewn around an alcove serving as a dressing room, among them the blue-green dominatrix gown she had worn to play the "Hammerklavier." This was the dress finally chosen for the portrait. The hair-and-makeup man, with whom Yuja had established laughing rapport, revised something in her hairdo at her request. "My cheeks are too fat," she said as she looked in the mirror. She ate a few forkfuls from a plate of salad that her friend Carlos Avila, a pianist who teaches at Juilliard, brought her from the catering table. Then she slipped into the blue-green dress and stepped into stiletto heels, and the photo shoot began. Yuja went to the designated piano, and Dukovic—a handsome young man, with a warm and charming manner—began circling around it, snapping pictures with a handheld camera, as she played bits and pieces of repertoire. At first, she played tentatively and quietly, starting a piece and trailing off—and then she worked her way into a horrible and wonderful pastiche of Rachmaninoff, Chopin, Beethoven, Mozart, Gershwin, Horowitz, Tchaikovsky, all mushed together, playing louder and louder and faster and faster, banging with mischievous demonic force, as Dukovic continued his circling and snapping, like the photographer in the famous orgasmic scene in "Blowup." Yuja ended with a parodic crescendo as Dukovic shouted, "I love you!" and she burst into laughter.
The arresting photograph that was chosen out of the hundreds, possibly thousands, of pictures Dukovic took of Yuja at the piano and, later, in the first-floor showroom, posed full figure in front of a piano with its lid up, represents her as no concertgoer has ever seen her. The wild disorder of the hair has never been seen in a concert hall. (Yuja's hair tends to stay in place throughout the most rousing of her performances.) And the foreshortened, oversized hand is an obvious deviation from the consensus we call reality. Will Yuja cringe when she looks at the photograph? Or will she see it as expressive of her impudent, defiant nature and find in it, almost hear in it, an echo of her incomparable musicality?
_Susan Orlean_
Susan Orlean is the bestselling author of eight books, including _The Bullfighter Checks Her Makeup_ , My Kind of Place, Saturday Night __, and _Lazy Little Loafers_. In 1999, she published _The Orchid Thief_ , a narrative about orchid poachers in Florida, which was made into the Academy Award-winning film, _Adaptation_. Her 2011 book, _Rin Tin Tin: The Life and the Legend_ , was a _New York Times_ bestseller and a _New York Times Notable Book_.
Orlean has been a staff writer for _The New Yorker_ since 1992. Her subjects have included umbrella inventors, origami artists, skater Tonya Harding, and gospel choirs. Her work has also appeared in _Esquire_ , _Rolling Stone_ , _Outside_ , _Smithsonian_ , and _The New York Times_.
Orlean graduated with honors from the University of Michigan and was a Nieman Fellow at Harvard University in 2003. In 2012 she received an honorary Doctor of Humane Letters from the University of Michigan. In 2014, she was awarded a Guggenheim Fellowship in Creative Arts/Nonfiction. In 2016, she was selected as the Bernardine Kielty Scherman Fellow at the MacDowell Colony. She has lectured at Yale University, New York University , University of California, Berkeley, University of Michigan, Kenyon College, the Bread Loaf Writers Conference, Goucher College, and Harvard University, among others, and has been awarded residencies at the MacDowell Colony and at Yaddo.
She is currently writing a book about the Los Angeles Public Library.
La Matadora Revisa Su Maquillaje (The Bullfighter Checks Her Makeup)
Every time Cristina Sanchez kills a bull, she presents an unforgettable tableau—a self-possessed young woman elegantly and lethally playing out an ancient masculine ritual.
I went to Spain not long ago to watch Cristina Sanchez fight bulls, but she had gotten tossed by one during a performance in the village of Ejea de los Caballeros and was convalescing when I arrived. Getting tossed sounds sort of merry, but I saw a matador tossed once, and he looked like a saggy bale of hay flung by a pitchfork, and when he landed on his back he looked busted and terrified. Cristina got tossed by accidentally hooking a horn with her elbow during a paa with the cape, and the joint was wrenched so hard that her doctor said it would need at least three or four days to heal. It probably hurt like hell, and the timing was terrible. She had fights scheduled each of the nights she was supposed to rest and every night until October—every night, with no breaks in between. It had been like this for her since May, when she was elevated from the status of a novice to a full _matador de toros_. The title is conferred in a formal ceremony called "taking the _alternativa_ ," and it implies that you are experienced and talented and that other matadors have recognized you as a top-drawer bullfighter. You will now fight the biggest, toughest bulls and will probably be hired to fight often and in the most prestigious arenas. Bullfighting becomes your whole life, your everyday life—so routine that "sometimes after you've fought and killed the bull you feel as if you hadn't done a thing all day," as Cristina once told me.
When Cristina Sanchez took her alternativa, it caused a sensation. Other women before her have fought bulls in Spain. Many have only fought little bulls, but some did advance to big animals and become accomplished and famous, and a few of the best have been declared full matadors de toros. Juanita Cruz became a matador in 1940, and Morenita del Quindio did in 1968, and Raquel Martinez and Maribel Atienzar did in the eighties, but they all took their alternativas in Mexico, where the standards are a little less exacting. Cristina is the first woman to have taken her alternativa in Europe and made her debut as a matador in Spain.
There was a fight program of three matadors—a corrida—scheduled for the Madrid bullring the day after I got to Spain, and I decided to go so I could see some other toreadors while Cristina was laid up with her bad arm. One of the three scheduled to perform was the bastard son of El Cordobes. El Cordobes had been a matador superstar in the sixties and a breeder of several illegitimate children and a prideful man who was so possessive of his nickname that he had once sued this kid—the one I was going to see— because the kid wanted to fight bulls under the name El Cordobes, too. In the end, the judge let each and every El Cordobes continue to be known professionally as El Cordobes.
The kid El Cordobes is a scrubbed, cute blonde with a crinkly smile. Outside the rings where he is fighting, vendors sell fan photos of him alongside postcards and little bags of sunflower seeds and stuffed-bull souvenirs. In the photos, El Cordobes is dressed in a plaid camp shirt and acid-washed blue jeans and is hugging a good-looking white horse. In the ring, he does some flashy moves on his knees in front of the bull, including a frog-hop that he times to make it look like he's going to get skewered. These tricks, plus the renown of his name, have gotten him a lot of attention, but El Cordobes is just one of many cute young male matadors working these days. If his knees give out, he might have nothing.
On the other hand, there is just one Cristina, and everyone in Spain knows her and is following her rise. She has gotten attention far outside of Spain and on television and in newspapers and even in fashion magazines; other matadors, even very good ones, fuse in the collective mind as managainst- bull, but every time Cristina kills a bull she forms part of a singular and unforgettable tableau—that of an attractive, self-possessed young woman elegantly slaying a large animal in a somber and ancient masculine ritual—and regardless of gender she is a really good matador, and she is being painstakingly managed and promoted, so there is no saying where her celebrity will stop. This is only her first season as a full matador, but it has been a big event. Lately El Cordobes or his publicist or his accountant has been igniting and fanning the rumor that he and Cristina Sanchez are madly in love, with the hope that her fame will rub off on him. She will probably be more and more acclaimed in the four or so years she plans to fight, and she will probably be credited with many more putative love affairs before her career is through.
Before the fight in Madrid, I walked around to the back of the bullring and through the _patio de caballos_ , the dirt-floored courtyard and stable where the picadors' horses and the donkeys that drag away the dead bull after the fight relax in their stalls and get their hair combed and get fed and get saddled. I was on my way to the bullfighting museum—the Museo Taurino—which is in a gallery next to the stalls. It was a brilliant day with just a whiff of wind. In the courtyard, muscle men were towing equipment back and forth and unloading a horse trailer. Another twenty or so men were idling in the courtyard in the few pockets of shade or near the locked door of the matadors' chapel, which is opened before the fight so the matadors can stop in and pray. The idlers were older men with bellies that began at their chins and trousers hiked up to their nipples, and they were hanging around just so they could take a look at the bulls for tonight's fight and see how they were going to be divvied up among the three matadors. Really, there isn't a crumb of any piece of bullfighting that goes unexamined by aficionados like these men. I lingered for a minute and then went into the museum. I wandered past the oil portraits of Manolete and Joselito and of dozens of other revered bullfighters, and past six stuffed and mounted heads of bulls whose names were Paisano, Landejo, Mediaonza, Jocinero, Hermano, and Perdigon—they were chosen for the museum because they had been particularly mean or unusual-looking or because they had killed someone famous. Then I stopped at a glass display case that had in it a picture of the matador Juanita Cruz. The picture was an eight-by-ten and looked like it had been shot in a studio. Juanita Cruz's pearly face and her wedge of a chin and her pitch-black hair with its tiny standing waves were blurred along the edges, movie-star style. She looked solemn, and her eyes were focused on middle space. In the case next to the picture were her pink matador knee socks and her mouse-eared matador hat and one of her bullfighter suits. These are called _traje de luces_ , "suit of lights," and all toreadors wear them and like to change them often; Cristina has half a dozen, and Juanita Cruz probably owned twenty or so in the course of her bullfighting career. This one was blush-pink with beautiful gold piping and sparkly black sequins. It had the classic short, stiff, big-shouldered, boxshaped toreador jacket but not the capri trousers that all matadors wear, because Juanita Cruz fought in a skirt. There is no such thing as a matador skirt anymore—Cristina, of course, wears trousers. I looked at the skirt for a while and decided that even though it looked unwieldy it might actually have been an advantage—in a skirt, you can bend and stretch and lunge with a sword unconstrained. On the other hand, a skirt would have exposed so much fabric to the bull that in a fight it would have gotten awfully splashed and smeared with blood. Every matador has an assistant who is assigned to clean his suit with soap and a toothbrush after every fight. Juanita Cruz was popular and well accepted even though she was an anomaly, but late at night, as her assistant was scrubbing her big bloody skirt, I bet he cursed the fact that she had been wearing so much fabric while sticking swords into bulls.
I went to visit Cristina at home the morning before she was going to be fighting in a corrida in a town called Mostoles. It was now a week since her injury, and her elbow apparently had healed. Two days earlier, she had tested it in a fight in Cordobes and another the following day in Jaen, and a friend of mine who reads Madrid's bullfight newspaper told me Cristina had gotten very good reviews. It turns out that I was lucky to catch her at home, because she is hardly there during the bullfighting season—usually she keeps a rock star schedule, leaving whatever town she's in with her crew right after she fights, driving all night to the next place on her schedule, checking into a hotel, sleeping until noon, eating lunch, watching some television, suiting up, fighting, and then leaving again. She was going to be at home this particular morning because Mostoles is only a few miles from Parla, the town where she and her parents and sisters live. She had come home the night before, after the fight in Jaen, and was planning to spend the day in Parla doing errands. The corrida in Mostoles would start at six. The assistant who helps her dress—he is called the sword boy, because he also takes care of all her cutlery—was going to come to the apartment at five so she could get prepared and then just drive over to the bullring already dressed and ready to go in her suit of lights. Parla is an unglamorous place about forty minutes south of Madrid; it is a kernel of an old village that had been alone on the wide open plains but is now hemmed in by incredibly ugly high-rise apartment buildings put up in the midsixties for workers overflowing the available housing in Madrid. The Sanchez apartment is in a slightly less ugly and somewhat shorter brick building on a busy street, on a block with a driving school, a bra shop, and a bank. There is no name on the doorbell, but Cristina's father's initials are barely scratched into a metal plate beside it. These days it is next to impossible to find Cristina. The nearly unmarked doorbell is the least of it. Cristina has a magician press agent who can make himself disappear and a very powerful and self-confident manager—a former French bullfighter named Simon Casas—who is credited with having gotten her into the biggest bullrings and the best corridas in the country but is also impossible to find and even if he were findable he would tell you that his answer to your request to speak to Cristina is no. He is especially watchful of her international exposure. Simon Casas didn't know I was coming to see Cristina in Parla and he might have disapproved simply to be disapproving, and after I saw him later that afternoon in Mostoles, prowling the perimeter of the bullring like an irritable wild animal, I was that much gladder I'd stayed out of his way.
Anyway, Cristina wasn't even home when I got there. I had driven to Parla with my translator, Muriel, and her bullfighter husband, Pedro, who both know Cristina and Cristina's father, Antonio, who himself used to be a bullfighter—if it sounds like just about everyone I encountered in Spain was or is a bullfighter, it's true. We needed to see Cristina for my interview and then get right back, because Pedro had a bullfight that night in a town on the other side of Madrid. No one answered the doorbell at the apartment. Cristina's car wasn't around, so it looked like she really was gone. A car is the first thing matadors seem to buy themselves when they start making big money—that is, when they start getting sometimes as much as tens of thousands of dollars for a major fight. The bullfighter car of choice is a Mercedes, but Cristina bought herself a bright red Ford Probe, which is much sportier. She also bought her mother a small business, a gift store. We decided to wait a bit longer. Pedro killed time by making some bullfight business calls on his cellular phone. Just as we were debating whether to go looking for Cristina at her mother's store, Mrs. Sanchez came around the corner, carrying a load of groceries; she said Cristina was at the bank and that in the meantime we could come upstairs. We climbed a few flights. The apartment was tidy and fresh-looking and furnished with modern things in pastel tones, and in the living room there were a life-size oil painting of Cristina looking beautiful in her suit of lights, two huge photographs of Cristina in bullfights, one of her as a civilian, a large photograph of the older Sanchez daughter getting married, and a big-screen TV. On almost every horizontal surface there was a bronze or brass or pewter statuette of a bull, usually bucking, its withers bristling with three or four barbed harpoons called _banderillas_ , which are stuck in to aggravate him before he is killed. These were all trophies from different corridas and from Cristina's stint as a star pupil at the Madrid bullfighting school. Lots of Cristina's stuff was lying around the room. On the dining table were stacks of fresh laundry, mostly white dress shirts and white T-shirts and pink socks. On the floor were a four-foot-long leather sword case, three hatboxes, and a piece of luggage that looked like a giant bowling-ball bag, which is a specially designed case for a matador's $20,000 suit jacket. Also, there was a small black Kipling backpack of Cristina's, which cracked me up because it was the exact same backpack that I was carrying.
Mrs. Sanchez was clattering around in the kitchen, making Cristina's lunch. A few minutes later, I heard the front door scrape open, and then Cristina stepped into the room, out of breath and flustered about being late. She is twenty-five years old, and has chemically assisted blond hair, long eyelashes, high cheekbones, and a tiny nose. She looks really pretty when she smiles and almost regal when she doesn't, but she's not so beautiful that she's scary. This day, she was wearing blue jeans, a denim shirt with some flower embroidery, and white slip-on shoes with chunky heels, and her hair was held in a ponytail by a sunflower barrette. She is not unusually big or small. Her shoulders are square and her legs are sturdy, and she's solid and athletic-looking, like a forward on a field hockey team. Her strength is a matter of public debate in Spain. The weakest part of her performance is the very end of the fight, when she's supposed to kill the bull with one perfect jam of her sword, but she often doesn't go deep enough or in the right place. It is said in certain quarters that she simply isn't strong enough, but the fact is that most matadors mess up with the sword. Later, when we were talking, I brought it up, and she shook her head and said, "People who don't understand the bullfighting world think you have to be extremely strong, but that's not the case. What is important is technique and experience. You have to be in good shape, but you don't have to match a man's strength. Besides, your real opponent is the bull, and you can never match it in strength."
Her mother came in and out of the room a few times. When she was out, Cristina said in a low voice, "I'm very happy with my family, but the time comes when you have to be independent." The tabloids have reported that she has just bought a castle on millions of enchanted acres. "I bought a small piece of property right near here," she said, rolling her eyes. "I'm having a house built. I think when I come back from my winter tour in South America I'll be able to move in."
What I really wanted to know was why in the world she decided to become a bullfighter. I knew she'd grown up watching her father fight, so it had always been a profession that seemed normal to her, even though at the ring she didn't see many girls. Plus she doesn't like to sit still. Before she started training to be a matador, she had worked in a beauty parlor and then as a typist at a fire-extinguisher factory, and both jobs drove her crazy. She is a very girly girl—she wears makeup, she wants children, she has boyfriends— but she says she could only imagine doing jobs which would keep her on her feet, and coincidentally those were jobs that were mostly filled by men. If she hadn't become a matador, she thinks she would have become a trainer at a gym, or a police officer, or perhaps a firefighter, which used to be her father's backup job when he was a bullfighter, in the years before he started advising her and became a full-time part of her six-person crew. She didn't become a woman matador to be shocking or make a feminist point, although along the way she has been shunned by some of her male colleagues and there are still a few who refuse to appear in a corrida with her. Once, in protest, she went to Toledo and instead of having a corrida in which three matadors each killed two bulls, she took on all six bulls herself, one by one. She said she wants to be known as a great matador and not an oddity or anecdote in the history of bullfighting. She simply loves the art and craft of fighting bulls. Later that day, when I saw her in the ring, I also realized that besides loving the bullfight itself she is that sort of person who is illuminated by the attention of a crowd. I asked her what she'll do after she retires from the ring in three or four years. "I want to have earned a lot of money and invested it wisely," she said. "And then I want to do something in the movies or on TV."
She mentioned that she was eating early today because she had a stomachache. With a fight every night for months, I suppose there would be nights when she felt crummy or wasn't in the mood. Cristina laughed and said, "Yeah, sometimes you do feel like, oh God, I don't have the slightest desire to face a bull this afternoon!" Personally, I'm not a huge coward, but the phrase "desire to face a bull" will never be part of my life, any afternoon, ever. I figured that nothing must scare her. She shook her head and said, "Failure. My greatest fear is failure. I'm a woman who is a fighter and I always think about trying to surpass myself, so what I most fear is to fail."
Just then, Mrs. Sanchez came into the room and said the sandwiches were ready, so Cristina started to get up. She paused for a moment and said, "You know, people think that because I kill bulls I have to be really brave, but I'm not. I'm a sensitive person, and I can get super-terrified. I'm afraid of staying home by myself, and I get hysterical if I see a spider." I asked if bulls ever haunted her dreams, and she said, "I don't dream much at all, but a few times I've dreamed that a bull was pursuing me in the ring, up into the stands. And the night before my debut in Madrid, I did dream of bulls with huge, twisted horns."
I had seen the first bullfight of my life a few days earlier, on that night in Madrid, and it was a profound education. I learned that I should not eat for several hours beforehand and to start looking away the minute the picadors ride in on their stoic-looking blindfolded horses, because their arrival signaled that the blood and torment would begin. At first, in Madrid, I had been excited because the Plaza de Toros is so dramatic and beautiful, and also the pageantry that began the corrida was very nice, and when the first bull galloped in, I liked watching it bolt around the ring and chase the matador and his assistants until they retreated behind the small fences around the ring that are there for their protection. The small fences had targets— bull's-eyes, actually—painted on them. The bull would ram into them with its horns and the fence would rock. The more furious bulls would ram again and again, until the matador teased them away with a flourish of his cape. The bulls were homely, with little heads and huge briskets and tapered hips, and they cornered like school buses and sometimes skidded to their knees, but they had fantastic energy and single-mindedness and thick muscles that flickered under their skin and faces that didn't look vicious at all and were interesting to watch. Some of the fight was wonderful. The matador's flourishes with the shocking pink and bright yellow big cape and his elegance with the small triangular red one; the sound of thousands of people gasping when the bull got very close to the cape; the plain thrilling danger of it and the fascination of watching a bull be slowly hypnotized; the bravery of the picadors' horses, which stood stock-still as the bull pounded them broadside, the flags along the rim of the ring flashing in the late-afternoon light; the resplendence of the matador's suit in that angling light, especially when the matador inched one foot forward and squared his hips and arched his back so that he was a bright new moon against a sky of sand with the black cloud of a bull racing by. I loved the ancientness and majesty and excitement of it, the way bullfighting could be at once precious and refined yet absolutely primal and raw. But beyond that I was lost and nauseous and knew I didn't understand how so many people, a whole nation of people, weren't shaken by the gore and the idea of watching a ballet that always, absolutely, unfailingly ends with a gradual and deliberate death. I didn't understand it then, and I doubt I ever will.
In the little brick bullring in Mostoles, Cristina killed two bulls well but not exceptionally—for the first kill the judge awarded her one of the bull's ears, but for the second she got no award at all. A once-in-a-lifetime sort of performance would have earned two ears, a tail, and a hoof. After that second fight Cristina looked a little disgusted with herself, and she hung back and talked for several minutes with her father, who was standing in the crew area, before she came out and took the traditional victory walk around the ring. She was clearly the crowd favorite. People wave white handkerchiefs at bullfights to indicate their support; in Mostoles it looked like it was snowing. As she circled the ring, men and women and little kids yelled, "Matadora! Matadora!" and "Olé, Cristina!" and tossed congratulatory sweaters and flowers and shoes and blazers and sandwiches and a Levi's jacket and a crutch and a cane, and then a representative of a social club in Mostoles stepped into the ring and presented her with an enormous watermelon.
After the fight, Cristina left immediately for Zaragoza, where she would have her next fight. I went back to Madrid to have dinner with Muriel and Pedro. Pedro had just finished his fight, and he looked very relaxed and his face was pink and bright. The restaurant, Vina P, was practically wallpapered with old and new fight posters and photographs of bullfighters and some mounted bulls' heads. Its specialty was slabs of beef—since the animals killed in bullfights are butchered and are highly sought after for dining, the specialty of the house might occasionally be straight from the bullring. Pedro said Vina P was a bullfighters' restaurant, which means it is the rough equivalent of a sports bar frequented by real athletes in the United States. Before I got to Spain I imagined that bullfighting was an old and colorful tradition that was preserved but isolated, a fragile antique. Cristina Sanchez would be honored, but she would be in the margins—it would be as if she were the very best square dancer in America. Instead, she looms, and bullfighting looms. There are tons of restaurants in every city that are bullfighter- and bullfight-aficionado hangouts, and there are pictures and posters of bullfights even in the restaurants that aren't, and there is the bullfight newspaper and regular television coverage, and every time I turned around I was in front of the headquarters of some bullfight association. At a gas station in a nowhere place called Otero de Herreros the only bit of decoration I saw was a poster for an upcoming fight; it happened to have a picture of Cristina on it. The biggest billboards in Madrid were ads for Pepe Jeans, modeled by Francisco Rivera Ordonez, Matador de Toros. Mostly because of Cristina, bullfight attendance is up and applications to the Madrid bullfighting school are up, especially with girls. The Spanish tabloids are fat with bullfighter gossip, and they are really keen on Cristina. That night while we were eating dinner, Pedro noticed a gorgeous young man at another table and whispered that he was a Mexican pop singer and also Cristina's old boyfriend, whom she'd recently broken up with because he'd sold the story of their relationship to the celebrity press.
I had planned to leave Spain after the fight in Mostoles, but when I heard that Cristina was going to fight soon in a town that was easy to get to, I decided to stay a few more days. The town was called Nava de la Asunción, and to get there you head north from Madrid over the raggedy gray Sierra de Guadarrama and then onto the high golden plain where many fighting bulls are raised. The occasion for the fight was the Nava town fair. According to the local paper, "peculiar and small amateur bullfights used to be done in the fenced yards of local houses until for reasons of security it was recommended to do away with these customs." The bulls were always chased through the fields in the morning so the townspeople could see what they were like. The paper said, "Traditionally there are accidents because there is always a bull that escapes. There is maximum effort put out to be sure that this does not occur, even though it is part of the tradition." It also said, "To have Cristina Sanchez in Nava is special." "The Party of the Bulls—Cristina Sanchez will be the star of the program!" "Cristina Sanchez will show her bullfighting together with the gifted Antonio Borrero Chamaco and Antonio Cutino—a great bill in which the star is, without a doubt, Cristina Sanchez."
Nava is the prettiest little town, and on the afternoon of the fight there was a marching band zigzagging around and strings of candy-colored banners hanging along the streets, popping and flapping in the wind. Just outside the bullring a few vendors had set up booths. One was selling soft drinks, one had candy and nuts, one had every manner of bullfighter souvenir: T-shirts with matador photos, pins with matador photos, photo cigarette lighters and key chains, autographed photos themselves, and white hankies for waving at the end of the fights. Of the nine photo T-shirts, seven were of Cristina. Six were different pictures of her either posing in her suit of lights or actually fighting. The other one was a casual portrait. She was dressed in a blue blouse trimmed with white daisy embroidery, and her blond hair was loose and she appeared to be sitting in a park. A nun came over to the souvenir booth and bought a Cristina photo-hankie. Big-bodied women with spindly little daughters were starting to gather around the booth and hold up first one Cristina T-shirt and then another and finally, sighing, indicate that they would take both. Skittery little boys, sometimes with a bigger boy or their fathers, darted up and poked through the stuff on the table and lingered. After a while, a couple of men pushed past the throng, lugging a trunk marked C. SÂNCHEZ toward the area under the bleachers where the matadors and picadors were getting ready. Now and then, if you looked in that direction, you could catch a glimpse of someone in a short sequined jacket, and until the band came thundering by you could hear the hollow clunking of hooves and the heavy rustling of horses and donkeys.
The tickets were expensive whether you bought one for the sunny side or the shade, but every row was packed and every standing-room spot was taken. The men around me were smoking cigars and women were snacking on honey-roasted peanuts, and every few minutes a guy would come through hawking shots of Cutty Sark and cans of beer. Young kids were in shorts and American basketball-team T-shirts, but everyone else was dressed up, as if they were going to a dinner party at a friend's. At 5:30, in slanting sunlight, the parade of the matadors and their assistants began. Each of them was dressed in a different color, and they were dazzling and glinting in the sun. In a box seat across the ring from the entrance gate were the sober-looking judge and three girls who were queens of the fair, wearing lacy white crowns in their hair. Antonio Borrero "Chamaco" fought first, and then came Cristina. She was wearing a fuchsia suit and had her hair in a braid and had a look of dark focus on her face. When she and her assistants entered the ring, a man stood up in the stands and hollered about how much he admired her and then an old woman called out that she wanted Cristina to bless a little brooch she had pinned on her shawl.
The bull came out. He was brownish-black, small-chested, widehorned, and branded with the number 36. Cristina, the other two matadors of the day, and Cristina's picadors and banderilleros spread out around the ring holding hot-pink capes, and each one in turn would catch the bull's attention, tease him into charging, and then the next person would step forward and do the same. It was like a shoot-around before a basketball game. Meanwhile, the matadors had a chance to assess the bull and figure out how fast he moved and if he faked right and passed left or if he seemed crazy. This bull was a sprinter, and all around the ring the capes were blooming. Then two picadors rode out and positioned their horses at either end of the ring, and as soon as the bull noticed one, he roared toward it, head down, and slammed into the padding that protected the horse's flank. The picadors stabbed the bull with long spears as he tangled with the horse. After he was speared several times by each picador, he was lured away by the big capes again. A few moments later, the ring cleared, and a banderillero sprinted into the ring carrying a pair of short, nicely decorated harpoons. He held them high and wide. Eventually the bull lunged toward the banderillero, who ducked out of the way of the horns and planted the banderillas into the bull's withers. Then a second banderillero did the same thing. The bull was panting. The band burst into a fanfare, and then Cristina came out alone, carrying a small red heart-shaped cape. She stood at attention and tipped her hat to the judge—asking permission to kill the bull—and then turned and glanced just slightly toward her father, who was standing between the seats and the ring. The bull stood motionless and stared at her.
For ten minutes or so she seduced him toward her, and just as he thought he was about to kill her, she diverted him with dizzying, rippling, precise swings of her cape—first a windmill, then a circle, then a chest pass, where the bull rushes straight toward and then under the cape. As the bull passed her, Cristina's back was as arched as a scythe. When the bull was swooning, she stood right in front of him, rubbed his forehead lightly with the flat of her sword, and then spread her arms, yelled something, and dropped down on one knee. The bull looked like he might faint.
Then she started getting ready to kill him. She walked over to her sword boy and traded him for her longest, sharpest blade. The band was toodling away on some brassy song, and after a moment she glowered and thrust her hand up to stop it. She drew the bull toward and past her a few more times. On one pass, she lost her grip on her cape and her father shot up from his seat and the crew raced in to help her, but without even looking up she waved them away. Then the bull squared up and she squared up. His fat beige tongue was now hanging out, and a saddle-blanket of blood was spreading from the cuts that the picadors and the banderilleros had made. Cristina's eyes were fixed with a look of concentration and command, and her arm was outstretched, and she lined up the bull, her arm, and her sword. She and the bull had not seen each other before the fight— matadors and bulls never do, the way grooms avoid brides on their wedding day—but she now stared so hard at him and he at her that it looked as if each was examining the other through and through.
When it was over, she got flowers, wineskins, berets, bags of olives, loafers, crutches, more wineskins, hundreds of things shoved at her to autograph, and both of the bull's black ears. The bull got two recumbent laps of the ring, hauled around by a team of donkeys, and there was a butcher with a five-o'clock shadow and black rubber hip boots waiting for him as soon as the team dragged him through the door. When the whole corrida was finally over, a leftover bull was let loose in the ring, and anyone with nerve could hop in with him and fool around. Most people passed on that and instead filed out of the stands, beaming and chatting and slapping backs and shaking hands. Just outside the front gate was a clean white Peugot van with CRISTINA SÂNCHEZ stenciled in script on the front and the back, and in it were a driver and Cristina and Cristina's father and her crew, still dressed in their sumptuous fight clothes, still damp and pink-faced from the fight. Cristina looked tremendously happy. The van couldn't move, because the crowd had closed in around it, and everyone was waving and throwing kisses and pushing papers to autograph through the van's windows, and for ten minutes or so Cristina signed stuff and waved at people and smiled genuinely and touched scores of outstretched hands. It was such a familiar picture of success and adoration and fame, but it had a scramble of contradictory details: here was an ancient village with a brand-new bullring, and here was a modern new car filled with young and able people wearing the uniforms of a sport so unchanging and so ritualized that except for the fresh concrete and the new car and the flushed blond face of Cristina it all could have been taking place a hundred years in the future or a hundred years ago.
At last Cristina whispered "no mas" to the driver, and he began inching the van down the driveway and then out toward the highway, and soon you could only see a speck in the shape of the van. The town of Nava then returned to normal. Cristina was going on to fight and fight and fight until the end of the European season, and then she planned to fly to South America and fight and then to Mexico and fight and then to return to Spain and start the season again. Once someone suggested that she try to get a Nike contract, and once she told me that she would love to bring bullfighting to America. But it seems that bullfighting is such a strange pursuit and the life bullfighters lead is so peculiar and the sight and the sound and the smell of the whole thing is so powerful and so deadly that it could only exist where strangeness is expected and treasured and long-standing and even a familiar part of every day.
It was now deep evening in Nava, and the road out had not a single streetlight. Outside town the road cut through huge unlit pastures, so everything in all directions was pure black. No one was on the road, so it felt even more spooky. Then a car pulled up behind me, and after a moment it sped up and passed. It was a medium-size station wagon driven by a harried- looking man, and there was a shaggy dappled-gray pony standing in the back. The man had the interior lamp turned on, maybe for the pony, and it made a trail of light I could follow the whole way back to Madrid.
_Lillian Ross_
Lillian Ross joined the staff of _The New Yorker_ in 1945, during World War II, and worked with Harold Ross, the magazine's founder and first editor. She began as a Talk of the Town reporter and, over the course of her career, has written hundreds of pieces, contributing to nearly every section of the magazine.
Several of her pieces for _The New Yorker_ were collected in two books: _Talk Stories_ , published in 1966, and _Takes_ , published in 1983. She has published more than a dozen books largely based on her work with the magazine, including _Portrait of Hemingway; The Player: A Profile of an Art_ ; _Reporting_ ; and her memoir, _Here But Not Here: A Love Story_ , where she details her somewhat unconventional romance with William Shawn, the longtime editor of _The New Yorker_ who died in 1992. Ross was also close friends with the notoriously reclusive J. D. Salinger for more than fifty years, according to her article "Bearable," which ran in February 2010.
Ross's most recent book, _Reporting Always: Writing for The New Yorker_ , was published in 2015.
She lives in Manhattan.
Takes on the Town
For decades, Lillian Ross's inspired short gems have set the gold standard for The New Yorker's iconic Talk of the Town section.
_EDITOR'S NOTE: While staff writer Lillian Ross has written numerous full-length articles for The New Yorker, among them her famous profile of Ernest Hemingway, she has a special fondness for doing Talk of the Town pieces. She explains why in the introduction to Takes, a 1983 collection of her Talk stories._
I love reporting for The New Yorker's Talk of the Town. In many ways, it's the most challenging kind of reporting: the most demanding, the most interesting to do, the most fun, and the most open to humor. Like the longer pieces of reporting for The New Yorker—Profiles and A Reporter at Large, for instance—Talk stories can take many forms. From time to time, some people have said of Talk of the Town that "it all sounds alike." As it happens, the opposite is true. For one thing, Talk stories, like any other kind of writing, vary tremendously in style from writer to writer. A Talk story reveals the very spirit of the writer, and in any given year Talk of the Town has been written by anywhere from twenty to thirty writers, each of whom has a unique, strongly personal style and feeling. Talk stories usually run to about a thousand words, but sometimes, for one reason or another, they may run twice—even five times—as long. Whatever their length, the writing requires that in a short space one build and reveal a character or a situation or an event or a moment and do so with honesty, with humor, with clarity, with freshness, and with truth.
Talk of the Town deals for the most part with life in or related to New York, and it is constantly discovering the excitement and fun in that life. Where there's life, there's humor. And usually the humor presents itself naturally, because it's part of the truth; there is no need to alter the truth to fit one's preconceptions or preoccupations. For me, the greatest opportunity for humor in writing is presented by Talk stories. In fact, they offer, on a small scale, an opportunity for working toward everything I respect and admire about reporting—what it can and should be.
Talk stories are not usually "news" stories, as many reporters define the word, and they are not "feature" stories, as newspapers use the term. Nor are they in the realm of "gossip," which seems to pervade so many books and magazines as well as newspapers these days—to such an extent that the basic meaning of "reporting" has become fuzzy.
A Talk story brings out some aspect of the truth about a person, a situation, or an event. If it is about a person, I try to build the character as one might do in fiction. Quotations help to provide insight. There are three key limitations that make this kind of factual writing difficult:
1. A reporter must have a strong sense of what should and what should not be revealed—a sense of responsibility toward the person he or she is writing about.
2. In quotations, both what is said and how it is said must be conveyed accurately.
3. What is said must be said in the presence of the reporter. It cannot be "reconstructed" from what the reporter thinks may have been said or from what someone else thinks he _remembers_ having been said. Memory is notoriously undependable. A reconstructed quotation is not a fact; it is a pseudo-fact.
So you are limited in being bound to the _facts_ about a real living person and by your responsibility not to venture too far into the privacy of that person. You must never assume that you have a right to intrude on that person's thoughts and feelings, to violate the inherent secrecy of what is going on in that person's brain. If what you select to report about the person reveals what you _think_ is going on in that person's brain, that is another matter. But you must always be aware of the difference between what you know and what you merely _think_ you know about a person.
The roots of good Talk reporting are to be found in the tradition of good reporting in general. Good reporting can be long or short, and I love it whatever its length. A Talk story is a specific yet mysterious literary form, of which brevity is just one element. A Talk story is not a shorter version of something else. It has its own dimensions, and is complete in itself, often compressing into inches the substance of an article of great length. It contains the essence of something. It is a distillation. For me, the definition of good reporting is good writing about real people, real situations, real events. When the reporting, whether short or long, is very good, I find it thrilling to read. Without exception, it directs my attention as a reader to _what_ has been written, not to _who_ has done the writing.
Davis on Dogs
Our dog-groomer, dog-partisan friend Mel Davis gave a few unsettling cries of anguish about his profession the other day while he worked—in his establishment, a cozy, warm, tidy place on East Forty-Ninth Street near the East River—on a ten-year-old small miniature apricot poodle named Goldie, who stood on a grooming table as though transfixed by the shop's windows, which were opaque with steam against the icy blasts outside. "I'm so upset about what's happening with dogs," said Mr. Davis, who is, as half the poodle owners in town know, a small, faithful, puppylike man, with a small, faithful, puppylike face, who believes that every dog has the right to look naturally beautiful, and who measures all people by the way they recognize that right. Wearing pastel-green slacks and a gaudy flower-patterned sports shirt open at the collar, Mr. Davis sat on a bar stool, his heels hooked over the top rung, and clipped away at Goldie's fluff, shaping a kind of halo on top of her head. "Every time I go out on the street and see the way dogs look, I feel so depressed," Mr. Davis said. "All around me, I see long, low dogs with haircuts shaped to make them look like walking coffee tables. Short-backed dogs look as though their heads were pushed up against their behinds. Has mass production reached the point now where all dogs are made to look alike, no matter what the breed, and with no artistry and no styling? I know it's been happening with books and with clothes and with television programs and with movies and with medical doctors and with politicians and with the faces of people. _But dogs!_ "
Goldie gave Mr. Davis a look of sympathetic gloom.
"Don't the owners see the travesties that are made of their dogs?" Mr. Davis asked. "Don't they see how badly their dogs are patterned? I feel so _sorry_ for the dogs—and for the owners, too. I see so many shaved bodies, with full legs but with a small, scissored head, making the dog look so unbalanced, as though he came out of a barber school on the Bowery. Can you imagine schnauzers and other terriers not being made to look like the precisely tailored creatures they should be? The other day, I met a woman whose schnauzer looked like a poodle. She didn't know how her dog should look. But the _dog_ knew. He came to us so unhappy. Well, we did a complete restyling. The dog looked so serene, and the woman was hysterical with joy."
Goldie wagged her tail.
"Who knows or cares these days about how to enhance the personality or conformation of the dog?" Mr. Davis asked. "The trouble is, first of all, with the grooming schools. They appeal to the students to learn and to go out fast and earn a living in grooming. The students are dog lovers, and are sincere in their efforts. But how much can they learn in a crash course in a few months? Can an artist be taught to be an artist in the first place? We interview and audition quite a few of these graduates, but we rarely find anybody with real talent. We try to encourage the youngsters to start at the bottom, doing rough clipping, washing, drying, and brushing, and leave the styling to our talented groomers—at least, until the youngsters _learn_ something. But no, these youngsters want to start immediately at the _top_. Their attitude is, once they are graduates they are immediately qualified to do it _all_."
"Sounds familiar," we said.
"To commercialize an artistic profession is to me an outrage," Mr. Davis went on. "Our clients are very fussy. When we have a new client, at first we observe the animal and suggest the correct cut. Some of these cuts are very intricate and require more than just a basic style. A Dutch clip, or a modified Dutch clip, must give the effect of a poodle wearing a jacket or pants. If you mess up on _that_ , the poor dog looks as though the pants were falling down. If only these would-be groomers would attend _one_ all-breed American Kennel Club–sanctioned dog show and observe the charges of the professional handlers, they would learn more in one day than in an entire course taking months."
Mr. Davis picked up clippers to trim Goldie's nails, and she trustingly held up one paw. "Poodles are treated the worst," he said. "The textures of their coats are so varied that the formula of the shampoo has to be chosen with care. We happen to have developed our own, with high-protein content. It makes thin-coated poodles look virtually heavy-coated, while our woolly woollies become manageable, so that we can scissor and style them easily. Each poodle has an individual look. Nobody should ever forget that."
Mr. Davis looked Goldie in the eye, and she gave him a fervent wag of the tail.
"Heads have to be designed and cut to capture the true personality of each poodle," Mr. Davis said. "Every day, I go out and see dozens of horrible pinheaded dogs. When these dogs flex their ears, a horrible point appears on the top of the head. Horrible. This is what you get when you let a book-learning groomer do it. I'm so miserable when I see that Happy Hooligan look."
Goldie looked alarmed.
"It's OK, Goldie, not you," Mr. Davis continued. "I have seen legs clipped starting from incredible angles, so that they look as though they were just hanging there, defying the laws of gravity. Tails are made to look like witches' brooms. And the mustaches! Some of them only people of Diamond Jim Brady's time would wear."
Goldie cringed.
"When I do Goldie, I start from the base of the neck and work toward the shoulder line and blend it in down to the full leg," Mr. Davis said. "You're done, Goldie. I'll give you your yellow bows."
As Mr. Davis tied the bows on Goldie's head, behind her ears, he almost smiled. "I took a walk yesterday," he said. "The unhappiest walk of my life." He looked solemn again. "I started from the Forties, where every dog looked either matted or the object of a different kind of indifference on the part of the owners. I saw one Afghan hound who actually had a tied-up topknot, à la Yorkie. I saw a freshly groomed schnauzer sporting a closely trimmed poodle face and a hula skirt with legs to match. Then I saw a beautiful Yorkshire terrier with so many ribbons and bows its whole head was hidden. One head bow is sufficient for a Yorkie. Only an idiot doesn't know that. In the Fifties, I found a glut of Lhasa Apsos and Shih Tzus that had been 'economically clipped,' so the regal look of these beautifully coated breeds was ruined. When I reached the Sixties, I saw a cocker spaniel with a completely shaved skull, instead of a topknot blended into the dome. And again the hula skirt, like the one on the poor little schnauzer in the Forties, _plus_ sloppy, unrounded paws. Then I saw some poodles with full puppy clips without shapings. They resembled little cigars. In the Seventies, I saw more guess-what-we-are poodles and pulis."
"All right," we said. "Specifically, what are the main faults you found with the poodles?"
"The legs were clipped so high the dogs looked as if they were walking on stilts," Mr. Davis replied. "The heads were too small and tightly cut, so the dogs looked as if they were wearing caps. The bodies were shorn too close—usually with injury to the animal, revealed by discolored hair over the skin wound. Tails sometimes looked like bananas instead of nice little pom-poms. The ears weren't blended to the side of the head, so I knew there had been deep scissoring, cutting away precious hair that had taken years to grow."
Goldie was ready to leave, looking satisfied with herself.
_"She_ knows," Mr. Davis said.
Remembering Picasso
Last week, in a glassed-in office of the West Fifty-Seventh Street Gallery of Sidney Janis, eighty-five-year-old art exhibitor extraordinaire, who was one of Pablo Picasso's first American friends, and who helped bring Picasso's "Guernica" to this country from France in 1939, Carroll Janis, the younger of Sidney's two sons, fortyish-looking, wearing blue jeans, tan running shoes, and a tan velour sweater over a pink tennis shirt, was sitting on a corner of a desk talking on the telephone about arrangements for the gallery's opening, on October 25th—the hundredth anniversary of Picasso's birth—of an exhibition of two hundred and fifty photographs, all by David Douglas Duncan, of Pablo Picasso. In the gallery's four exhibition rooms, the white-painted walls were blank. On the floors, carpeted in beige, were rows and rows of photographs, mostly in black and white, measuring ten or twelve inches by fifteen, and mounted on rag board. Standing upright in a corner was a big photograph, in color, about forty inches by sixty; a mounted dye transfer on a thick panel, it showed Picasso in 1962, from the rear, seated in a chair in the large salon of his château at Vauvenargues, the room bare of usual furniture but with a couple of unframed Picassos on easels and one in the fireplace.
Duncan, wearing a blue-and-red checked sports shirt open at the collar, brown corduroy pants, brown socks, and no shoes, was tiptoeing among his photographs, rearranging them, studying them, straightening them, getting them ready for the walls. He picked up one showing Picasso and his second wife, Jacqueline, eating at their kitchen table and laid it beside a photograph of Picasso kidding around as he embraced his wife. Duncan, a Midwesterner who has lived for many years in the South of France, near where Picasso lived, looked the same as he has for decades: deeply tanned, thin, boyish, talkative, exuberant, and eager, as always, to share some of his endless enthusiasm for Picasso as both man and artist, which had led him to bring out four books of photographs of Picasso— _The Private World of Pablo Picasso_ (1958), _Picasso's Picassos_ (1961), _Goodbye Picasso_ (1974), and _Viva Picasso_ (1980).
An elevator door opened. Into the gallery stepped Paloma Picasso Lopez Sanchez, the thirty-one-year-old daughter of the artist, who is a jewelry designer now working in New York City. Trim, attractive, energetic, with straight black hair hanging almost to her earlobes, Mrs. Lopez Sanchez is about an inch taller than her five-foot-two father. Except for a red wool French beret with metal grommets, she was all in Saint-Laurent— navy-blue leather jacket, straight skirt of navy with white pinstripes, red-white-and-blue silk print blouse with a bow at the collar, red leather belt, black stockings, navy leather pumps trimmed with red around the instep. She also wore dark-tinted Italian glasses and a single piece of jewelry, of her own design—a gold ring set with rubellite and amethyst. She glanced at the photographs spread out on the floor and then, with a pleased smile, embraced Duncan, who grinned at her with unrestrained approval.
"This is what I wanted you to see," Duncan said. "This is our birthday present to your pop." He led her closer to the photographs on the floor. "This is not really an exhibition of photographs," he said. "The work of art here is your father."
She gave a little laugh. "Dave, it is very impressive," she said. "Very impressive." She looked up at him with large, Picasso-duplicate brown eyes.
Duncan looked as though he had just been presented with a dozen Picassos.
"And Sidney Janis is not charging one dime to get in here," he said. "This is going to be a _celebration_."
Mrs. Lopez Sanchez carefully walked along a row and looked down at a photograph of a spectacularly beautiful little black-haired girl sitting with Picasso in a café and eating an ice-cream cone. "He was always buying me ice cream," she said.
"Paloma, you know your English is astonishing," Duncan said. "Extraordinary. Perfect."
"I can do the British accent if I want to, but it's pretentious," she said seriously.
"Do you remember where that photograph was taken?" Duncan asked.
"Café des Belges, Juan-les-Pins," she said. "It was 1957."
"What a memory!" Duncan said. "You have the same memory as your pop."
"I remember it because my brother Claude and I spent so much time looking up at the Sputnik," she said.
Carroll Janis ambled over and was introduced. "It's all starting to shape up," he said.
"There are so many here I've never seen before," Mrs. Lopez Sanchez said.
"I had such an opportunity," Duncan said. "Nobody was ever self-conscious in front of the camera."
"This picture," Mrs. Lopez-Sanchez said, going over to one of the few photographs in color. "This one wearing the Indian war bonnet."
"Gary Cooper brought him that," Duncan said, with delight. He pointed to the photograph of Picasso-mugging-as-Indian-chief. "Remember when he grabbed that bull's tail that someone had given him and put it on as the Indian's hair?" he asked, with further delight. "He was the greatest mime of them all."
"I would be sitting with him, and suddenly he would put on a funny hat or a plastic nose," Mrs. Lopez Sanchez said. "They were always sending him things."
"Every day was Christmas," Duncan said.
They walked slowly and carefully between the rows of photographs.
"Here I am making faces out of leaves," she said. "He was always working on the dinner table. When I came, he would say, 'It's OK. Stay. But don't open your mouth.' I always had a second breakfast with my father. His breakfast was always the same: dry toast and Caro, a coffee without caffeine. He was homeopathic. He was careful with foods."
"He never drank anything alcoholic," Duncan said.
"I liked the big vicuña quilt he had on his bed, summer and winter," she said. "And his moccasins, with the fur inside. I'd walk in his shoes."
They looked at photographs taken at the beach. "A driver would take us all to the beach," she said. "My father never learned to drive a car. He would say that when he was poor he couldn't afford a car and then when he was rich he could afford a driver, so he never drove himself. We'd stay at the beach till five. We'd take pedal boats out."
"And here he's permitting himself to be sketched by one of those beach artists," Duncan said. "They never left him alone."
"He used to go to that beach, La Garoupe, for years before it became popular," she said. "One day, I read a book that said that my father and the Gerald Murphys and Hemingway and Chanel started going to that beach in the summer. Before that, nobody ever went there in the summer. It was too hot. They went to Deauville. Now La Garoupe is too crowded."
They looked at photographs of Picasso and his children with a goat.
"Esmeralda," Duncan said.
"I don't know how it arrived," she said. "One day, a _bull_ arrived—a big fighting bull. He said, 'I don't want it here.' We sent it back." Pointing to a picture of a large boxer lying on marble steps at the front door of Picasso's house, she said, "Ian! I was sort of raised by this dog. I called him my milk brother."
"He put that dog in many paintings," Duncan said. "Do you have this one?" He indicated—in a photograph—a painting of a girl jumping rope.
"Yes," she said. "When we divided the paintings, I chose this rope-jumping painting of me, and the rope-jumping sculpture of me cast in bronze. I love it. We divided the paintings among us. I think we're six: Jacqueline, then Maya, Claude, and me—we're the children—and Marina and Bernard, the grandchildren. And the French government. It took twenty-five percent of everything." She pointed to a photograph showing a Picasso bronze sculpture of a baboon. "That is the one he made of Claude's cars," she said, laughing. "Claude's two plastic cars. My father made it so fast. He stuck the cars together to make the head. He put on a broken spring of a toy to make the tail. He made ears of the broken handles of two coffee mugs. Then he sent it all off to be cast in bronze. My brother was furious, because his toys were taken away from him."
"You can see the cars," Duncan said, outlining them with a forefinger, as Carroll Janis bent closer to see the sculpture. "The top of the baboon's head, and the bottom."
"I never knew about that," Janis said. "Toy cars," he said. "Sculptured in bronze."
"My brother has it now," Mrs. Lopez Sanchez said. "He loves it."
"The man had a perfect boxer's stance," Duncan said as she turned her attention to a photograph of Picasso pretending to box. "He was in constant motion. And here's how pure his vision was." Duncan was now pointing to a photograph of a painting of Jacqueline. "He made at least thirteen great portraits of Jacqueline, all dedicated to her. She has all of them. And there's another one. On a thin sheet of steel. He painted her profile on it and then cut it out as a sculpture. In this photograph, it's very late at night. Picasso and Jacqueline are sitting here, with the profile in metal of Jacqueline illuminated in the background."
"Did she come to the exhibition last year at the Museum of Modern Art?" Carroll Janis asked.
"Without anyone's knowing about it," Duncan said. "The week before the show closed, she came here, and we went every day."
"There's _your_ dog!" Mrs. Lopez Sanchez said to Duncan.
"Do you remember Lump?" he asked as they looked at a photograph that showed Picasso, a small dachshund nearby, making one of his plates with a representation of the dog on it.
"I liked that dog," she said.
"That's the day he asked me, 'Does Lump have his own plate?'" Duncan said. "I said yes. So he asked me, 'Does he have a plate with his _name_ on it?' I said, 'He can't read.' So your father said, 'How do you know?' And he made that plate for him."
"Here I am, when I was seven, at the dinner table again with my father," Mrs. Lopez Sanchez said, picking up one photograph. "He was making a linogravure with a gouging knife. And I was drawing at the same time. With crayons."
The Golden Ladies of the Golden Door
The ladies, traditionally limited to a quota of thirty-nine, arrive on Sundays at the Golden Door, a spa situated on a hundred-and-seventy-seven- acre tract of land in the Peninsula Range, about an hour's drive north of San Diego. Mostly, they come wearing designer jeans and designer blazers and carrying small bags containing their bathing suits (near-bikinis) and their sneakers (aerobic and running) and their underwear and socks and nighties. Everything else they will need for their $4,250 week is provided by the spa—shorts, warm-ups, T-shirts, terry robes, sun hats, and assorted creams and potions and pastes and a fresh toothbrush. They go quickly to their rooms, each with private bath, a perfectly mattressed queen-size bed, good lamps, a radio, stationery inscribed with the guest's name in gold letters, and reading material ( _Chop Wood_ , _Carry Water_ , _Fifty Simple Things You Can Do to Save the Earth_ ). All the guests receive, on request, bottles of Evian and the daily newspapers of their choice. They never refer to what, if anything, they have read in the news.
The ladies emerge quickly, uniformly democratized in aquamarine- colored warm-ups and gray T-shirts. They are mostly fortyish or fiftyish, with husbands and children, but there are also a few nervous singles and hopeful fatties, and a scattering of heavy oldies (thigh-covering bathing suits), hoarse-voiced, with Barbara Bush hair, icy blue eyes, false teeth, and scads of grandchildren. Children—anybody under the age of sixteen, in fact—are banned from the premises. "We don't want to remind anybody of what she left behind," one of the spa's supervisors explains.
Most of the guests have heard that exercise increases sexuality, and they think a lot about sex and youth and health. They are immediately eager to mention their husbands, quite a number of whom are power figures in the entertainment business. The Hollywood wives have Vogue-cover faces and, as revealed in the Jacuzzi, large and mathematically correct breasts. The New York power wives are often good-egg types, sort of old-fashioned tomboys. There are many wealthy Eastern do-gooders with or married to a lot of money, and they quickly bring up their good works as a counterbalance to other revelations—about the country house, the tennis court, the horse that jumps, brilliant children who are going off to do admirable and hazardous work in developing countries. (The West Coasters don't bother counterbalancing anything.) As the ladies hungrily register with one another what they have in common, they still seem to be searching wistfully for something indefinable, perhaps a surprise.
The visual backdrop is inspired by Japanese culture—koi pool, rain chains, stone lanterns, sand gardens, water gardens, Zen meditation gardens, a wooden bridge that juts crookedly because "evil spirits go in a straight line." The overall ambience reflects the passions and the interests of Deborah Szekely, a cherubic, pink-cheeked, Brooklyn-born dynamo, and the widow of an ahead-of-his-time Hungarian philosopher and nature lover, Edmond Szekely. In 1939, the couple settled in the tiny village of Tecate, in Baja California, and started a health camp. People brought their own tents and paid seventeen dollars and fifty cents a week for the privilege of chopping firewood and helping to milk goats for cheese. Professor Szekely gave radical lectures lambasting cholesterol. The popularity of the camp grew especially with people from Hollywood, who then wanted a retreat with more privacy. So, in 1958, the Szekelys founded the Golden Door, with weeks limited to men only, women only, and couples only. Today, most of the weeks are for women only.
At the first dinner, the ladies introduce themselves and state why they've come to the Golden Door: "I came because I couldn't get into my bras." "I came to get away from my four children." "I came because my husband gave me this for my birthday." "I came to bond with my sister." They leave the first dinner feeling high on one another, high to be with women only, high to be away from their men, high to boast about those men. They learn an amazing lot about one another's husbands in only minutes, and then at 9:30 p.m. each goes to her room and to sleep.
The next day, reporting for the daily hike at dawn, the ladies are drinking mugs of freshly brewed coffee (black) and gulping Vitamin C tablets. They talk briefly about the howling coyote heard in the mountains. Their talk has the semblance of dialogue, but there is no back-and-forth. "Did everybody sleep through the coyotes?" one lady asks a companion, and the companion says, "I've got to call my husband's office, but it's too early in New York." "Nothing bothers me when I sleep," the first one says.
Their preoccupation with "I" and "me" seems to rule out any interest in hearing answers to questions. In pairs, as they do their hamstring and quadriceps stretches, arms around each other's shoulders, one lady says to her partner, "I was going to do the five-mile hike, but now I think I'll do the three." "I never have jet lag," the partner replies. The warm-up exercises end with an ever cheery, seemingly euphoric fitness guide reading one of Deborah Szekely's favorite Japanese-flavored aphorisms: "Success is not an end; it is a journey," or "The intellect is always fooled by the heart." The ladies listen blankly to the aphorisms, make no comment, and turn quickly to the business at hand.
As they head up the mountain, they pay little attention to the surrounding purple daylilies, the orange poppies, the baby daisies. They walk fast, seriously, heel to toe, and they continue talking, still adhering to the first-person singular. They identify further aspects of themselves. They talk more about their husbands. They outline for one another their husbands' routines—getting up early, being driven to work, outmaneuvering the other guy, obtaining courtside seats at basketball and hockey games, flying here and flying there, and running offices that efficiently deliver plane tickets and chauffeured cars to the ladies on request. Confidences abound. A West Coast lady tells an East Coast lady that her husband is Jewish and that being married to him is nice, because she's Irish-and-German, and Irishand- German husbands are not as kind or as warmhearted as her Jewish husband. The East Coast lady responds by telling a long story about how her husband is afraid of cats, but she _understands_ his fear of cats. Both ladies look as though they are communicating. Talking all the way up the mountain seems to take no toll on the ladies' wind or stamina. At the top of the trail, high-spirited fitness guides offer water and slices of oranges from the Golden Door's own trees. On the way back down, the ladies talk about food, about the breakfast that awaits them in their rooms on their return, at seven- thirty—today it is Baked Apple with Seven Grains, Raisins, Cinnamon, and Honey Served with Orange Slices. They talk about the lunch ahead of them—Chilled Cucumber Soup with Mint and Parsley, Green Lentil Salad on Bibb with Green Beans, Sweet Corn, Diced Vegetables, and Balsamic Vinaigrette, and Long Stemmed Garden Strawberries. "I miss chewing," one of the bonding sisters, both Golden Door repeaters, says to the other.
By Wednesday, the ladies are demonstrating much affection for and pride in one another. In fact, quite a lot of bonding is going on. Everybody seems to share in a general exhilaration; even the oldies and the fatties are getting in on it. Nobody has worn any makeup for three days. Everybody respects potassium. Everybody looks cleaner. Everybody has had a daily herbal wrap, a daily facial, and a daily body massage. Everybody will do more: country-line dancing, yoga, ballet, slide aerobics, tap dancing, self-defense, wreathmaking with herbs and flowers, nutrition lectures, and a tour of the herb garden. Digestive systems are working splendidly. Adrenaline is flowing. The atmosphere is bursting with good will. Both Easterners and Westerners express pride and joy in the oldie giving her Ruby Keeler all in aerobic renditions of "There's No Business Like Show Business" and "Give My Regards to Broadway." Nobody has as much as mentioned a husband all day.
By Thursday, though, the ladies are letting go with some negative remarks about men in general and husbands in particular. They are saying things like "Some men think that having a baby is very glamorous. Hah hah." They are saying things like "Men are from Mars; we are from Venus." Quite a few ladies are saying yoga things to one another, like "The light in me sees the light in you." Some of the powerhouse husbands on the West Coast and on the East Coast are coming in for complaints from their wives. An oldie who is also a fatty as well as a toughie says at the diner table, "At home, we have a condo. We have a big pool. Everything. Does my husband use it? No!" "The light in me sees the light in you," one of the fortyish ladies says. "You'd better believe it," the old toughie says.
On Friday, however, everybody is all business re the body. The talkers on the morning hike are muted. The bonding has become visibly loosened as each lady tends to herself. Saturday night, the ladies, manicured, pedicured, hairdressed, tighter, lighter, firmer, are having their farewell dinner of California Gold Salad with Grapefruit, Avocado, Edible Flowers, and Garden Greens and Grilled Prawns with Garlic and Ginger on Portobello Mushrooms, Wilted Spinach, and Couscous. Prizes are given out—for Hiking Maniac, for Pumping Iron Icon, for Most Active Newcomer, for Ms. Fred Astaire. There is a lot of laughter. The ladies hug, kiss, go through motions of exchanging addresses and telephone numbers. But they are retreating from one another. Almost every lady makes a point of saying loudly that waiting impatiently at home is her husband.
_Susan Sheehan_
Susan Sheehan graduated from Wellesley College in 1958, worked as a fact checker for Esquire for a year and a half, and started writing book reviews for _The New Republic_ in 1959 and light pieces for _The New Yorker_ in 1960. She became a staff writer for the latter publication in 1961 and wrote her first nonfiction series in 1963.
In 1965, Sheehan flew to Jakarta to marry Neil Sheehan, a _New York Times_ foreign correspondent she had met in New York a few months earlier. In the summer of 1965, he was transferred to Saigon, where Sheehan wrote her first book, _Ten Vietnamese_. Her husband was transferred to the Washington bureau of the _Times_ in 1967.
Sheehan continued to write Talk of the Town stories and occasional pieces for such publications as the _Times_ , _The Boston Globe_ , and _Washingtonian_ , but spent most of her time writing books, which were printed in their entirety in _The New Yorker_. _A Welfare Mother_ , published in 1976, won a Sidney Hillman award. _A Prison and a Prisoner_ , published in 1978, received the American Bar Association's Silver Gavel Award. _Is There No Place on Earth for Me?_ , published in 1982, won the Pulitzer Prize for general nonfiction. Her subsequent books include _Kate Quinton's Days_ , _A Missing Plane_ , and _Life for Me Ain't Been No Crystal Stair_.
She has received fellowships from the Guggenheim Foundation, the Woodrow Wilson Center for Scholars, and the Open Society Institute. She served as the chair of the Pulitzer Prize nominating jury for general nonfiction in 1988 and 1994, and as a member of that jury in 1991.
Kid, Twelve
The Tomberlind family represents millions of America's working poor. They're not on welfare, don't live below the poverty line. And yet they struggle every day to get by.
On a brisk Saturday afternoon in March, Brian Tomberlind, age twelve, sets off for the Hickory Hollow Mall just outside Nashville, with his mother, Penny Proctor, and his best friend, Carl. Brian's father, Nathan Tomberlind, stays at home, lying on the living-room sofa wrapped in a blanket. He has hepatitis C. Five months of interferon shots, the only approved treatment for the grave illness, have not put it in remission; they just made him feel weaker. "Nathan probably wouldn't go to the mall with us even if he felt normal," Penny says. "He don't like shopping when he ain't got money to spend. Me and Brian don't go often enough to get our hopes up, but we like looking around." Brian enjoys going to places like malls with his mother, and he gives her money for gas from his earnings, if she needs it, to get there. Penny is more at ease with herself than Nathan is, and this quality translates into ease with her son. When Brian is away from Nathan, he relaxes and seems to forget the troubles of his house.
Penny parks the family's black Mazda pickup and walks with the boys to the front of the Castner Knott department store; they giggle as they pass through the women's lingerie section on their way to a tier of small shops that line the mall's interior. At Pass Pets, the boys admire a variety of pedigreed dogs in glass cages and a selection of rats; they try in vain to coax the store's parrot to talk; they stop at a showcase of Teddybear hamsters and an adjacent showcase of green iguanas. The showcases are open on top. Ignoring numerous "Do Not Pick Up" signs, Brian reaches in, plucks a hamster out of its cage, and suspends it over the iguana cage. "Brian, don't be doing that," Penny says. "If you drop it in with them lizards, the hamster could have a heart attack and die." The hamster wriggles, and Brian accidentally does drop it in among the iguanas. Penny looks scared.
"Them iguanas is vegetarians," Carl assures her, and then says to Brian, "Their tongues don't hurt you. They would get you with their tails." Brian reaches in for the hamster and puts it back where it belongs.
At a store called the World of Science, Brian and Carl inspect talking jigsaw puzzles, kaleidoscopes, Spin Master devil sticks, and glow-in-thedark stars. "This is my favorite store in the mall," Brian says. "It's got cool stuff." Brian is a lanky kid (he's five feet two inches tall and weighs ninety-eight pounds) with indigo eyes, angular features, and fine long hair that has a home-cut look—Penny snips it with her scissors.
At Brian's second-favorite mall store, Spencer Gifts, he tries on a purple cap with a visor in front and a blond ponytail in back. "You look cute," Carl says, and tries on a headpiece with spiky "Wayne's World" hair in back. _"You_ look cute," Brian retaliates.
The boys proceed to a shelf filled with Old Fart Slippers, X-rated greeting cards, Whipped Creme Body Topping, and bottles of PMS pills. "You-all know you can't look in there," Penny says, and then asks, "Brian, what does PMS mean?"
"Poor Man's Suffering," he replies. "I heard it on TV." He knows that PMS really stands for "premenstrual syndrome," because he studied sex ed last year in fifth grade and because his parents are forthright with him about the biological facts of life. When Brian was ten, Penny thought she heard him ask her what a condo was. "It's like a town house," she replied. "No, Mama, a condom," he said. She catalogued its virtues.
Brian asks Penny for five of the ten dollars she owes him. He earned the money two weeks ago at a construction site by picking up pieces of drywall and insulation for Sam, a man his father worked for when he was still healthy enough to work. Brian lent the ten dollars to Penny because she needed "money to run on." He buys a can of green Super String 2 for three dollars. He buys a space-age ice cream for two dollars, and offers Carl and Penny pieces of his treat.
"It ain't cold," Penny says. "You could of saved the money."
"But I wanted it," he says.
"Stuff like that amazes Brian," she says. "He ain't had money lately, so when he gets some he spends it quick."
When Penny and Brian return home, Nathan is still wrapped in a blanket on the sofa. Always a thin man—he is five feet nine and used to weigh a hundred and fifty-five pounds—he is down to a hundred and thirty-five pounds. "I got fever and chills real bad," he says. "That medicine ain't making me better, it's making me worse." He shifts to his son. "Brian, what about your multiplication tables?"
Penny Proctor, who is thirty-three, and Nathan Tomberlind, who is thirty-six, did a lot of "drinkin' and druggin'" from the time they were in their early teens until Brian was six. In March 1990, while Nathan was in jail for driving under the influence, he decided to sober up; he went through a treatment program while incarcerated. Penny stopped using on her own. She had done less drinking than Nathan after she discovered she was pregnant and after Brian's birth. Moreover, she had never injected herself with drugs. "I'll be honest with you, I think I got hepatitis C from dirty needles," Nathan says.
Nathan had always had a bad temper and was prone to moodiness, but his temper got worse after he learned that he had the illness, in the fall of 1994. His moods darkened as well, and Brian felt even more edgy around his father. Nathan's illness and his inability to work had given him more time to focus on his son. Several months ago, part of Brian's homework was to put thirty words "in 'ABC' order" and then define them. He borrowed a classmate's completed assignment, "not because I couldn't do it but to get it done faster," and went into his room after school. He sat on his bed— he doesn't have a desk—copying away. When Nathan entered his bedroom, Brian covered the paper. "Give it to me, Bubba," Nathan said. It was the second time Brian had borrowed his classmate's homework but the first time he had been caught. Nathan telephoned the school principal, who summoned Brian to his office. "You're one of our best students," the principal told him. "If we didn't have people like you at school, the school would be bad. You know your daddy loves you or he wouldn't have done this. Don't do it again, or you'll be in major-league trouble." Nathan kept Brian indoors for several afternoons after the cheating episode. Asked if he had learned his lesson, Brian answered, "Yeah. I know if I cheat I'll be punished."
Nathan's current obsession is Brian's failure to memorize his multiplication tables. He is determined that Brian will master them before the last day of sixth grade. Brian has trouble with math and is lazy about learning multiplication. On this late-March afternoon, when Nathan asks him what six times six is, he doesn't know. He seems to hope that Nathan will forget about multiplication between March and the end of May.
On Sunday morning, the Tomberlinds dress casually. Brian is wearing a green T-shirt with a white Nike swoosh logo, which he bought with most of the money Sam gave him for his birthday the previous month. Nathan drives from their house, in south Nashville, to Cumberland Heights treatment center in the city's rural outskirts, where the family attends spiritual services.
In December 1994, the Tomberlinds had a brief relapse. Nathan started smoking marijuana, Penny smoked a joint and, after an operation on her foot, started taking more pain medication than the doctor had prescribed for her; Nathan helped himself to Penny's pills. "I went back to AA on December 29—that's my birthday—and picked up a desire chip," Nathan says. "I'd been over four years sober, from 1990 to 1994, but in 1995 I couldn't get past my thirty-day chip." There was a crisis in August, 1995. Nathan hit Brian with a belt—hard. Penny left the house, taking Brian with her to the home of one of her sisters. Nathan smashed three glass tables and two lamps, broke some dishes, and drank for three days.
"I hit bottom," Nathan acknowledges. His remorse led him to spend two weeks in treatment at Cumberland Heights. After the first week, he telephoned Penny and begged her to join him. "Baby, this is our opportunity to get straightened out and work through our problems," he said. Penny's boss gave her a week off so that she could commute to Cumberland Heights for family treatment. "I learned a lot about myself at Al-Anon," Penny says. "I learned how sick I was, how much I needed to be my own person, that I had to stand up for myself when Nathan yelled at me or tried to switch words around in my head, and that I couldn't put my sobriety on his."
After two weeks, the Tomberlinds continued treatment at a Cumberland Heights center in their neighborhood. Penny asked their counselor, Joy, if she could bring Brian with them. "At first, Joy said she didn't think an eleven-year-old would have the patience to sit still for two hours, but he was real quiet and he participated," Penny says.
"When I was asked to give a 'feeling word,' I did," Brian says. "I said I was scared when my parents started back using after four years, especially of Dad's cussing and yelling, and angry when they didn't listen to me, and glad when Dad went into treatment. I thought it would have a good result, because when he did it before it worked for a long time. And I learned that a lot of other people had troubles."
Joy told Penny that Brian astounded her—that he had handled a rough situation with a wisdom beyond his young years, and with a forgiving nature. "He's a kid hero," Joy said.
Sometimes the Tomberlinds eat a big buffet dinner at Cumberland Heights, but it costs six dollars apiece, and on this mild Sunday they don't have the money, so they go home and eat a big dinner, most of which has been simmering in Penny's Crock-Pot for five hours. Brian has an ample plateful of pot roast, turnip greens, carrots, potatoes, onions, tomatoes, peaches, and bread. "I like pickles, lemons with salt, Brussels sprouts, squash, okra, cabbage, corn, green beans, strawberries, and coconuts, and Nana's breakfasts," he says.
Nana is Nathan's mother. "She's the only relative I got who ain't messed up," Brian says. This spring, two of Penny's sisters, Nathan's only brother, and a cousin of Brian's were in jail for drug-related offenses.
Some of Brian's Super String 2 has got caught in the blades of the living- room ceiling fan. "Brian, get it off," Nathan says. "I told you not to do it, you don't listen, and you're stubborn, just like me." After Brian steps up on the coffee table, reaches up, and removes the strands, Nathan appends a "Thank you." Brian cleans his bathroom (a weekly chore), wheels his racing bike out of the storage room, and pedals across the street to Carl's house.
The Tomberlinds live in a new eleven-hundred-square-foot ranch house with gray vinyl siding, which they are buying under the auspices of a Nashville community organization that is trying to keep their neighborhood from turning as commercial and desolate as an adjacent section. The house is on a street with other modest homes, public housing, and trailers. They live within earshot of the Nashville Speedway USA, and on weekends they hear the roar of NASCAR races. Saturday nights, two neighbors tend to drink, brawl, and drive their own souped-up cars up and down the street, and other neighbors use and sell drugs. Brian has lived in ten small houses and apartments ("holes in the wall") and trailers since he was born. The worst one was a trailer where he slept on the top bunk of a double-decker bed and the first of Nana's two ex-husbands slept on the bottom bunk. "He drunk and he stunk," Brian says. His current home is "the biggest and best place," and the first house his parents have tried to buy. It has three bedrooms (one is being used as a storage room), two bathrooms, a living room/ dining room, and a kitchen. The walls, which Brian helped his parents paint, are white. A glass dining-room table and four velour chairs originally belonged to Nana, as did the living-room sofa and a large chair. The few books the family owns are on the bottom rung of the TV stand, next to the VCR.
On Sunday afternoon, Brian and Carl ride their bicycles until Penny and Lisa, Carl's mother, are ready to go walking. They load the bikes— Brian's Mongoose Menace and Carl's Mongoose Expert—into the back of the black truck and drive a short distance to a track at an abandoned school. Penny, who is five feet seven and has long, fluffy, layered brown hair, has been overweight since her teens. Since she and Lisa started walking for thirty minutes a day several months ago—most weekends and most weekdays after Penny gets home from work— her weight has dropped from 220 pounds to 207. Brian welcomes a brief change of scene. Sunday evening his supper is crackers with peanut butter from a large can labeled "Donated by the people of the U.S.A. for Food Assistance Program," a surplus-food commodity that Nana's first ex-husband receives and passes along to him. Brian watches a video, one of four that Penny, after attending Overeaters Anonymous, rented for the weekend for four dollars and eighty-six cents. Although Penny is more concerned about her weight than about relapsing into drugs, pills, or alcohol, she and Nathan both go to a Cumberland Heights aftercare program one evening a week, and Nathan goes alone to three AA meetings a week. At nine, Brian looks at the clock, hugs and kisses his parents, and says, "Good night, I love you, sweet dreams, and say your prayers."
On Monday morning, Nathan drives Penny to work. Since Brian was three, she has operated machines for a company that manufactures T-shirts, working "from eight-thirty to four-thirty and any overtime I can get." Then Nathan drops Brian off at Nana's apartment for breakfast and goes on to keep a doctor's appointment. Brian quickly and quietly puts away three fried eggs, two strips of bacon, three biscuits, and a glass of milk, carries his plate to the sink, and lies down on the living-room sofa to watch TV. He likes watching TV at his grandmother's. "We don't got a cable or a remote control at home," he says, hopscotching the channels from _Family Challenge to Let's Make a Deal_ and on to _Rugrats_. At eight-fifteen, Nana drives Brian to Glenview Elementary School, which has 363 students in kindergarten and grades three to six. Grades one and two are held at another school, because of a complicated court order desegregating Nashville's schools.
At the end of the 1994–95 school year, Glenview was 47 percent white, 44 percent black, 6.6 percent Hispanic, and 1.6 percent Asian. Approximately 25 percent of the students are bused from an inner-city housing project, and the majority of them are from single-parent homes. One of the four excuses that Glenview deems legitimate for an absence is "illness in the family requiring temporary help from the child." Glenview is a Title I school, one that qualifies for congressionally funded assistance, because it is among the schools that have a high concentration of children in poverty, meaning children eligible for free or reduced-price lunches—63 percent of the students, in Glenview's case. (Title I money and programs date back to 1965, as part of Lyndon B. Johnson's War on Poverty.)
Brian Lee Tomberlind is one of a 1.908 million twelve-year-old boys living in the United States. Like the overwhelming majority of them, he is white (80 percent) and he attends a public school (89 percent). In 1995, Penny earned $18,023 and Nathan $3,700. Those earnings, of $21,723, put the Tomberlinds in the bottom quarter of American wage earners but above the poverty line, which was $12,156 for a family of three last year. The Tomberlinds receive no subsidies from the federal government, but since Nathan's illness was diagnosed he has been on TennCare, a program that Tennessee created as an alternative to Medicaid, the federally financed health insurance for the poor and the diabled.
As a member of a three-person household with an income below $23,292, Brian qualifies for reduced-price breakfasts and lunches at Glenview, but Penny prefers to give him full-fare lunch money—a dollar and fifteen cents per day— to spare him the embarrassment that she felt as a child about receiving free school lunches. When Penny is short of cash, Nana provides lunch money. "I wouldn't mind getting free lunch if we needed it," Brian says. The kids at Glenview don't care who does or doesn't get reduced-price or free meals at school, but Brian says some of them taunt a boy in his class about his shabby clothes, because his mother spends most of her money on drugs. "The teachers are nice," he says. "They buy him clothes." Brian is not a teaser of others, because he is sensitive and good-hearted. "I wouldn't want no one talking about my clothes," he says. "It would make me feel bad." Brian is self-conscious about his wardrobe and was pleased that all his twelfth-birthday presents from his family were clothes: Nana bought him a gray sweatshirt with a hood and a pair of shiny navy nylon pants, and Penny bought him a pair of jeans. At any given time, he has about a dozen outfits in his closet and three pairs of sneakers—usually last season's model of a name brand bought for forty dollars or less at a discount shoe store.
There are twenty-nine students in Brian's homeroom. Two are currently in foster care—one girl because her father is in prison for murdering her mother; another girl spent last year in foster care. Several live with grandmothers, several with mothers and "bad stepfathers," many with mothers and "mean boyfriends," and many with single mothers. A few have been sexually abused. Very few live with their biological mother and father, as Brian does now. He was born Brian Lee Proctor and delivered by a midwife at Metropolitan Nashville General, a public hospital. He lived with his mother's family off and on until he was three. His parents married when he was four and a half, and his surname was then changed to Tomberlind. His parents occasionally separated, Brian always staying with Penny, but they got back together quite soon. Glenview calls itself a "high-transit" school. Of the twenty-nine boys and girls now in Brian's sixth-grade homeroom, only six others have been there since third grade, as he has.
Although Brian's situation is singular, his life and the lives of his classmates are representative of the few-frills lives of children of the working poor. The farthest that Brian has ever been from Nashville is Myrtle Beach, South Carolina, where he, his parents, and Nana spent a weekend in 1993. Middle-class customs are not ones he takes for granted. So far, he has had only three "cake birthdays"—birthday parties when he could invite other kids over for cake, ice cream, and fruit punch. And yet Brian Tomberlind is one of the three or four most fortunate youngsters in his homeroom, in that he has two parents who, despite the serious mistakes of their young years, have overcome their addictions: two parents who have stayed together and want Brian to have a better life than they have, and are working toward this goal—Penny by holding down a steady job and providing as many good times for him as she can, and Nathan by keeping him from such temptations as cheating and by trying to motivate him, sometimes by persuasion and sometimes by threats or punishment, to be a more diligent student. To Nathan, making Brian learn his multiplication tables is a symbol of the small victories that are possible in his world, and that are no less important for being small and thus perhaps attainable.
On a breezy Monday morning, Ms. Grant—an attractive fifty-three-yearold transplanted Californian, with a daughter who works as a lawyer in Pasadena—who is Brian's sixth-grade homeroom teacher and is also his reading and spelling teacher, greets her students as they arrive, with "Hi, honey" and "Hi, sweetie." She asks how their weekends went and where their homework assignments that were originally due on Friday are. Brian and a number of his classmates hand theirs in. Others make excuses: "My mother made me clean my whole room," "I had to take care of my little sister," "My mother said I didn't have to do all of it," "I don't got none."
"You don't got _none_?" Ms. Grant asks.
"I don't have any," the last kid says as Ms. Grant pencils check marks next to the names of the non-homework-doers on a piece of paper. The students know the rules: ten homework checks (or behavior checks) and they're ineligible to attend the Incentive Party held every other Friday for the virtuous. If they turn the homework in late, though, the checks will be erased. "The bottom line is I want them to do their work," Ms. Grant says.
"I can't believe a parent wouldn't let a child do his homework," Brian says, almost wistfully, in a low voice to a friend. Nathan has recently told Brian that he can no longer go out to play after school on Mondays ("He wants me to settle in after the weekend") or Wednesdays ("It's the night my parents go to aftercare and I go along and sit in the cafeteria watching TV and drawing while they're in meetings"). Less play equals more time for homework in general and math in particular.
At eight-forty-five, a student in the Glenview office asks the students over the intercom to please get ready for morning announcements. First, the students observe "a moment of silence," mandated by the State of Tennessee. During the silence, some crack their knuckles, some doze off. After they recite the Pledge of Allegiance, they chant the Glenview Attitude Pledge, fifteen inspirational lines ("I know I'm capable of success / I am a learner and I can achieve / It's in myself that I must believe") composed by Ralph M. Thompson, who has been the school's principal since the fall of 1992. Next comes the vocabulary word of the week ("'lavish'—hard work can lead to a lavish life style"), and a few reminders ("Please bring your school-picture money to the cafeteria"). By eight-fifty-five, Ms. Grant has taken attendance, and she now asks her class to line up. The students move single file, in alphabetical order, along a hallway wall to their first Monday class—Computers.
Glenview's computer room has twenty-nine Packard Bell computers, but ten of them are in disrepair, so the kids usually "partner up." They are not proficient on the computers; like Brian, twenty-four of his classmates have no computers at home, and they attend Computer class only twice a week. Today, they are not going to use the computers, because their teacher believes they need help with capitalization more urgently. In a few weeks, they will be taking annual achievement tests that are part of the Tennessee Comprehensive Assessment Program—the TCAPs, the tests are called—and Glenview wants its students to perform well. In a booming voice, the computer teacher reminds the class emphatically that days of the week, months of the year, street addresses ("like Animal Lane"), and proper names begin with capital letters.
Tuesdays through Fridays, Brian has Math (all academic subjects are taught four times weekly), but on Mondays his next period is Reading. Ms. Grant says she wants everyone to choose a subject for a research report on an African American man or woman—"and forget Michael Jordan or Michael Jackson and any other figures from the world of sports or entertainment." She goes through the five steps of the report, which range from "brainstorming" to "bibliography." When Brian has trouble coming up with a subject, someone proposes Thurgood Marshall, the first African American on the Supreme Court. He asks what the Supreme Court is. He has heard of President Clinton but doesn't know when he was elected or which political party he belongs to. He has no idea where Bosnia and Israel are.
The students' discipline is good in Reading. If Ms. Grant doesn't like something she notices—for example, a girl wearing a low-cut dress, exposing cleavage—she tells the student that she is in violation of Glenview's dress code. The girl goes to the school office without protest, borrows a T-shirt and wears it over the offending dress the rest of the day. (Most of the girls in the class are tall, a few are overweight, all wear bras.) When Ms. Grant spots a boy chewing gum, she pronounces his name deliberately and says, "You need to spit the gum out." He complies. (The boys come in a wider range of sizes, from small and slight to tall and heavy.)
Ms. Grant, who also taught Brian in third grade, is his favorite teacher, and he considers her a friend. This February 14, he spent all the money Nathan could afford to give him on a big Valentine's Day balloon for her. On Brian's birthday, she handed him five dollars and a card in a report- card envelope. Brian doesn't realize that she gives five dollars to each of her sixth graders on his or her birthday.
On Monday, Reading segues into DARE, which stands for Drug Abuse Resistance Education, a nationwide program. It is taught by Officer Bumpas, a uniformed Metro Nashville police officer, packing gold jewelry and a .38-caliber revolver. One Monday, she leads the class—she addresses the children as "sir" and "ma'am"—in a discussion of the ways in which people are pressured by the advertising media. She mentions products that are endorsed by celebrities and moves on to those with snob appeal. A student cites an ad for Grey Poupon mustard. Brian loves commercials. He raises his hand and offers another example—a commercial for Viennetta ice cream. "It shows fancy people's hands, and the women wearing pearl bracelets," he says. "There's fancy plates on the table, like china. And crystal cups, like wineglasses. And silver—it looks real. I'd like to try the ice cream, but not for the snob appeal, just because it looks good." In the category of "Having Fun," the DARE workbook shows a boy on skis, with the caption "Have a beer." Officer Bumpas asks what the picture shows and what it fails to show. Brian's hand is air-bounced again. "It shows that in order to have fun you have to have a beer," he says. "It doesn't show if you're up on a ski slope with a can of beer you wouldn't ski well. Beer would make you mess up more or make you slower. You'd be better off without it." It is not a coincidence that Brian is more vocal in DARE than in any other class.
The last morning class is Language. One Monday, the teacher struggles to teach prepositions; another Monday, she wages a losing battle to teach the difference between "can" and "may." The students don't pay attention, and start talking to one another.
"Is there any reason you're acting like this?" the teacher inquires of a girl who has been restyling another girl's hair.
"Yeah, because we're bored," the girl answers.
The lunchroom is noisy. Brian eats his corn dog and baked potato, drinks his milk and his orange juice ("Sometimes there's Jell-O, but today orange juice is the dessert"), and talks to two classmates about basketball.
In Social Studies, the class is studying the Renaissance. Brian and his classmates read that Leonardo da Vinci painted _The Last Supper_ , and Michelangelo painted the Sistine Chapel. Asked if learning about new artists in sixteenth-century Italy has meaning for him, he says "Knowing about the Renaissance makes me feel better, because it makes me feel smarter."
Science, Monday's last academic subject, has always been Brian's favorite. "I like making things and experimenting, and I collect rocks," he explains. The teacher distributes compasses, many of which don't point north. "I think some fifth graders stuck magnets in them and messed them up," Brian suggests.
From two-thirty to three o'clock every day, Brian's class goes to Physical Education. One Monday, the P.E. instructor, Mr. Majors, who is a former college football player, explains cardiovascular fitness. Brian excels at P.E. "In fifth grade, I won an award for being second-fittest in the whole school," he recalls. "We did pushups, pullups, situps, and we ran. I think I'm fit because I play a lot."
If Brian sees a fight on the way home, he avoids it. He has the gift of minding his own business—a useful gift for gliding through a Glenview day.
At the Tomberlinds' dining table in the afternoon, Brian's approach to homework is to dispose of it as quickly as possible. When his spelling assignment entails putting thirty vocabulary words into sentences, he spells the vocabulary word correctly but is careless about the spelling of other words he uses in his made-up sentences. "That pieace of cake is 'irresistible,'" he prints. (He has trouble writing in cursive.) And "Jason allways has to 'complicate' things." He avoids thinking up ways to use difficult words in sentences, and often puts the vocabulary word into a simple question, as in "What does 'predominant' mean?" and "What does 'gallant' mean?" He says he sees no point in trying harder: "Ms. Grant just looks at the homework and if you've used the spelling words you get an A. If you haven't, you get an F."
Nathan doesn't check Brian's homework, but he takes out the multiplication flash cards and asks Brian what seven times eight is. Brian hesitates, then gives an incorrect answer. Though Nathan appears angry, he doesn't preach to Brian, as he often does. Nathan was supposed to have an interferon shot yesterday and postponed it until today. Now, still dreading its side effects, he goes into the bedroom and prepares to give himself the shot.
Later, father and son discuss the fact that Glenview permits corporal punishment. Penny and Nathan have given their consent for Brian to receive it, if necessary, but Mr. Thompson has cut way back on it—from more than two hundred paddling to twenty in his first year as principal, and thereafter to below a dozen. Mr. Thompson sometimes prevails upon the parents to come to school and do the paddling, which is limited to two licks with a nine-inch-long oval wooden paddle, kept in the school office. Brian has never been paddled, but he is well informed about the ritual. "I've heard the screams from the principal's office," he says. "The kids are usually hollering before they get there. You have to bend over and put your hands on the desk. They hit you on the butt. One boy was paddled by Mr. Thompson, because his mama wouldn't do it. She tooken up for him even though her child needed correcting."
Eakin Elementary School is six miles—and a social universe—away from Glenview. It has 709 children in grades from kindergarten through six and classrooms in two old brick buildings situated near Vanderbilt University. Real-estate ads attempt to attract homebuyers to the neighborhood with the words "Eakin school district," because Eakin is among the city's most sought-after public elementary schools. There aren't quite enough local children to fill it, so parents with educational ambitions for their children enter a lottery for the out-of-zone slots. Sixteen of the twenty-nine students in Mrs. Hyde's sixth-grade class at Eakin have attended Eakin since kindergarten, twenty since third grade.
Both schools offer friendly environments and employ highly regarded principals and teachers—Glenview's Ralph Thompson was selected Nashville's Principal of the Year for 1993–94, and an Eakin teacher was chosen Tennessee's Teacher of the Year for 1994–95. The schools are distinguished in part by their differences. Eakins is less racially diverse than Glenview (it is 24.9 per cent black), and it describes the parents of its students as "middle income," with only 17 percent of the students receiving free or reduced-price meals. There is no corporal punishment at Eakin, no moving from class to class or building to building in alphabetical lines. Discipline isn't a problem at Eakin. Twenty-two of Mrs. Hyde's twenty-nine sixth graders have home computers; most are knowledgeable about current world events; and a few are already talking about going to Ivy League colleges. By seventh grade, some will be attending private schools and others selective magnet schools. Only three of Glenview's seventy-five sixth graders will go to a magnet school.
Perhaps the most precise measurement of the socioeconomic advantages that Eakin's student body has over Glenview's shows up in the annual scores on the standard nationwide achievement tests, which are part of the TCAPs. In the spring of 1996, 9 percent of Eakin's sixth-grade students and 28 percent of Glenview's tested below average in Reading, Language, and Math (the national norm was 23 percent); 47 percent of Eakin's sixth graders and 65 percent of Glenview's tested average (the national was 54 percent); and 44 percent of Eakin's sixth graders and 7 percent of Glenview's tested above average in these skills (the national norm was 23 percent).
When Brian Tomberlind was in kindergarten, most of his grades were satisfactory. A teacher commented on his report card in April 1990, that he was still "a little unsure of his letters and some numbers," and urged, "Please work with him on this. He is a precious little boy." In first grade, when Brian first took TCAPs, he had mostly 4s and 5s in each skill the TCAPs tested, although in Science he scored a 9. (Scores in stanines 1 to 3 are below average, 4 to 6 average, 7 to 9 above average.) First grade is memorable to Brian "because I discovered there was no Santa Claus and I told all the other kids." In second grade, Brian's stanines, including Science, were mostly 4s and 5s. Toward the end of fifth grade, in April 1995, his scores were all average except in Language (top of the third stanine) and Reading (bottom of the seventh). A sentence at the bottom of Brian's Tennessee Comprehensive Assessment Program report in 1995 states, "The student's total battery score is better than about 53 percent of the national sample, and falls in the average range." This spring, he scored better than 66 percent of the national sample—still in the average range.
There are two children in Brian's homeroom who test above average and may therefore have a chance to overcome their disadvantaged situations and compete with the more privileged Eakin students. One is a first-generation Vietnamese boy with a gift for math and a cultural heritage of striving for academic success. The other is a black girl who loves to read. "I read after school, I read to the kids I babysit, and I read at night," she says. "My mother's worried about the electric, so I read in bed with just a night-light until eleven o'clock—my bedtime on school nights. My mother stays up until she sees my light go off. Weekends and summers, I can read as late as I want. I'm going to go to Vanderbilt."
While Brian isn't keen on reading, and his math TCAPs, in the thirty- ninth percentile, are subpar, Mr. Thompson thinks he has a number of qualities that will enable him to succeed—with or without a college education. "Most of our kids are in bad situations none of them are responsible for, and they show it," Mr. Thompson says. "Brian senses a lot of love from his father, his mother, and his grandmother. He's a consistent and resilient child, and he knows how to insulate himself from disappointments. He's a survivor, and he's got character. Conventional wisdom has it that this generation isn't going to do as well as its parents. Brian is going to be the exception."
Asked if he expects to finish high school, Brian says "Yes," with calm but firm conviction. And what about college? "It's a long time off. But I would like to go to art school."
A conversation with Brian on a recent Friday afternoon in the Tomberlinds' living room:
"Brian, do you know what bad words are?"
"Yes. My parents use them around the house, but they told me not to."
"Are you interested in girls?"
"Yes." He turns deep red.
"In girls you see on TV?"
"Mostly in girls I know. I took one girl to the movies when I was in third grade. Me and her sat in the front of the theater, my parents sat in the back. Last year, I took another girl to a roller rink, but I stopped asking girls on dates, because I'm shy."
"Do you like to talk on the phone?"
"No. It makes me tired. It makes my ear hot. Maybe I'm not used to it. We got our first phone maybe three years ago."
"Where would you like to travel?"
"To Florida to see the Pacific Ocean."
"What effect has your father's illness had on you?"
"We don't do a lot, because he don't feel like it. And I worry he'll have to go to a hospital."
"What do you like to watch on TV?"
"Reruns of _Seinfeld_ and _Home Improvemen_ t with Mama when she gets home, and _Roseanne_ , _Martin_ , and _America's Funniest Home Videos_. Sports."
"How do you feel about black people?"
"Same as I do about white people. There's some good and some bad."
"Who are some of the good black people you know?"
"My friend Jonathan across the street, lots of boys in my class, Mr. Thompson, Ms. Grant, Mr. Majors, and Officer Bumpas."
"What makes you happy?"
"When someone says nice things about me. When I get finished with my homework. When we go somewhere."
"What makes you unhappy?"
"When I can't go outside, but I'm not too unhappy, because I know it's not going to help me."
"How do you like being twelve?"
"I like it. I'm older. It's better than being eleven. You're taller. You can ride the go-carts at Snookers, out at Hickory Hollow Mall. At first when I got older and taller I didn't like not being able to play on the bars at Chuck E. Cheese no more."
"Are you looking forward to being thirteen?"
"Yes. I'll be a teenager."
"Will your life be better as you get older?"
"Yes. I'm going to work at Value Plus. I think you have to be fourteen or fifteen. I'll have money, and I could get more things that I want."
"What things are you having trouble getting right now?"
"A go-cart, but they cost between six and seven hundred dollars, and my mama still ain't paid Nana back the $240 she put on her credit card for my bike. And an _NBA Jam_ game for the Super Nintendo that Sam gave me one Christmas. I only have two games for it— _Super Mario World_ and _Super Mario All-Stars_."
"Will your life be better than your parents' lives?"
"I think so, because I'm never going to drink or do drugs. It's dumb. I seen what they done to my parents and most of my grandparents."
"Are you worried about the future?"
"Not really, because I'm going to go to art school and be an artist, and I hope I get to do it."
Penny, who has been fixing tacos for supper, has heard part of the conversation. "Nobody ever says they want to grow up to be a drug addict," she puts in. "I wanted to take care of mentally retarded people. I love my job, but my first two years there I didn't know how to use the machines right and I had two carpal-tunnel operations. It's so important for me to have Brian graduate from high school. Me and all six of my siblings dropped out. Brian's said he wants to be an artist since he was three and a half and done drawings and took them to his grandfather and said, 'Pay me.' But he needs something to fall back on. He'll find his way. There's decent jobs for good people with high-school diplomas. Brian's Nana graduated high school, and she's earning eleven dollars and twenty-one cents as a security officer on the two-to-ten shift. That's good money for a woman in Nashville. I'm going to get the mortgage on this house paid off and give the house to Brian and his wife. I have a 401K savings plan. I let the company hold back 3 percent of my pay—that's nine dollars a week. The company puts in twenty- five cents for every dollar I save. The man who explained the plan to us said the money would be reinvested and would grow to fifty-seven thousand dollars before I retire."
Nathan, who is smoking a cigarette after taking a shower, has also been listening. "I ain't never had no goals at twelve, but I didn't think of sitting around AA meetings," he says. "Roofing was OK but not steady. One job finishes, you got to find you another one. I don't want Brian to do that— Lord, no. I want him to be where he's warm in winter and cool in summer. I have dreams for Brian to get the best education he could—maybe, if he wanted, to go to college. I think there's better-paying jobs if a man can use his head instead of his physical strength."
After dinner, Brian is at the cleared-off dining table. He has paper, colored pencils, and the latest issue of the magazine _Lowrider_ in front of him. He is looking at a drawing of a '47 Chevy and drawing his own version of it—on a slightly smaller scale and with tires that are square rather than round. "I guess my car has flat tires," he says.
Nathan is watching a video, and Brian goes into his room. It's furnished with a double-decker bed (Brian sleeps in the lower bunk and uses both mattresses on it, to make it more comfortable), a chest of drawers, a glass coffee table (its surface is cluttered with "my stuff"), a small TV, a stereo, and a radio. The room has a wide closet and a narrow window that faces a dog pen (occupied by the family dog until it got out and disappeared in February).
In April, Nathan stops giving himself interferon shots; his doctor agrees that they have not put the hepatitis C in remission. He doesn't regain any of the twenty pounds he has lost, but the chills and fever caused by the drug diminish, and his face is less pallid. He begins to receive monthly disability benefits, for which he filed in 1995, and which cover the family's phone and utility bills.
In the spring, Nathan no longer drives Penny to work and Brian to school—some mornings he can't get out of bed in time—but when he feels up to it he starts tilling a garden, goes fishing, or does some cooking. On Good Friday, he fixes a ham, and the day after he cooks "white beans with a bunch of ham and garlic in it." For Easter, he gives Penny money. "I bought me some underthings," she says. Penny's last clothing purchase, in late February, had been "three T-shirts, brand-new, and two pairs of jogging pants, at Goodwill, for a dollar eleven each." Penny grew up hard, with a father who worked, drank, and "tried to give us kids something for Christmas, even if it was a piece of fruit." Penny is caring and likeable—qualities that Brian has inherited. When money is tight, Penny does without and doesn't complain, and when there is a little extra she is grateful for it, as Brian is, too. In late March, an acquaintance of Nathan's gives Nathan and Brian some scrap aluminum, which Penny and Nathan sell at a recycling place for thirty dollars. Nathan and Brian give Penny the money for her birthday. She buys a pair of black Lycra bicycle shorts and a black T-shirt with Garfield the cat on it, at Wal-Mart. Brian earns twenty additional dollars doing jobs for Sam, the remodeler for whom Nathan used to work. Penny drives Brian to a flea market—Brian loves flea markets. He buys Nathan a lighter on a key chain for two dollars. "I hate it when Dad smokes, but he's always losing his," he says. He spends ten dollars panning for gold flakes and gems. "I really like to go panning, and this time I came up with a cut gem, black with a little orange on it, that's ready to be mounted on a ring for Mama," he says.
In May, Sam goes out of town for a week and offers the Tomberlinds the use of his houseboat. Nathan takes Brian out of school for two days, and Penny joins them on the weekend. Nathan nags him about his multiplication tables, but Brian studies them only halfheartedly. On May 30, after Glenview's last full day of school, Penny tests Brian on his multiplication tables. She looks at the answers on the reverse side of her flash cards; her multiplication is iffier than Nathan's, and she is embarrassed that her ability to read is so limited. Brian gets most of the answers right. "I ain't as nervous around my mama," he says.
On May 31, there is school from 8:45 a.m. to 11:30 a.m. One boy's mother fetches him at ten-thirty, and he waves goodbye. His classmates say nothing. At eleven-thirty, Ms. Grant wishes her sixth graders a good summer and asks them to visit her classroom next year. They leave nonchalantly.
Nana fetches Brian, and they go out to eat at a Po Folks. Nathan has been napping at his mother's, and they bring him back the hamburger he requested. He eats it, and she leaves for work. The television set is on, and Nathan is watching a science program. "What's seven times nine?" Nathan asks Brian as the show ends and a _Columbo_ rerun begins. "Seven times nine is..." It is obvious that Brian is stalling for time, but time doesn't buy him the correct answer. "Go sharpen a pencil and write down your times tables," Nathan says.
Brian goes to the kitchen and starts opening drawers, in search of a knife. Nathan thinks that he is taking too long—that he is watching _Columbo_ instead of searching with due diligence—and slaps Brian hard and noisily on the leg. "Next time, it will be with a belt, Brian," he says. Brian is sobbing. Nathan flies into a rage. "You didn't sharpen the pencil fast enough! I saw you looking at the TV! You can con your mother, but you can't con me!" he shouts.
Brian answers that he wasn't watching TV and he had trouble finding the right knife.
Back talk angers Nathan. "I've asked you your multiplication tables for the last few months," he says, and he continues firing words at his son: "You know the consequences. I'm tired of talking. I'm tired of being so patient with you. I'm tired of your crying all the time. You just talk, you don't take no action. I'm tired of your bluff. You won't go outside and play." He sends Brian, still sobbing, off to a bedroom to write the multiplication tables.
"I'm tired of Brian's bullcrap," Nathan says. "I don't think he takes me seriously. It hurts me when I hurt him. I hurt my left hand with my thumb."
Asked if he'll hit Brian with his right hand next time, Nathan answers, "Yes." Asked if he doesn't think he hit Brian a little too hard, Nathan says, "A whipping never killed me. I think kids take advantage of their parents, just like I did. They take your kindness and your generosity for weakness." And the appropriateness of slapping Brian on the last day of school? "The timing was just the way it was supposed to be. As far as I'm concerned, he's out of sixth grade and started the seventh today. Seventh and eighth grade are important years, and rough ones. Brian has to settle down and do his work hisself. The teachers won't have the time to work with him. My dad didn't care. He was only around a little when I started seventh grade. That's when my grades went down and I started getting high. I don't want Brian turning out like me. When Brian gets eighteen, it's his choice, but he's going to mind me as long as he lives with me. Brian may convince my mama or Penny that he's not capable of learning his multiplication tables, but I think he's very capable of learning them."
Asked if the only important thing in life is that eight times nine is seventy-two, Nathan, who is half-watching Columbo and is calming down, says "It's hard being a parent."
Half an hour passes.
"Brian," Nathan calls to his son.
"What?" Brian asks, emerging from the bedroom with several pieces of paper, on which he has printed the multiplication tables more neatly than ever before.
"Brian, did I hurt your feelings?"
"Yes."
"I'm going to ask you those multiplication tables on Monday, and if you don't know them you'll stay indoors all summer."
Brian nods.
On Friday, at 4:30 p.m., Nathan and Brian pick up Penny at work, stop briefly at a drive-in bank where she cashes her weekly paycheck—her takehome pay is two hundred and eighty dollars—and drive quickly downtown. Nashville's annual four-day Summer Lights festival started Thursday, and the Tomberlinds, Nathan included, all wanted to go. They have a nice time listening to bands, watching street performers, and eating corn dogs, roast corn on the cob, fried peppers, and sausages. Brian goes from Penny to Nathan and back easily. He is not afraid of Nathan. They get home around eleven o'clock. Brian, who is double-jointed, does marvelous acrobatic stunts in the living-room floor. All three Tomberlinds enjoy his performance. "We often have good times with each other," Penny says. "We've learned how to argue less."
On Saturday morning, after going fishing, Nathan says, "I should have counted to ten and cooled off a little bit before I hit Brian yesterday. But on Monday if he doesn't know the multiplication tables he's going to stay indoors all summer."
On Monday morning, Nathan drives Penny to work and fixes breakfast for Brian—lots of pancakes and crisp bacon. "OK, Bubba," he says as they wolf down the meal. "What's seven times nine?"
"Seven times nine is sixty-three," Brian says, with conviction.
"And six times eight?"
"Forty-eight."
"What's nine times nine?"
"Eighty-one."
Victorious, Nathan hugs Brian. Victorious, Brian goes outside to play.
_Gloria Steinem_
Gloria Steinem is a writer, lecturer, political activist, and feminist organizer, who in 2013 was presented the Medal of Freedom, this country's highest civilian honor, by President Barack Obama.
Steinem cofounded _Ms_. magazine in 1972, and remained one of its editors for fifteen years. She continues to serve as a consulting editor and was instrumental in the magazine's move to join the Feminist Majority Foundation. She also helped found _New York_ magazine, where she was a political columnist and feature writer. As a freelance writer, she has been published in _Esquire_ , _The New York Times Magazine_ , and publications around the world. She produced a documentary on child abuse for HBO and a feature film about the death penalty for Lifetime, and has been the subject of profiles on Lifetime, Showtime, HBO, and Makers: Women Who Make America. She is a co-founder of the Ms. Foundation for Women, the Women's Media Center, and a co-convener of Donor Direct Action. In 2009, she was awarded the Medal for Journalism by the James Weldon Johnson Institute for the Study of Race and Difference.
Her books include the bestsellers _Revolution from Within_ ; _Outrageous Acts and Everyday Rebellions_ ; _Moving Beyond Words_ ; _Marilyn: Norma Jean_ ; _As If Women Matter_ (published in India); and _My Life on the Road_. Together with Amy Richards, she has produced for Viceland a 2016-17 series of eight documentaries on violence against women in countries around the world. Her writing has appeared in many anthologies and textbooks. She lives in New York.
Mrs. Kennedy at the Moment
An intimate portrait of Jackie Kennedy as she confronts difficult decisions in the year following the assassination of her husband, President John F. Kennedy.
After a privileged childhood; a reign as Debutante of the Year; an education at Vassar, the Sorbonne, and George Washington University; a brief career as an inquiring photographer; a long courtship with a man twelve years her senior who she feared "couldn't be less interested in me"; marriage to that same ambitious young senator; his several illnesses and near death; his campaign for the vice-presidency; the bearing of four children and the death of two; four years as the wife of a presidential hopeful; two years and ten months as the thirty-first First Lady of the United States; the witnessing of her husband's murder, and a full year of the rituals and restrictions of mourning, Jacqueline Kennedy, at thirty-five, must decide what to do with her life.
If Jack Kennedy had lived, the problem of how to spend the post–White House years (a problem he rarely discussed) would have been left to him, but his wife's solution was simple. "I'll just retire to Boston," she said, "and try to convince John Jr. that his father was once the president."
Her own special charisma plus the continuing political ambitions of the Kennedy family might have kept her from disappearing as quickly as a Mamie Eisenhower or Bess Truman (Robert Frost predicted after the Inauguration that Kennedy would be one of the few presidents in history who could never be thought of without also thinking of his wife), but there was nothing to indicate she would follow the Eleanor Roosevelt tradition either. Mrs. Kennedy's influence was that of a great beauty, a hostess, and a woman of taste, but she retained a certain boarding-school air ("She's the very best of that sheltered group," said a friend, "but she's still of the group") that made it difficult to take her seriously. No one was surprised when she confided breathily to a reporter: "Housekeeping is a joy to me, I feel this is what I was made for. When it all runs smoothly . . . when the food is good . . . the flowers look fresh . . . I have such satisfaction." Or when she sat, bored and unhappy, leafing through a copy of Vogue while her husband campaigned. Or when, in the White House, pregnancy kept her from attending a breakfast given in her honor by congressional wives, but not from going to New York that evening to see the ballet. Or when she explained at a press conference, "I really do not think of myself as the First Lady, but of Jack as president."
She seemed to be a lovely, well-bred girl who painted a little and wrote a little; whose early heroes—Diaghilev, Oscar Wilde, and Baudelaire— valued style at least as much as content; who, in the first winter she was married, took a course in American history to please her husband, but who much preferred eighteenth-century France ("American history is for men," she said); who, while First Lady, vacationed with some of Europe's lessloved capitalists; whose taste was good enough to bring a rare distinction to the White House, but no more eclectic than polite society would allow; who was, in short, the most worthwhile kind of ornament.
Sometimes, being apolitical was an asset. After a few halfhearted denouncements for her lack of involvement in women's peace groups, even the Soviets left her alone. White House correspondent Marianne Means described the galvanic effect she had on anti-American Venezuelan farmers "by appearing like a vision in a lime sheath and greeting them with a warm, simple speech in Spanish." The president was proud of "her concentration on giving historical meaning to the White House furnishings," and a little surprised that she managed the whole thing so well: "Mrs. Kennedy displayed more executive ability," he told a reporter, "than I imagined she had." (And she was working hard. Her mother, Mrs. Hugh Auchincloss, remembered a younger Jackie who wasn't interested in picking out so much as a chair for her room.) Pope John, who had been coached by his secretary to address her as "Madame," opened his arms and said "Jacqueline!" when he saw her; and Ludwig Bemelmans christened her "Cleopatra of the Potomac" after General de Gaulle put his glasses on to look at her during a state dinner, "and he is so vain that he doesn't know who is in front of him until his aide tells him." She held about twelve formal receptions a year more than Mrs. Eisenhower, entertained about twice as many state visitors, and got Pablo Casals to play in the White House in spite of our aid to Franco's Spain. "There hasn't been such a born giver of feasts in the White House," wrote Katherine Anne Porter, "since Dolley Madison."
But even when her nonpolitical accomplishments turned to pure political gold, she was still regarded as an ornament, a _salonnière_ at heart.
Until the assassination.
From the moment she appeared in the bloodstained clothes she refused to change, she became a symbol of high tragedy and courage. "I'd always thought," said one rather cynical lady of the Washington press corps, "that there was nothing Mrs. Kennedy did that I couldn't have done better. I was wrong. I couldn't have gone through that funeral. For the first time, I find myself writing words like 'heroine' with a straight face."
The ceremonies ended, and she moved out of the White House and into retirement, but even her friends looked at her differently. "She's not," said one, "the brittle flower we all thought she was." ("It's just possible," said a journalist who had always scoffed at the value of good schools and "good" families, "that this country actually bred her. There's hope for us all.") Her house was surrounded by patient crowds who waited hours to catch a glimpse of her and then looked away when she appeared, as if ashamed to be caught intruding. "They just waited and waited," recalled a neighbor. "It was as if they were waiting for her to tell them what to do." More than Robert Kennedy or any other member of the family, the odd thaumaturgy of the Kennedy Administration seemed to have passed to her. "When the Kennedys lived here," said a White House guard, "nobody walked by without looking—they couldn't resist it, like people going by a mirror. It wasn't like that with the Trumans or the Eisenhowers, and it isn't with the Johnsons. Now, it's her house in Georgetown that they look at."
The thought of all that power going to waste was more than politicians could bear. Her year of mourning included wearing black and canceling all public appearances and official functions, but within a month after the assassination Mrs. Kennedy's future was being discussed nearly as much as President Kennedy's past. Dean Rusk suggested that she become a touring Goodwill Ambassador, and a Michigan lady legislator passionately advocated her appointment as ambassador to France, though the "biggest problem might be that . . . other nations might want someone of the same caliber." Clare Boothe Luce, with the questionable sincerity of a Goldwater Republican advising a Democrat, wrote an article proposing that Mrs. Kennedy rise to the podium at the Democratic Convention, make a dramatic plea, and force the delegates to give Robert Kennedy the vice-presidency. A politician from Boston suggested that Mrs. Kennedy be vice president herself. Civil rights leaders spoke of her becoming a kind of good-looking Eleanor Roosevelt who would lead Negro children into school, and give fireside, mother-to-mother talks on integration; and a few liberals daydreamed aloud, without much hope, that her marriage to Adlai Stevenson might make him a candidate again. Politicians who feared the power of her endorsement advised that she retire, as widows of other presidents have done, or stick to culture, or, at the most, run an international salon. Even those who thought her power was more moral than political had ambitious plans: an international newspaper column, a weekly show on Telstar, a campaign to beautify America, a foundation to aid young artists, an appointment as head of UNESCO.
In a pre-assassination article comparing the role of Mrs. Kennedy, then in the White House, to that of Mrs. Roosevelt, sociologist Margaret Mead wrote: "American society accords far greater leeway to widows than to wives—even permitting them to carry on activities initiated by their husbands, in whose shadows they were supposed to live quietly as long as their menfolk were on stage." The same Mrs. Kennedy who had been dismissed as an ornament was being urged to be a leader.
"This November twenty-second, when her retirement is over," said a State Department official, "Jackie could become, if she wanted to, the most powerful woman in the world."
In contrast to the attention lavished on her, the life Mrs. Kennedy was living in retirement seemed to be very simple, or at least very private. Newspapers, desperate for news of her, ran front-page photographs of her walking the dog, taking her daughter to school, or her son to his playgroup. Books and one-shot magazines on her life sold by the millions— _Jacqueline Kennedy: From Inauguration to Arlington_ , and _Jacqueline Kennedy—Woman of Valor (Her Dreams As A Girl, Her Prayers As A Woman, Her Fears As A Mother)_ —and a nationwide poll showed that she was the most admired woman in the world. On the suspicion that she might have been pregnant when the president died, at least one magazine delayed a special issue for months, and gave up only after she was photographed in ski clothes in late March. _Time_ , _Life_ , and _Newsweek_ wrote all they could find out about her daily routine which, minus the adjectives and padding, boiled down to the fact that she saw the children in the morning, answered correspondence or worked on plans for the Kennedy Library, played with the children in the afternoon, and sometimes had dinner with friends. The cover of a television semiannual promised "Jacqueline Kennedy: Her Future in TV" (the article said she didn't want one), and a movie magazine headlined "The Men Who Love Jackie Kennedy" (they turned out to be Lyndon Johnson, Dean Rusk, and others who had paid her sympathy calls). When she took John Jr. to a horse show where he was to ride his pony in the lead-line class, well-bred society ladies crowded around in an effort to hear what she was saying to her son. One finally managed it and reported back. What Mrs. Kennedy said was, "Keep your heels down."
Reporters were notified of what her press secretary, Pamela Turnure, called "milestone occasions"—visits to the president's grave with foreign dignitaries, an occasional appearance for the Kennedy Library, her appointment by President Johnson to the White House Preservation Committee—but the rest of her life was, as Miss Turnure said, "pretty insulated." She was surrounded by her protectors: all the members of the Kennedy family plus such old friends as artist William Walton, British Ambassador Sir David Ormsby-Gore and his wife, Mr. and Mrs. Ben Bradlee, Secretary of the Treasury Douglas Dillon and his wife, Franklin Roosevelt, Jr., Mr. and Mrs. Charles Bartlett, Secretary of Defense and Mrs. Robert McNamara, Michael Forrestal, and others.
"We all invited her to small dinners," explained one of that group, "but no matter which one of us gave it, the guest list was always pretty much the same—all friends from the old days—and the conversation always ended in reminiscing."
In the beginning, visitors noticed that her only photograph of the president was one taken while he watched a ceremony on the White House lawn shortly before he died. It showed him from the back. But, by midwinter, she was able to go through her husband's mementos to select "things I hope will show how he really was" for a touring exhibit, and to watch _The Making of a President_ , a private screening of a television documentary of the 1960 campaign. (A witness said she looked teary, but "When the film showed Kennedy making wisecracks, she laughed. When he made serious points, she nodded in agreement.")
Still the small dinners continued. ("I never liked them too much before," she said frankly, "and I like them even less now.") Those close to her who felt she should be distracted as well as protected—especially her sister and almost full-time companion, Lee Radziwill—invited outsiders to entertain her, but the new guests were often so fearful of saying the wrong thing that they hung back and said nothing at all. A man described as a "young embassy type" spent an evening as her partner, and found himself reduced to asking politely if she had ever seen a bullfight. ("You just can't go through all that again," observed a guest. "It was like being right back at college.") A New York journalist who sat next to her at dinner found her starved for information. "She wanted to know if Richard Burton's _Hamlet_ was good," he said, "and what Shepheard's was like, and which of several current books were worth reading and wouldn't make her sad."
By way of distraction, Mrs. Radziwill invited movie producer George Englund to stop in for a drink and some discussion of a Kennedy Foundation Dinner, on which he was an adviser. For further distraction, Mr. Englund brought along one of his actors, Marlon Brando. The three decided to go out for dinner, and asked Mrs. Kennedy to come along. ("It was very simple and spur-of-the-moment," explained Pamela Turnure. "They just didn't want to leave her there alone.") They chose what was later described by the press as "an Embassy Row Restaurant," The Jockey Club, because it was the only one in which Mrs. Kennedy, during her three years in the White House, had been able to have a private, reporterless meal. (She had lunched there with John Kenneth Galbraith, then ambassador to India, at the president's suggestion.) But this time it was not so private. Newspaper reporters were everywhere, Brando inquired about backstairs escape routes, and the restaurant manager offered to loan them his car, but it was too late. The story—down to the fact that Brando "smiled slightly as he lit cigarettes"—was carried by newspapers here and abroad.
The fact that she had chosen to make her first social appearance since the assassination with Mr. Brando (which was pretty much the way the no-explanation newspaper report made it appear) lifted the lid on other criticisms. And among those who believed that John F. Kennedy had not yet proven himself as a president, or that the Kennedy family was using the country's grief to continue its power, or that its power was not being used to advance the right causes, or that Mrs. Kennedy should have invited them to one of those small dinners, there had been a kind of underground current of criticism for some time.
That both Kennedy and Lincoln had been assassinated, for instance, was regarded as too small a basis for their identification; and some critics felt there was a clear and faintly sinister effort to identify them. (On her return from Dallas with her husband's body, Mrs. Kennedy had asked Chief-of-Protocol Angier Biddle Duke to find out how Lincoln had been buried, and the precedents he discovered during an all-night investigation were used as guideposts for the Kennedy funeral. Later, she arranged to have a Kennedy inscription added to that in the Lincoln Bedroom of the White House.) There was some resentment of the Eternal Flame she requested for his grave. When schools, bridges, airports, Cape Canaveral, a fifty-cent piece, and the National Cultural Center took on the Kennedy name, resentment grew. Fundraising for the Kennedy Library, the Telstar program celebrating what would have been Kennedy's forty-seventh birthday, the touring exhibit of Kennedy memorabilia, and the designation of his Brookline birthplace as a National Historical Landmark were all greeted as new proofs of excess.
One Washington newspaperman insisted that Kennedy himself— like the Roman Emperor who said on his deathbed, "I fear they will make a god of me"—would not have approved.
Not all the criticism was directed at Mrs. Kennedy. Some felt that the political ambitions of the Kennedy clan made them more her exploiters than her protectors; and others were apprehensive that the family's continuing devotion, however well-meaning, would bolster the group that Washington columnist Mary McGrory had dubbed "the Kennedy irreconcilables," and make the new administration's work more difficult. Robert Kennedy had tried to slow down the epidemic of renaming, and his sisterin- law had said in her first public statement that it was time people paid attention to the new president and the new First Lady, but, to some, those efforts seemed insincere or not enough.
Some complaints were more picayune. When, on January 14, Mrs. Kennedy made her first television appearance to thank the thousands
who had sent messages of sympathy, it coincided with Mrs. Johnson's first White House dinner. As a result, newspapers were full of Mrs. Kennedy and printed little about the dinner. Ladies of the Washington press corps grumbled about Mrs. Kennedy's thoughtlessness, but no one could determine who had planned what first.
President Kennedy was thin-skinned to political criticism but not to personal barbs. (A writer for a national magazine recalled his reaction to a rumor, printed by Dorothy Kilgallen, that Jackie was not really pregnant during the 1960 campaign but was being kept under wraps for political reasons. "He just laughed about it," said the writer, "and added a few amusing comments of his own about what one could expect from that quarter.") For Mrs. Kennedy, the areas of sensitivity seemed to be reversed. She was disturbed by the incident of the Brando dinner and took time to explain to friends how it happened, but she seemed perfectly confident of the ways in which the president's memory would be served best. ("She's less upset by those criticisms than her staff is," said Pamela Turnure. "Once she decides something is right, she just does it.") For a woman whose combativeness, while First Lady, was usually limited to writing letters to _Women's Wear Daily_ protesting the hard time given her by that trade paper about where and how she bought her clothes, Mrs. Kennedy was remarkably tough-minded in the face of charges that she was overestimating her husband's historical influence. "He changed our world," she said firmly, "and I hope people will remember him and miss him all their lives."
"She's not political," explained a former Kennedy adviser, "but she is immensely loyal. She didn't like campaigning or being a political wife, but she did it because it mattered to Jack. If he hadn't become president, it would have taken him a while to recover; the whole force of his tremendous energy was concentrated on that goal. If Jackie hadn't become First Lady, she would have smiled gently and said, 'All right. What do we do now?'"
Her loyalty was transferred to her protectors, especially to Robert Kennedy. (He is, as she explained in a now-famous quote, "the one I would put my hand in the fire for.") As a nondriver can relax with someone else at the wheel, Mrs. Kennedy seemed able to act on the political counsel of her protectors and not worry about it.
Her controversial "endorsement" of former Press Secretary Pierre Salinger, then running in California's Democratic state primary, was given at Salinger's request, but only with Robert Kennedy's permission. There had been a suggestion that she do more than make a statement: Under the guise of spending a few days at Elizabeth Arden's "Maine Chance" beauty farm in Arizona, she was to make a personal appearance in the campaign. ("There was a point there," said one of his supporters, "when Pierre was really worried that he wouldn't make it.") The trip was turned down on the grounds that the public appearance would be inappropriate. ("Can you see Jackie," said an amused friend, "at Maine Chance?") Instead, a telephone interview was granted to California newspaperman Robert Thompson, and the result for Salinger was a helpful statement that "President Kennedy valued his advice and counsel on all major matters." Reportedly, Robert Kennedy approved her giving that support only after it was apparent that Salinger was, in effect, running as a Kennedy. (His slogan was, "Let the man who spoke for two presidents speak for you," and his staff sent out four million postcards bearing a photograph of President Kennedy, the words, "In His Tradition," and a sample ballot with an "X" after Salinger's name.) But getting her to do it seemed to take very little persuading. "She feels it's natural," explained a writer friend of the Kennedys, "to serve her husband's memory by helping his men, the men who will carry on his work. I'm sure she'll keep right on doing it, but of course only if Bobby says so."
Shortly before moving to New York, Mrs. Kennedy apologized for not allowing a picture magazine to do a story on her children. She was sorry to be difficult, she said, but she really hoped her children could grow into their teens without publicity. The editor pointed out that a photograph of Caroline and John Jr. with Robert Kennedy and his children had recently appeared on the cover of _Life_. "Oh," she said, and smiled, "but that was for Bobby." In their temperament and background, Mrs. Kennedy and her brother-in-law have very little in common. (A recent guest at a Kennedy family dinner noted that they had nothing to talk about. "They share a common loss and a basic kind of guts," he said, "and that's about it.") But Robert Kennedy is head of the family now, and Mrs. Kennedy—who instructs her children that "Kennedys don't cry"—seems determined to be loyal.
Her sister, Lee Radziwill, three and a half years younger, often serves as a lightning rod for social criticism of Mrs. Kennedy in much the same way that Robert Kennedy, on political issues, often did for his brother. Any association with so-called frivolous social types (i.e., The Best-Dressed List, The Jet Set, Marlon Brando, or Europeans who don't work) as opposed to worthwhile friends who befit a First Lady (i.e., André Malraux, John Steinbeck, Members of the Cabinet, and Americans who work) has been blamed, traditionally, on Mrs. Radziwill. As a doyenne of Paris fashion, a sometime expatriate, and the once-divorced wife of a Polish prince, she is generally regarded as the representative of café society in the woodpile.
"Lee is what Jackie would have been if she hadn't married a Kennedy," explained one intimate, "a charming, witty, intelligent woman who, as the daughter of a rich stockbroker, never acquired much experience of real life or much social conscience." Another theory is that the existing difference between the sisters was only dramatized by their choice of husbands, that Mrs. Kennedy always had been more frank and serious-minded. But they are loyal to each other. (After their joint trip to India in 1963, Mrs. Kennedy said her sister had been "marvelous. . . . I was so proud of her—and we would always have such fun laughing about little things when the day was over. Nothing could ever come between us.") "They've been through a lot together; their parents' divorce, their mother's remarriage and all that," said an old friend. "When the chips are down, it's still the Bouvier girls against the world."
It was partly due to her sister's urging that Mrs. Kennedy decided to leave the Georgetown house with the perpetual crowds outside. She spent more and more weekends in New York, and the press tried vainly to trace her movements. ("When something is published about one of them," explained an anonymous friend, "it's a game with the Kennedys to figure out who told.") She was seen inspecting a cooperative apartment on Fifth Avenue, having lunch with novelist Irwin Shaw, talking to cartoonist Charles Addams at a dinner party, walking on Madison Avenue, and going to church at Bedford Hills, New York. A local newspaper reporter spent much of that Sunday phoning around to the country homes of wellto- do Manhattan families, including that of Mr. and Mrs. James Fosburgh (whom President Johnson had appointed to the White House Preservation Committee), where Mrs. Kennedy was spending the weekend. He also phoned Broadway producer Leland Hayward and his wife—with whom Mrs. Kennedy, Mr. and Mrs. Bennett Cerf, Truman Capote, and others had just had lunch—but no one told. It was difficult for her to be anonymous anywhere, but clearly it might be a little less difficult for her and her children in New York.
In July, Mrs. Kennedy announced that her Georgetown house, whose redecoration she had halted a few months before, and Wexford, her
Virginia estate, were being put up for sale. The Washington Post regretted "losing a longtime resident and foremost tourist attraction." "She came among us like some wildly unexpected fairy queen," wrote _The Washington_ Star, "and with her goes the heart of everyone who had lived in this place when she did." She was breaking her initial resolve to "live in the places I lived with Jack . . . Georgetown, and with the Kennedys at the Cape," and New York rejoiced.
"Of course, the social echelons are excited," said society press agent Mrs. Stephen Van Rensselaer Strong. "Her presence is a signal honor and will have primary salient impact; she would make any party." "She'll dress up New York," said a fashion writer. "People will go more formally to the good restaurants and theater because she might be there. It will be chic to be cultural." A spokesman for the Gray Line New York Tours promised that buses would not go off their routes to pass her apartment, "but I see no reason why, if it's on the route, our guides—we call them lecturers— shouldn't point it out." A rumor that east-side bookmakers were taking bets on Mrs. Kennedy's choice of a school for her daughter was squelched by the news that it was a sure thing: "She's enrolled in the second grade at the Convent of the Sacred Heart on Ninety-First Street," said an alumna; "just think how many tickets we can sell for our benefit next year!" With the exception of a few dissenters (a querulous writer in the "Voice of the People" column of the _Daily News_ wanted to know "what part of . . . Harlem she will live in as a symbol of her husband's civil rights bill"), New Yorkers were pleased to receive her, if for rather selfish reasons. But Mayor Wagner was optimistic. "We will give her every opportunity," he said stoutly, "to have as much privacy as she wants."
In anticipation of her emergence into public life in November, some large charity events have already been postponed. (The December benefit performance, in Washington, of _My Fair Lady_ was rescheduled in the hope that she could come. It is for the Kennedy Center for the Performing Arts: Mrs. Kennedy is one of the patrons.) Her social secretary, Nancy Tuckerman, has so many requests for Mrs. Kennedy to be sponsor or honorary chairman of charity-social events that she can't list them all. New York hostesses who are not optimistic enough to think that she will come to their parties ("The competition," said one, "is going to be absolutely cutthroat") are trying to spot in advance the functions she might attend. (Any Lincoln Center benefit, any premiere sponsored by the Kennedy Foundation for Retarded Children, the Polish Ball, the National Horse Show, and the Convent of the Sacred Heart's alumnae benefit are the current favorites.)
"I hope," remarked _Glamour_ editor Kathleen Casey, "that she has some good advisers and serious friends, because she's going to be set upon by café society and social climbers who will try to attach themselves to her just as they did to the Windsors."
In fact, Mrs. Kennedy has been surrounded by friends, serious and otherwise—by exploiters, distractors who vie to amuse her, and protectors who insulate her from the world—for much of her life, but she is, as William Walton said, "a strong dame." She has survived. "Jackie has always kept her own identity," said Robert Kennedy admiringly, "and been different."
Married to a strong-willed man twelve years her senior, plunged into the Kennedy clan and the role of senator's wife at twenty-four and the White House at thirty-one, Mrs. Kennedy was often in danger of being submerged. ("I feel," she said after her husband's election, "as though I have just become a piece of public property.") She was so uncertain of being able to remain "a private person" that she rarely cooperated with a press who, for the most part, adored her (she signed a photograph to Pierre Salinger, "from the greatest cross he has to bear"), and once disguised herself in a nurse's uniform and a wig in order to take her daughter out unnoticed. Some of the journalists who went along on her Indian trip had also accompanied Ethel Kennedy and Queen Elizabeth, and they complained that they had got to know "not only Ethel, but the Queen much better than Mrs. Kennedy." A writer who was a personal friend managed to get a few words with her only on the plane going home, and a photographer commented that "She barely said hello to any of us; it was hot as hell and she didn't sweat or let a hair get out of place; she didn't feel well through much of the trip but she never showed it; she was playing the great lady and total stoic."
Yet, at the end of the trip, she saw each newspaperwoman individually and presented her with a note of appreciation and a hand-painted box she had picked out herself. And on the last days NBC reporter Barbara Walters remembered that she relaxed for the first time, as if she had been let out of school: "We met the camel driver whom Lyndon Johnson had invited for a visit, and someone asked if Mrs. Kennedy would like to ride the camel. She was hesitant about it and said, 'No, but Lee would.' Her sister said something like 'Thanks a lot.' It was the first time we had seen that kind of banter between them, and it was obvious that they really liked each other's company. When Lee got off, Mrs. Kennedy got on, riding sidesaddle with her skirt up over her knees. She looked kiddish and charming and as if, finally, she was relaxed and enjoying herself."
Much of the demure, soft-voiced image she presented as she sat, hands folded and immobile through countless public functions, was evidence of the seriousness with which she took her role as First Lady. ("I'm getting good at it," she said. "I just drop this curtain in my mind. . . .") The pose concealed shyness, but it also hid a sharp wit—"It's the unexpectedness of it," said John Kenneth Galbraith, "that makes her so fascinating"—and a strong will—"I wouldn't dream of telling Jacqueline what to do," said her mother; "I never have." At her first press conference after the election, Mrs. Kennedy apologized, with a touch of whimsy, for being unable to speak "Churchillian prose," but when a reporter asked a little condescendingly if she thought she could do "a good job as First Lady," her reply was firm. "I assume," she said, "that I won't fail [him] in any way." Referring, of course, to her husband.
An interesting case might almost be made for the transference of some of the president's qualities. Robert Kennedy—who was always "the active one while Jack was the sick kid who read the books"—has taken to reading political theory and quoting Thoreau and Emerson. Ted Kennedy— the gay, slightly pampered one—is convalescing with a painful back injury and is said to be using the time to research a book. For Mrs. Kennedy, the change seems to be in attitude: she has acquired a new sense of history and a sense of her own place in it. ("Once, the more I read of history, the more bitter I got," she told writer Theodore H. White. "For a while I thought history was something that bitter old men wrote. But then I realized history made Jack what he was. . . .")
The change is not so great that she plans to accept any formal political responsibility: she has limited herself to work having directly to do with her husband's memory. Her children are still her first concern. ("I was reading Carlyle," she once told a reporter, "and he said you should do the duty that lies nearest you. And the thing that lies nearest me is the children.") She is no longer the dilettantish Bouvier girl, and New Yorkers who hope she will become the center of a glittering social scene are likely to be disappointed. She still has no real interest in politics or activism, but if she or her advisers feel that her role calls for her to act politically, indications are she will do it.
"Her public and private images of herself," explained a former classmate, "are not as different as they used to be. After a few years on her own—if she can survive all the problems and pressures—she just might emerge as the person who, up to now, only her friends have known."
_Mimi Swartz_
Mimi Swartz is an executive editor of _Texas Monthly_. Previously, she was a staff writer at _Talk_ , _The New Yorker_ , and _Texas Monthly_. In 1996 Swartz was a finalist for two National Magazine Awards and won in the public interest category for "Not What the Doctor Ordered." She was also a National Magazine Award finalist for her November 2005 story on tort reform, titled "Hurt? Injured? Need a Lawyer? Too Bad!" and won the 2006 John Bartlow Martin Award for Public Interest Magazine Journalism, for the same story. In 2013 she won her second National Magazine Award, again in the public interest category, for "Mothers, Sisters, Daughters, Wives."
Over the years, Swartz's work has appeared in _Vanity Fair_ , _Esquire_ , _Slate_ , _National Geographic_ , and _The New York Times_. It has also been collected in _The Best American Political Writing_ , 2006, and _The Best American Sportswriting_ , 2007. She is the author, with Sherron Watkins, of _Power Failure: The Inside Story of the Collapse of Enron_. She has been a member of the Texas Institute of Letters since 1994.
Swartz grew up in San Antonio and graduated from Hampshire College, in Amherst, Massachusetts. She now lives in Houston with her husband, John Wilburn, and son, Sam.
Mothers, Sisters, Daughters, Wives
What happens to women in Texas as the state wages war on Planned Parenthood, cuts family planning funding, and passes new sonogram laws?
There are things about women that most men would just as soon never discuss. The stirrups in a gynecologist's office, for one; the tampon aisle at the grocery store, for another; and pretty much any matter involving words like "cervix," "uterus," and "vagina." At least, that's how it was until March 2, 2011. Back in January of the same year, at the start of that legislative session, Governor Rick Perry had pushed as an emergency item a bill requiring all women seeking an abortion to have an ultrasound twenty-four hours beforehand. As Sid Miller, the legislator who sponsored the bill in the House, put it, "We want to make sure she knows what she is doing."
At a public hearing on the bill the following month, Tyler representative Leo Berman took the mic and insisted that fifty-five million fetuses had been aborted since _Roe v. Wade_ —or, as he called it, "a Holocaust times nine." The author of a book on abortion rights gave a somewhat overwrought speech about the differences between "a zygote and a baby." A woman named Darlene Harken described herself as "a victim of abortion" because, she maintained, she wasn't warned about the mental and physical fallout from the procedure; Patricia Harless, a representative from Spring, thanked her for her "bravery" and "strength." Alpine's Pete Gallego countered by expressing his resentment of "people who stop caring after the child is born."
In March the bill reached the House floor, where debate raged for three days, as much as ten hours a day. Tensions ran high in the chamber, which was lit by a benevolent winter sun that glinted off the manly oak desks and supersized leather chairs. On the first day, March 2, Miller, a burly man with white hair and a sun-lined face that wrinkles into a bright, inviting smile, explained the legislation. A former school board member from Stephenville, he has a loamy Texas accent and favors a spotless white Stetson. If you stare at him long enough, you might easily forget that it's the twenty-first century.
Miller described his bill in a matter-of-fact tone, as if he were pushing a new municipal utility district. "What we're attempting to do is to provide women all available information while considering abortion and allow them adequate time to digest this information and review the sonogram and carefully weigh the impact of this life-changing decision," he began. Miller then listed everything his bill would require before an abortion could be performed. A woman would have to review with her doctor the printed materials required under the 2003 Woman's Right to Know Act. While the sonogram image was displayed live on a screen, the doctor would have to "make audible the heartbeat, if it's present, to the woman." There was also a script to recite, about the location of the head, hands, and heart. Affidavits swearing that all of this had been properly carried out according to Texas law would have to be signed and filed away in case of audits. A doctor who refused could lose his or her license.
As soon as Miller finished, Houston Representative Carol Alvarado strode up to the podium. There could have been no clearer contrast: her pink knit suit evoked all those Houston ladies who lunch, its black piping setting off her raven hair. Her lipstick was a cheery shade of fuchsia, but her disgust was of the I-thought-we'd-settled-this-in-the-seventies variety.
"I do not believe that we fully understand the level of government intrusion this bill advocates," she said tersely. The type of ultrasound necessary for women who are less than eight weeks pregnant is, she explained, "a transvaginal sonogram."
Abruptly, many of the mostly male legislators turned their attention to a fascinating squiggle pattern on the carpet, and for a rare moment, the few female legislators on the floor commanded the debate. Representative Ana Hernandez Luna approached the back mic and sweetly asked Alvarado to explain what would happen to a woman undergoing a transvaginal sonogram.
"Well," Alvarado answered helpfully, "she would be asked by the sonographer to undress completely from the waist down and asked to lie on the exam table and cover herself with a light paper sheet. She would then put her feet in stirrups, so that her legs are spread at a very wide angle, and asked to scoot down the table so that the pelvis is just under the edge."
"What does this vaginal sonogram look like?" Luna asked, ever curious.
"Well, I'm glad you asked," Alvarado answered, "because instead of just describing it, I can show you."
And so the state representative from Houston's District 145 put both elbows on the lectern and held up in her clenched fist a long, narrow plastic probe with a tiny wheel at its tip. It looked like some futuristic instrument of torture. "This is the transvaginal probe," Alvarado explained, pointing it at her colleagues as she spoke, her finger on what looked like a trigger. "Colleagues, this is what we're talking about. . . . This is government intrusion at its best. We've reached a"—she searched for the word—"climax in government intrusion."
Those who could still focus gaped at Alvarado. No one spoke. The silence seemed to confirm for Alvarado something she had long suspected: most of the men in the House chamber didn't know the difference between a typical ultrasound—the kind where a technician presses a wand against a pregnant belly and sends the happy couple home with a photo for their fridge—and this. She locked Miller in her sights. "What would a woman undergo in your bill?" she asked.
Miller seemed confused. "It could be an ultrasound, it could be a sonogram," he began. "Actually, I have never had a sonogram done on me, so I'm not familiar with the exact procedure—on the medical procedure, how that proceeds."
"There are two different kinds of sonograms," Alvarado said, trying again to explain. "The abdominal, which most of our colleagues may think [of as] 'jelly on the belly'—that is not what would be done here. A woman that is eight to ten weeks pregnant would have a transvaginal procedure." Miller stammered a response, but Alvarado was not done with him. She continued the grilling for several more minutes, keeping Miller on the ropes with a sustained barrage of icky female anatomy talk. Ultimately, however, the room was stacked against her.
On March 7 Miller's bill passed 107–42.
Over the next few months, as the Senate passed its version of the bill, which was sponsored by Houston senator Dan Patrick, and as Governor Perry signed the legislation into law at a solemnly triumphant ceremony, the exchange between Alvarado and Miller stood as a glaring reminder of the peculiar way in which women could be largely boxed out of decisions that were primarily concerning them. (A number of female Republican legislators supported the bill too, but the overwhelming majority of the votes cast in its favor were from men.) Of course, women have rarely held the reins of power in Texas, but there has also seldom been a season as combative on the subject of women's health as the one we have experienced in the past eighteen months.
Miller's bill was only the beginning of what turned out to be the most aggressively anti-abortion and anti-contraception session in history. In the words of one female reporter who covered the legislature, "It was brutal." Not only did the sonogram law pass, but drastic cuts were made to statewide family planning funds, and a Medicaid fund known as the Women's Health Program was sent back to Washington, stamped with a big "No thanks." When the dust settled, Texas had turned down a $9-to-$1 match of federal dollars, and the health care of 280,000 women had been placed in jeopardy. And that wasn't all. Earlier this year, around the time that the new laws began to take effect, an epic, if short-lived, fight broke out between Planned Parenthood and the Susan G. Komen Foundation, pitting two of Texas's most powerful women against each other and highlighting the agonizing, divisive nature of the debate over women's health. No sooner had this conflict subsided than the legislature's decision to kill the Women's Health Program was dragged into the courts for a series of reversals and counter-reversals that is still not resolved.
These conflicts could all be seen as the latest in a long struggle, as women in Texas try to gain control over not just their own health care decisions but their own economic futures and those of their families. This is the state, after all, from which the modern abortion wars originated in 1973 with _Roe v. Wade_ , a case, let's not forget, that pitted a twentyone- year-old Houston woman and two upstart lady lawyers from Austin against formidable Dallas County district attorney Henry Wade. It's a decades-old battle between the sexes over who knows best and, more importantly, who's in charge. And over the past year, the fighting has intensified. On the one side are the Carol Alvarados of the world; on the other, the Sid Millers. The outcome will determine nothing less than the fate of Texas itself.
For most of Texas history, even during the seemingly halcyon period that was Ann Richards's governorship, the goal of Texas women to achieve parity with Texas men has been out of reach. The men who settled the state were a tough bunch. They had to survive a harsh, unforgiving climate; murderous Comanche; soil that was in many places relentlessly resistant to cultivation; rattlesnakes; bandits; long, lonely cattle drives; and more. But women—to paraphrase Richards—had to do most of that barefoot and pregnant and without any of the liberties or rights that men enjoyed. As the saying goes, "Texas is heaven for men and dogs, but it's hell for women and horses."
Many frontier women learned quickly that they were effectively on their own—the downside to hooking up with a rugged individualist far more comfortable with his cattle than with his wife. They bore, raised, and, too often, buried their kids. They figured out how to make do in the face of cruel poverty. Women had to contend with a challenging contradiction: on the one hand, the clearly defined sex roles of the nineteenth century dictated a courtliness and paternal protectiveness on the part of Texas men that survives to this day. On the other hand, the state was settled in most cases by force, fostering a worship of physical strength and a visceral contempt for anyone too weak to make it on his or her own.
Modern Texas history is filled with stories of women who were held down by what academics like to call "the patriarchy" and the rest of us might simply call "macho white guys." When trailblazing federal judge and legislator Sarah Hughes ran for reelection to the House in 1932, for instance, her opponent suggested that her colleagues "oughta slap her face and send her back to the kitchen."
Governor John Connally's Commission on the Status of Women, established in 1967, found numerous inequities in education and the workforce—but also noted that "overly enthusiastic soapboxing oratory can do the feminine cause more harm than good." It has been frequently pointed out that Kay Bailey Hutchison, one of the most successful females in recent Texas history, became a television reporter in the sixties because, after finishing law school, she couldn't find work as an attorney. During Barbara Jordan's entire term in the legislature, which lasted from 1967 to 1973, she was the only woman in the Senate; across the hall, there was only one female in the House: Sue Hairgrove, followed by Sissy Farenthold.
What their male counterparts seemed slow to grasp was that, having endured the same adverse frontier environment as their husbands, fathers, and brothers, Texas women developed many of the same characteristics: the indomitable independent streak, the persistent optimism in the face of lousy odds. But instead of speculating in cotton or oil or real estate, women focused on sneaking power from men.
The pseudonymous Pauline Periwinkle campaigned for improved food inspections in 1905 by suggesting that a woman lobby her otherwise uninterested husband after he "has broken open one of those flaky biscuits for which your cuisine is justly famous." During the Depression, when contraceptives were among the obscene materials the Comstock law deemed illegal to send through the mail, one Kate Ripley, from Dallas, used boxes from her husband's shirt company to disguise the contents of illicit packages that she shipped to women all over Texas.
As the twentieth century advanced and women began to win seats at more influential tables, several distinct types emerged. For many years, before it was considered politically incorrect, a woman in the political arena was known as a good ol' girl or a man's woman. This complimentary description meant she could drink, cuss, and cut a deal and probably never cried in public. Many of these women were what today we'd call liberals, people like Jordan, Farenthold, Sarah Weddington, Richards, and Molly Ivins. They may have endured the hollow loneliness of public scorn, but they managed to get the Equal Rights Amendment ratified in Texas in 1972. It's probably no accident that these particular heroines came from the liberal tradition—it's the one that has been most likely to let women talk, even if they weren't always heard.
But conservative women made their presence felt as well. The most successful ones, like Hutchison or Harriet Miers, played an inside game, making nice—or at least appearing to make nice—while quietly accumulating power. Beauty helped, especially when combined with a rich husband, as Joanne Herring has demonstrated. Barbara Bush took a page from sturdy Republican club women and made herself a common-sense heroine in low heels and pearls. In other words, there were various ways to get around men and grab the steering wheel, and over the years Texas women used them all.
Regardless of their politics, both Democratic and Republican women used their power to advance the cause of family planning. During the time when abortion was both dangerous and illegal—before 1973—volunteering for Planned Parenthood was a socially acceptable, even admirable, thing for many middle- and upper-middle-class women to do. It isn't surprising that Farenthold and Richards were big family planning advocates, but so were the very social Sakowitz and Marcus families, the archconservative Hunt family, and George and Barbara Bush (at least until he joined the anti- abortion Reagan team in 1980). Partisanship just wasn't in the picture.
"Over the years, everyone wrote a check," said Peggy Romberg, who worked for Planned Parenthood in Austin for seventeen years. The issues seemed very different in the sixties and seventies: women had husbands who made them remove their IUDs, or who made them quit school to tend to their babies. A woman's sexual history was allowed to be admitted in court during rape cases. Married women who wanted credit cards in their own name needed their husbands to cosign for them.
Then came that landmark moment in the history of women all over the United States. The story of _Roe v. Wade_ is, in many ways, the story of Texas women. Norma McCorvey (a.k.a. Jane Roe), raised in poverty in Houston, a high school dropout at fourteen, beaten by her husband, and pregnant with her third child in 1969, tried first to lie in order to get a legal abortion—she claimed she had been raped, which would have permitted the procedure in Texas—and then she tried to get an illegal abortion, but her clinic of choice had been shut down by the authorities. Eventually, two Austin attorneys, Weddington and Linda Coffee, filed suit on her behalf, arguing that her right to privacy included her right to have an abortion. (A San Antonio oil heiress, Ruth McLean Bowers, underwrote the legal costs of the case.) In 1973 the US Supreme Court agreed that state laws banning abortion were unconstitutional. The vote was 7–2.
Nearly forty years of legal abortion have followed, along with an endless stream of bitter arguments and toxic political strife. McCorvey, who wound up having her baby as the case progressed through the courts, later did an about-face, becoming an activist with the pro-life group Operation Rescue. Weddington went on to become an icon of the women's movement. In time, the case they launched emerged as one of the most divisive and politically expedient issues in American politics. Maneuvering from the governor's mansion to the White House, George W. Bush used it to successfully solidify conservative Republicans around his candidacy. Though Bush said he was against overturning _Roe v. Wade_ , he talked about promoting a "culture of life," signed the Abortion Ban Act, in 2003, and campaigned vigorously on the issue, using it to draw a sharp distinction between himself and both Al Gore and John Kerry.
In Texas the past decade has seen a sharp turn in the rhetoric of the issue. Some of it is the result of the ferocious GOP primary wars that are now a fixture in what has essentially become a one-party state. Since there is only one election that matters anymore, it has tended to become a contest over who can move furthest to the right. Being labeled a RINO—a "Republican in Name Only"—is a fate worse than death, and what better way to establish one's conservative bona fides than by passing laws limiting abortion?
A parental notification law for minors seeking abortions passed in 1999 (later, legislators passed a law requiring that minors get permission from their parents before getting the procedure). The 2003 Woman's Right to Know Act, sponsored by Representative Frank Corte, of San Antonio, required doctors to give pregnant women a booklet—tinted pink with a daisy on the cover—that includes information about the growth and development of "the unborn child" and color photos of the fetus from four to thirty-eight weeks of gestation. (This booklet is also infamous for erroneously linking abortion with difficulties during future pregnancies and higher rates of breast cancer.)
By far the most important change came as a result of a 2005 lawsuit called Planned Parenthood of Houston and Southeast Texas v. Sanchez, which required the separation of all family planning facilities into two entities: one would distribute birth control and perform women's wellness checkups and cancer screenings, while the other would provide abortions exclusively. Government audits were mandated annually to make sure that no state money—no tax dollars—could ever be used to fund abortions.
"I think we all thought this was harassment—it wasn't going to improve public health. But we said OK, we'll get through this too," said Peter Durkin, who was the president and CEO of Planned Parenthood Gulf Coast for twenty-seven years. Still, despite all the conflict over abortion, there remained some restraint in the legislature over family planning. It was a given that reasonable people could differ over abortion, but most lawmakers believed that funding birth control programs was just good policy; not only did it reduce the number of abortions, but it reduced the burden on the state to care for more children.
That changed dramatically after 2010, when Republicans won twenty-five seats in the House, giving them a supermajority of 101 to 49 and total control over the law-making process. (The male-female split is 118 men to 32 women.) As the Eighty-Second Legislature began, a freshman class of right-wing legislators arrived in Austin, determined to cut government spending—a.k.a. "waste"—and push a deeply conservative social agenda. At the same time, Governor Perry was preparing to launch his presidential bid, burnishing his résumé for a national conservative audience. It wasn't a good time to be a Democrat, but it wasn't a great time to be a moderate Republican either. Conservative organizations turned out to be as skilled at social media as your average sixteen-year-old, using Twitter and Facebook to chronicle and broadcast every move of the supposed RINOs. A climate of fear descended on the Capitol. "Most people in the House think we should allow poor women to have Pap smears and prenatal care and contraception," an aide to a top House Republican told me. "But they are worried about primary opponents."
The result, in Texas and beyond, was a full-scale assault on the existing system of women's health care, with a bull's-eye on the back of Planned Parenthood, the major provider of both abortions and family planning in Texas and the country. As Representative Wayne Christian told the _Texas Tribune_ , in May 2011, "Of course it's a war on birth control, abortion, everything. That's what family planning is supposed to be about."
For those with institutional memory, the most striking thing about Cecile Richards is how unlike her mother she is. The president of the Planned Parenthood Action Fund possesses none of the folksiness and none of the bite that helped make Ann Richards an icon, maybe because neither quality is really necessary or useful anymore (and could actually be considered a hindrance for the head of a national women's organization in 2012). In fact, on the rainy day I met with Richards at Planned Parenthood's headquarters in Manhattan, she looked like someone who had come into her own. Long gone was the awkward perm she once sported. Tall and willowy, she wore a deep-purple sheath with matching peep-toe heels, a combination of chic understatement with just a hint of flash.
Clearly, she had learned her political skills not just from her mother but also from her father, labor lawyer David Richards. Before coming to Planned Parenthood, in 2006, Cecile was a labor organizer and the founder of two progressive groups: America Votes, a nonprofit designed to promote liberal causes, and the Texas Freedom Network, an organization designed to combat the Christian Right. In other words, to Planned Parenthood's opposition, she's the Antichrist.
Richards long ago learned to modulate her anger for public consumption. But when she gets to talking, she can be extremely frank. "The equity that women have now in education and wages is because of family planning," she told me, leaning forward, her voice hardening just a little. "For women, it's not a social issue. It's not political. It's fundamental— fundamental to their economic well-being." She went on, seemingly unable to believe that she was being forced to restate the obvious: birth control enables women to stay in school instead of dropping out, and to get a degree that boosts their economic status for life. It allows them to control the size of their families so that they can afford the kinds of futures they envision for themselves and for their children. A woman who once had five kids might now only have two—and send them both to college. And so on.
But why, I asked, should taxpayers be on the hook to pay for it?
"Why should we pay for Viagra?" she responded. "Why should men be treated differently? We pay for all other medications. Birth control is the most normative prescription in America. Ninety-nine percent of women use birth control. It's 2012, for God's sake!"
As for the changes in Texas, she was deeply disappointed. She had worked on the border with women who have since lost access to cancer screenings. She didn't think Governor Perry was taking the majority of Texans where they wanted to go. "It's hard to go back home," she told me. "That heartlessness does not track with the Texas I grew up in."
Indeed, in the Texas of Richards's youth (she was born in 1958), lieutenant governors like Bill Hobby and Bob Bullock worked with Planned Parenthood to set up a network of clinics all over Texas, in both small towns and big cities. Texas Health and Human Services offered funding through a federal grant for communities willing to open new clinics for the underserved, and Planned Parenthood provided everything from breast and cervical cancer screenings to abortions. "We were encouraged to open new locations," Durkin recalled, "and the state sat right next to us when the extremist furor erupted—and it always died down."
One reason for the tolerance, Durkin said, was that twenty-five years ago there was a greater tendency to "keep out of a lady's business." "In the good old days," he explained, "the Texas Department of Health was managed by retired military doctors who focused more on afternoon golf than reproductive health care issues. And the governor's office didn't interfere either."
The expansion of family planning was crucial to the general health and future of the state itself. Texas has the second-highest birth rate in the nation, behind California. Historically, it has also had one of the highest rates of uninsured women in the country. Today, more than half the babies born each year are to mothers on Medicaid. Since the cost differential between a Medicaid birth plus postnatal care and a year of birth control pills is huge (around $16,000 for the former versus $350 for the latter), the notion that publicly funded birth control was good public policy had never been a subject of debate in the past. Prior to the last legislative session, the state's family planning program was serving close to 130,000 clients who had no form of health insurance, the poorest of the poor. And according to the nonpartisan Legislative Budget Board, the state's investment in family planning saved $21 million a year by averting more pregnancies. Ironically, before the last session began the LBB advocated for more money to be spent on family planning in order to save on the cost of pregnancies and births, which last year totaled $2.7 billion.
But that's not exactly what happened.
"We're going to be making bad decisions all day." It was the morning of April Fool's 2011, a day of important debate in the House over HB 1, the budget bill, and Wayne Christian was just getting started. Christian, from Center, is one of the more ebullient House members, and despite his grim prediction, his mood seemed upbeat. He knew that, thanks to his party's supermajority, power would continue to rest in the heart of the Republican caucus, a place he felt very much at home. A past president of the Texas Conservative Coalition and a successful gospel singer, Christian qualified as a true believer, and on this day he was calm. He was, after all, a man with a plan.
The plan had emerged from several years of strategizing by Texas pro-life groups, and it had as its central goal the demise of Planned Parenthood. To those who oppose abortion, the separation of health care clinics and abortion clinics that the Legislature mandated in 2005 had not gone far enough. Even though organizations like Planned Parenthood are audited annually by the state to ensure that no taxpayer dollars go to pay for abortions, this arrangement remains suspect to pro-lifers.
"The separation agreement is not really enforceable," said Elizabeth Graham, an attractive, sharp-tongued brunette who is the director of Texas Right to Life. "The Legislature has never been comfortable with giving money to 1200 Main Street and 1201 Main Street isn't getting that money. The funds are fungible." So Graham's organization had been working with legislators like Christian, diligently preparing, waiting for the right opportunity. It had finally come. The tactic was to eviscerate Planned Parenthood through the family planning budget. Lawmakers, Graham later told me, "were prepared and understood where funds could go. They had assistance from agencies and information that helped them to redirect funds."
House Republicans also had a clever procedural maneuver up their sleeves. Ordinarily, budget amendments are vetted by the Appropriations Committee, which may hold public hearings on controversial issues. This time, however, the GOP legislators kept mum, intending to present these amendments from the floor, circumventing the traditional vetting process. (Unlike his iron-fisted predecessor, Tom Craddick, current House Speaker Joe Straus has proved less inclined to prevent such tactics.) This meant the amendments would come with no advice from Appropriations, so members were left without guidelines on how to vote.
Indeed, on April 1, when the family planning section of the budget came up for review, conservative legislators began attaching a blizzard of new amendments, each one designed to shrink the size of the $111.5 million budget from which Planned Parenthood drew support. First up was Representative Randy Weber, who wanted to move $7.3 million out of family planning and allocate it to an organization that seeks alternatives to abortion.
In support of his amendment, Weber, a conservative Republican from Pearland, cited a journal article from 2002 that asserted that in addition to contraception not eliminating pregnancies, it also correlated to a higher rate of pregnancy among women who use it. (In fact, the article stated the opposite.) Representative Mike Villarreal, a Democrat from San Antonio, asked Weber if he thought that birth control simply didn't work.
"Not for those that get pregnant," Weber quipped.
"Have you ever used contraceptives yourself?" Villarreal shot back.
"Well, you know, I don't think I know you well enough to go down this road," Weber cracked.
Villarreal shifted tactics, insisting that Weber's plan would do nothing to reduce abortions. Further, if they did what Weber asked, members would be moving money from programs that would save the state around $60 million into one that would not save it a cent.
Weber's amendment passed 100–44. Next up was Christian, who proposed an amendment that would move $6.6 million from family planning into a program to help autistic children. After consideration of additional proposals, Christian's amendment passed 106–34. Two more Republican representatives came forward and laid out amendments to move $20 million into early-childhood intervention programs and the Texas Department of Aging and Disability Services. Those passed too. Representative Bill Zedler asked to move funding from "the abortion industry" to services for the deaf and blind and those with mobile disabilities. Representative Jim Murphy wanted to move money to EMS and trauma care, which was operating with a $450 million surplus at the time. Representative Warren Chisum followed with an amendment to move family planning money to more-generalized medical clinics.
As the night wore on, tempers flared; sometimes it was hard to hear over the members' shouting at one another. Even a staunch Republican like Beverly Woolley found herself moving to the microphone in solidarity with the Democrats. But on and on it went. By one in the morning, the House had slashed the family planning budget from $111.5 million to $37.9 million. The final vote passed with 104 ayes. On May 3, the Senate passed its budget, with the same cuts in place—partly because House Republicans had threatened to hold up the entire budget process if they did not.
By this point, the tenor of the session was clear. As the chair of the Senate Democratic Caucus Leticia Van de Putte said, "Texas is going to shrink government until it fits in a woman's uterus." A little over a month later the sonogram bill went to the governor's desk. "This will be one of the strongest sonogram bills in the nation," declared an exultant Sid Miller. "This is a great day for women's health. This is a great day for Texas," said Dan Patrick, who had tried twice before to get such a bill passed.
Needless to say, not everyone agreed. "I went to an event with Senator [Kevin] Eltife," Patrick told me some months later, "and I parked and a car pulled up behind me and a woman started screaming at me. I've never had that happen. I've had some interesting emails too. Just amazing. But I'm a big guy. I can take criticism, because this is the right thing to do to save a life."
By his account, over time the sonogram bill will save up to fifteen thousand lives. "There will be people alive in ten to twenty years who wouldn't be alive without this bill," he told me. To Patrick, the legally mandated ultrasound isn't an invasive procedure. Critics of the bill further contend that its ultimate purpose is to limit access to abortion, especially for low-income women who may not be able to take off more than one day of work to accommodate the twenty-four-hour waiting period. Patrick rejects this argument too. As he puts it, "The purpose in sponsoring this bill was to improve women's health care." His political opponents, he says, "don't know the facts. They are dealing from emotion." He thinks the claim that most women have made up their minds long before they reach the door of an abortion clinic is "nonsense."
"Most of these women don't know," he said. "No one is trying to embarrass them, but we are trying to save a life. You want the woman to have a choice to have a baby or not, but you don't want them to have a choice to look at a sonogram? That makes no sense to me." (In fact, prior to the sonogram bill, women seeking an abortion at Planned Parenthood could elect to look at a sonogram.)
As the session reached its halfway point, many female legislators grappled with the magnitude of what had happened. Democratic women could at least enjoy the full-throated support of their male colleagues, but moderate Republican women frequently found themselves all alone, treated to a front-row seat from which to view their own powerlessness. To speak up was to be targeted for defeat in the next primary, after all. They dragged through the Capitol with heads down, making apologies to staffers and colleagues for their votes. Ultimately, both Beverly Woolley and Florence Shapiro announced their retirements. The latter told a lobbyist, "These are no longer the people who elected me."
Shapiro and Woolley weren't the only veteran Republicans to find themselves in an awkward position. Take the case of Robert Deuell and the Women's Health Program. Senator Deuell, a physician from Greenville who has held office since 2003, was known to be both pragmatic and conservative when it came to public health. He supported programs like needle exchange for addicts, but he was strongly opposed to abortion. In fact, he had worked tirelessly since 2007 to toss Planned Parenthood from the network of providers included in the Women's Health Program, a Medicaid fund for poor women started in Texas in 2006. Yet unlike many of his new comrades-in-arms, Deuell favored taxpayer-supported birth control.
If the state doesn't make birth control available, he told me, "we are going to be providing prenatal care. It's the lesser of two undesirables, and that's the point I've tried to make. Do I wish women waited? Yes, but they don't."
Deuell has always favored shifting the services provided through the WHP from Planned Parenthood to community-based health organizations and clinics known as Federally Qualified Health Centers. There were some obstacles to this, among them the question of whether the FQHCs could deliver the same quality of care. Many FQHCs were already overrun with very sick people. Jose Camacho, the head of the Texas Association of Community Health Centers, which oversees FQHCs in Texas, had insisted that, despite what Deuell wished, the FQHCs could not absorb the overflow, given Texas's soaring birth and poverty rates along with the vast number of uninsured. "We served one million patients this year, at least," he told me. "To think that any health system can ramp up to take, in effect, 20 percent more patients is not realistic."
Deuell didn't give up. The rules of the WHP had been written to exclude providers affiliated with organizations that perform abortions. This was in conflict with federal law, so in 2008, a waiver was granted that allowed Planned Parenthood to participate. In 2010 Deuell asked Attorney General Greg Abbott, who also fervently opposes abortion, to check on the constitutionality of the waiver, and when the Eighty-Second Legislature rolled around, Deuell was prepared with a rider to the budget bill that would reauthorize the WHP while explicitly preventing Planned Parenthood from ever taking part in it. But by May, he had a problem. He could see a disaster looming—the health care of 130,000 women was already at risk because of cuts to the state's family planning budget, and now, as a result of the political climate, he saw that he didn't have the votes in the Senate to get his version of the WHP reauthorized.
"I guess what took me by the most surprise was an overall opposition to family planning," Deuell told me. The fact that such programs were statistically proven to save money by the Legislative Budget Board was not enough to change hearts and minds, even in a budget-slashing session. "My feeling is that ['the program will save money'] is what you hear every time they want to increase the size of government," said Representative Kelly Hancock, the policy chairman of the Republican caucus. He added that the caucus's opposition to such programs "had nothing to do with the women's health issue."
With time running out, Deuell found himself in the surreal position of joining forces with ultraliberal Garnet Coleman, who was trying to push a bill to save the Women's Health Program in the House. (Back in 2001, it was Coleman, the son of a prominent Houston doctor, who first carried legislation to create the WHP, which Governor Perry vetoed.) This did not go over well, especially with the folks at Texas Right to Life. After a particularly nasty budget committee hearing, Elizabeth Graham compared Deuell to Margaret Sanger, the founder of Planned Parenthood.
Finally, the bill was saved at the end of May by some last-minute politicking— it was attached as an eleventh-hour budget rider. But the victory for women was a hollow one: Planned Parenthood was no longer allowed to participate. It promptly filed suit, as many who had kept their frightened silence in the Legislature had hoped it would. By then, nearly three hundred thousand Texas women were facing the loss of birth control, wellness checkups, and cancer screenings.
And Deuell, for his part, was still stinging from Elizabeth Graham's attack. "For her to compare me to Margaret Sanger," he told me, "it's beyond the pale."
This past January, one year after the start of the Eighty-Second Legislature, the U.S. Fifth Circuit Court of Appeals ruled that the sonogram bill was legal and could stand. (The opinion was written by Edith Jones, a female judge from Texas who has never made her opposition to abortion a secret.) Many women in Texas who had perhaps not been closely following the moves of the legislature were now discovering the fruits of their representatives' labors. Others were beginning to realize that, because of the cuts to health care, they couldn't even get in to see a doctor for annual pelvic exams. Clinics were already closing, or cutting hours, or charging fees for services that had previously been covered.
The session had made it clear that Republican legislators and prolife groups were intensifying their fight against Planned Parenthood, not just in Texas but across the country. If there was anyone who still didn't get it, the news of January 31 made it impossible to miss. That was the day that the Associated Press reported that Susan G. Komen for the Cure, originator of the pink ribbon, had decided to cancel the $700,000 annual grant it had been contributing to Planned Parenthood since 2005 for breast cancer screenings. (None of Komen's money ever went to abortion services.)
The news erupted nationwide, but in Texas it detonated like an atomic bomb. Komen, after all, was based in Dallas and was worshipped there in almost cultlike fashion. What's more, the organization's founder, Nancy Brinker, was a role model for many Texas women, a radical reformer who back in the early eighties had, as one of her oldest friends put it, "brought breast cancer out of the closet." Before she took on the cause, promising her dying sister in 1982 she'd find a cure, most people wouldn't even say the word "breast," much less "breast cancer," in polite conversation. Brinker, a former PR woman originally from Peoria, Illinois, who had married well, to the late Dallas restaurateur Norman Brinker, built Komen into a $1.9 billion philanthropic powerhouse in a relentless, but very feminine, way. She was also a highly visible moderate Republican woman, and a friend of George and Laura's who was rewarded with an ambassadorship to Hungary in 2001 and a position as White House chief of protocol in 2007.
What happened, in brief, was this: anti-abortion groups had been harassing Komen (and the Girl Scouts of America and Walmart) for years over its support of Planned Parenthood. A very vocal, if small, faction was alarming affiliates with threats to disrupt the footraces that have long been Komen's major source of funds. John Hammarley, Komen's senior communications adviser, found himself fielding more and more phone and email inquiries about the relationship between the two organizations. "It took up a sizable amount of my time," he told me.
A few years earlier, Brinker, who is sixty-five, began to step away from running the organization. She brought in a new president, who in turn brought in former Georgia Secretary of State Karen Handel. Handel, who is strongly opposed to abortion, was hired as chief lobbyist and asked to work on the problem of the protestors. Along with Hammarley, she came up with several options that included everything from doing nothing to defunding Planned Parenthood in perpetuity. Hammarley warned Komen that doing the latter would cause severe problems, so the board elected to cancel funding for one year and then re-evaluate.
Komen notified Planned Parenthood, who issued a press release decrying the decision. Immediately, social media exploded with anti-Komen messages—1.3 million on Twitter alone—that ranged from irreverent to near homicidal. Komen seemed utterly gobsmacked by the response. A campaign called "Komen Kan Kiss My Mammogram" sprang up, designed to raise $1 million for Planned Parenthood to replace (and then some) what Komen had withdrawn. Someone hacked a Komen online ad and changed a fund-raising request to say "Help us run over poor women on our way to the bank." What may have been worse were all the blog posts and mainstream media reports that exhumed negative stories about Komen's business practices—how much it spent to aggressively protect its For the Cure trademark, how much of its money actually went to research, whether the organization was supporting the right kind of scientific research, whether its pink nail polish might contain carcinogens, and so on.
There was something very retro about Komen's response—as if they didn't know how to fight like modern women. First, they hid, shutting down all interview requests. Then they tried to cover their tracks, issuing a press release that claimed their decision regarding Planned Parenthood was part of their new "more stringent eligibility and performance criteria" that eliminated any group that was the focus of a congressional investigation. (At the time, Planned Parenthood was the only Komen beneficiary to have such a problem; it had been the focus of a trumped-up investigation, spearheaded by anti-abortion forces, that had come to nothing.) On February 2, a glamorous if somewhat stressed-out Brinker appeared in a video posted on YouTube. Even though stories of internal discord and resignations were already leaking to the press, she reiterated that her decision to end the funding for breast cancer screenings for Planned Parenthood was not political but simply a way of maintaining their standards. "We will never bow to political pressure," she insisted. "We will never turn our backs on the women who need us the most."
In this particular fight, however, another Texas woman, Cecile Richards, would get the upper hand. As the head of an organization under constant attack, Richards was adept at keeping her emotions in check. At every press conference, she was the picture of empathy and calm. "Until really recently, the Komen Foundation had been praising our breast health programs as essential," Richards told _The New York Times_. "This abrupt about-face was very surprising. I think that the Komen Foundation has been bullied by right-wing groups." Meanwhile, Planned Parenthood was churning out fund-raising emails, eventually raising $3 million, far more than it usually got from Komen.
Just four days after it all began, Komen reversed itself, and Brinker, looking even more drawn, appeared before the cameras again, this time to apologize and say that the funding to Planned Parenthood would be reinstated. Handel subsequently resigned, berating Planned Parenthood for its "betrayal" in making public Komen's decision to remove their funding. Both organizations now say they are very happy to be working together again.
Other battles have not turned out the same way. In February the Texas Health and Human Services commissioner—who works at the behest of the governor—signed a rule banning from the Women's Health Program any organizations that provided abortions themselves or through affiliates. Perry declared that if the federal government didn't like it, he would find the spare $30 million for poor women elsewhere, regardless of the state's budget shortfall. In March Kathleen Sebelius, U.S. Secretary of Health and Human Services, stood among Houston's poor at Ben Taub General Hospital and announced that unless Texas relented, the WHP would not be renewed. Federal law required that women have the right to choose their own providers.
The Perry administration was still determined to stop women from being treated by abortion providers, however, so the Health and Human Services Commission distributed a flyer to clients in the WHP, saying they might have to find new places to go—even though there was an injunction in place at the time allowing Planned Parenthood to continue as a provider while the organization's case against the state made its way through the courts.
In May district judge Lee Yeakel blocked Texas from keeping Planned Parenthood out of any women's health program receiving federal funds. "The record demonstrates that plaintiffs currently provide a critical component of Texas's family planning services to low-income women," he noted in his twenty-five-page opinion. "The court is unconvinced that Texas will be able to find substitute providers for these women in the immediate future, despite its stated intention to do so." The state is currently appealing.
In June I went to a Planned Parenthood clinic in the Gulfton section of southwest Houston. Like most of the organization's ten local affiliates, the Gulfton Planned Parenthood is a modest place. It sits in a strip shopping center near a ninety-nine-cent store, a pawn shop, and an appropriately bicultural restaurant offering "Sushi Latino." Which is to say, it's about as far removed from the clubby halls of the legislature or the plush headquarters of the Komen Foundation as possible.
For more than a year, Planned Parenthood, and women's health generally, had been the subject of withering attacks and intense controversies, but the scene inside the clinic was mundane. A television on a wall of the sun-streaked waiting room played some kind of Judge Judy variation. By eleven in the morning, the place was filled with people of all backgrounds— African, Guatemalan, Vietnamese, browns, blacks, and whites—as well as both sexes and multiple generations, not only mothers and their teenage daughters with toddlers, but mothers and their teenage sons. Almost everyone was wearing T-shirts and jeans and staring at their smart phones.
With its encouraging posters depicting happy couples and happy families, the clinic is supposed to be a cheerful place, but the atmosphere was like any doctor's office where bad news might have to be delivered about an HIV test, breast exam, or pregnancy test. And lately, the information that clinic director Maria Naranjo has to share with her patients includes the fact that, because of the drastic cuts to the family planning budget, the clinic has had to raise its fees. The tab for a wellness checkup, formerly covered by state and federal funds, now costs $133—a prohibitive amount for someone having to choose between paying that or an electric bill. She explained to me that most people think the family planning funds have just run out until the next fiscal year, something they are accustomed to. Most do not understand they are gone for the foreseeable future.
Naranjo, who has worked for Planned Parenthood and other family planning agencies for twenty-seven years, is a bustling, efficient woman with soulful eyes and a lined face. She is the child of migrant workers and was a mother at seventeen. "This is where I can do the best service," she told me. "I know where they are coming from, and I know how difficult it is."
Naranjo has established, on her own, a pay-as-you-go program to keep the clients from staying away entirely. But some do anyway. Those are the ones who keep her up at night—the young immigrant who wanted to get birth control for the first time after having her third child, and another, not yet thirty, who couldn't afford to see a doctor about the growing cancer in her breast. "She doesn't have anyone," Naranjo said of the woman, who is also an immigrant. (Every patient has to present proof of legal status.) Naranjo found a private organization willing to provide treatment, but she doesn't know for how long—or how many more she can continue to impose on their goodwill.
And, of course, there are all the teenagers who no longer have access to free birth control: they now have to come up with $94 for an initial visit and a month's supply of pills. "That's where we are seeing a higher incidence of pregnancy," Naranjo said. She tries to work her sliding scale. She offers condoms, which are cheaper than pills, and then, she said, "you cross your fingers that their partners use them. You know they are going to be sexually active, no matter what you say."
The cycle Naranjo predicts is this: the state government prevents poor women from getting affordable health care and birth control, so there will be more abortions, more Medicaid births, more expensive complications, and more illnesses caught too late. This doesn't seem like a good outcome for anyone, much less fiscal conservatives or those who oppose abortion.
"We are going backward instead of forward," Naranjo said with a pained shrug. And then, like generations of Texas women before her, she got back to work.
_Joyce Wadler_
Joyce Wadler is a New York City humorist who writes the "I Was Misinformed" column for _The New York Times_ , where she was a staff reporter for 15 years. Her previous newspaper and magazine work included New York correspondent for _The Washington Post_ , reporter for _The New York Daily News_ and the _New York Post_ , and contributing editor to _New York_ and _Rolling Stone_.
Her memoir about breast cancer, _My Breast: One Woman's Cancer Story_ , originally a two-part cover story for New York magazine, was later expanded into a book. A television movie of My Breast, written by Wadler and starring Meredith Baxter and Jamey Sheridan, was broadcast on CBS and won the American Women in Radio and Television Excellence in Programming Award in 1995.
Five years after beating breast cancer, Wadler was diagnosed with advanced ovarian cancer, leading her to write _Cured_ , _My Ovarian Cancer Story (Plucky Cancer Girl Strikes Back Book 2)_. It was first published as a two-part cover story in New York magazine.
Wadler is the author of _Liaison: The True Story of the M. Butterfly Affair_ , about Bernard Boursicot, the French diplomat who was seduced into Chinese espionage and who granted her wide access to the story behind his affair with Shi Pei Pu, a male opera singer Boursicot believed to be female.
My Breast
A heartfelt, engrossing, and unexpectedly humorous account of one woman's victorious battle with breast cancer.
I have a scar on my left breast, four inches long, that runs from the right side of my breast to just above the nipple. Nick, whom I no longer see, once said that if anyone asked, I should say I was attacked by a jealous woman. The true story, which I prefer, is that a surgeon made the cut, following a line I had drawn for him the night before. He asked me where I wanted the scar, and I had put on a black strapless bra and my favorite party dress and drawn a line in ink just below the top of the bra, a good four inches below the tumor. The surgeon took it out using a local, and when he was done, I asked to see it. It was the size of a robin's egg, with the gray brainlike matter that gives it its name: medullary cancer. It rested in the middle of a larger ball of pink-and-white breast tissue, sliced down the center like a hard-boiled egg, an onionlike layering of whitish-gray tissue about it, and I looked at it hard, trying to figure it out. We did not know it was cancer until twenty minutes later, when they had almost finished stitching me up and the pathology report came back, and then I was especially glad I had looked. Mano a mano, eyeball to eyeball. This is a modern story. Me and my cancer. I won.
Whom do I introduce first, me or my breasts? Formerly, I thought of my body as a unit, indivisible, with my breasts in some small way contributing to my notion of who I am. Now that they have shown the ability to destroy me, I regard them with new respect, thinking perhaps they deserve not only separate but higher billing. As this is a breast-cancer story, maybe they should have it.
They are, anyway, good-size breasts, and though they are fibrocystic, which means the milk-producing tissues thicken and form fluid-filled sacs, and though I have what some people claim may be other predisposing factors for cancer—menstruation at an early age, no children—I did not worry about the disease. There is no history of breast cancer in my family; I do not smoke; I go to the gym. My father, the year before my diagnosis, died of prostate cancer, but I viewed this as a separate thing. Also, because I knew it would be difficult for me to spot a malignant lump given the cystic condition of my breasts, my gynecologist always examined them, and I had regular mammograms. I had my first when I was thirty. For the past five years, I had gone to the Guttman Breast Diagnostic Institute, which had been recommended to me by my gynecologist as being as good as a private service and a whole lot cheaper. In 1986, it was $45, as opposed to $125, and if a woman couldn't afford to pay, it was free. The wait was long, but there was a cozy female camaraderie, sitting in your paper hospital shirt next to ladies of all ages and seeing how many shapes we come in. One morning, when the room was exceptionally crowded, I counted and figured out there were 140 breasts ahead of me. I had a mammogram once a year, and every year the letter I got afterward began the same:
"Dear Ms. Wadler,
"We are pleased to inform you that the results of your examination were satisfactory and within normal limits. . . . "
Who I am is a journalist, forty-four, Jewish, never married, which, as everybody in New York knows, thanks to our one million collective hours of analysis, is a whole other category than single. I was raised in the Catskills, in a boarding house, in a large, noisy, opinionated family headed by my father's mother, who, rather than leaving the Russian shtetl of Molov Guburney, brought it to America with her. It enclosed her like a capsule, the Bubbie in the Bubble; she never learned to read English and spoke to me in Yiddish, a language I did not entirely understand. I came to New York, to the Village, at seventeen and have lived here since, working for newspapers and magazines. My closest friend is Herb, a comedy writer. We hang out so much that when I am seeing somebody, we joke about how to explain about Herb. Herb's idea is that I throw a sheet over him when he is lying on the couch reading the newspaper, and after each date I pull back the sheet a little bit, and by the time it gets serious, the guy's got the picture.
By the time this story begins, last year, I had a lot of serious dates and a lot of jobs and was working as a writer at _People_ magazine. If, as research claims, tension contributes to disease, I was a good candidate: I had been working, for three years, on a book about a French espionage case, juggling six-month stays in Paris with a job in New York. Though the story, which inspired the play _M. Butterfly_ , was wonderful, Paris, when I arrived, was hard: I had two friends in the city; I did not speak French; I sometimes went entire Sundays speaking only to waiters. Soon after I returned from my first stay in France, my father died an ugly death, hooked up to a life-support machine. I was a bad fit at _People_ and always had been: I like forty inches just to say hello; the style at _People_ , which I had come to respect as one does a skill that does not come easily, was somewhere between sausage and haiku: Reduce _War and Peace_ to a snappy two-pager, and then, if Photo can't get a home take of Pierre and Natasha in the hot tub, they kill the story anyway. I was tired all the time: On weekends and evenings, I wrote my book; during the day I went to the magazine.
Also, I was in a difficult relationship. His name was Nick Di Stefano, he was a sportswriter I had known for years, and I had been seeing him, on and off, for eight months. He was Italian, which in my family is considered practically Jewish, except that (1) as children, Italians don't talk back to their parents, and (2) as adults, the men Run Around. Naturally, being so troublesome, we find them very appealing, and anyway, I had always liked Nick. He was smart; he knew all the lyrics to _The Pajama Game_ ; he dressed like a forties sharpie; he had the requisite newspaper Up-Yours Attitude toward authority. Also, there is something very nice about a relationship in which you have known each other a long time and are in the same business. We watched old movies from his collection, and he cooked and told me how much he loved his mother and took me dancing. Then he waltzed off to Miami for a weekend with an old girlfriend, and that was the end of Nick, Chapter One. She, it turned out, just wanted to be friends. Now when Nick is with me he is often petulant, seeing himself as the tragic hero of a doomed love affair, a role I have traditionally tried to reserve for myself.
"Why does it always have to be so serious with you?" he says. "Why can't we just live in the moment?"
And also, "You don't want me to work it out and decide what's right for me. You think if you give me enough time I'll get her out of my system."
"That's what you want in a woman, to be that selfless, you should be dating Mother Teresa," I say. "Why don't you call her up in Calcutta and see if she's available? From what you tell me, she's the only single woman you haven't nailed."
Then we break up and I go to bed for the weekend and lose two days out of my book.
That's where we're at, broken up, the morning I discover the lump. It is the first week in March, Monday, a crazy day at _People_. I am feeling particularly tense because I'm taking another leave of absence and have one week in which to finish my stories. I am so frantic I have canceled my mammogram at the Guttman, figuring I'll do it when my leave begins.
Then, as I'm showering, I feel it: a large, oval swelling on the upper inner part of my left breast. I have always wondered how women who discover lumps find them, but there is no missing this; it seems to be, as I move my hand around it, the size of an egg, slightly raised, sore to the touch. My breasts, since my mid-thirties, have been sore and swollen before my period, and as I've gotten older the soreness has increased—but I had my period two weeks ago. Another strange thing, this lump seems so big, and I don't remember it being there yesterday. I decide I should get it checked out, but am not very concerned. What I have heard about breast cancer is that except for a lump, it is asymptomatic; you don't have pain. I figure it's just another one of my fibrocystic lumps, which come and go. I'll call the Guttman and make that appointment for next week.
I go to work and forget about it. Then, in the afternoon, my breast starts to ache. I remember People has a staff doctor and call him. I feel a little silly about this; I am sure it is nothing, but I figure a doctor is right there in the building, so why not? He doesn't seem worried, either, until he examines me. Then his face tightens up. In the bright light of the examining room, where there is a small mirror, I see why: There is a pink flush on my breast over the lump, as if there is an inflammation, which I did not see at home. There is definitely something there, the doctor says. What it is he cannot say, but he thinks I should see a specialist. If I like, he'll be glad "to expedite it." I tell him I'm planning to go to Guttman next week. "I think it would be better if you saw somebody sooner than that," he says.
I burst into tears.
Boy, I think, I really must be strung tight today, and to him, though he hasn't mentioned the word that is now as much a presence in the room as another human being, I say, "Sorry. My father died last year of cancer."
He makes a call. An hour later, I am outside of the Time-Life Building, hailing a cab for the Upper East Side offices of a surgeon we'll call Luke. I am scared. Before I leave his office, the doctor asks if I will have health coverage during my leave, and that has added to my feeling that this is serious. I am now flip-flopping between telling myself I am overreacting and a giddy hysteria. Standing on Sixth Avenue, I have turned into Zorba the Greek. I want to _live_. The things I haven't done flash before me, a long list of "But wait, I wanna. . . ." But wait, I wanna finish my book; but wait, I wanna get married; but wait, I wanna make some money and take Nick to Paris to meet my friends; but wait, I'm just getting started. . . . I think about Nick and the time we've wasted fighting and make a deal with myself: If everything's OK, I won't worry about monogamy; I won't _hok_ him about moving in; I will make the most of every moment. As unwittingly as Newton discovered gravity, I have stumbled upon the key to making me the dream girl of every uncommitting man in Manhattan: breast cancer.
In the doctor's office, there are a dozen women. They seem older than I, and oddly, they all look alike. They look like a truck ran over their faces, I find myself thinking, which I know, as soon as it crosses my mind, is an ugly thought and not correct. Then I realize what I am looking at: fear. I have never seen so much of it sitting together. It's a good thing there's nothing wrong with me, I think. Then, as I have to wait, I go for a walk. I have already called Herb, but now I find I want to talk to Nick too. He tells me it is probably nothing and is very sweet.
"Just tell me what you want me to do, baby," he says.
The doctor, when I get in to see him, is my age, a good listener, with the kind of Waspy calm I like to see in airline pilots and other people to whom I am entrusting my life. Speaking to him, I remember something: In the past few months, in addition to soreness before my period, my breasts have been sore afterward—so much that it was uncomfortable if Nick rested his head on my chest, and I wondered if I had had a false period and was pregnant. Though I had called my gynecologist's office and asked a nurse if that was possible, it never occurred to me to make an appointment and have the doctor check my breasts—she had examined them four months before.
Now Luke examines me.
"I don't think this is anything to worry about," he says, and I feel relief rushing over me like a warm bath. "Malignancies tend to be hard, almost stony. You can't manipulate them. This you can. I'm 98 percent sure this is not malignant."
What he believes I have, Luke says, is an inflammation of some sort, perhaps a cyst. To find out, he would like to aspirate the lump: take out some liquid with a hypodermic, and send it to be analyzed. It's a painless procedure: All I'll feel is a needle prick. When a cyst is aspirated, a lot of liquid usually comes out, generally clear. It is painless, but it doesn't go as planned.
"Huh, that's odd," Luke says, and he shows me: He has been able to draw out very little liquid. What there is is thick and puslike, though that could be consistent with infection.
I get dressed. Luke tells me he still sees no reason for concern; the signs point to an inflammation, and he's prescribing Dicloxacillin, a form of penicillin. We'll try that for a week or two, and see if it reduces the swelling. If not, he will remove the lump. I am concerned: If it's a cyst, I say, how come more liquid didn't come out? And if it's not a cyst, what is it?
"I don't know," says Luke. "That's why we're doing the tests."
I go meet Nick at the Lion's Head, downtown. He's wearing his fedora low on his head and gives me that cocky Bronx grin that has always knocked me out.
"See, I knew it would be nothing," he says, and within hours we are un–broken up.
I am not a hypochondriac. I lean toward the other extreme, associating sickness with weakness and therefore denying being sick. This, I believe, is the legacy of my mother, Milly, who ran off to Florida at seventeen to paint flamingos on glass, in my childhood stole trees from state preserves insisting they were hers because her tax dollars had paid for them, and at sixty-five is still one of the great forces of nature.
"I've never been sick a day in my life," she says. "One hour after I had you, I was eating. The other women in the hospital were screaming their heads off. I made up my mind, 'How it went in, it will go out,' and that was "I've never been sick a day in my life," she says. "One hour after I had you, I was eating. The other women in the hospital were screaming their heads off. I made up my mind, 'How it went in, it will go out,' and that was 332 The Stories We Tell that. This worrying you have about every little thing, _that_ you got from your father. He was the worrier. Him and his mother. The Aspirin Addict."
Also, before going off at sixty-two as a volunteer in the Israeli army, "I don't fear death. Death to me is just another adventure. I can think of no greater honor than dying for the state of Israel, the Jewish homeland."
"You're an old dame, Ma," I say. "You think they're gonna put a machine gun in your hand and send you to the front? You're gonna be cleaning toilets."
"Don't even bother to bring back the body," she says.
I do fear death. Even more, I fear a bad death, strapped to machines in a hospital like my father. "Joyce," he had taken to telling me from the mountains, when I called once a week from Paris. "Your father is a very sick man. Your father is dying." I did not entirely believe him. I knew he was sick, very sick. I had been there for the early operations in the city and the last-minute flights to Florida. I knew the cancer was creeping up his spine and down his legs and was eventually going to kill him. But his blood counts were good, he was going to his business every day. It is a rotten thing to admit, but a voice in me, hearing him, was satirizing him; "Joyce, your father is dying"—Hebraic Dramatic Third Person, now replacing that previous family favorite, "You realize, of course, you are killing your father." He was a worrier, and critical and angry. Worrying how he and his mother and his two younger brothers would survive on a small dairy farm when he was nineteen and his father died; worrying about making a business out of nothing when he was in his thirties; worrying once he was successful it would all disappear. Then, when I got home from Paris, I saw the worrying was real: My father was sixty-seven and got up from his desk at the office like a man of eighty-five, his weight down thirty pounds, shaking, and supporting himself on a cane. Seeing me, he started to cry. "I never thought I'd see you again," he said, and I was filled with self-loathing. What the f--- was I doing in Paris all that time? I didn't even need all of that stuff. Why wasn't I here with my father? Two weeks later, he fell and broke his hip, and after that operation, his heart started to fail and they put him on life support. "You're not getting enough oxygen, Bernie," the doctor said. "Your lungs are exhausting your heart. If we don't put you on this machine, you're going to die. Do you give your consent?" My father nodded yes. Nobody in the family had any idea what life support meant, but in an hour, when they let us in to see him, we found out. An oxygen tube had been stuffed down his nose, his hands were strapped to the side of the bed, and he was pulling against the straps like an animal at auction, trying to speak but unable to because of the tube down his throat.
"We had to tie his hands to the bed because he already pulled the tube out of his nose once," one of the doctors said. "He's a little out of it now because we sedated him."
He was on the machine for two months. A few days into it, they gave him a tracheotomy so he would be more comfortable, but he could never again speak. I knew it was his life and going on the machines had been his decision, but I never changed my mind about it. I thought he would have been better off dead.
I do not, however, dwell on that memory the week of the scare. I trust Dr. Luke, and I know he's good—a friend was a patient; his reputation is excellent. I do mention the lump to my mother, who is in Florida for the winter, but I tell her I don't think it's serious, and I believe it.
That changes a little on Thursday when I talk to Luke about the test results. He tells me, in a tone indicating there is nothing to be concerned about, the results are pretty much what he expected, though there are "a number of atypical cells which could be consistent with an inflammation."
My old reporter's bell goes off.
"What do you mean atypical?" I ask. "How many cells?"
He seems a bit irritated, as if I'm worrying for no reason.
"A number," he says. "But that could happen with an infection. Keep taking the penicillin and come in in ten days, and we'll see what to do then."
The next week, I start my leave. Though _People_ and I have our problems, they have been extraordinarily good to me. This is my third leave; it includes health benefits. I am perhaps eighty pages away from finishing the book, and it is a wonderful section. My twenty-four-year-old assistant, Stefan, back in Paris, has cornered a particularly evasive source. The story is reaching its climax: My hero, Bernard, a member of the French Foreign Service, has been charged with espionage and made the terrible discovery that the woman for whom he became a spy is a man.
The only thing is, I am distracted by this thing in my chest. It's so sore I cannot sleep on my stomach. The penicillin doesn't seem to be doing much after ten days. There is a very bright light in my gym locker room, and I see how vivid and delineated the area around the lump still is.
"I'm starting to feel this thing has a life of its own," I tell Herb one night, as he's stretched out on the couch. "Like it's gonna come flying out
of my body any minute, like that thing in _Alien_ , and run around the living room and put on a sports channel and tell me to get it a beer."
I decide it's time for independent research and pull out my medical reference books. My old standby, the AMA Family Medical Guide, is not very comforting: It defines "breast abscess" as a pus-filled infected area but it says that it is uncommon and usually affects women who are breast-feeding. It says, starting to make me nervous, that a cancerous lump "may or may not be painful," occurs most often in women in their forties and fifties, and is "slightly more common in women who have never breast-fed a baby." The only good news is it is also "slightly more common" in women whose families have a history of the disease. The _Professional Guide to Diseases_ is worse. It adds white middle- and upper-class women to the higher-risk list, as well as those "who are under constant stress or undergo unusual disturbances in their home or work lives."
Eleven days after the discovery of the lump, I go back to see Dr. Luke. He examines my breast and in less than a minute makes a decision.
"This has to come out," he says.
I am not scared now—I am relieved. I don't think it's cancer—I'm too healthy for cancer—I just want this thing out of my body, the sooner the better. I'd be happy if Luke could do it right now in the office. He says that's out of the question. It will be done with a local anesthetic at a hospital and will take maybe half an hour. I ask if I can watch: I saw breast surgery when I did a story on a plastic surgeon in Beverly Hills, and I also hung out for a month at the New York City morgue, so I figure I won't be squeamish. Luke says he'll be glad to explain as he cuts but that most people do not want to watch when theirs is the body involved. I decide he's right.
I also realize I am concerned about a scar. I've never considered myself particularly vain; I have always thought of scars as a badge of honor, a sign of an enemy vanquished, but those, I now realize, were scars on other people. Luke says he can reach the lump from any number of spots—just show him where to make the cut. He books the surgery for five days later, at St. Luke's-Roosevelt Hospital Center, one of the hospitals where he has privileges. I have one last problem.
"I've got this deadline on this book," I say. "This isn't going to hang me up time-wise, is it?"
"Listen," Luke says, "this comes first. This is your life."
A few friends by now know I have a lump in my breast and am a bit worried about it, but it is Herb I ask to come to the hospital. He is not simply my best friend but a free lance, while Nick is on a staff. Also, after the initial enthusiasm that accompanies all our reconciliations, Nick is preoccupied with his own problems: an apartment he cannot sell; his unrequited romance. He also hates doctors. Who goes to the doctors? asks Nick. Women. Something is wrong, the best thing you can do is leave it alone, and it will fix itself. In my case, we don't even know that anything is wrong, so let's just quit thinking about it. I think I know what the real story is: his first wife, the mother of his twenty-two-year-old son, who developed schizophrenia in her late twenties. She was a nurse, she had some idea what was going on, and when a doctor confirmed it, she killed herself. I give up trying to talk to Nick and take a stroll by myself to Barnes & Noble, to the section where they have the medical textbooks. The most comprehensive seems to be _Breast Cancer, Conservative and Reconstructive Surgery_ , by Bohmert, Leis, and Jackson, a surgical atlas. It's $129, too much to spend if I don't even know I have a problem, but I skim it, looking at the pictures. There are a lot of women squishing a breast like they are squeezing the Charmin. I figure it's to show how lifelike reconstructions are, but it strikes me as a man's notion of what is important to a woman. I have never squeezed my breast that hard, and if a man did it I would holler. I flip through the studies. Every one seems to include a five-year survival rate. I put the book back.
I spend the night before surgery alone. Nick calls three times, asking when I am leaving for the hospital so he can call and wish me good luck. I remember I have to make my decision about the scar. I put on a bandeau bra that is the skimpiest I own and a skinny little Nicole Miller dress, deep purple, with spaghetti straps, that I wore when Nick took me dancing at the Rainbow Room. I loved that night. I had a thirties evening bag that I had got for forty francs at a flea market in Paris and a Deco rhinestone bracelet from an estate sale in New York, and as I get dressed I wonder about the women who had owned the bag and the bracelet, and where they had worn them, and if they had been as happy as I. Then I take off the dress and turn down the top of the bra a little bit and trace the edge with a ballpoint pen. As I do, I start to cry. I don't have a perfect body by model standards, my breasts are different from what they were in my twenties, but they are my breasts, it is my body, and I like it very much. Now I am making a mark that says, "Cut me."
Next morning, I talk to Nick. "Call me with the good news as soon as you get out of surgery," he says. Then I go to the hospital with Herb. In the taxi, I remember all our strange trips; Kenya, where we eyed the lions from an open Land Rover and were scared they were seeing two New York Jews and getting an urge for delicatessen; Paris, where we went looking for Jim Morrison's grave at Père-Lachaise Cemetery and had no idea where to find it until we spotted a girl with pink hair. I tell Herb what we should do is regard this as just another weird adventure.
"You sure you don't want to ask to watch the surgery, because it could be kind of interesting," I say.
"Pass," says Herb.
Roosevelt is gloomy. A group of homeless people has set up housekeeping on the Fifty-Eighth Street side, a sofa and two armchairs arranged in a traditional living-room style. Inside, the hospital needs painting. On the third-floor short-term-stay center, Herb parks himself in a reception area, while I go to a large room, which is partitioned with curtains, and change into baggy hospital clothes. Taking off my bra, I see that the line I have drawn is very low, nearly halfway down my breast. Wonderful, I think. Now the doctor is going to think I'm fast. A few minutes later, the surgical resident who will be assisting Luke drops by.
"Whoa! You can't miss that!" he says when he examines me.
Luke comes to get me. He looks very preppy, sockless in clogs, and is very sweet, putting an arm around me as we walk to the operating room. I have a feeling this is politically incorrect behavior and I am not supposed to like it, but I do. The operating team includes a male and a female nurse, as well as the resident and Luke. Seeing the line on my breast, Luke laughs.
"You've sure made this idiotproof," he says.
They paint my breast with a red-brown ointment that smells like iodine and cover the rest of my chest with sterile cloth. I can't see the surgery, because Luke has asked me to turn my face to the right, but he has promised to tell me what I will feel and what he is doing. The anesthetic is Xylocaine. He injects it around my breast, waiting for the area to numb, then makes a cut. I have a feeling of warmth and wetness. Then there are strong sensations of tugging as he pulls back tissue and starts tunneling up to the lump, in the inner upper quadrant of my breast. Sometimes I feel a bit of pain, almost a burning sensation, and he gives me more Xylocaine.
The tunneling goes on for twenty minutes, and while it is not as unpleasant as a dentist's drilling, the more tissue that is pulled apart and clamped, the more uncomfortable I become. I am having second thoughts about being so concerned about looking good in a low-cut dress. Luke tells me they've reached the lump, but they're going to go beyond it and take a margin of healthy tissue. I'm getting worried again. I don't know whether the room is cool or I'm feeling a nervous chill, but Luke seems to be cutting a lot of flesh—I know the lump is high, but I feel he is burrowing up toward my collarbone. Then I feel some final tugging and the thing is out, and I see out of the corner of my eye a metal tray and they are cauterizing blood vessels. Luke moves away from the operating table and a few minutes later comes back. It's a tumor all right, he says, sounding serious, but what sort he cannot say. He's sending it to the lab now. I tell him that before he does, I'd like to see it.
"You sure?" he says.
"Yeah," I say. He picks it up. I am astonished at how big it is. The excised flesh is the size of a tangerine and has been sliced down the middle to expose the cross-section of the tumor—that must be what Luke did when he left the table. The tumor, which is the size of a robin's egg, is grayish white, with a layer of whitish-pink tissue. Around that is what appears to be normal breast tissue, pink and white, like very fatty, coarsely ground chopped meat. Luke points out the layering around the tumor, saying it appears to be encapsulated, and that is good. I don't think any of this is good. I can't believe this big gray glob came out of me. I have a bad feeling, a sense of unreality, as if I am in a dream or a place I had no intention to be.
"How soon will we know the results?" I say, as they start stitching me up.
"About twenty minutes," Luke says.
And then, more to myself than to anyone else.
"How am I going to tell my mother?"
"Don't get yourself worked up," the male nurse says. "We don't even know that it's anything, yet," and I try to hold on to that thought. But another part of me thinks he's patronizing me; maybe they don't want to deal with a flipped-out woman on the table if they've got to stitch up her chest. I feel lonely, unable to say what I'm thinking, and scared. I concentrate on being calm. In fifteen minutes, just as they've finished bandaging me, somebody comes into the room.
"Well, it is a tumor, and it is malignant," Luke begins briskly, as if he's giving a lecture to a group of medical students. "It's what's called a medullary carcinoma; it's . . ."
I am having trouble following. Thoughts are going through my head faster than I was aware thoughts could travel: This can't be real. Is he telling me I'm going to die? Should I ask for a rabbi? No, wait, I'm not a religious Jew, I'm more like an ethnic Jew—that would be hypocritical. But maybe rabbis in hospitals are more like therapists. Why is he telling me this stuff here, where I'm alone? Wasn't that the point of bringing Herb?
I interrupt him.
"Do you think we could hold off on this until we get upstairs and you can talk to my friend too?" I say.
And, as we head to the third-floor waiting room, "I think I could use a drink."
They offer me a wheelchair, but I don't want it—it is very important for me to be on my feet. Herb is where I left him. I have been formulating the idea that it will be bad to be negative, that I'm under attack and it's got to be all systems go, but as I see Herb, I give him a thumbs down and shake my head. Luke shows us into one of the little curtained-off cubicles.
"It's, like, malignant," I say.
Herb looks dazed. We find chairs. A nurse, hearing what is going on, brings me a cup of coffee, a small act of kindness that is enormously comforting. Luke starts his talk from the top. I had remembered from my father's illness that it is important to take notes when you see the doctor, because in times of stress you do not remember all you hear. My notebook is in a locker with my clothes, but I see Herb, stunned as he is, pull his little notebook from his blazer and start writing, as if it's the old days and he's at a press conference. I feel a wave of love. He's so solid. I focus in on Luke. He is saying that they've removed a medullary cancer, which is a relatively infrequent type, with "a better than average prognosis." It was "a well-circumscribed mass," 2.8 centimeters, with seemingly clean tissue around it—he'll have more detailed results in a few days. It has been caught early; clinically, it's a stage-two cancer. Provided there is no cancer in the lymph nodes under the arm, it is "quite curable." I do not entirely believe him. I was in the room with my father when a New York specialist told him that prostate cancer was curable. Four years later, he was dead. On the other hand, this is all so weird, I don't know what to believe. I don't even know, when I say what I say next, if it is me or something I picked up from the movies. I just feel it's important to get it straight.
"Look," I say, "I have no plans of dying of this thing. That's just not how I see my life. So what's the next step?"
Luke runs through them: The next thing to do is remove some lymph nodes from under my left arm and see if the cancer has spread. That's very important, the key diagnostic tool. We also have to decide how we want to treat the breast: with lumpectomy and radiation or with mastectomy and reconstruction. Lumpectomy is removing the tumor and leaving the breast, which is what he has just done, except that he would reopen the incision to take another look. The success rates for lumpectomy and mastectomy are the same. Whichever I choose, the lymph nodes have to come out.
I have another terror besides death—general anesthesia.
"Lymph-node surgery, can it be done under a local?" I ask.
"Impossible," he says.
I remember lymph nodes. When they took a sampling from my father's groin, there was cancer in eight out of eleven. I didn't know what that meant, exactly, but I knew it was bad: The surgeon, calling Dad's room after the operation, asked to speak to me, not my mother.
"What are the chances it's in the lymph nodes?" I ask.
"Twenty to 30 percent," he says.
I'm feeling dreamlike again. I don't get it, I tell Luke. I had mammograms, I had checkups, this thing was enormous; how was it missed? He says medullary is not like other cancers—it may not calcify and can appear on a mammogram as a cyst.
"So how do we know there's not another one of these things somewhere inside me? I say.
"We don't," he says. "Your breasts are a breast surgeon's nightmare. They're large and dense and full of lumps.'
I remember the pictures in the medical book. The real ones are gonna be this dangerous, let them make me a fake.
"Take it off," I say.
He explains a bit about breast reconstruction. I had assumed it was like the breast-enlargement surgery I had seen in Los Angeles, an operation in which the doctor put a silicone implant under the muscles or tissue of the breast and the patient woke up with a new breast—except that in the case of cancer, you would remove the breast tissue first. Luke says it is not that simple: You don't wake up with a new breast; they put an expander in your chest; it takes a few months. If I go with lumpectomy, in which radiation is required, it takes six weeks. Lymph-node removal involves three or four days in the hospital. There are no shortcuts—this is cancer. I do not have to decide about mastectomy or lumpectomy in a few days, but I should have surgery within four weeks. Whichever procedure I choose, lymph-node removal will be done at the same time. After that, there's a good chance I'll need chemotherapy for six months. Meanwhile, they'll be doing more tests on the tumor: DNA analysis, hormone receptors. I can take the cotton dressing off my breast tomorrow and come to his office Friday; he'll have those results and take out the stitches. I'm having trouble assimilating all this; so is Herb. We're two liberal-arts guys suddenly thrown into Columbia Medical School. I'm still back with the idea that reconstruction is a long-term process and I may be walking around lopsided for five months. Luke recommends _Dr. Susan Love's Breast Book_.
I have one more question. I am afraid to ask it, but I have to anyway.
"What am I looking at here?" I ask. "Statistically?"
He isn't any happier answering than I am asking. He doesn't care much for statistics, he says. You can still have a cancer that has a high cure rate, and if you're in the percentage that is not cured, it doesn't matter. In my case, I have a cancer that has a favorable prognosis and is "more curable than average."
I need something harder.
"When my father was diagnosed with prostate cancer, it was something like a 60 percent survival rate at four years, a 40 percent survival rate at seven years," I say.
"I would say the statistics, in your case, are considerably better than that," he says.
"How much better?" I ask.
"For breast cancer, the overall cure rate is 70 percent. For medullary, it's above that. I would say 80 percent, 90 percent."
I feel better. I like these odds. I don't entirely believe them, but I like them. This leaves me with one immediate problem: how to tell my mother. Herb has the solution: "lead with the positive." I find my notebook, and
we work out the lead and phone it in. The last time I did this, I remember, I was filing a breaking story for _The Washington Post_ on a Concerto for Piano and Dog at Carnegie Hall. The dog had stage fright, which was good for me, as it gave me a new top. There is a reason people hate reporters. The phone rings, and I begin the performance.
"Well, Ma, I'm out of surgery, and I'm here at the hospital and everything went great," I say.
"Oh, thank God, I'm so relieved, I don't know what to say, I was so nervous I couldn't sit still, my friends called, I told everybody, 'Get off, get off, I can't talk, my daughter is right now this minute having surgery in New York. . . .'"
I break through the wall of words, power-talking, a skill I developed from forty-three years of training with champions.
"The lump turned out to be malignant, but it's the best kind you can have," I say. "It's called medullary, it tends not to spread, they seem to have got it all. It was in one lump, I saw it, it looks like it was encapsulated, that's a good sign."
Silence. She believes me like I believe the doctors.
"I'm coming north," she says.
I tell her she is staying put, I probably won't be having surgery for at least a month, and hit her with all the other positive stuff I can think of. This cancer is very rare, hardly anybody gets it, and it has a very, very good prognosis. Yeah, it was big, but this kind grows very fast, and the doctor says we caught it early. The longer I talk, the harder it is. I am hearing my mother and my dead grandmothers and all the aunts in the family. "The worst thing in the world that can happen, the very worst thing, is for a parent to survive a child." they are saying.
"Talk to Herb, Ma," I say and walk down the hall.Then I call Nick. Most of our relationship, I've wanted him to be more expressive. Often, when we are together, he withdraws and watches two or three old movies in a row—if he doesn't, he says, he'll think about his life, which he can't bear. In the morning, he moves the television so that he can watch _Lucy_ reruns from the shower. Right now, however, I have this feeling that if he falls apart I will fall apart, and I need him to be strong.
"I'm going to tell you something, and I don't want you to get emotional, because it's going to sound worse than it probably is," I say.
I have the feeling, at the other end of the line, of a man who has been slugged in the stomach.
"You just got to give me a minute. I wasn't expecting this," he says.
Then Herb and I head downtown. Normally, a glass of wine puts me to sleep. Now we ago to the back room of the Lion's Head, where we are known as The Ones Who Only Eat, and I order a margarita. I get a second one. Then I talk tactics. The position I am taking, I say, is not that I have cancer, but that I _had_ a cancer and they cut it out. I am not doing an avoidance number, we will research the hell out of this and get the best people in the business, but until it is established otherwise, I consider myself healthy. I go tottering off to my place. I am not sure whether the sense of unreality is coming from the news I have received or the drinks. The Xylocaine is wearing off, and with every step, even in a bra and bandages, my breast bounces and hurts. Luke had offered me a prescription of Tylenol 3, but I'm a little afraid of drugs, and I didn't think I needed it. Now I see I do. I call up the pharmacy to have the drug sent over. Even with the Tylenol my breast feels as if someone has stabbed me. I know I should talk to my brothers, who by this time have probably had thirteen conversations with Ma, but I am too tired. I go to bed, exhausted, wanting to be taken care of. I think of my grandmother Wadler, round, warm, and cushiony, the one member of the family who thought I was perfect just as I was, and wish she were still around. I think about Nick, who has said he will get out of work as soon as he can and pick up supper, and wonder what is keeping him. He calls, eventually, from the street near his bar. The bank must have messed up, he says, he can't get any money from the machine, he's got maybe three dollars. I go to meet him at Balducci's, bumping into Sigmund Freud on the way. "You understand the message he is sending you," Freud says. "You vill not depend on him for nossing." I banish him from my consciousness by taking him to the deli department and giving him a number and telling him to pick up some derma. There is no derma in Balducci's. By the time Freud figures it out, I'll have lost him.
When Nick and I get back to my place, just on a point of pride, I set the record straight.
"I'm still the same person I was yesterday," I tell him. "If we break up every three weeks, we break up every three weeks. I don't want you to treat me any differently."
Which, as it turns out, is the stupidest thing I will say in the course of this whole illness.
And also, as far as Nick is concerned, the least necessary.
If this were ancient Egypt, and people were buried with the things they used most often, the executors of my estate would have no problem making a decision: They would plant me with a phone in one hand and a Diet Pepsi in the other, and if it turned out there was life after death, I would be on the phone, talking to one of my girlfriends or having an emergency session with my shrink. It being late when I am prone to anxiety attacks, I would probably reach a machine:
"It's Joyce. It isn't a question of life and death—well, actually, it is, but I mean I can handle it—um, anyway, this death thing has turned out to be a little more stressful than I thought, and if you have some time, can you give me a call? If it's not inconvenient. Otherwise, I'll see you at the regular time Thursday. One good thing about this, you won't have any trouble getting me to lie down."
But when I get a diagnosis of cancer, it changes. It isn't just that I am numb from the news and the surgery. It isn't even that I need to be alone to sort this stuff out. I have spent a lifetime sorting things out with my friends. But now, I feel, I am under serious attack, and when the Scud missiles are raining on your head, you don't have time to get on the phone with your girlfriends and say you are terribly depressed. Also, there is something else—I am afraid of negativity. _Cancer_ is a scary word; people hear it and think "death," and I don't want that sort of energy around me. I also don't want to hear, however well-meaning, other people's stories. Until now, I thought breast cancer was breast cancer. I had no idea there were different kinds, some more dangerous than others. I also realize that everyone's body is different. I love my friends, I want their support, but hearing a story about a friend of a friend who "had it" and is now doing fine will be a waste of my time—what I need is hard facts about medullary and information about the options. I'll tell some close friends the diagnosis, but they have to keep it to themselves. Just on a professional level, I don't want this around. Journalists are the biggest gossips in the world and the least reliable—one lunch at Orso, and three hours later word will be all over town that Wadler is dying, and I'll never get another book. I'm also making a rule: Information goes out, but unless I ask, it doesn't come in. Herb and I also ask friends to let me call them. If they want to know the details of what's happening medically, they can call Herb for briefings. Herb calls them Breast Conferences.
Wednesday, the day after the surgery, I get organized. I have an advantage: I am a reporter, and so are a lot of my friends. I call up two or three and give them a task: Herb looks for Dr. Love's book and checks on Luke's credentials (they're excellent); Heidi, a magazine editor I have known for twenty years, will call the American Cancer Society and the National Institutes of Health; Max, who is the bureau chief for an out-of-town paper, will call his contacts: we'll all get names for second opinions. There is no way, with a life-threatening disease, I am not getting a second opinion. The reference I trust most comes from an old friend who is a doctor and researcher. "You'll go to Jeanne Petrek at Sloan-Kettering for the surgery, Norton as the oncologist. He's the head of the breast-cancer department at Sloan, very sharp. He's a friend, our wives are friends. Make the call and tell him I sent you. No, wait, I'll make the call myself."
I've got other problems, too: my job and my book. My publisher has paid a bundle for this story—"Let's face it, now they own you," Ma had said when I signed the contract, and I have no idea how long this breast business is going to hang me up. I have the same concern about People. Neither is a problem. The publisher tells me to concentrate on my health. People editor Lanny Jones changes my unpaid literary leave to a medical leave— on full pay—and says the resources of the company are behind me. Within days, I've got four people from Medical calling me with the names of cancer support groups and specialists. It's a relief. But I wonder, What happens to poor women in New York who don't have medical insurance, and don't have families that can help them, and don't have friends to get them to the head of the department at Sloan-Kettering?
But I have another ongoing problem closer to home: Ma. They have an interesting way of dealing with illness in my family. They form little whispering cabals, deciding who can "take it." Or, if they must deliver bad news, they hit you in a roundabout way. "You know your uncle Murray, in the hospital in Kingston, he's not doing very well," my aunt Shirley had told me, in a phone conversation years ago. Then she asked to speak to my boyfriend. A few minutes later, he passed back the phone. "Actually," said Shirley, "he's dead."
I have never understood this, but now I do: You don't tell the people you love, because you want to protect them. But in doing that, you cut yourself off. I talk to Nick about it. He says mothers are stronger than you think, and anyway, I owe my family the full story. The day after the biopsy, I call her.
"I figured you might be worrying, and I was just wondering if you had any questions," I say.
"Yeah," she says. "What aren't you telling me?"
Trick question. Damn, these mothers are smart. I tell her there is a small possibility "it" may be in the lymph nodes, but if it is, it's not the end of the road. I say because I am concerned another lump might one day be missed, I am leaning toward mastectomy and reconstruction, but that might not be so bad—it would be fun to be able to wear cute little camisoles, and maybe, at forty-three, I could use a perkier pair.
She's scared. I can tell because she hits me with Second-Generation Wadler Cure-All One:
"You know money is not an issue."
"I know that, Ma," I tell her. "It's OK. I got insurance."
"New underwear, anything cosmetic, that's on me," she says.
"Well, I don't know, Ma," I say. "My bras are very expensive. I don't know if a poor old widow like you can afford them."
"Thirty-four B is a good size," she says. "I'll bring cash. I'll put a thousand in your account." She starts upping the amount, bargaining with some unseen force. "Three. No, five. Six. For the things that aren't covered by insurance. Taxis for back and forth to the hospital. New underwear. A wig." I'm suddenly peeved.
"What makes you think I'm gonna need a wig?" I ask her. "I didn't say anything about chemotherapy. I'm healthy. I _had_ cancer. I'm just giving you some remote possibilities, because you asked. Anyway, that stuff about chemotherapy has changed—not everybody loses their hair."
"A blonde one," she says. "On me."
This is another strange thing about breast cancer: Though I have just been told I have a life-threatening disease, it's not like a cold or the flu, where you feel sick. Physically, the day after the biopsy, I feel as strong as I've ever been. My breast aches, but only mildly, and I can take care of it with the Tylenol. I can't see the cut on my breast when I take off the cotton pads, because it's covered with a row of fancy bandages, but my left breast, despite the amount of tissue that's been removed, looks the same size as the right, and somehow I knew it would. Medically, however, we're all still very confused. Herb is having trouble finding Dr. Love's book; NIH doesn't know of any medullary experts. Also, we don't understand why you would do a mastectomy at the same time as the lymph-node surgery. If the lymphnode surgery is to see if the cancer has spread, wouldn't you do that first? If it has spread, why take off a breast?
I go back to Barnes & Noble. They don't have Dr. Love's book, either, but they do have my old pal, _Breast Cancer, Conservative and Reconstructive Surgery_. I plunk down the $129 and get it. I also pick up _The Pill Book: The Illustrated Guide to the Most Prescribed Drugs in the United States_ , one or two paperbacks on breast cancer, and a book by Norman Cousins, the former editor of _Saturday Review_ magazine: _Heard First: The Biology of Hope and the Healing Power of the Human Spirit_. I remember hearing about Cousins's work a few years ago; he had a serious illness and cured himself by laughing. Thursday evening, before going to Luke's, I start reading the medical books. What they say is a lot stronger than what Luke said:
Cancers are classified in stages, depending on size, whether they are in the nodes, and whether they have spread to other parts of the body. There are four stages and stage two is not that great: According to one study, the five-year survival rate is 65 percent. Medullary is rare, accounting for perhaps 7 percent of breast cancers, but it can spread, and if it does, it can kill you. The worst kind of breast cancer, accounting for perhaps 2 percent, is inflammatory. The skin is flushed and has a _peau d'orange_ texture—exactly what I saw the day my lump was discovered. Very few people live beyond five years with inflammatory cancer. I am petrified. I don't care that the lab reports have classified my cancer as medullary. What if they made a mistake? And even if it's only medullary, these statistics are hell. I call up Nick, convinced I am doomed. "You're driving yourself crazy," he says. "What do you care what some book says? Maybe it's out of date. Your doctor says you have the best kind." I am not interested in anything Nick has to say. I just want to be next to him in bed and hold him.
At nine the next morning, I meet Herb at Dr. Luke's. My breast, when Luke pulls out the sutures, has a thick pink scar, but I think I heal great. The problem is, I'm so frightened by what I've read in the medical books, I'm almost stuttering. When I tell Luke about my research, he is not happy. He knows some patients do this—lawyers, usually—but it's not a great thing to do if you're not a doctor because you can easily misinterpret things. I do not have inflammatory cancer. On the basis of size I have a stage-two cancer, but medullary is not the average breast cancer. I have, he repeats, a very favorable case.
Herb wants to know the statistics on recurrence. Luke says with lumpectomy, it's 15 or 20 percent; with mastectomy, it's down to 4 percent. My chances of getting cancer in the other breast is higher than other people's, 7 percent for the next ten years, but Luke does not recommend a prophylactic mastectomy. He's sending me for a mammogram, but he sees no indication of trouble in my right breast.
I want to know about reconstruction. Luke says at the time of the lymph-node surgery, he'd remove the breast tissue, leave most of the skin, but remove the nipple—it's safer, because in one out of four times, the cancer is in the nipple. Then a plastic surgeon puts in an implant and constructs a new breast. It will look good, he says, but it won't _feel_ like a breast. It's an artificial implant. I try to imagine what it will feel like. A contact lens which at first you are always aware of, then never feel? A football?
I am lost. I ask, since the current rate is the same with mastectomy and lumpectomy, what the doctor recommends.
"I think mastectomy is the better treatment for you," Luke says. "You've got difficult breasts, large, lumpy, and you're worried about recurrence. Lumpectomy is for people who say, 'I don't want to lose the breast no matter what.' That wasn't your response. The only advantage of lumpectomy is that it preserves the breast. But it's your decision."
It is true, I think, that my first reaction was "Take off the breast"— but that was before I knew what reconstruction involved. Now I'm not certain. I ask Luke if, aside from statistics and my case, he has a personal bias. He says he has had three medullary patients, and since one had a recurrence, he leans toward reconstruction. He also says that since he took so much tissue out of the breast, reconstruction will probably give me the better cosmetic result. I tell him I don't think that will be a problem; my breasts are still the same size.
"That's swelling from the surgery, and some pockets of air," he says. "When it goes down, it may be much smaller."
He suggests I talk to a plastic surgeon—there is one he thinks would be temperamentally suited for me, because he's an artist and a doctor. I take this to mean that Luke is clarifying me as a patient who is not so stable and is likely to cut off an ear or that he has been influenced by Herb's beard, but I'm happy to be seeing the artist-doctor. Maybe when we get to his office, he'll offer us an espresso. I haven't had any breakfast. I could use it.
My mammogram, which we have taken across the street, is normal, except for what the report calls "a large radiolucency, in the left breast, consistent with residual air." Apparently, Luke is right: My nice plump breast is pumped up like a Macy's balloon and may deflate at any moment.
Then, in what's turning into a cancer triathlon, Herb and I rush to the office of the plastic surgeon, Dr. Frank Veteran, in the Eighties, off Fifth. I'm a little worried about Herb. He's the sort of man who feels uncomfortable in the lingerie department at Saks; I'm remembering photos of mastectomy from my medical books and wondering how graphic this consultation is going to get. But at the same time I'm excited. In the taxi, I have come up with a wonderful idea: Rather than mastectomy, why not, after treating my breast with radiation, do a reduction? If I get rid of, say, 30 percent of breast, I remove 30 percent of potentially dangerous, cancer- bearing tissue. I won't have to run around nipple less or with a football in my chest. I could also end up with a very pretty pair of breasts. I do like my body, basically; there are times I look at myself naked and think I'm gorgeous, but as I've gotten older, or have seen skinny women with high little breasts at the gym, I have sometimes felt bad, looking at my sag, and wondered what it would be like to have a lift. I like Dr. Veteran, too. He's not slick; there's an air about him that suggests he has had personal experience with serious illness. Now, after Dr. Veteran examines me, I hit him with my idea. It's original, all right—Veteran doesn't know of anyone who's done it—but he also says it's not a good idea. Radiated tissue is difficult to work with: Some of the smaller blood vessels are destroyed; it doesn't heal as well as normal skin. If one must operate on radiated tissue, one does, but he would prefer not to. Doing the reduction before the radiation is not a good idea, either. Reduction is a major surgery; it takes time to heal, and that could delay radiation treatment. This is cancer; the medical considerations have to come first. My skin is good; I'm young; I can get "a very good cosmetic result" with reconstruction. The words "very good cosmetic result" disturb me. Is it a suggestion that I really could use a new pair? Is he saying, in a roundabout way, that what he has seen is awful? I have a sudden image of Joan Rivers, in an off-camera booth, feeding the surgeon lines. "She takes off her bra," she says, "she could nurse SoHo."
He explains reconstruction: At the time of the mastectomy, he would put an expander, made of silicone, under the muscles of my chest. You couldn't put it directly under the skin, as you would with breast augmentation, because all the breast tissue is gone, and there would be nothing to serve as a cushion between the implant and the skin. Over a four-to-sixweek period, a saline solution would be injected into the implant, enlarging it. The muscles on top of the implant would stretch, as in pregnancy, but, as in pregnancy, you couldn't stretch them all at once. After two or three months, after the tissue around the implant has "settled down," there would be a second operation and a permanent prosthesis would replace the expander. If you're having chemotherapy, you have to wait longer because chemotherapy usually brings down the white-blood-cell count, increasing the risk of infection. Finally, in the case of large-breasted women like myself, there would be a third operation, a reduction of the healthy breast, to make it match the first and to build a nipple for the reconstructed breast.
To somebody who is terrified of general anesthesia, this is awful—I'm looking at three extra operations, not one. Then the doctor shows us the pictures and it's worse: a color Polaroid of a woman whose breast looks like a halved grapefruit. The shape is perfectly round; a thick red scar runs from one side to the other; the woman has no nipple and no areola.
"I can't walk around for six months looking like that," I say. "It's like a nuclear catastrophe."
Then I feel terrible.
"What I mean is, it just sort of throws me, the idea of walking around like that with no nipple. I'm sure when it's all finished it looks really nice," I say.
The doctor shows us more Polaroids, including women with their finished breasts, who look much better. He says there are implants he can use for a more natural look, but while they have the same texture, all breast tissue is different, so my breast may not feel the same. His patients say, however, that after a while they are not aware of the implant—it just becomes their breast. He's a lovely guy. It's running on two-thirty; we were booked at the last minute, and I'm sure he hasn't had any lunch, but he acts like he has all the time in the world.
We leave. I know now, I tell Herb, I don't want reconstruction. There is no way I am going to do those things to my body.
I also realize something else.
"You know how we're always saying we miss things," I say. "We weren't around for Paris in the twenties; we weren't reporters in New York in the forties; I had tickets to Woodstock, too much mud, I didn't go. It just hit me: All these stories about breast cancer—for this trend I'm right on time."
A week after I am diagnosed with breast cancer, I have a nightmare: I am in France with Stefan, my twenty-three-year-old assistant and translator on my book, and we are driving in my father's car, a big Chevy, in a part of the countryside that is new to us. The people are suspicious and unfriendly; the terrain is strange: encroaching on the road, perilously close, are ditches of water and irregularly shaped little lakes, some covered with ice. I don't know whether Stefan or I am driving, I only know that we have to be careful. Then, suddenly, we are on a lake. At first we are OK, driving on top of the water, and then we start to sink. I am trying to figure out how to get out and still save my father's car, which is fairly new and expensive. Should we roll the windows shut, so the car doesn't sink as fast and maybe even will float? But if we do that and the doors stick, we'll be trapped, so maybe we should just forget the car and get out. Somehow, as the car sinks, we escape, but I am left in a strange country without anything. I am supposed to be taking care of Stefan—he is younger than I, and my responsibility—but suddenly I have no money, I am more dependent than ever on Stef in this strange territory where I do not speak the language, and I am thinking that my father's car is at the bottom of this lake, and he will blame me. Then I wake up, and instead of reality being better, it is worse:
Oh, s---, I have cancer. Excuse me, that is a negative thought, cancel it. I had cancer. I had a cancer in my breast. Now it is out.
I am trying to be exceptionally positive because some of the cancer books I've been collecting claim that cancer cells are always popping up in people; on average you may "get" cancer as often as six times a year, but a healthy immune system kills the cells. Some writers, including Norman Cousins, have this theory, which normally I would consider crackpot, that stress weakens the immune system, while positive energy and laughter create, uh, endorphins or something that helps the immune system battle the disease—so don't worry, be happy. But how can you be happy when, according to some doctors, by the time a cancerous lump is big enough to be felt, around one centimeter, it contains one hundred billion cells and has been in your body ten years, and chances are there are millions of other cancer cells sloshing through your bloodstream, looking to own, not rent?
"Yes, we were very happy in Miss Wadler's left breast for a long time. It was a wonderful space—you've seen it yourself, in a T-shirt—I mean, it's enormous. But now the family is growing, and we're looking for something with a little more air and light—like maybe her lungs and the top of her head."
If there's a chance positive thinking can work, I'll try it, but I am skeptical. If my being a worrier has contributed to this disease, am I going to be able to change my outlook fast enough to stop it? I've had fifteen years of therapy trying to get married, and I still haven't been able to pull off the one space in journalism I really want: the _New York Times_ wedding page.
There is also another problem: I am dealing with a growing group of specialists— breast surgeon, plastic surgeon, oncologist, radiologist. Each puts his or her field first and is somewhat ignorant, at times even disdainful, of the others.
The doctor who removed the malignant lump from my breast, Dr. Luke, has told me that the cure rates for lumpectomy and mastectomy are the same. He stresses that the decision is mine and my prognosis is good, because medullary cancer—a rare type, which accounts for 7 percent of breast cancers—tends not to spread quickly. His personal feeling, however, is that it may be safer for me to have a mastectomy and reconstruction. My breasts, he said, are a surgeon's nightmare: They are dense, large, and full of lumps, one of which was already missed on a mammogram. Also, he has had three medullary-cancer patients, and one had a recurrence. I have come up with my own ideas about treatment: If my breasts are full of potentially dangerous tissue, why not reduce the danger by reducing the size of the breasts? Instead of lumpectomy followed by radiation, the common procedure, why not do a lumpectomy and breast reduction and then have radiation? The plastic surgeon Dr. Luke sends me to see, Dr. Frank Veteran, tells me that he knows of no one who has done this, and that it is a bad idea. Reduction surgery might delay the time at which my breast could be treated with radiation. Irradiating the breast first and then doing a reduction won't work, either—radiated tissue is difficult to work with. Dr. Luke, when I speak with him later, is not aware of this problem—he suggests radiation treatment, then a reduction. He also does not mention that if I have a lumpectomy and radiation and the cancer comes back, I will not be able to have another radiation treatment—the breast can only take so much. If I decide I want to have a mastectomy then, reconstruction will be an extremely complicated procedure in which muscle and skin have been transferred from my stomach or back. Even if I give up my idea of a reduction, there will be three to four people involved.
"I understand that I don't see the oncologist until after the lymphnode surgery, and the radiation doctor comes in after that," I tell a nurse at Memorial Sloan-Kettering, when I make an appointment to see surgeon Jeanne Petrek for a second opinion. "But who's in charge?"
"There will be a group of doctors, who will each be monitoring you at various steps along the way," the nurse says cheerfully, "but as far as who's in charge, in a sense you are."
In a sense, that's good—I should be the one to have the final say about what happens to my body. In another, it's terrifying. I know nothing about biology or chemistry or science; I'm one of those people who still aren't sure why planes stay up in the air. What this disease needs, I decide, is a contractor—or at least one room where we could get all the doctors together.
On the positive side, it turns out that while the United States is lousy on health care, we are a great nation for booklets. Call 1-800-4-CANCER, the Cancer Information Service at the National Cancer Institute, and you can speak to a cancer expert, get fact sheets, and receive free pamphlets on everything from treatment to psychological support. Their Physician Data Query service, which is also free, sends information on cancer treatment to patients and doctors, so that even if you live in Podunk, you can still be on top of the research. I also get a lot of information from the American Cancer Society's Cancer Response System, at 1-800-ACS-2345. According to its fact sheet—which, like NCI's, is reviewed and updated frequently— medullary cancer (which I now think of as my brand) seems to grow in a capsule within the duct, and although it can become quite large, "it does not metastasize as frequently as others and has a better outlook." For the first time since the diagnosis, I have a strong sense of relief: This is not just one doctor trying to snow me. The best overview of breast cancer and therapy options comes from _Dr. Susan Love's Breast Book_ , which Dr. Luke, the surgeon who removed the tumor and made the diagnosis, recommended. I wish I could have found it earlier.
Meanwhile, I'm beat. I run around town, picking up medical records and my past four years of mammograms. The mammograms are important to me. I'm hoping someone will be able to look at them and spot the tumor that was missed a year ago, when I had my last, and use it to find any other dangerous lumps. I change my mind daily on whether to have reconstruction or lumpectomy, mostly because Dr. Veteran, who has told me reconstruction involves three operations and knows I am afraid of surgery, called me with a way to reduce the number of operations from three to two: rather than use an expander that will later be replaced with an implant, he can use an expander that will remain inside the body. It sounds good, but the shape my breast will have if we use this device—"grapefruit," according to Dr. Veteran—does not. My breasts may not be saluting the sun, but they have their charm, and I'm not about to replace them with a set of citrus. I can't believe that I'm the only woman in America who's had the idea of treating cancer with lumpectomy and reduction, either. I continue reporting. I call the American College of Radiology, the American Society of Plastic and Reconstructive Surgeons. I hear of no one who has done reduction and lumpectomy to treat and prevent the disease.
I read. Though I returned my expensive breast-reconstruction text after talking to Luke, discarding research, as he has suggested, is an idea I've discarded. It's my body and my life. I read about drugs, I read about nutrition, I read about alternative therapies like visualization, where you picture cancer cells in your body and tell them to drop dead. Searching for an expert on medullary cancer, I call SHARE, a support-and-information group. They know of no one, but the volunteer I speak with does, when I ask, give me a lot of positive case histories involving women who have had cancer in the lymph nodes. My big fear, which I try not to think about, is that the cancer will be in the lymph nodes. Finally, though, our conversation, in which we discuss the possible side effects of chemotherapy—including hair loss, mouth sores, early menopause, and possibly vaginal infections— disturbs me. I understand that people with a common experience can offer a kind of special support, but at the same time, I see a club to which I do not want to belong and yet have been drafted, a fellowship of sick people. I make another decision: Rather than go to strangers, I will ask my friends to get Dr. Love's book so we will all be able to speak Cancer. If the going gets really rough psychologically, I will have a talk with the two friends who have had run-ins with the disease. I would, however, prefer not to. They both had tumors the size of peas. It's making me nervous to be the girl with the biggest one on the block.
I'm also having trouble with Nick.
"What's the good news?" he asks whenever I come back from the doctor.
When he hears rough news—the possibility of an ugly reconstruction that could go on for months, the possible problems with chemo—he brushes it off.
"The doctor says you have the best kind," he keeps saying. "You've got nothing to worry about. I wish I could exchange my financial problems for your medical problems."
I think part of this is my fault. After all, the day I was diagnosed, I told him that we had to be positive. I also know this is his way of keeping me from being overcome by terror.
"I could have a very strange look for a few months," I tell Nick one night before bed. "Scars, no nipple. . . ."
"So what's a few months?" he says. "When it's over, you have a great new pair. Maybe better."
"I thought you liked these," I say.
"I _love_ them," he says. "But you know what they say—variety is the spice of life."
Meanwhile, his demand that I lead with "good news" is making me feel that he just doesn't want to hear the bad. One day, I blow up.
"There is no good news," I say to him. "This is cancer. I could lose a breast; I could die; I could be spending the summer with a hole in my chest. If you want the good news, get yourself a twenty-four-year-old California girl. With no health problems."
Another time, I get him together with Herb.
"Could you explain to Nick this is a difficult, life-threatening disease?" I say.
"It's a difficult, life-threatening disease," says Herb, who feels that a man who spends as much time worrying about a tan as Nick cannot be taken seriously.
"Could you also say that I'm scared, especially of general anesthesia?" I say.
"What's to be scared?" says Nick. "You die on the table, you never know what hit you. I keep telling you, it's the best way to go."
"You're just saying that because you're a guy and you don't like to admit to fear of dying," I say.
"My mother isn't a guy; she wouldn't be scared," Nick says.
"Your mother is seventy," I tell him.
"Eighty," says Nick. "But she wouldn't be scared if she was forty. If she was twenty, she still wouldn't be scared."
"What is it with Italian guys and their mothers?" Herb says. "Jewish guys insult their mothers and make jokes about them. Say anything to an Italian guy about his mother and he's ready to get in a fight."
"Why would anyone say anything about my mother?" says Nick. "She's an incredible woman."
I give up and go back to research. The week before Easter, I go to Roosevelt, where they took out the lump, to pick up my biopsy slides for the new doctors. The slides are the most important medical records I have— the only specimens of my cancer—and they will be critical in determining my treatment.
Roosevelt, when I return, is even more depressing than I'd first found it. The street people who had set up housekeeping on Fifty-Eighth Street are still there in their sofa and armchairs and seem to have settled in for the long haul. Their possessions are about them in garbage bags; the sidewalk is littered with dirty fast-food containers—just what you want to see next to a hospital. They are very nice, though, complimenting me on my fedora and waving like we are old friends. I wave back. For all I know, by the time I come back for surgery, they'll be living in the waiting room. Maybe one will have a fellowship, and I'll look up in OR and he'll be there, assisting.
The pathology department, where I have to pick up my slides, is on the ground floor, behind a door that opens onto a reception desk and a rat's run of intersecting halls. I am stumbling through them, searching for pathology, when suddenly I am hit by the smell. It is a smell I will never forget: sweet and rotting, with an overlay of formaldehyde. I see the sign just as I am on top of it: MORGUE. The door is opening, and I feel like I have suddenly split into two people, one of whom is physically turning the other's face away, so she cannot see. The corridors are like a maze in a nightmare, and although I don't believe in fate and don't believe in folk stories, I feel this is a precursor: Death has seen my face near the door of the morgue, and if I come back to this hospital for surgery, he will recognize me and think that's where I belong. If I come back to Roosevelt, they will wheel me into surgery and wheel me out to a refrigerated compartment. I manage to find the pathology department, which is, as one would expect it to be, next door, but I am now so frightened I am gulping air. A lab technician gives me a three-by-four-inch padded envelope, which she says includes my biopsy slides and the medical report. I sign for them, then beat it out for the door, looking for a phone. There is an old-fashioned one in a wooden booth. I duck into it, shutting the door behind me, crying and hiding. Hide from the Angel of Death. I call Herb. The machine answers. I call Nick.
"Oh, Jesus, you almost gave me a heart attack. I thought something had happened; you just got scared," he says.
"But something did happen," I want to tell him. "I felt my death foretold." But he is at work, and I am not a crazy person, so we just talk for a while, and I leave.
I don't know what to do. On one hand, I think I should get away from this hospital and this ugly, depressing neighborhood and go to my house and have a nice, hot bath and get sane. On the other hand, I know that when I am not dealing with cancer, I like this neighborhood a lot. It has great food stores and demented, high-energy street life and, most important, a lot of thrift shops I usually don't get to. I once got _Bells Are Ringing_ around here for ninety-nine cents. I love show tunes. I took a bunch with me to Paris, and when I got scared and had to do an interview with a fancy diplomat, I used to psych myself up with _Chicago_ or _Guys and Dolls_. By the end of my first hitch, my French was good enough that I could sing the first verse of "I got the horse right here" to my newsstand dealer, who always had his nose stuck in the racing form. If I stick around and hit the thrift shops, I decide, it will be much more life-affirming than panicking. Also, maybe I'll be able to find a new _Pal Joey_.
Then, walking down Ninth Avenue, I realize what I'm holding in my hand: my cancer. It seems sort of creepy, going shopping with it. If I go to the Forty-Sixth Street Salvation Army, my favorite thrift shop, will they make me check it? I could put it in my bag, but if my purse is snatched, I'll lose it. On the other hand, if somebody sees me carrying an envelope, they could think it holds something valuable and snatch that. It's the most important thing I own; if somebody steals it, what will I do? Put signs on the trees like they do in my neighborhood when somebody loses their cat? "Cancer missing; last seen in small brown envelope, vic. Ninth Avenue and Fifty-Fifth. Reward—no questions asked." Then, if I get it back, the Daily News will do a story like they do when a Taiwanese music student leaves his cello in a taxi and somebody returns it. They'll run a picture of me clutching my cancer, and I'll have to look grateful and teary-eyed, while all across town cabbies will be making fun of me. "Putz. Lookit this. She's carrying her cancer in her hand, and what does she do? She goes shopping. Just like a woman." I do go shopping, just to spite them, but it is unsatisfying; I stuff my cancer into my purse so I don't have to check it, and make an incredibly bad buy: a pair of navy-and-white Ferragamo flats a size too large. When you buy shoes you know do not fit, it is time to go home. When I do, I find the package with my slides unnerving me; I have a sense of them powerful and glowing, like Kryptonite, sending killer rays around the apartment. I don't know where to put them. The flowered hatboxes, where I keep my fancy underwear? The drawer with my foreign currency and passport? I'm also still shook up by the trip to the morgue. My shrink has said if I need her, call her. I do. She says she has one minute; her group is coming in. "I'm not sure I can do this in a minute," I say, but I try: my sense of death foretold; my father diagnosed with cancer at exactly this season five years ago; my fear that I will die like he did, my bones eroded by cancer, a tube down my throat. "You're not your father," she says. "This is all in your mind. You're doing this to yourself. Snap out of it." Wonderful. I get cancer—excuse me, I had cancer—and my shrink turns into Cher in _Moonstruck_. I call up my friend Heidi, who's less pressed for time. She understands why I am upset: It is insensitive and appalling that a hospital would make one pick up records next to the morgue, she says. I feel better. When I get off the phone, I decide I will demystify the cancer pack. I saw it whole, before I knew it was cancer; am I going to let it bother me now just because I know what it is? What kind of a take-charge attitude is that? I open the package: just two little slides, with a bit of translucent tissue that is stained a purple-pink, and a one-page pathology report. "Frozen Section Diagnosis: Medullary carcinoma," it says at the top. And later on, "Adeno carcinoma, infiltrating ductal carcinoma, medullary type completely excised." I give the slides some death-ray visualization, like I have read about in the books. Then I put on _Bells Are Ringing_ , remembering that Judy Holliday died young, of cancer. She left her voice, anyway, that gorgeous mixture of bubble and sass, and who can't smile hearing it? In no time at all I am dancing with an invisible partner to "Long Before I Knew You," so lost in dreams it takes me a while to notice Judy has dropped in and plopped on the couch.
"So whadidya do with the slides, finally?" she asks.
"I put them in my sock drawer," I tell her.
"Oh, no," she says. "That's the first place they look.
You see the difference between Sloan-Kettering and Roosevelt the moment you walk in Sloan's York Avenue entrance: money. The reception desk is banked with pots of white narcissus; the walls are hung with tasteful prints; I haven't had so many smiling people asking if they could be of assistance since I was in Los Angeles. "This place is like Disneyland, they're so perky," I whisper to Herb. "Cancerland." A pretty nurse, warm and pneumatic, with shoulder-length blonde hair, comes out to get us. "Cancer hostess," I tell Herb after she takes my history and goes off with my cancer slides. We wait an hour and a half for Dr. Petrek, but I like her the minute I see her. She's a great-looking redhead who wears a Mickey Mouse watch with her white doctor's coat and has a smart, hyper, full-disclosure style. She enters the room with my slides in her palm, holding them as if they were just another report and not the signpost of my possible early demise. Then, cheerful as you please, she challenges my greatest-little-cancer-youcan- get diagnosis.
"This looks like medullary, but medullary carcinomas are very rare," she says. "And this report is ambiguous. 'Infiltrating duct medullary type.'"
I can't believe what I'm hearing. Neither can Herb. We exchange a frightened glance. Infiltrating ductal, I know, spreads far more quickly than medullary. I have the sinking feeling I had when I was diagnosed. Dr. Petrek doesn't seem to notice. She's picked up a pencil and straying the outline of the tumor on the slide, like a teacher who wants to make sure you get the lesson, though the lesson to me is that I may shortly drop dead.
"Cancers are very dense, packed edge to edge, and infiltrating ductal cancers have edges that look like pointy little stars—very irregular, shooting out," she says. "Medullary has a very smooth, round outline, like this, see? It looks like medullary, but I'm going to have our own people look at it."
The fear recedes. She's said "looks like medullary" twice; until the lab guys tell us different, I'll keep believing I've got it.
On mastectomy versus lumpectomy, she is far less equivocal than Luke—she sees no reason to do a mastectomy. The top medullary man in the country, Dr. Paul Peter Rosen, is at Sloan, she says. He's done a study of hundreds of breast-cancer patients over a twenty-year period and believes that medullary is much less likely to recur than other cancers. It is also less likely to be in the lymph nodes. Whether my cancer is medullary or infiltrating ductal, Dr. Petrek thinks it is best to save the breast. Given the clean margins surrounding my tumor, I am a perfect candidate for lumpectomy. She also paints a different picture of breast reconstruction from Dr. Veteran's—it can be a very uncomfortable procedure for large-breasted women, she says.
The notion of combining breast reduction with lumpectomy doesn't appeal to her at all. Nor, I gather, does reduction alone.
"Breast reduction combined with lumpectomy is a large operation," she says. "If you were ever to walk in and see it—ugh. What they do is remove a section of tissue from the lower part of the breast. Then, because they have to bring the nipple up higher, they cut the tissue all around it, and the nipple is hanging there, like on a stalk."
She pauses, giving us all time to visualize. I see from Herb's graygreen complexion he's got the picture and may never unbutton anyone's blouse again. Dr. Petrek moves along.
"There's a lot of rearranging of tissue and scarring with reduction, which could interfere with spotting another tumor on your mammogram. It would take a lot longer to heal than lumpectomy, you might have to put off radiation treatment. Combining reduction with lumpectomy . . . I don't know if you want to be the first."
She wouldn't do it, she says. But she will discuss it with the head of the breast-radiation unit, Dr. Beryl McCormick, at the next department meeting. She also suggests I get in touch with Dr. Susan Love, an "ardent, zealous" believer in lumpectomy.
Dr. Love's secretary, at her office in Boston, knows of no one who has combined lumpectomy with reduction. Still, willing to be first, I consult a prominent Manhattan plastic surgeon. Unlike Dr. Veteran, he is willing to do the surgery, as long as the reduction is done after the radiation. After talking with this surgeon, Dr. Luke is willing, too.
I decide I will do it. Then, an hour later, I decide that I will not. Then, I change my mind again. Then I talk to Ma. She has been an uncharacteristically good listener during this crisis—in her forty-three years of Jewish motherhood I've never known her to be so lacking in opinion—but five thousand years of genetic baggage happen to be working against her, and now she snaps.
I decide I will do it. Then, an hour later, I decide that I will not. Then, I change my mind again. Then I talk to Ma. She has been an uncharacteristically good listener during this crisis—in her forty-three years of Jewish motherhood I've never known her to be so lacking in opinion—but five thousand years of genetic baggage happen to be working against her, and now she snaps.
"I know you said news goes out and it doesn't go in, but I'm the mother and I got a right to an opinion," she says. "My opinion is, I don't like this business of first. First is good for the doctor; he gets to write it up and be a big shot. First is good for the hospital; they get money from the government. But how is it good for you? They don't know what they're doing, they never did it before, and if something goes wrong, it's good-bye, Charlie— you're the one that has to live with it.
I give it a few days, but ultimately I agree. If reduction could result in scarring and a failure to spot recurrent cancers, it's not a good idea. If the cancer turns out to be in the lymph nodes, and reduction puts off immediate treatment, it's not a good idea either. I can't spend months trying to find someone who's had this procedure—there's a disease to beat; I'm up against time. I will stick with the basics: lymph-node removal to see if the cancer has spread; re-excision of my left breast.
That leaves me with deciding whom to go with as my surgeon. I like both Luke and Petrek; Herb has gone to the library and checked their credentials; they're both sharp, good guys. But I want the resources of a cancer-research center behind me. Luke has had three medullary patients. Rosen did a study of more than seven hundred breast-cancer patients, about 5 percent of whom were medullary. I've wanted a place where all my records, all the specialists involved, would be under one roof. I have that at Sloan. I want to know that the people who read my biopsy slides are all cancer specialists. I set the date for surgery with Petrek, for the second week in April—three weeks from the day I was diagnosed. I also ask her if she can give me some positive statistics on survival if the cancer turns out to be in the lymph nodes. She says that finding cancer in the nodes is not uncommon— it occurs in slightly less than half of all breast-cancer patients. One of those was Betty Ford. She was diagnosed in 1974, with cancer in two nodes. She is doing just fine.
Dr. Petrek also has good news about my biopsy slides. As is policy at Sloan, two doctors have examined them under the microscope. They both say I have a medullary carcinoma.
I don't spend Passover with my family. My mother, who made plans to close the Florida house the hour I told her the lump was malignant, is getting ready to drive north; I don't want to be with the clan in the Catskills. My aunt Shirley is the hostess of that one, and I haven't told Shirley the full story because I think she Can't Take It. If I go home, we may hit the next step in the Wadler serious-illness scenario: Shirley will Figure It Out.
I end up, then, spending the holidays with Nick and his family. It works out nicely, Easter falls on the same weekend as Passover, and I like Nick's family a lot, particularly his mother. She hugged me the first time she met me. She is big and soft and cushiony. She grew up in Italy and speaks with such a heavy accent that I sometimes cannot understand what she is saying, which makes me feel like I have my grandmother back. "Iiiii, maron!" she says, when she gets fed up. "What's this 'maron' your mother is always talking about?" I ask Nick finally. "Madonna," he says, giving it the Italian pronunciation. "Madonna, idiot." Since the death of her husband, Mrs. Di Stefano has lived with her children, spending a month here, a month there, with each one. When she stays at Nick's, he moves out of his bedroom into the den and sleeps on a futon on the floor. She arrives with a statue of the Blessed Virgin and six suitcases, one of which holds only medicine, and prays three hours in the morning and two hours at night. She prays for the dead, she prays for the living, she prays for Bush—not because she likes him but because he's president and she feels she should. When Nick tells her I have cancer, she starts praying for me too. In between, she knits socks, and says what everyone in the family is doing wrong, and tells stories about sickness.
"Whata you got, my daughter-in-law got, too," she says, lying on Nick's bed, knitting. "First she gotta one side, they cutta it out. The next year it come to the other side, they cutta it out. A cousin of mine, she got it, too."
'Oh, yeah?" I say. "You pray for her too?"
Mrs. Di Stefano nods yes.
"How'd she do?" I ask.
She does this thing with her head and her shoulder that suggest both the mortal coil and the limitations of prayer.
_"Eehhhh,_ " she says.
Easter over, I move back to medical concerns. I have a bone scan and liver ultrasound to show the presence of cancer cells. They come back negative. I go to a lawyer and make a will. I sign a health-care proxy, sent to me by Sloan, giving Herb the right to make medical decisions on my behalf in case I am incapacitated during surgery, and write an outline of the last chapters of my book, assigning rights again to Herb. I have been going to the gym a lot, to strengthen my heart and lungs for surgery; now I focus on a psychic attack for my hospital stay. Most of the patients I see in hospitals, except new mothers, shuffle around like depressed shlubs. Analyzing it, I think I know why: bathrobes. Only Rex Harrison could look good in a bathrobe. Hospital pajamas aren't that great, either, though the floppy drawstring pants, matched with oversize shirts, have possibilities. Casual clothing, I decide, is what is called for. I pull out a bunch of Hawaiian shirts and bright, oversize beach tops, and flowered tights, and pink sandals. I have a presurgical session with the shrink, telling her that even if cancer is in the nodes, I will try to beat the disease, and if not, or something goes wrong on the table, I have an interesting life behind me. I have friends who love me, I got to have two grandmothers, I've been in love in Paris, I drank champagne with a spy at ten in the morning, I wrote a few things I liked, I had Herb, I had really wanted to stand with someone under a wedding canopy, but maybe it is silly to think you can have everything.
"You have a good life ahead of you," the shrink says. "It isn't finished. And if you want me, you call, and I'll come to the hospital. We've been together a long time."
The last big decision is who to take to Sloan. Though my mother is now in striking range in the Catskills, I do not want her with me the day of surgery. Part of this is superstition again. Every time my father went into surgery, the family gathered around, and I do not want death or his messenger service, seeing a bunch of Wadlers around a hospital bed, to think there's another one waiting for pickup. Another reason, the stronger one, is that I may be frightened, and if my mother is there, I will have to pretend I am not, so she won't feel worse.
I don't want to ask Nick either. The more I expose my breast to strangers, the more I am pricked with needles and scanned by machines, the more I want to be held and cherished—and Nick has made it very clear that illness or no, I am not the woman of his dreams. At night, even after making love, he still puts on a video of _Rope_ and uses it to fall asleep, his face as frozen and angry as that of a little boy who feels he's been denied and always will be. I know this look because for most of my life, I have been Nick Di Stefano's psychological twin—discarding the lovers who wanted me, idealizing the ones who did not—but it's a crazy way of life and I don't want it anymore, because now I value time. "Why don't you turn off the movie and try to fall asleep with me?" I say to Nick, meaning, "The best movie is here, right under your nose; it is real, and it is me. Love me, you jerk, because now is a gift and I don't know what I'm looking at and soon I could be gone." It does no good; I can't break through the wall. I try not to rock the boat, either, with what Nick considers to be excessive demands. I have seen posters at Sloan for stress-reduction clinics. I can't risk asking Nick to come with me to the hospital. I could get too upset if he says no. I ask Herb and Heidi instead. Nick will come in the evening, after work; my mother will arrive the next day. Then, the weekend before surgery, I change my mind. I want Nick to be with me. If Heidi, who has two young children, can take a day off, so can a fifty-two-year-old sports reporter. I ask him to come. He refuses. "What is this, a test?" he says. "You know what kind of week I'm looking at at work." I try not to get too upset. I remember the posters at Sloan. Stress kills.
Mrs. Di Stefano, anyway, is solid. Sickness is her game; when it comes to life-threatening illness, she's the MVP of the Eastern Conference. She knits me a pair of socks and asks me what time surgery is scheduled, so she can launch a special prayer. Nick stays at my place the night before surgery. Next morning, as he is sleeping, Heidi and Herb pick me up. We hang out in a hospital room together, and, waiting, I am impressed with Sloan. A nurse comes by to see if I have any questions. When another nurse and an orderly come with a gurney to take me to pre-op and I tell them I would feel very silly being pushed down the hall on a bed, they go into a huddle, bend the rules, and alert pre-op on a walkie-talkie. "Patient walking," they say, and we form a nice little soaring-hospital-costs tableau: a nurse, followed by me, Heidi, and Herb, followed by an orderly and an empty gurney, parading down the hall. Heidi and Herb haven't marched since we all protested Vietnam; they think it is hilarious I am remaking the rules. "Go, Joyce!" they cheer. At the pre-op room, which is adjacent to the operating rooms, I have to tell them goodbye. The room is dim, with a row of eight or nine gurneys and three other patients. I reluctantly climb onto a bed, feeling very scared and alone, knowing there are three other people feeling exactly the same, a few feet away. It would be nice, I think, if we could all hold hands, but maybe, finally, that wouldn't help—we would each still be going it alone. I try to think how to handle the fear—and find myself, though I am not religious, silently saying the _Shema_ , the prayer I was taught Jews are supposed to say when they may be facing death. If God exists and turns out to be a reefer-smoking Kerouac bebop hipster, I think, I've probably just made another disastrous career decision. An orderly comes in and wheels me into the operating room. Almost everyone is a woman, which gives me a good feeling, but I am still terrified of the anesthesia. The more scared I am, the faster I joke. "You know the movie _Coma_?" I say to the nurse anesthetist when she puts the IV in my arm and starts a saline drip. "That's my big fear. I have this feeling I'm going to, like, wake up as the featured vegetable in Gristede's." She hits me with the statistics: Twenty years ago the mortality rate from anesthesia was one in twenty thousand cases; these days it's roughly one in two hundred thousand. Then, she suggests something: a sedative called Versed, which feels like a very strong Valium. Next thing I know, I am groggily opening my eyes in a dimly lit recovery room, and then I'm in my own room with Heidi and Herb, and soon after that, Nick arrives. He has an armful of red roses and the sad look he gets when he is looking at his son or watching a Disney film. I am so doped up with morphine I can barely move my head when he bends down to kiss me, but I remember the anesthetist clearly. That lovely woman. She slipped me a mickey.
I will not know the results of lymph-node dissection for a few days. Until the lab report comes back, the surgeon herself does not know how many lymph nodes she has taken. Lymph fights infection; the nodes in the armpit lie embedded in fat like a cluster of grapes, and everyone has a different number. When nodes are removed, lymph continues to collect in that area of the body until, after a few weeks, it finds other channels. To siphon off the lymph near my left breast, I have, four inches under my armpit, a Reliavac drain coming out of my side: five feet of plastic tubing that empties into a transparent plastic canteen. The amount of pale yellow fluid that collects is the literal watermark of my release from the hospital. When I get down to fifty ccs over a twenty-four-hour period, they will remove the drain and let me go. I'm cautioned against packing my bags. Big-breasted women, the nurses tell me, have a lot of fluid.
Meanwhile, the mood on the eighteenth floor, which is known as the breast-cancer floor, is surprisingly up. Women come and go, sharing stories. After the first day, I have very little pain. I can't raise my left arm straight up over my head or touch the middle of my back, but Sloan has stretching classes to get the arm back to normal as quickly as possible, an estimated four to six weeks. My roommate, a research biochemist in her mid-fifties who had multiple lesions in her breast, has had a mastectomy but seems unconcerned: Maybe she'll have a reconstruction, maybe she won't, she says. There is no guy in her life, but if one comes along, she figures if he's a good man, he'll love her for who she is; if not, the hell with him. My mother arrives and says the food at Beth Israel is better and talks two hours at a stretch without paying much attention to anything I have to say, which makes me feel things are back to normal; Nick comes every night. Bernard, the French spy who is the hero of my book, calls from Paris; Stefan sends chocolates; my friends bring flowers and themselves. Every few hours, a different volunteer arrives. The most helpful is a former breast-cancer patient exactly my age, who also had a lumpectomy—somewhere, somebody is doing some careful matching. She's a strikingly pretty woman, in a flowered, V-neck dress, and she asks if I understand the next step in my treatment, radiation therapy. When it comes to cancer treatment, I consider myself the smartest kid in the class:
"Takes about fifteen minutes a session; you have five treatments a week for six weeks. Some people get tired; your breast can get a little swollen and pink, like you're sunburned. They mark where they're going to radiate you with ink."
"Wrong," she says. "In most places they mark you with ink. At Sloan- Kettering, they give you tattoos."
I stare at her. A tattoo? What's it gonna be? A picture of a single- breasted mermaid and SEMPER CARCINOMA on my breast?
"They're very small," she continues. "They give you four or five of them; it's better than ink, which comes off on your clothes. You can have them removed after the treatment, but I know only one person who's bothered. Can you see mine?"
I look closely at her chest. She has a sprinkling of freckles, like me, and after a minute, I can see one tattoo, above her cleavage. It's the size of an ink dot. But if someone hadn't shown it to me, and I had just heard the word tattoo going into radiation, I would have been upset.
Friday, though I'm still throwing off a lot of lymph fluid, they show me how to empty the drain, and let me go home. I wrap the Reliavac pouch in a red silk scarf, pin it to my sweater, and head out with Ma and Herb to Quatorze, on Fourteenth. I must be very distracted not to enjoy lunch, but I am not in a great mood. I want to get Ma back to the mountains as soon as I can. Dr. Petrek has told me to call her office at three, and she'll have the report on the lymph nodes. If the news is bad, I won't be able to bear my mother's face. At two-thirty, Herb and I send Ma off to Port Authority. At three, I call Dr. Petrek. Then I call my brother who is meeting Ma at the bus station, and Herb and Nick and Heidi and Max and everybody else I can think of.
"They took out twelve nodes," I say. "They're all clean."
Schlepping around with a plastic pouch coming out of my side does not make me feel particularly beautiful, but it's not a major problem. I discover that if I tuck it into a runner's pouch around my waist, no one knows it's there. It's more difficult in bed, where I pin the pouch to my T-shirt, but Nick is sweet. "You're embarrassed about the thing," he says. "You know you don't have to be embarrassed with me," and he puts his arms around me, and contented as a kitten, I go to sleep. Eight days after the surgery, the drain is pulled out. There is no pain. The area around my underarm is numb from the lymph-node surgery and will be for at least a year.
A few days later, I see Dr. Larry Norton, the oncologist who is the chief of breast-cancer medicine, to discuss the next step in my treatment. He's fortysomething, relaxed, amiable, and balding, and he has wonderful news for me: Since 1990, he says, it has been common practice to give chemotherapy to breast-cancer patients even if cancer is not in the lymph nodes At Sloan, unless a tumor is smaller than one centimeter, they follow that protocol. But "true" medullary cancer, Norton says, is "a very, very, benign disease," and Dr. Rosen, Sloan's expert in breast pathology, has done a study showing that medullary patients whose tumors are smaller than three centimeters have about a 90 percent chance of no recurrence. If a medullary carcinoma is smaller than three centimeters and the nodes are negative, the doctors at Sloan see no reason for chemotherapy. Dr. Norton is ordering additional tests on my original biopsy slides from Roosevelt to establish that I did have a true medullary carcinoma. If so, since my tumor was just under three centimeters, I will not need chemotherapy.
The tests confirm the diagnosis. I am not feeling 100 percent recovered from surgery. I still have to go to the hospital two or three days a week, so that the lymph fluid that collects under my arm can be drained. As it collects, I have a swollen, uncomfortable feeling, as if there's a volleyball under my skin. My left side and breast are also sore, but I'm very happy I won't need chemotherapy. I celebrate by going off with Nick to the Mohonk Mountain House, a rambling Victorian hotel upstate. The room has two double beds. We shove them together and open some champagne and don't get to the woods till the next day. And yet, the weekend is not a success. There is sex and there is communion, and I have the feeling that Nick is losing himself in the first, not making love to me. When we go walking in the woods, stopping to rest in one of the gazebos that circle the lake, I am not altogether surprised to run into Freud. He is on his hands and knees, carving a heart and two names into the bench. DEPENDENT LOVES NARCISSIST, it says.
"You're telling me I should break up with him, right?" I say.
"What?" says Freud. "And miss watching the two of you in bed? I wouldn't think of it. The energy you put into your lost causes."
"You may have noticed, since you've been studying everything so carefully, he doesn't exactly lie there like a lox," I say.
"Exceptional vigor for a man his age," says Freud. "And why do we think that is?"
I don't care what the answer is, I'm mad.
"I know you say there are six people in every bedroom, but you dogging me everyplace I go with this guy is too much," I say.
"In your case, darling, it's more like eight in the bedroom," says Freud. "You also got the lawyer and an agent. Plus the West Coast office of I.C.M. I'm amazed the little boulevardier has room to take off his pants."
Nick and I go back to the city. Monday morning, he switches on Lucy. We fight. I go off to Sloan to have a hypodermic stuck in my side. It's impossible to find a phone where you have any privacy at Sloan. I wait till I'm in the street to call Nick.
"It may seem to you like this cancer thing is over, but it's not," I tell him. "My breast hurts, my side hurts; every week I see another doctor. I just sometimes need to be held."
"Then get yourself another guy," Nick says. "We don't have that kind of relationship."
I dial Donal, my old boyfriend, at his studio. He's a big man, with a honey-colored mustache and the air of an Oxford don.
"Need a lap," I say.
I go to the studio, and Don brews a cup of tea and brings me Humphry, the fat white studio cat, who years ago lived with Donal and me. I love Humphry, though he makes me sneeze. We all curl up together.
"I know you; I know what you need," says Donal. "You're never going to get it from that man."
"I know that," I say. "It's just these treatments are making me such a baby."
Nick and I break up.
Radiation therapy, which begins in mid-May, goes off without a hitch. I have some tiredness, but I attribute that mostly to psychological factors and feel well enough to take myself off medical leave and return to my book. By the end of June, my cancer treatment is over. The medical bills come to $32,300, but almost all of it is covered by insurance. Life returns to normal. Herb and I watch TV and complain about editors. My mother re-ups in the Israeli army, as her way of giving thanks for my health. Before she flies off, we all go out to dinner and Ma and Herb reminisce about the motor pool and army life. Ma is much more enthusiastic than Herb. In September, I go back to _People_. One month later, I am among the 605 Time-Warner staffers to be laid off. _"Arrivederci_ , hot tubs," I think and go back downtown, take off my pearl earrings, and finish my book. For New Year's, in celebration of life, I make a bet with Bernard that California champagne can match French, throw a few bottles in my carry-on luggage, and go to Paris for the weekend. Bernard picks me up at the airport, waving down from the mezzanine with his great felonious grin. When I get through Customs and he kisses me formally on the cheek, I start to cry.
"I didn't know if I was ever going to see you again," I say.
Between me and Nick it's over. Nick did, a while back, say, "Enough craziness; let's get rid of the other people and live together already," but it was a mood, and it passed. His mother, however, is still praying for me. She is also, Nick says, praying for Magic Johnson.
That was going to be the end of my cancer story. Then, as I was writing this piece, an extraordinary thing happened: I discovered not all of the slides of my original biopsy had been sent from Roosevelt to Sloan-Kettering.
The discovery was made by accident. Calling up Dr. Petrek, I asked if she might pull out my slides and review what she had originally discussed with me. She agreed but told me that I would have to get the slides from Roosevelt—hospital protocol, after reviewing specimens, is to return them to the institution after a procedure has been performed. Soon after picking up the slides from Roosevelt, I received a call from Dr. Norton. He told me they were two new slides—unknown to Dr. Petrek, the pathologists at Sloan-Kettering had gotten permission from Roosevelt to keep my original slides. Norton said that on the basis of the slides he had just received, he was thinking of changing my treatment. There was "an important but subtle" distinction between medullary carcinoma and infiltrating ductal carcinoma with medullary features, he said. Four pathologists at Sloan had looked at the slides; it was a close call, but on the basis of the new material, Dr. Norton felt my cancer was more likely to be an infiltrating ductal with medullary features than pure medullary. I still had, he stressed, an excellent prognosis. Nonetheless, purely as a precaution, he felt it best to give me a little extra treatment and was advising chemotherapy.
I am horrified. Infiltrating ductal is more aggressive than medullary, and if there were any cancer cells in my body before, for one year they had been getting a free ride. I am also upset that all my slides were not sent the first time. Dr. Norton says my cancer doesn't seem aggressive at all. "At the very worst, you're looking at a cancer with a very high cure rate," he says. "At the very best, you're looking at a 100 percent cure." He also explains that hospitals rarely send all their biopsy specimens—they select those that they consider diagnostic. Asked how he felt about my case, he struggles to be diplomatic.
"I'm not happy; what can I say?" he says. "I don't think there was any malicious intent. It's a difficult distinction to make, based on your pathology. Clearly, having more tissue helps you make the distinction. . . . They're supposed to send over slides which are characteristic. . . . I don't know why they chose those two slides."
I go to Roosevelt, ask for all of my slides, and talk to Dr. Bozidar Lazarevic, chief of anatomic pathology at Roosevelt. He says he stands by Roosevelt's diagnosis: I had a medullary cancer. Nor is he defensive about having sent only two slides—as Dr. Norton had said, it was protocol.
"I send the key slides I would use to make the diagnosis," he says. "I'm afraid to send all the slides—half the time I don't get the slides I send back. If there was disagreement about the diagnosis and the doctor asked to see all the slides, we would send them, but when a patient goes to see a doctor for a second opinion on therapy, we send only the diagnostic slides."
I see not only a problem with priorities here but also a Catch-22: A doctor will get all the slides only if he or she questions the diagnosis, but how will he know the diagnosis is in question if he doesn't get all the slides? I suggest to Dr. Lazarevic that he change the policy and let doctors know how many slides exist. He says doctors do know how many slides there are from notations on the pathology report, that his people may get only one of two slides from other institutions, and that in the fifteen years he's been at Roosevelt, this is the first time a doctor has challenged a diagnosis over their slides.
It all seems crazy to me, but there's nothing I can do; I have two doctors with differing diagnoses. I never, however, question Dr. Norton's recommendation for chemotherapy. If there is any risk of recurrence, I say, hit me with your best shot.
In early March, I begin a course of chemotherapy called CMF—a combination of the anticancer drugs Cytoxan, Methotrexate, and 5-Fluorouracil, as well as an antinausea sedative called Ativan. The CMF is given in an intravenous drip, which takes about forty-five minutes and is administered once every three weeks over a six-month period. My main concern about chemotherapy is that it will make me so sick it will disable me, but it does not. The first day after treatment, I am so wiped out by the Ativan tablets I get from the Sloan pharmacy that I sleep all day; the next day I am speedy and tense, but after that everything is fine. I take my Walkman with me for my chemo treatments and listen to _Beauty and the Beast_ and _La Cage aux Folles_ and, because Ativan has a mild narcotic effect, get pleasantly high. In my five weeks of treatment to date, I have increased my weights at the gym to prove that I am not a sick guy. I have not had hair loss or any other physical problems. My breasts are the same size. While the doctors have said the drugs will probably put me into early menopause, I am for once going against statistics and taking the position that is unlikely to happen to me.
And even if it does, I consider myself a lucky guy. Not just because in the time of the great breast-cancer uprising, mine was so benign, but for the terror of the ride. Nothing is real until you are close to it, and for a few weeks, I was given something few people have: a dress rehearsal of my mortality. And though cancer has not made me a model of mental health, though I remain tempted by the drama and danger of gangsters and ladies' men and continue to worry about every little thing, my experience with serious illness has changed me. Death, I now see, may not come when I am eighty-five and weary, or after I have solved all of my problems or met all my deadlines. It will come whenever it damn well pleases; all I can control is the time between. So when I see something I want, I grab it. If the tulips are particularly yellow, I buy them. If I hear Pavarotti is in town, I make a run to the Met and work the crowd for a scalper. I make time for my friends the way I used to make time for work. If someone treats me disrespectfully, I leave.
As for the mark on my left breast, I am happy to have it. It is the battle scar over my heart, and if no one but my doctor and the girls at the gym have seen it lately, I am certain, believing as I do in musical comedies, that somebody will soon.
"So, how'd ya get that?" he'll ask, our first lazy morning, and I'll say, delighted he has found me, listening for the bells to ring, "Glad you asked, 'cause it's a wonderful story. . . ."
_Isabel Wilkerson_
Isabel Wilkerson is the Pulitzer Prize–winning author of _The Warmth of Other Suns_ , the _New York Times_ bestseller that follows the fifty-five-year migration of black Americans from the South.
Wilkerson won the Pulitzer Prize for her work as Chicago bureau chief of _The New York Times_ in 1994, making her the first black woman in the history of American journalism to win a Pulitzer Prize and the first African American to win for individual reporting. Wilkerson has also won a George Polk Award and a John Simon Guggenheim Fellowship, and she was named Journalist of the Year by the National Association of Black Journalists.
She has appeared at universities across the country and in Europe and on national programs such as CBS's _60 Minutes_ , PBS's _Charlie Rose_ , NPR's _Fresh Air with Terry Gross_ , NBC's _Nightly News_ , MSNBC, the BBC, C-SPAN, and others. She has taught narrative nonfiction as Ferris Professor of Journalism at Princeton University, as Cox Professor at Emory University, and as Professor of Journalism at Boston University.
In 2012, _The New York Times Magazine_ named _The Warmth of Other Suns_ to its list of the best nonfiction books of all time. In 2016, President Barack Obama awarded her the National Humanities Medal for "championing the stories of an unsung history."
Soul Survivor
Civil rights activist Kwame Ture, formerly known as Stokely Carmichael, was a fiery leader of the Black Power movement during the 1960s.
The telephone rings, an ordinary ring in an ordinary Florida condo, and a long, lanky man you recognize from the grainy footage of the you-remember-the-civil-rights-days, the double-edged-sword-tongued soldier in the Ray-Bans and the pencil-lapel suits like the original Man in Black, spouting "Power to the people!" when the people had no power; who organized, organized, organized in the wicked Mississippi sun for the right to vote before you maybe even were born; who was rolling out of Chevy Impalas to cluck white-sheriff bullets so you could walk into a booth and check off who should be the water commissioner or the president; who had a god's face sculpted from brown velvet and could have been on the cover of GQ if that had been his bag, which it wasn't; who went by the unusual if patrician-sounding name of Stokely Carmichael, is now seated on his mother's white sofa in front of me, if you can believe it, unclasping his arms to pick up an impatient receiver.
He goes by Kwame Ture now, and his story unfolds between telephone rings. You rewind the spools of history. And you realize that the reason thirty-four million people call themselves black is because of the man sitting serenely in front of you, that before he got the nerve to shout "Black Power!" to a civil rights crowd in a Mississippi square in the belly of Klan country in 1966, we were neither black nor beautiful, but Negroes or worse. He was an undercover Malcolm working under Martin Luther King, Jr., professing nonviolence by day but packing a .38 just in case. Of all the civil rights generals—Malcolm, King, Medgar Evers—he is the last icon of the sixties left to set the record straight and, some wish, lead us to the Promised Land, although that in itself was never his intention.
He is fifty-six now, and sick with cancer that the doctors cannot cure. He lives in Guinea, West Africa; has not lived in the United States for nearly thirty years. Every visit here is a miracle that grows out of a patchwork budget, last-minute shuttling of boarding passes by followers around the world and the grace of God. He is here in this country for another round of chemotherapy, but never knowing how he will get here, happened to arrive this time through Cairo, where a brother from the Nation of Islam ran into him and gave him the money to get to Rome, and then New York for his chemo and then on to Miami, where his mother lives in a two-bedroom condo. He has no money, never did—got only ten dollars a week when the Student Nonviolent Coordinating Committee (SNCC, called Snick for short) could come up with the cash. Even when he was prime minister of the Black Panther Party, he was never on what you would call a payroll.
Now after all that rolling out of cars and jumping out of windows, and all those beatings in southern jails, he returns to a country that—between the drugs and the violence and the assaults on everything he worked for—may seem worse off than when he was on active duty. He heard about Rodney King and O. J. and the white backlash against affirmative action. He knows about the young brothers killing one another over rocks somebody cooked up in a basement and the bourgeois betrayals of the Clarence Thomases of the world. It has not worn him down. He had to be crazy or idealistic or both to go down South in the first place as a smart-mouthed college boy with the nerve to think he could change the world. And he does not waver.
The telephone rings again. He reaches over to answer it. His voice is West Indian molasses and rises up like a melody. He does not say hello. He greets the caller with his thirty-year-old mantra, uttered quickly and matter- of-factly as if it were a single word: "Ready for revolution."
He says it like it was the most natural thing in the world, sometimes rushed, sometimes weary, the way you would if you'd been working for years at a company with a long and convoluted name, the way you'd say Consolidated Edison. You could say that revolution is his line of work. He still has faith in his people, no matter what's happening. "We always make progress in spite of ourselves," he tells me. "We gon' win. We gon' win. In spite of ourselves, we gon' win." He lets out a hearty laugh that rolls deep like the ocean.
Scroll back to 1966. It is June. It is hot. Malcolm is dead. Medgar Evers is dead. Countless others have been killed or beaten just because they want to vote or eat in a restaurant. And even though the Voting Rights Act is now law, most black people still can't vote in Mississippi. The foot soldiers in the Movement—the college boys running SNCC who went around trying to sign up sharecroppers too scared to even be seen with them—are restless now.
Tension was rising between SNCC and the old heads like the Southern Christian Leadership Conference (SCLC), which King ran. King preferred big marches that showed black people's perseverance and attracted national attention. SNCC liked working behind the scenes, liberating people one by one.
They also disagreed on the question of nonviolence. "Most of us in SNCC did not accept nonviolence as a principle," Ture says. "King did. We saw it as a tactic. As a principle, you have to use it all the time under all conditions. As a tactic, we could use it today if it was working fine. If it wasn't working, tomorrow we could toss hand grenades."
In every demonstration King had, he says, people would be throwing bricks and bottles. The front lines with the suited reverends were righteously nonviolent, but "the back would be snipers shooting," he says. "They'd say, 'Listen, we respect you all. Y'all nonviolent, but we ain't going for that. These honkies in here shooting, we gon' shoot back. We gon' just let you know.'" Sometimes, he says, SNCC would negotiate with the local people to hold off until after the march was over. Then, he says, "you be hearing shots all night long."
Everyone understood that guns were a part of life in the Movement. King refused to keep a gun even after his house was bombed in Montgomery. But when he came to Carmichael's turf in the Mississippi Delta, guns were everywhere to make sure he was safe. "When he came I had cars to accompany him," he says. "The first car was full of guns, and the last car was full of guns. Cars throughout were full of guns. And I had a gun."
Carmichael and others in SNCC were more militant partly because they worked behind enemy lines, deep in Klan territory. Their job was to rally sharecroppers to stand up against the caste system they lived under and register to vote. The students were like guerrillas in a war. Death was always right around the corner.
They had code names for themselves (Carmichael went by "Greenwood Sweets," after the Mississippi town). They learned how to roll out of their cars in case of a shooting and traveled their districts in souped-up sedans, each one outfitted like a poor man's James Bond. Driving along the Delta's dry-dust roads, they realized that the only way the police could track them at night was to follow their taillights. So they devised a way to switch off the taillights at a curve. "The car following us inevitably has to go off the road," he says.
They made another discovery, albeit too late to save Medgar Evers. The Mississippi NAACP leader was fatally shot by Byron de la Beckwith while he was getting out of his car at night. "It was the car light that gave de la Beckwith the target," Ture says. After that, "we made sure the bulbs were taken out."
And so by the summer of 1966, the students were growing impatient with the passive "We Shall Overcome" and "Freedom Now" rhetoric that had brought great strides but, to them, had not finished the job. They plotted a more militant approach, talking up the idea of a radical new battle cry for their followers, not knowing for sure where it might lead. It would be not only a call to take their rightful place but also a sea change in identity from polite Negroes to proud black people embracing what the world had ridiculed.
They had secretly tested the idea for years, convincing the sharecroppers of its merits. But it wasn't until June of 1966, when a major march from Memphis to Jackson was scheduled to sweep through the heart of Carmichael's territory, that SNCC thought the time was right to change the tone of the Movement. So they set out to take over the march. They sabotaged the more conservative groups like the National Urban League and the NAACP—he refuses to say exactly how—shutting them out when the march hit Greenwood. That left mainly SNCC and SCLC. Then an astonishing thing happened. "Dr. King had gone to Memphis to tape a show, I think _Meet the Press_ ," he remembers. "He wouldn't be there. We couldn't believe it."
With King out of the picture and the mainline groups silenced, SNCC had the march and the podium virtually to itself. It could drop the new rallying cry on the public without dissent. But a few hours before the march, Carmichael was arrested, he says, as a ploy by white officials to disrupt the protest. Supporters bailed him out just in time for the rally, where he clenched his fist in the air and for the first time publicly shouted "We Want Black Power!" to the world.
Three years later, he was gone. He moved to Guinea in 1969 after a dizzying sequence of events: He had married Miriam Makeba, the South African singer; been ousted from SNCC over divisions he believes were fomented by the FBI; and become prime minister of the Black Panther Party, which, you should know, did not start in Oakland as most people think but in Alabama with SNCC, he says. It began as an act of defiance against the Democratic Party and the white rooster that symbolized white supremacy. By the late sixties, Carmichael had become a true-blooded revolutionary. Resented by some for his celebrity and targeted by the FBI, he left the United States and joined Kwame Nkrumah's pan-Africanist All African People's Revolutionary Party in Guinea.
By the mid-eighties, he would be divorced from Makeba (the FBI pursued them, he says, and the spying took its toll) and caught up on the losing side of a military overthrow in Guinea that killed most of his comrades. The military even put him in jail for several days. He survived.
He now has two sons—one seventeen by his second ex-wife, a Guinea-born physician who now lives in Washington, and a four-year-old in Guinea by a woman he never married. He admits he is not an especially attentive father. "If I can be of help to them, I will," he says. "But they know the people are more important than them."
In between speeches and meetings with pan-Africanists around the world, he is now working on a book about his life, called _Dancing in the Fire_ , with the writer Michael Thelwell, a friend from his SNCC days. The title comes from something he used to say to the ladies who caught his eye. "I'd ask them if they could inhale smoke," he says, "because I dance with fire."
The radical left-wing black militant leader of the youth arm of the Civil Rights Movement grew up middle-class in an Italian section of the Bronx and played piano at Sunday school at Westchester Methodist Church. This is not a printer's error. He was also a Boy Scout—"believe it or not," his mother, Mabel Carmichael, says grandly in her sweet West Indian brogue.
The family migrated from Trinidad to New York City in waves, completing the process in 1952 when Stokely was eleven. His mother dreamt that her brilliant only son would grow up to be a doctor like he said he would. She might have had a cushy retirement and a pool and a wealthy surgeon son she could brag about, the way middle-class mothers do. What she got was a revolutionary who was thrown into southern jails too many times to count and whom numerous entities—was it the FBI, J. Edgar Hoover himself, Cointelpro, enemy capitalistic forces, rival comrades jealous of his charms?—wanted dead and all but forced out of the country.
Through the sixties, while he was plowing through southern backwoods, trying to outrun the Klan and sign up wary black people to vote, she would sleep with the radio for word on whether he was still alive.
It was not supposed to be this way. This was an immigrant family with middle-class strivings, headed by a carpenter father who refused to let his wife work and who had big hopes for their son and four daughters. Stokely went to predominantly white schools, where, in the eighth grade, a white teacher actually spat grape seeds at him from her desk. (Oddly enough, neither he nor his family complained because they felt no good would come of it.) He made it into the prestigious Bronx High School of Science, one of few blacks at the time, where he started reading up on Lenin and Marx and feeding the charming defiance that runs on his mother's side and would carry him through The Struggle.
He went off to Howard as a premed student in September 1960, and soon got involved with the SNCC chapter there. In May 1961, Mabel Carmichael got a call from her son, the one who was going to be a doctor, announcing that he was joining one of those freedom rides, the ones you saw on the news where black and white students would board a bus or a train and ride around the South defying segregation and getting southern whites all riled up.
"I almost went crazy," his mother says. "He called me from Howard and he said, 'I'm going on one of those freedom rides,' and hung up."
"I didn't hang up," he says from across the room.
"You hang up," she says, her voice deep Caribbean now.
"No, you even said to me, 'Oh, I know you're joking. I know you're coming home.' I said, 'I'm really getting on this plane. I'm going down there.'"
"Could been," she concedes. "Then after that, he hung up."
"She hung up," he insists.
"I thought he was fooling me," she says. So on he went to New Orleans to board a train to Jackson, Mississippi, with other students, to challenge segregated train stations. "That was rough, boy," he says, laughing hard. "It was wild. It was just wild. White people were fighting each other trying to get to us. Wherever the train stopped, they broke windows. What didn't they do! Every train station we got to, it was just mobs of people."
Soon after, he broke the news to his parents that he wouldn't be a doctor after all. "He say he doesn't want to help people when they get sick," his mother recalls. "He want to help them before they get sick. I didn't know what the hell he was talking about. How you going to help people before they get sick?
"And you know what else he says?" she remembers later. "He says, if we make him become a doctor, he's going to the furthest part of Africa and help poor people. So if we're sending him to school to be a doctor 'cause we think we're going to get money out of him we're wrong."
"At least I was right there," he says. "Up to now you haven't got a penny!" They both shake with laughter.
"Money don't mean a thing to that boy," she says. "He goes around the world without a dime in his pocket."
I ask if his speeches are a source of income. "We don't speak for money," he says. His mother is doubled over now and all but rolling on the floor.
"You thought he worked for money!" she says, shrieking with laughter. She mentions a book he wrote years ago about the Movement. "All the earnings went to SNCC, and his mother was out starving. I used to tell him, 'Child, would you bring that home? How you could do this?' He leave college and never make a dime." She bursts out laughing at both of their fates.
All she asked was that whatever he did, just be the best. And he was. That first time down South, he landed in jail like a good revolutionary. His mother heard it over the radio.
By now, the neighbors were talking. "I didn't know how to face people," she says. "One friend called and says, 'You talk about everything else, but you never say your son was in jail.' I didn't know how to respond to that. Very quickly I said to her, 'It wasn't for stealing, you know.'"
This was only the beginning. "He spent the whole summer in jail," she says. "This nigger spent the whole summer in jail! When he come home, we have to send him to Harlem to cut his hair—full of lice! That," she says, pointing at Ture, "my nice son, my doctor son."
Even the death of his father in 1962 did not pull him from the revolution. "I ask him to come home; say, we're only girls here," his mother remembers. "He has to go for voter registration. Whatever he was doing down there, I don't know. But he did not come home." Her voice breaks apart. "I don't have to tell you the things which I said, which I'm very, very sorry for now."
"She didn't understand how I could sacrifice the family for the people," he says, trying to explain the situation.
Nor could she understand, until she saw Mississippi for herself years later, why he and the other would-be doctors and lawyers were down there in the first place. "You're in a class, you've got your own house, you've got everything," she says, "and I just couldn't see him coming here, going to fight for black people. Why don't they go to work? Why in heaven don't they go to work?"
It wasn't, on the face of it, the son she raised. "He's down there in Farmer Brown overalls and a pair of boots his cousin gave him, making believe he's poor." She turns to him. "What you was trying to prove?"
"Not making believe," he tells his mother, "just blending in with the South."
"Don't pay him any mind, honey," his mother says. "If you give this boy here six underwear he gives away all of them, come back with one. Say this one told me he didn't have none. That's the type of thing he was before he even got into SNCC."
Because the idea of Black Power is so central to her son's legacy, I ask her did she use the term black at home while raising her children. "At home?" she asks. "I called them niggers all the time."
She tells me it was a way of toughening them up for what the outside world might do to them. "How would you use that?" I ask her.
"I'd say, 'Nigger, come up here,'" she says. "'Didn't you hear me calling ya?'"
His own name change is still a sensitive subject. She gave him the lordly name Stokely Standiford Churchill Carmichael for a reason. It was at the suggestion of his godfather, who admired a college professor called Stokely. The Stokely we know changed his name to Kwame Ture (after Kwame Nkrumah and Sekou Toure, the respective leaders of Ghana and Guinea) in the late 1970s.
"It hurt for a while, and up to now I still curse him," she says, smiling. She calls him both names now.
His life as a revolutionary made her one by association. As he was stalked, so was she. Her house in the Bronx was under round-the-clock surveillance until he left the country. FBI agents followed her wherever she went and tapped her phone, she says.
Sometimes agents would call and ask her where her son was. "I would say that's your problem," she says. They sent black agents posing as Stokely's friends to fish information out of her. They followed her to the market when she bought her fish and green bananas.
Soon, she had had enough. She picked up the phone and spoke into the receiver. "'Mr. Hoover,'" she would say to the man who ran the FBI, "'I know you are listening to our conversation right now. But I wish you the slowest, eating-up cancer that will take time to die.' Every single day I pick up my phone, I used to curse Mr. Hoover. And here now my son have cancer. How you like that? God, how I'm sorry for that one."
Ask Ture what happened to the Movement and you get a dose of radical speechifying about the corruption of the petite bourgeoisie with their self-serving class-action suits, FBI sabotage, and the relationship of the people to the means of production. He holds in particular contempt the beneficiaries of the Movement who forget that people died so they could be a judge or an assemblyman or senior vice-president of such-and-such. He calls people like Ward Connerly, the black conservative who led the charge against affirmative action in California, "reactionary garbage." He says, "My grandmother used to say, 'When you boil dirty water, the first thing to rise to the top is the scum.'"
On politics and The Struggle: "The enemy seeks to take us out of the streets to the ballpark of electoral politics, which solves nothing. . . . We have more mayors and worsening conditions. We've had to admit as a people that reform cannot solve our problem; only revolution can."
On drugs in the black community: "In revolutionary history, drugs are usually the last go-round before the defeat of imperialist forces. . . . I think most people don't recognize we're in a war. . . . At certain times, they heat up certain fronts, so the drug war is heated up the last two decades."
On the disproportionate number of black men in prison: "It's to be expected. The system is going to do everything possible to suppress us. But those prisons are going to turn out Malcolm X in mass."
On the perils of integration: "It's an insidious subterfuge for the maintenance of white supremacy. Because it makes you think that for things to be better, you gotta go to whites. You want a better school, you gotta go to a white school. You want a better neighborhood, you gotta go to a white neighborhood. It instills an inferiority complex in the minds of Africans."
On his infamous quote about the position of women in the Movement being prone: "You can't judge a person's position on women by a phrase. It was a private joke inside of a private SNCC meeting during a debate on the role of women within SNCC. We laughed. We were all laughing. You couldn't say it and really mean it. This phrase was taken up by white women who were put out of SNCC when I was chair. They were mad at me and trumpeted it everywhere."
On the need for unity: "I want Julian Bond to meet with Farrakhan. I want a united front."
On why we keep looking for a savior: "It says we're disorganized therefore powerless, so we look for shortcuts, whether a King or a Mandela."
One night last fall, Kwame Ture sat in a den in Miami surrounded by teachers and librarians and community activists and other working people with children and car notes talking about the situation in South Africa. They listened to the master dissect the state of race relations, suggesting that here, like there, blacks in position is not the same as blacks in power.
The hour grew late and still the people drank in his words. Then the master, weary, went to another room. And then as often happens when black people meet, the subject turned to why black people can't get themselves together, and the discussion toured all the suspected reasons—from self-hate to apathy. The people in the room, their leader departed, were left groping, searching, arguing and sometimes agreeing, on their own. It was Black America itself—Dr. King dead, Malcolm X dead, Medgar Evers dead, Stokely Carmichael gone to the Motherland, the people left wandering alone, convinced they need a leader. The man who left the country left the group. The group would have to figure out the situation on its own. There would be no Messiah with a miracle.
In December 1995, Kwame Ture went to visit his mother in Miami. She had just gotten the new condo and, since he didn't have any money and she was so anxious to see him, she paid for his plane ticket herself. When he got there, she noticed he walked with a limp. And his son noticed he walked with a limp. And his nephew noticed it, too. "Yes, Uncle Kwame, you not walking so good."
The doctor in Guinea said it was a pinched nerve. His mother wasn't convinced. She told him to see another doctor. He saw one in New York. He was hospitalized right away. It was the prostate. It was cancer. The prognosis was not good. "The cancer was already started in his bones," his mother says. She still found reason to be glad. "I was really thankful to God that he was letting him stay on this time with us."
He has an everything-is-everything calm about him on these days. He laughs loud and easily. Seems at peace. The smooth face has white stubble now, and a straight white buzz cut above. He rests his swollen legs and feet on his mother's glass coffee table during hours of interviewing, leaning back against the cushy spine of his mother's white sofa, the one with the lace doilies on the armrests, tired not so much from the fight for your civil rights but from a cellular invader.
His line of work means no medical insurance, no pension. There is a resignation in his mother's voice when she speaks of how others seemed to make the most of their civil rights fame. "Every last one of those boys used that freedom ride to climb," she says. "So Stokely could have climbed, too. So this is really what he wanted."
He lived his life hoping his devotion would bear fruit. "If you struggle for the people, the people won't let you starve," he insists.
The fruit falls in unlikely places. One black doctor treating him in New York volunteered his services out of respect and gratitude. He said he might not have been a doctor had Stokely Carmichael not done what he did. "I told him, 'Don't ever worry about a bill from me,'" said Dr. Gerald Hoke, his urologist. "The honor of being associated with him is payment enough. This is the least I can do."
Ture believes that the FBI gave him the cancer—how exactly, he doesn't know. "The cancer still has not gone into remission—the FBI was serious with this one," he says soberly, but managing to smile at the paradox. "The chemotherapy and the other treatments have been able to calm the pain. That's all. They can't stop the cancer."
In the meantime, he's still talking about revolution, making calls from his bedside if he has to, giving speeches when he can, meeting with his band of believers on several continents, talking about straws and slingshots, and how eventually there will be enough of them pointed and piled together to end injustice. "We never know which straw will break the camel's back," he says. "But we do know a straw will break the camel's back. Just keep putting the straw on. The camel's back will be broken sooner or later."
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"They Were Commandos" by Madeleine Blais, first published by _The New York Times_ , April 18, 1993. Reprinted with permission of the author.
"The Cheerleaders" by E. Jean Carroll, first published by _Spin_ , June 2001. Reprinted with permission of the author.
"Some Dreamers of the Golden Dream" by Joan Didion, first published by _The Saturday Evening Post_ , April 1966. Reprinted with permission of the author.
"Wonder Dog" by Melissa Fay Greene, first published by _The New York Times Magazine_ , May 2, 2012. Reprinted with permission of the author.
"Holy Days: The World of a Hasidic Family" by Lis Harris. A version of this story was first published by _The New Yorker_ , September 23, 1985. Reprinted with permission of the author.
"The Last Day" by Robin Marantz Henig. This story was first published with the title "The Last Day of Her Life" by _The New York Times Magazine_ , May 14, 2015. Reprinted with permission of the author.
"On the Bus with B.B. King" by Gerri Hirshey, first published by _Rolling Stone_ , December 25, 1998. Reprinted with permission of the author.
"Nureyev Dancing in His Own Shadow" by Elizabeth Kaye, first published by _Esquire_ , March 1991. Reprinted with permission of the author.
"The New Face of Richard Norris" by Jeanne Marie Laskas, first published by _GQ_ , August 27, 2014. Reprinted with permission of the author.
"The Troubled Life of Boys" by Adrian Nicole LeBlanc, first published by _The New York Times Magazine_ , August 22, 1999. Reprinted with permission of the author.
"Prodigal Daughter" by Jill Lepore, first published by _The New Yorker_ , July 8, 2013. Reprinted with permission of the author.
"The Split" by Suzannah Lessard, first published by _The New Yorker_ , December 8, 1997. Reprinted with permission of the author.
"Yuja Wang and the Art of Performance" by Janet Malcolm, first published by _The New Yorker_ on September 5, 2016. Reprinted with permission of the author.
"The Bullfighter Checks Her Makeup" by Susan Orlean, first published by _Outside_ , May 2, 2004. Reprinted with permission of the author.
The following stories have been reprinted with permission of the author:
Excerpts from "Introduction" by Lillian Ross, from _Takes: Stories from the Talk of the Town_ , published March, 1983 by Congdon and Weed. All rights reserved.
"Davis on Dogs" by Lillian Ross, first published by _The New Yorker_ , January 26, 1981. All rights reserved.
"Remembering Picasso" by Lillian Ross, first published by _The New Yorker_ , October 19, 1981. All rights reserved.
"The Golden Ladies of the Golden Door" by Lillian Ross, first published by _The New Yorker_ , June 2, 1995. All rights reserved.
"Kid Twelve" by Susan Sheehan, first published by _The New Yorker_ , August 19, 1996. Reprinted with permission of the author.
"Mrs. Kennedy at the Moment" by Gloria Steinem, first published by _Esquire_ , October 1964. Reprinted with permission of the author.
"Mothers, Sisters, Daughters, Wives" by Mimi Swartz, first published by _Texas Monthly_ , August 2012. © 2012 by _Texas Monthly_. Reprinted with permission of _Texas Monthly_.
"My Breast" by Joyce Wadler, first published by _New York Magazine_ , April 13, 1992, and April 20, 1992. Reprinted with permission of the author.
"Soul Survivor" by Isabel Wilkerson, first published by _Essence_ , May 1998. Reprinted with permission of the author.
About the Editor
Patsy Sims is the author of three nonfiction books, including _The Klan_ and _Can Somebody Shout Amen! Inside the Tents and Tabernacles of American Revivalists_ , named a noteworthy book of 1988 by _The New York Times Book Review_. She is also the editor of _Literary Nonfiction: Learning by Example_ and coauthor of the narration for Academy Award–nominated documentary _The Klan: A Legacy of Hate_.
Prior to writing books, she worked as a staff writer and editor for the New Orleans S _tates-Item_ , _The San Francisco Chronicle_ , and _The Philadelphia Inquirer_. Her work has appeared in _The New York Times Book Review_ , _The Washington Post Magazine_ , _Oxford American_ , _Texas Observer_ , among other publications. She has been a recipient of creative writing fellowships from the National Endowment for the Arts and the District of Columbia Commission on the Arts and two Associated Press Awards for investigative-interpretive reporting. "No Twang of Conscience Whatever," about the 1964 murders of three civil rights workers in Mississippi, was named a Notable Essay of 2014 by _The Best American Essays_. She directed Goucher College's MFA program in creative nonfiction from 2001–2014.
About the Publisher
The Sager Group was founded in 1984. In 2012 it was chartered as a multi-media artists' and writers' consortium, with the intent of empowering people who create—an umbrella beneath which makers can pursue, and profit from, their craft directly, without gatekeepers.
TSG publishes books; ministers to artists and provides modest grants; and produces documentary, feature, and commercial films. By harnessing the means of production, The Sager Group helps artists help themselves. For more information, please see www.TheSagerGroup.net.
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 713
|
\section{Introduction}
The Galactic plane is known to be an intense emitter of X-rays and $\gamma$-rays\
from keV to at least 100 GeV.
At X-ray energies, it has recently been claimed
\citep{2006A&A...452..169R}
that the 2-10 keV emission can be explained entirely by a population
of weak sources, mainly CVs.
Above 50 keV, unresolved sources also appear to be required to explain the hard-power law emission observed by INTEGRAL
\citep{2005A&A...444..495S,
2005ApJ...635.1103B};
AXPs and/or pulsars are potential candidates.
This raises the question of the situation
for $\gamma$-rays\ , in particular the range observed by EGRET, 30 MeV - 100 GeV.
$\gamma$-ray\ telescopes are relatively insensitive and reveal only the `tip of
the iceberg' of the sources, so most of them will go undetected unless it happens
that only strong sources exist in the Galaxy.
Conventional wisdom is that the source contribution to the unresolved ridge emission is at the few percent
level,
The problem of the GeV excess in the diffuse emission
compared to the expected spectrum from interstellar processes
\citep{2004ApJ...613..962S},
is however a
hint that the emission may have other components whose contribution
is also energy-dependent. The failure to explain the 1-30 MeV emission measured by COMPTEL
from interstellar components is another pointer to a source population.
This topic has wide ranging implications: for example it has been claimed that the GeV excess
is a signature of dark matter decay
\citep{2005A&A...444...51D
\footnote{but see critique by
\citet{2006JCAP...05..006B}.}
, but this is only plausible if all viable alternatives are
excluded, and source populations provide one important such candidate.
\section{$\gamma$-ray\ source properties}
The 3rd EGRET Catalogue contains 271 sources, including 66 well-identified extragalactic sources (AGN),
leaving about 200 for the present study. A subset of these may also be extragalactic.
Almost the only definite Galactic identifications are 6 pulsars (+ some other pulsar candidates) and the Crab nebula,
which have isotropic luminosities $L_\gamma$($>$100~MeV)
ranging from $10^{37}$ph s$^{-1}$\ ( 10$^{33}$ erg\ s$^{-1}$\ ) (Geminga) to $10^{39}$ph s$^{-1}$\ (10$^{35}$ erg\ s$^{-1}$\ ) (Crab pulsar).
A summary of $\gamma$-ray\ pulsar observations is given by
\citet{1999ApJ...516..297T}.
The Crab at 2 kpc is perhaps the most luminous and distant Galactic source detected,
while Geminga at 160 pc the least luminous and nearest. The Vela pulsar is intermediate
in distance (300 pc) and luminosity $10^{38}$ph s$^{-1}$\ (10$^{34}$ erg\ s$^{-1}$\ ) although it is the brightest source on the sky
(8 $10^{-6}$ cm$^{-2}$ s$^{-1}$\ compared to 2 $10^{-6}$ cm$^{-2}$ s$^{-1}$\ for the Crab).
Other detected or candidate pulsars also lie in this luminosity range.
Apart from pulsars,
plausible identifications exist e.g. for the microquasars LS5039 and LSI +61 303,
but these do not significantly help the present investigation.
Most sources with $L_\gamma$($>$100 MeV)$<$ 10$^{33}$ erg\ s$^{-1}$\ are invisible to us with present instrumentation.
The data are too sparse to construct a luminosity function directly, so we leave this as a parameterized input.
We approach the problem using population synthesis, comparing the models both with
source counts and the intensity of diffuse emission
\footnote{We will often refer to unresolved emission generically as `diffuse' regardless of its true nature}.
Comparison in various sky regions increases the discrimination power of the comparisons.
Thus source counts in high-latitude regions constrain the
low--luminosity, nearby source population
while counts in low-latitude regions constrain the high-luminosity sources,
and diffuse emission in low-latitudes constrains both low and high-luminosity populations.
As usual in such studies a major uncertainty is the spatial distribution of sources,
but with plausible assumptions this can be modelled.
Once the spatial distribution is assumed, the observed (source-produced) gamma-ray sky
depends only on the luminosity function.
For the truly diffuse (interstellar) emission we make use of the {\it galprop} models
\citep{2004ApJ...613..962S}
.
The present study is designed to be independent of the physical details of the sources,
making use of, for example, theoretical $\gamma$-ray\ luminosity functions of pulsars only for guidance.
We do not want to restrict attention to known classes of objects.
The essential input is only geometry, the inverse square law,
simple luminosity functions,
the EGRET source catalogue and EGRET skymap data.
\section{Previous studies}
Most previous population studies have been aimed at deducing the nature of the EGRET
unidentified sources, either generally or in particular as pulsars.
Although the nature of the sources is not our goal here, these studies are still relevant.
A population synthesis study of the 2nd EGRET catalogue
\citep{1996A&AS..120C.461K}
showed that the unidentified sources had luminosities $>$100 MeV in the range
$6\ 10^{34} - 3\ 10^{35} $erg\ s$^{-1}$\ , with 700--3400 objects in the Galaxy.
Flux distributions of EGRET unidentified sources were constructed by
\citet{2001ICRC....6.2566R}
and \citet{2000Natur.404..363G}
in attempts to deduce
the properties of unidentified sources, including division by variability and angular distribution.
\citet{2001AIPC..587..663C}
made a population synthesis study of the 3EG catalogue, including disk and isotropic source distributions.
Other studies considered pulsars specifically.
Very detailed physical pulsar population synthesis studies by
\citet{2002ApJ...565..482G}
and
\citet{2004ApJ...604..775G}
using their polar-cap model were aimed at establishing
the relation between radio and $\gamma$-ray\ properties and predicting detections for future missions like GLAST.
\citet{2001ApJ...548L..37H,2003pasb.conf..115G,2004AdSpR..33..571H,2005Ap&SS.297...71G}
\citet{2004ApJ...608..418C}
considered the possibility that gamma-ray sources in the
Gould's Belt are nearby pulsars.
\citet{1998MNRAS.301..841Z}
proposed that pulsars can account for the diffuse GeV excess.
\citet{2000A&A...357..957Z}
made a detailed population synthesis based on the outer-gap model, and also studied the properties of 38 low-latitude unidentified sources, proposing an
a correlation with SNR and OB associations.
\citet{1997ApJ...476..347Y}
proposed that the EGRET unidentified sources are compatible with young pulsars,
while in contrast
\citet{2003A&A...404..163B}
claim they are better traced by spiral arms and molecular clouds.
In view of the uncertainty in the nature of the unidentified sources, a flexible approach to modelling is desirable,
as described in this paper.
\section{Population synthesis}
A general-purpose population synthesis code has been written.
Sources are assigned a density $\rho(R,z,L_\gamma)$ and sampled by standard Monte-Carlo techniques.
Oversampling is used to reduce statistical fluctuations.
The density is normalized to $\rho$ at $R = 8.5$ kpc, in units of sources kpc$^{-3}$.
Power-law luminosity functions within given $L_\gamma$ limits can be generated.
The $(R,z)$ source distribution is here based on pulsars
\citep{2004IAUS..218..105L}
as representative of $\gamma$-ray\ sources, but other distributions are also possible and will be
addressed in future work.
The resulting source list is analyzed to generate differential
source counts N(S) and the total emission spectrum both above and below a given detection threshold.
The total spectrum is then combined with interstellar emission models from {\it galprop}
\citep{2004ApJ...613..962S}.
We use source counts for one energy range ($>100$ MeV) only; it would be preferable to consider
the energy-dependence of N(S) but the spectral information in the available catalogues is limited.
In future (e.g. for GLAST) this will be feasible.
A note about beamed sources, in particular pulsars. For the present purpose a populations of randomly-oriented beamed sources
is fully equivalent
to a population of unbeamed sources with a lower spatial density of sources
having an isotropic emission.
Therefore we do not explicitly include beaming in our population synthesis.
\subsection{Pulsar luminosity function}
We use pulsars just as an guide to the choice of luminosity function.
For this we use the luminosity as a function of spin-down power $\dot{E}$: $L_\gamma\propto\dot{E}^\beta$
for which a wide spread exists in the literature depending on the model.
The luminosity function can be then estimated as follows:
$N(L_\gamma)={dN\over dL}={dN\over d\dot{E}}\ {d\dot{E}\over dL} = {dN\over dt}\ {dt\over d\dot{E}}\ {d\dot{E}\over dL}
\propto {dt\over d\dot{E}}\ {d\dot{E}\over dL}$ for constant birthrate.
Since $\dot{E}\propto B^2/P^4\propto \dot{P}/P^3$, $d\dot{E}/dt\propto\dot{E}/P^2$ and hence
$N(L_\gamma)\propto B^{-1}L_\gamma^{-(1+2\beta)/2\beta}$
According to the polar-cap model of
\citet{2002ApJ...565..482G}
$\beta\approx {1\over 2}$, so $N(L_\gamma) \propto L_\gamma^{-2}$
The slot-gap model of
\citet{2003ApJ...588..430M}
gives $\beta={1\over 4}$ so $N(L_\gamma) \propto L_\gamma^{-3}$
The outer-gap model of
\citet{2004ApJ...604..317Z}
gives $\beta=0.4 - 1$ so $N(L_\gamma) \propto L_\gamma^{-2.3}- L_\gamma^{-1.5} $.
The dynamic range of $\gamma$-ray\ luminosity in these models is about 1000.
\ifthenelse{0=1}
{
According to the polar-cap model of
\citet{2002ApJ...565..482G}
, $L_\gamma\propto P^{-x} $, x= 27/14 or 9/4 depending on spin-down luminosity.
where the proportionality constant depends roughly linearly on the magnetic field and the cosine of the angle between rotation and dipole axes.
The luminosity function can be then estimated as follows:
$dN/dL=dN/dP\ dP/dL = dN/dt\ dt/dP\ dp/dL$
$ = P\ dP/dL= P\ P^{1+x} = P^{2+x} = L^{-{2+ x\over x}} $
since $dP/dt=P^{-1}$.
Hence $N(L)\propto L^{-2.03,-1.89}$, for x=27/14, 9/4.
so a luminosity function index -2 is sufficient to approximate this model.
Obviously distributions of $P_o$, $B_{12}$ and $\alpha$ will modify this simple law,
but we anyway allow for a much wider range of indices in our synthesis since we want
to avoid dependence on a particular source model.
The outer gap model predicts $L_\gamma\propto P^{-2/7} $
(Zhang,Zhang and Cheng 2000) so $N(L)\propto L^{-8}$
which is not bracketed by our range of models.
But in new outer gap model Zhang et al 2004 604,317 population synthesis gives
$L_\gamma\propto P^{-1.5} to P^{-2} $ (i.e. $Edotsd^{.38, .46,1}$ depending on the sample.
This is again covered by our grid of models.
}
\subsection{Known populations}
We start with the EGRET catalogue sources (excluding AGN identifications),
attempt to reproduce their source counts by population synthesis
and hence estimate the contribution of unseen members of this population to the diffuse emission.
The luminosity function is assumed to be a power law, the index and limits being free parameters.
In our reference model the range of luminosities considered is $L_\gamma$($>$100 MeV) = $10^{36} - 10^{39}$ s$^{-1}$,
covering the range of detected pulsars as discussed in the Introduction.
The local density $\rho$ is fixed by the requirement that the low-latitude source counts are reproduced.
For EGRET we use a limiting flux ($>$100 MeV) of $ 10^{-7}$ cm$^{-2}$ s$^{-1}$; fainter sources are detected by EGRET at high latitudes
but in the plane the limit is higher:
$1.6\ 10^{-7}$ $|b|<10^o$, $0.7\ 10^{-7}$ $|b|>10^o$).
For a luminosity function index -1.5,
$\rho=37$ kpc$^{-3}$ (model 1b), the low-latitude source counts in $300\deg<l<60\deg,\ |b|<10\deg$ (hereafter region H) are reproduced (37 above threshold), the
high-latitude sources are very few (so they must be extragalactic in this case),
and about 6\% of the Galactic emission
( $2\ 10^{-4}$ cm$^{-2}$ s$^{-1}$\ ) in this region
comes from the 4000 sources below the threshold.
The fluxes above and below the threshold are about equal, $1.5,1.2\ 10^{-5}$ cm$^{-2}$ s$^{-1}$\ $>$100 MeV in the simulation,
compared to $1.4\ 10^{-5}$ cm$^{-2}$ s$^{-1}$\ for the 37 EGRET sources in region H.
For a luminosity function index -2.0,
model 1c,
again choosing $\rho$ so that the low-latitude source counts in region H are reproduced,
13\% of the Galactic emission comes from the 26000 sources below the threshold.
How critical is the luminosity function shape ?
Flattening the luminosity function index from -1.5 to -1.0,
about 4\% of the Galactic emission comes from the 728 sources below the threshold.
Steepening the luminosity function index to -2.5,
about 28\% of the Galactic emission comes from the $10^5$ sources below the threshold.
So even a major difference in assumed luminosity function shape does not change the conclusion that a significant
contribution to the diffuse emission must come from sources physically like those in the EGRET catalogue but below the detection threshold.
A steeper luminosity function or a lower minimum luminosity leads to a larger contribution from unresolved sources.
The high-latitude source counts are not very constraining, due to the low space density for these high luminosity sources,
but the predicted counts are consistent with (i.e. do not exceed) the observed counts.
For a steep luminosity function (index -2.0) or a low minimum luminosity ($10^{35}$~ph s$^{-1}$\ ~), a significant fraction of the high-latitude sources can be Galactic.
With the small statistics a quantitative comparison of observed and predicted shape of N(S) is difficult,
any index considered here reproduces the observed counts reasonably,
and no distinction between the various indices can be made.
\begin{figure*}
\centering
\includegraphics[width= 6cm]{strong_fig1a.eps}
\includegraphics[width= 6cm]{strong_fig1b.eps}
\includegraphics[width= 6cm]{strong_fig1c.eps}
\includegraphics[width= 6cm]{strong_fig1d.eps}
\caption{ Differential source counts in $300\deg<l<60\deg,\ |b|<10\deg$ (region H). Blue, connected points: models 1a-d,
luminosity index -1.0,-1.5,-2,-2.5 (left to right, top to bottom).
Red unconnected points: 3rd EGRET Catalogue, excluding AGN identifications.
}
\label{NS_1a}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width= 6cm]{strong_fig2a.eps}
\includegraphics[width= 6cm]{strong_fig2b.eps}
\includegraphics[width= 6cm]{strong_fig2c.eps}
\includegraphics[width= 6cm]{strong_fig2d.eps}
\caption{Spectra in region H ($300\deg<l<60\deg,\ |b|<10\deg$ ) for models 1a-d.
luminosity index -1.0,-1.5,-2,-2.5 (left to right, top to bottom).
Sources below (dotted, cyan) and above (dashed, cyan) the EGRET detection limit are also shown
together with sources below the limit added to the conventional interstellar model from
\citet{2004ApJ...613..962S}
(continuous, cyan).
Data: EGRET, COMPTEL.
}
\label{spectra}
\end{figure*}
\subsection{Unseen / dim populations}
We turn now to source populations mainly below the detection threshold,
with the aim of placing constraints on their properties. The relevant models are
2, 3 and 4 in Table 1.
The luminosity function is assumed to be a power law with index -1.5 and various luminosity ranges.
The normalization is chosen to give about a tenth of the EGRET diffuse emission, as an illustration of a plausible level.
Consider first $L_\gamma$($>$100 MeV) = $10^{36} - 10^{37}$ ph s$^{-1}$\ , model 2.
For $\rho= 500$, 15\% of the diffuse emission in region H is from the $5\ 10^4$ sources below the threshold and 44 of the EGRET unidentified high-latitude sources
are Galactic
and 17 (cf 3rd EGRET Catalogue: 37) sources are above the limit in region H.
Longitude profiles (not shown here) indicate that the sources below the threshold cause fluctuations in the emission
which however are not distinguishable from the interstellar emission.
For $L_\gamma$($>$100 MeV) = $10^{35} - 10^{36}$ ph s$^{-1}$\ , $\rho=2\ 10^3$, (model 3), 7\% of the diffuse emission in region H is from the $2\ 10^5$ sources below the threshold,
2 sources are above threshold in region H, and 19 at high latitudes.
For $L_\gamma$($>$100 MeV) = $10^{34} - 10^{35}$ ph s$^{-1}$\ , $\rho=2\ 10^4$, (model 4), 7\% of the diffuse emission in region H is from the $2\ 10^6$ sources below the threshold,
no sources are above the threshold in region H, and only 5 at high latitudes.
It follows that for $L_\gamma$($>$100 MeV) $<10^{35}$ ph s$^{-1}$\ all of the `diffuse' emission {\it could} come
from sources without violating source counts anywhere on the sky (e.g. by scaling up $\rho$ in model 4).
This is of course highly unlikely given our knowledge of interstellar emission processes but cannot be excluded from $\gamma$-ray\ data alone.
If we consider that all the high-latitude unidentified sources are Galactic then a dim dense population like in models 2,3,4 is
{\it necessary} in addition to the bright population. Strict limits are then set by the requirement of not violating
the observed diffuse emission considering plausible interstellar emission.
If instead they are extragalactic, the dim Galactic population is not required.
\begin{table*}
\label{models}
\caption{Summary of population synthesis models. Using thresholds $S_{EGRET}=1\ 10^{-7}$ cm$^{-2}$ s$^{-1}$, $S_{GLAST}=4\ 10^{-9}$ cm$^{-2}$ s$^{-1}$.
For comparison, the EGRET measured diffuse emission in region H ($300\deg<l<60\deg,\ |b|<10\deg$ ) is $2\ 10^{-4}$ cm$^{-2}$ s$^{-1}$\ }
\centering
\begin{tabular}{l l l l l l l l l l l}
\hline\hline
Model& & $L_{min}$ & $L_{max}$ & $\alpha$ &$\rho(R_\odot)$ &$N(>S_{EGRET})$ / &$N(>S_{EGRET})$/ &$F(>S_{EGRET})$ /&$N(>S_{GLAST})$ / &$F(>S_{GLAST})$ \\
& & & & & & $N(S<S_{EGRET})$ &$N(<S_{EGRET})$ &$F(S<_{EGRET})$ &$N(S<S_{GLAST})$ &$F(<S_{GLAST})$ \\
& & $>$100 & ph s$^{-1}$ & &kpc$^{-3}$ & Region H & high lat. ®ion H ®ion H \\
& & MeV & & & & & & $10^{-4}$ cm$^{-2}$ s$^{-1}$ \\
\hline
\hline
EGRET 3EG & & - & - & - & - & 37/1 & 47/70 & 0.14 / .01 \\
\hline
\hline
1a & & $10^{36}$& $10^{39}$ & -1.0 & 7.3 & 37.5/728 &1.4/0.5 & 0.16 /0.08 & 339/426 &0.23/0 \\
1b & & $10^{36}$& $10^{39}$ & -1.5 & 37 & 37/3899 &4.2/4.0 & 0.15 /0.12 & 578/3357 &0.24/0.01 \\
1c & & $10^{36}$& $10^{39}$ & -2.0 & 250 & 37/26550 & 18/39 & 0.14 /0.26 &1167/25420 &0.29/0.10 \\
1d & & $10^{36}$& $10^{39}$ & -2.5 &1000 & 36/106370 & 59/177 & 0.11 /0.56 &2003/104400 &0.34/0.33 \\
\hline
2 & & $10^{36}$& $10^{37}$ & -1.5 &$500 $& 17/$0.5\ 10^5$ & 44/74 & 0.06 /0.32 &1138/$5\ 10^4$&0.16/0.20 \\
3 & & $10^{35}$& $10^{36}$ & -1.5 &$2\ 10^3$&2/$2 \ 10^5$ & 19/490 & 0.001/0.14 &224/$2\ 10^5$&0.02/0.12\\
4 & & $10^{34}$& $10^{35}$ & -1.5 &$2\ 10^4$& 0/$2 \ 10^6$ & 5/4728 & 0.0 /0.14 &94/$2\ 10^6$ &0.007/.14 \\
\hline
\hline
\end{tabular}
\end{table*}
\section{Comparison with physical pulsar population synthesis}
It is interesting to see how our simple generic approach matches detailed specific models.
In their pulsar population synthesis based on their polar cap model,
\citet{2004ApJ...604..775G}
find 26 pulsars detectable by EGRET, which presumably would mean a substantial fraction of the unidentified sources are pulsars.
They predict that 600 pulsars will be detectable by GLAST for a threshold $2-5\ 10^{-9}$ cm$^{-2}$ s$^{-1}$.
This matches best our model 1b (index -1.5)
The outer gap model of
\citet{2000A&A...357..957Z}
predicts 32 pulsars detectable by EGRET, which again would mean a substantial fraction of the unidentified sources are pulsars.
This model predicts 1180 GLAST pulsars,
for assumed threshold of $4\ 10^{-9}$ cm$^{-2}$ s$^{-1}$
matching best our model 1c with luminosity function index -2.0.
\section{Sources can produce the MeV, GeV excesses}
The unresolved source fraction is energy-dependent, so that one can ask whether it can
produce the well-known GeV excess over the standard cosmic-ray interaction models.
Consider first model 1c (Fig 3c) with spectral index -2.0 and luminosity function index -2.0,
combined with the `conventional' interstellar emission model from
\citet{2004ApJ...613..962S},
but with the cosmic-ray source distribution from
\citet{2004A&A...422L..47S}.
The GeV excess is {\it not} reproduced.
However for this source spectrum the source contribution to the COMPTEL (1--30 MeV) and INTEGRAL (.02 -- 1 MeV) ranges
might provide an explanation of the excess above the interstellar emission at those energies.
Sources with a Crab-pulsar-like index of -2.1 with a break above 4 GeV
\citep{1998ApJ...494..734F,2001A&A...378..918K}
would be too steep to reproduce the GeV excess, but the COMPTEL diffuse emission {\it could} be fitted
\begin{figure*}
\centering
\includegraphics[width= 6cm]{strong_fig3.eps}
\caption{Sources like Geminga, Vela pulsars {\it can} produce the GeV excess:
spectra in $330\deg<l<30\deg,\ |b|<5\deg$ (region A of
\citet{2004ApJ...613..962S}
) for L($>$100 MeV) = $10^{36} - 10^{39}$ s$^{-1}$. luminosity index 2.0, spectral index -1.5, break at 2 GeV to -2.0
to match Geminga, Vela. $\rho=150$.
Sources below (dotted, cyan) and above (dashed, cyan) the EGRET detection limit are also shown
together with sources below the limit added to the conventional interstellar model from
\citet{2004ApJ...613..962S}
(continuous, cyan).
Data: EGRET, COMPTEL.
}
\label{spectra_Geminga_like}
\end{figure*}
Consider now sources with hard spectra like the Geminga (index -1.42, 30-2000 MeV) and Vela (index -1.62, 30-1000 GeV)
pulsars \citep{1998ApJ...494..734F}
(also B1706, B1055 have hard spectra with a GeV break);
we adopt a spectral index -1.5, with a break at 2 GeV to -2.0, luminosity index 2.0, $\rho=30$ (cf. model 1c);
the GeV excess is easily produced (Fig \ref{spectra_Geminga_like}).
Indeed these pulsars show maxima in $E^2\ F(E)$ around 2 GeV very reminiscent of the Galactic GeV excess.
This explanation of the GeV excess was also proposed by
\citet{1998MNRAS.301..841Z}
on the basis of their outer-gap pulsar model,
but our result is not dependent on a particular physical model.
However the diffuse emission below 30 MeV is not explained by such sources.
It is then tempting to propose a mixture of Crab-like and Geminga/Vela-like pulsars to produce both the MeV and GeV excesses
and reproduce the entire ridge spectrum.
The spectra of the unidentified low-latitude sources scatters broadly around -2 (-1.7 to -2.7)
\citep{2000A&A...357..957Z}
but the details of the spectra are not sufficient to determine whether this supports our hypothesis about the GeV excess. GLAST will contribute
significantly on this point.
\section{Will GLAST resolve the issue ?}
Although GLAST will not detect all the $\gamma$-ray\ sources in the Galaxy, it will resolve essentially all the source {\it flux}
of the bright populations. Thus for model 1c (luminosity index -2.0), 75\% of the source flux is above a threshold $4\ 10^{-9}$ cm$^{-2}$ s$^{-1}$,
compared to 35\% for the EGRET threshold. For model 1b (luminosity index -1.5), 96\% of the source flux is above the GLAST threshold,
compared to 55\% for the EGRET threshold.
For the dimmer populations ($L_\gamma$($>$100 MeV) $<10^{37}$ ph s$^{-1}$\ ) progressively less of the flux will be resolved.
Even for $L_\gamma$($>$100 MeV) $=10^{36}-10^{37}$ ph s$^{-1}$\ (model 2) only 44\% of the flux is above the GLAST threshold,
so the analysis will remain a challenge.
\ifthenelse{0=1}
{
\section{Hard X-ray energies}
Although not the main focus of this work, we can apply the same techniques to the hard X-ray range (20-500 keV)
with particular reference to the INTEGRAL results.
\citet{2005A&A...443..485D}
using the 1st IBIS catalogue find luminosities covering the range L(20-100 keV)= $10^{36-38}$ erg\ s$^{-1}$\ = $10^{43-45}$ ph s$^{-1}$\ ,
in particular
LMXB: L(20-100 keV)= $10^{36}- 5\ 10^{37}$ erg\ s$^{-1}$\ , HMXB: $3\ 10^{36}- 2\ 10^{37}$ erg\ s$^{-1}$\
or
BH : L(20-100 keV)= $3\ 10^{35}- 5\ 10^{37}$ erg\ s$^{-1}$\ , NS: $5\ 10^{35}- 3\ 10^{36}$ erg\ s$^{-1}$\ .
As before we attempt to reproduce the source counts by population synthesis and then predict the contribution from unresolved sources of the same type,
and assess the possible contribution from dimmer populations.
We use the analysis results of \citet{2005A&A...444..495S
which give both source and diffuse emission spectra based on INTEGRAL/SPI
using source positions from the 2nd INTEGRAL/ISGRI Catalogue
\citep{2006ApJ...636..765B}
Review of luminosity index:
\citet{2002A&A...391..923G
using RXTE survey data of the Galaxy found index -1.26 for LMXB, -1.64 for HMXB.
\citet{2005astro.ph.10049S
obtain the luminosity function over a very wide range using RXTE ........
\citet{2004MNRAS.349..146G}
using Chandra observations of 11 nearby galaxies found index -1 for LMXB and -1.6 for HMXB.
\citet{2005A&A...443..485D}
using the 1st INTEGRAL/ISGRI catalogue with source identifications and distances, give luminosity functions which
have roughly index -1.6 for both LMXB and HMXB, or -1.3 for BH systems,-2.3 for NS systems.
(see also Dean ISDC presentation).
Using these results as a guide to the expected range,
we use luminosity indices -1.0, -1.5 and $\rho=1-1.5$
which reproduces the source counts;
taking a mean spectral index of -3 produces the summed flux from detected sources (Fig X).
Then the 20-50 keV diffuse flux can be explained as unresolved sources, but above this the
diffuse flux is much too hard for these to make a contribution.
A steeper luminosity function with index -1.5 produces too much 20-50 keV diffuse flux and can be excluded for the steep spectrum sources (Fig Y)
but is allowed for the hard spectrum sources.
To produce the spectrum above 50 keV, we need lower luminosity sources to be below the detection threshold, and a hard spectrum;
we use $10^{41-43}$ ph s$^{-1}$\ , a spectral index -1.5 and a break at 1 MeV, representing AXPs, pulsars or microquasars (see Bosch-Ramon astroph/0601238)
is used
and a density $\rho=0.5$ or 1.0 depending on luminosity index.
This population would be invisible to INTEGRAL except for a few of the brightest objects,
and lies below the observed source counts. At the same time it produces the observed diffuse emission in 50--500 keV.
\begin{table*}
\label{models}
\caption{Summary of population synthesis models for hard X-rays. Using $S_{lim}=2\ 10^{-3}$ cm$^{-2}$ s$^{-1}$.
Plotted models indicated by *.
ADD fraction of diffuse ?}
\centering
\begin{tabular}{l l l l l l l l l}
\hline\hline
Model& spatial& $L_{min}$ & $L_{max}$ & $\alpha$ &$\rho$ &$N(>S_{lim})$ / &$N(>S_{lim})$/ &$F(>S_{lim})$ / \\
& distr. & & & & (R=8.5 kpc) & $N(S<_{lim})$ &$N(<_{lim})$ &$F(S<_{lim})$ \\
& & 18-28 keV& photons s$^{-1}$ & &kpc$^{-3}$ & Region H & high latitudes®ion H $10^{-4}$ cm$^{-2}$ s$^{-1}$ \\
\hline
\hline
SPI &- & - & - & - & - & 45/42 & - & 4088/454 \\
catalogue & & & & & & & & \\
\hline
\hline
5a &pulsars & $10^{42}$& $10^{44}$ & 1.0 & 1.5 & 61/95 & - & 5263 /526 \\
5b &pulsars & $10^{42}$& $10^{44}$ & 1.5 & 2.5 & 45/250 & - & 4756 /1049 \\
6a &pulsars & $10^{41}$& $10^{43}$ & 1.0 & 0.5 & 2/50 & - & 81 /104 \\
6b &pulsars & $10^{41}$& $10^{43}$ & 1.5 & 1.0 & 1/105 & - & 76 /119 \\
7 &pulsars & $10^{40}$& $10^{43}$ & 1.5 & 3.0 & 1/316 & - & 68 /117 \\
\end{tabular}
\end{table*}
}{}
\section{Conclusions}
1. Modelling the contribution from unresolved sources is essential to understanding the diffuse Galactic emission .
\noindent
2. The contribution from unresolved sources to the EGRET low-latitude emission is at least 5-10\%, and can be 20\% for steep luminosity functions.
\noindent
3. An arbitrarily large fraction of the diffuse emission could come from sources $L_\gamma$($>$100 MeV) $<10^{35}$ ph s$^{-1}$\
from sources without violating EGRET source counts anywhere on the sky.
\noindent
4. The GeV excess can be produced naturally by a sufficient population of sources like Geminga and Vela,
but this has to be studied with more detailed models.
\noindent
5. Crab-like sources can produce the COMPTEL excess but not the GeV excess.
\noindent
6. A combination of source populations combined with the conventional model of interstellar emission could explain
the full COMPTEL/EGRET Galactic ridge spectrum.
\noindent
7. Whether GLAST can settle these issues depends critically on the source luminosity function.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,559
|
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